View Full Version : autoequivalences of D-branes
Urs Schreiber
Mar20-05, 10:53 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>As the people over on sci.math.research have kindly informed me, the\ngroup of autoequivalences of the category D(X) of D-brane states on an\norbifold is well known. It is called the derived Picard group of the\npath algebra of the quiver associated to the orbifold.\n\nThe natural question is what the physical interpretation of this\nPicard group action is. The natural guess is that it represents gauge\nand duality transformations. Indeed, I have seen the Picard group\nmentioned in the context of mirror symmetry. Does anyone know any more\ndetails?\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>As the people over on sci.math.research have kindly informed me, the
group of autoequivalences of the category D(X) of D-brane states on an
orbifold is well known. It is called the derived Picard group of the
path algebra of the quiver associated to the orbifold.
The natural question is what the physical interpretation of this
Picard group action is. The natural guess is that it represents gauge
and duality transformations. Indeed, I have seen the Picard group
mentioned in the context of mirror symmetry. Does anyone know any more
details?
Aaron Bergman
Mar20-05, 11:38 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <206f2305.0503200853.531a7d12-100000@posting.google.com>, Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n> As the people over on sci.math.research have kindly informed me, the\n> group of autoequivalences of the category D(X) of D-brane states on an\n> orbifold is well known. It is called the derived Picard group of the\n> path algebra of the quiver associated to the orbifold.\n>\n> The natural question is what the physical interpretation of this\n> Picard group action is.\n\nThere are various monodromies around singular points in the moduli space\nwhich are autoequivalences.\n\nJust glancing at a paper or two, for the orbifold case, the group seems\nto be related to the orbifold quantum symmetry.\n\nAaron\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <206f2305.0503200853.531a7d12-100000@posting.google.com>, Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
> As the people over on sci.math.research have kindly informed me, the
> group of autoequivalences of the category D(X) of D-brane states on an
> orbifold is well known. It is called the derived Picard group of the
> path algebra of the quiver associated to the orbifold.
>
> The natural question is what the physical interpretation of this
> Picard group action is.
There are various monodromies around singular points in the moduli space
which are autoequivalences.
Just glancing at a paper or two, for the orbifold case, the group seems
to be related to the orbifold quantum symmetry.
Aaron
Urs Schreiber
Mar21-05, 03:46 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Sun, 20 Mar 2005, Aaron Bergman wrote:\n\n> In article <206f2305.0503200853.531a7d12-100000@posting.google.com>, Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n>\n> There are various monodromies around singular points in the moduli space\n> which are autoequivalences.\n\n\nThat makes sense. But since the Picard group is that of _all_ the\nautoequivalences, should it not contain more transformations than one gets\nby going around closed loops in moduli space? Or is that all there is?\n\nSince the triangulated Fukaya category describing A-branes is conjectured\nto be equivalent to the derived category describing D-branes, it seems\nnatural to guess that mirror symmetry is also part of the Picard group,\nwhich indeed is consistent with some hints I saw in the literature. But if\nthat\'s true, shouldn\'t all dualities be sitting inside the Picard group?\n\n\n> Just glancing at a paper or two, for the orbifold case, the group seems\n> to be related to the orbifold quantum symmetry.\n\n\nI am currently on the road and have a hard time making literature\nsearches. That\'s why you see me asking a lot of things that I could\notherwise try to look up. Your help is much appreicated.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Sun, 20 Mar 2005, Aaron Bergman wrote:
> In article <206f2305.0503200853.531a7d12-100000@posting.google.com>, Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
>
> There are various monodromies around singular points in the moduli space
> which are autoequivalences.
That makes sense. But since the Picard group is that of _all_ the
autoequivalences, should it not contain more transformations than one gets
by going around closed loops in moduli space? Or is that all there is?
Since the triangulated Fukaya category describing A-branes is conjectured
to be equivalent to the derived category describing D-branes, it seems
natural to guess that mirror symmetry is also part of the Picard group,
which indeed is consistent with some hints I saw in the literature. But if
that's true, shouldn't all dualities be sitting inside the Picard group?
> Just glancing at a paper or two, for the orbifold case, the group seems
> to be related to the orbifold quantum symmetry.
I am currently on the road and have a hard time making literature
searches. That's why you see me asking a lot of things that I could
otherwise try to look up. Your help is much appreicated.
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:\n> That makes sense. But since the Picard group is that of _all_ the\n> autoequivalences, should it not contain more transformations than one gets\n> by going around closed loops in moduli space? Or is that all there is?\n>\n> Since the triangulated Fukaya category describing A-branes is conjectured\n> to be equivalent to the derived category describing D-branes, it seems\n> natural to guess that mirror symmetry is also part of the Picard group,\n> which indeed is consistent with some hints I saw in the literature. But if\n> that\'s true, shouldn\'t all dualities be sitting inside the Picard group?\n>\n\n\nIf you have an example, maybe there are more equivalences\nthan are realized with loops. Maybe not. I can\'t see that\nit matters much one way or another. For example, take any\none-parameter family and then consider its universal cover.\nQuotienting by the group of equivalences generated by loops\nwill give you back your example (assuming the monodromy\nrepresentation is faithful, or some such jargon). Quotienting\nby anything less will give you a family with more equivalences\nthan are realized by loops.\n\nAs for mirror symmetry, to relate one category to another\ndoes little for you, practically speaking. Object Ralph is\nrelated to object Fred, Alice to Wilma, etc. Remember that\nequating one category to another is not an AUTO-equivalence.\n(Analogy: there are zillions of equivalent two-dimensional\nvector spaces, but what\'s interesting is the group GL(2) of\nequivalences of any given one.) Mirror symmetry involves\n*families* of categories. What buys you something is relating\none family (or a one-family deformation of your category)\nto another. The relation of the parameters is the mirror map,\nand if the different parameters carry symplectic or complex\ninformation, then you can get one from another.\n\n-ez\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:
> That makes sense. But since the Picard group is that of _all_ the
> autoequivalences, should it not contain more transformations than one gets
> by going around closed loops in moduli space? Or is that all there is?
>
> Since the triangulated Fukaya category describing A-branes is conjectured
> to be equivalent to the derived category describing D-branes, it seems
> natural to guess that mirror symmetry is also part of the Picard group,
> which indeed is consistent with some hints I saw in the literature. But if
> that's true, shouldn't all dualities be sitting inside the Picard group?
>
If you have an example, maybe there are more equivalences
than are realized with loops. Maybe not. I can't see that
it matters much one way or another. For example, take any
one-parameter family and then consider its universal cover.
Quotienting by the group of equivalences generated by loops
will give you back your example (assuming the monodromy
representation is faithful, or some such jargon). Quotienting
by anything less will give you a family with more equivalences
than are realized by loops.
As for mirror symmetry, to relate one category to another
does little for you, practically speaking. Object Ralph is
related to object Fred, Alice to Wilma, etc. Remember that
equating one category to another is not an AUTO-equivalence.
(Analogy: there are zillions of equivalent two-dimensional
vector spaces, but what's interesting is the group GL(2) of
equivalences of any given one.) Mirror symmetry involves
*families* of categories. What buys you something is relating
one family (or a one-family deformation of your category)
to another. The relation of the parameters is the mirror map,
and if the different parameters carry symplectic or complex
information, then you can get one from another.
-ez
Urs Schreiber
Mar30-05, 03:09 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Zaz" <zaslow@math.northwestern.edu> schrieb im Newsbeitrag\nnews:ba9ff483.0503291555.58f618de@pos ting.google.com...\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:\n>> That makes sense. But since the Picard group is that of _all_ the\n>> autoequivalences, should it not contain more transformations than one\n>> gets\n>> by going around closed loops in moduli space? Or is that all there is?\n>>\n>> Since the triangulated Fukaya category describing A-branes is conjectured\n>> to be equivalent to the derived category describing D-branes, it seems\n>> natural to guess that mirror symmetry is also part of the Picard group,\n>> which indeed is consistent with some hints I saw in the literature. But\n>> if\n>> that\'s true, shouldn\'t all dualities be sitting inside the Picard group?\n>>\n>\n>\n> If you have an example, maybe there are more equivalences\n> than are realized with loops. Maybe not.\n\nUnfortunately I do not have any examples. As you know, it is much easier in\nthis context to discuss the general setup than to compute any details in a\nspecific case. Like Berenstein and Douglas remark in a closely related\ncontext on p. 39 of hep-th/0207027:\n\n"This is rather formidable in general [...] we suspect that the physics\nreader who tries to follow this up will soon find himself seeking\nprofessional help."\n\n;-)\n\nBut in fact the same paper also offers at least a different interpretation\nof these equivalences, namely in terms of Seiberg duality. I am not sure if\nthey claim that every "tilting equivalence" can be interpreted as a\nmanifestation of Seiberg duality or just that Seiberg duality is an example\nof a tilting equivalence.\n\nOf course these tilting equivalences in general go between different derived\ncategories. Two algebras A and B are Morita equivalent if the category\n\nA-Mod\n\nof (left, say) A-modules\nis equivalent to the category\n\nB-Mod,\n\nof left B-modules\n\nwhich apparently is the case if and only if there is a weakly invertible A-B\nbimodule T which gives rise to the equivalence functor\n\n(T \\otimes_B o ) : B-Mod -> A-Mod .\n\n\nSimilarly, two derived categories of such modules are derived equivalent if\nand only if there is a tilting complex (a weakly invertible complex)\n\nT in D( A-Mod-B )\n\nand\n\n(T \\otimes^L o ) : D(B-Mod) -> D(A-Mod) .\n\ngives the equivalence.\n\nInterestingly, when A and B are derived equivalent in this sense it means\nthat "large" _sub_categories of A-Mod and B-Mod are Morita equivalent in the\nordinary sense.\n\nSo now let A = B. Wouldn\'t this mean that there are nontrivial "auto\nSeiberg-equivalences" in the derived Picard group? Do they all come from\nnontrivial monodromies in moduli space? Maybe they do, I don\'t know.\n\n\n> I can\'t see that\n> it matters much one way or another. For example, take any\n> one-parameter family and then consider its universal cover. Quotienting by\n> the group of equivalences generated by loops\n> will give you back your example (assuming the monodromy\n> representation is faithful, or some such jargon). Quotienting\n> by anything less will give you a family with more equivalences\n> than are realized by loops.\n\n\nNot sure what point you are making here. All I am trying to do is getting a\nbetter understanding of what all these equivalences in the derived Picard\ngroup are like, physically.\n\n\n> As for mirror symmetry, to relate one category to another\n> does little for you, practically speaking. Object Ralph is\n> related to object Fred, Alice to Wilma, etc. Remember that\n> equating one category to another is not an AUTO-equivalence.\n\n\nSure, yes. Unless of course the other category secretly "is" the former one.\n\n\n> Analogy: there are zillions of equivalent two-dimensional\n> vector spaces, but what\'s interesting is the group GL(2) of\n> equivalences of any given one.\n\n\nRight, and that in fact makes my point. When you find an invertible morphism\nbetween any two n-dimensional vector spaces you can always get an\nautomorphism from this by using any (other) isomorphism between the two to\nidentify them.\n\n\n> Mirror symmetry involves\n> *families* of categories.\n\n\nWhat do you mean by that? Mirror symmetry asserts that there is an\nequivalence between some derived category of coherent sheaves and some\ntriangulated Fukaya category. (Ah, of course there are different such\ncategories for different points in moduli space. Probably that\'s what you\nmean?)\n\nOr at least that\'s the idea. Apparently there is this and that ingredient\nwhich one has to throw into the triangulated Fukaya theory to make this\nreally work. But once it works, it could be that the resulting improved\nFukaya category actually has a strictly invertible functor to the derived\ncategory of coherent sheaves. This would make any equivalence between the\ntwo into an auto-equivalence of any one of them.\n\nI don\'t know if this is the case. But does there exist any indications that\nit is not the case?\n\nI\'d be content either way. I am just trying to understand what kinds of\ntransformations we find in the derived Picard group.\n\n\n\nBTW, something concerning this category A-Mod-B makes me wonder: We can\nthink of this as a 2-category with\n\nobjects being algebras A, B, C\n\nmorphisms from A to B being A-B bimodules\n\n2-morphims being bimodule homomorphisms.\n\nThis is reminiscent of the bicategory of von-Neumann algebras that Stolz and\nTeichner use in "What is an elliptic object?"\nhttp://math.ucsd.edu/~teichner/Papers/Oxford.pdf (pp. 59)\n\nThere, however, one uses confusion, er, I mean "Connes fusion", instead of\nthe ordinary tensor product over A,B,... to define the monoidal product.\n\nUnfortunately I am not completely sure about the definition of Connes fusion\nthat the authors give, but it seems to be essentially the ordinary tensor\nproduct refined in such a way that we can think of all bimodules involved as\nbeing Hilbert spaces.\n\nIt would be nice to better understand what this modification amounts to. A-B\nbimodules which are weakly invertible with respect to the Connes fusion\nproduct should give equivalences between A-Mod and B-Mod, too, it seems.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Zaz" <zaslow@math.northwestern.edu> schrieb im Newsbeitrag
news:ba9ff483.0503291555.58f618de@posting.google.c om...
> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:
>> That makes sense. But since the Picard group is that of _all_ the
>> autoequivalences, should it not contain more transformations than one
>> gets
>> by going around closed loops in moduli space? Or is that all there is?
>>
>> Since the triangulated Fukaya category describing A-branes is conjectured
>> to be equivalent to the derived category describing D-branes, it seems
>> natural to guess that mirror symmetry is also part of the Picard group,
>> which indeed is consistent with some hints I saw in the literature. But
>> if
>> that's true, shouldn't all dualities be sitting inside the Picard group?
>>
>
>
> If you have an example, maybe there are more equivalences
> than are realized with loops. Maybe not.
Unfortunately I do not have any examples. As you know, it is much easier in
this context to discuss the general setup than to compute any details in a
specific case. Like Berenstein and Douglas remark in a closely related
context on p. 39 of http://www.arxiv.org/abs/hep-th/0207027:
"This is rather formidable in general [...] we suspect that the physics
reader who tries to follow this up will soon find himself seeking
professional help."
;-)
But in fact the same paper also offers at least a different interpretation
of these equivalences, namely in terms of Seiberg duality. I am not sure if
they claim that every "tilting equivalence" can be interpreted as a
manifestation of Seiberg duality or just that Seiberg duality is an example
of a tilting equivalence.
Of course these tilting equivalences in general go between different derived
categories. Two algebras A and B are Morita equivalent if the category
A-Mod
of (left, say) A-modules
is equivalent to the category
B-Mod,
of left B-modules
which apparently is the case if and only if there is a weakly invertible A-B
bimodule T which gives rise to the equivalence functor
(T \otimes_B o ) : B-Mod -> A-Mod .
Similarly, two derived categories of such modules are derived equivalent if
and only if there is a tilting complex (a weakly invertible complex)
T in D( A-Mod-B )
and
(T \otimes^L o ) : D(B-Mod) -> D(A-Mod) .
gives the equivalence.
Interestingly, when A and B are derived equivalent in this sense it means
that "large" _sub_categories of A-Mod and B-Mod are Morita equivalent in the
ordinary sense.
So now let A = B. Wouldn't this mean that there are nontrivial "auto
Seiberg-equivalences" in the derived Picard group? Do they all come from
nontrivial monodromies in moduli space? Maybe they do, I don't know.
> I can't see that
> it matters much one way or another. For example, take any
> one-parameter family and then consider its universal cover. Quotienting by
> the group of equivalences generated by loops
> will give you back your example (assuming the monodromy
> representation is faithful, or some such jargon). Quotienting
> by anything less will give you a family with more equivalences
> than are realized by loops.
Not sure what point you are making here. All I am trying to do is getting a
better understanding of what all these equivalences in the derived Picard
group are like, physically.
> As for mirror symmetry, to relate one category to another
> does little for you, practically speaking. Object Ralph is
> related to object Fred, Alice to Wilma, etc. Remember that
> equating one category to another is not an AUTO-equivalence.
Sure, yes. Unless of course the other category secretly "is" the former one.
> Analogy: there are zillions of equivalent two-dimensional
> vector spaces, but what's interesting is the group GL(2) of
> equivalences of any given one.
Right, and that in fact makes my point. When you find an invertible morphism
between any two n-dimensional vector spaces you can always get an
automorphism from this by using any (other) isomorphism between the two to
identify them.
> Mirror symmetry involves
> *families* of categories.
What do you mean by that? Mirror symmetry asserts that there is an
equivalence between some derived category of coherent sheaves and some
triangulated Fukaya category. (Ah, of course there are different such
categories for different points in moduli space. Probably that's what you
mean?)
Or at least that's the idea. Apparently there is this and that ingredient
which one has to throw into the triangulated Fukaya theory to make this
really work. But once it works, it could be that the resulting improved
Fukaya category actually has a strictly invertible functor to the derived
category of coherent sheaves. This would make any equivalence between the
two into an auto-equivalence of any one of them.
I don't know if this is the case. But does there exist any indications that
it is not the case?
I'd be content either way. I am just trying to understand what kinds of
transformations we find in the derived Picard group.
BTW, something concerning this category A-Mod-B makes me wonder: We can
think of this as a 2-category with
objects being algebras A, B, C
morphisms from A to B being A-B bimodules
2-morphims being bimodule homomorphisms.
This is reminiscent of the bicategory of von-Neumann algebras that Stolz and
Teichner use in "What is an elliptic object?"
http://math.ucsd.edu/~teichner/Papers/Oxford.pdf (pp. 59)
There, however, one uses confusion, er, I mean "Connes fusion", instead of
the ordinary tensor product over A,B,... to define the monoidal product.
Unfortunately I am not completely sure about the definition of Connes fusion
that the authors give, but it seems to be essentially the ordinary tensor
product refined in such a way that we can think of all bimodules involved as
being Hilbert spaces.
It would be nice to better understand what this modification amounts to. A-B
bimodules which are weakly invertible with respect to the Connes fusion
product should give equivalences between A-Mod and B-Mod, too, it seems.
Urs Schreiber
Mar30-05, 06:48 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> schrieb im Newsbeitrag\nnews:3av8m3F5tfiqgU1@news.dfncis.de.. .\n\n> Wouldn\'t this mean that there are nontrivial "auto Seiberg-equivalences"\n> in the derived Picard group? Do they all come from nontrivial monodromies\n> in moduli space? Maybe they do, I don\'t know.\n\nI just found that in\n\nS. Franco & A. Hanany\nOn the fate of tachyonic quivers\nhep-th/0408016\n\nthis is reviewed in the introduction. It is well known that Seiberg duality\ntransformations form a proper subgroup of the group of so-called\nPicard-Lefschetz monodromies. Those monodromies which do not give Seiberg\nduality are called fractional Seiberg dualities -- as certainly everyone\nexcept me was well aware of...\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> schrieb im Newsbeitrag
news:3av8m3F5tfiqgU1@news.dfncis.de...
> Wouldn't this mean that there are nontrivial "auto Seiberg-equivalences"
> in the derived Picard group? Do they all come from nontrivial monodromies
> in moduli space? Maybe they do, I don't know.
I just found that in
S. Franco & A. Hanany
On the fate of tachyonic quivers
http://www.arxiv.org/abs/hep-th/0408016
this is reviewed in the introduction. It is well known that Seiberg duality
transformations form a proper subgroup of the group of so-called
Picard-Lefschetz monodromies. Those monodromies which do not give Seiberg
duality are called fractional Seiberg dualities -- as certainly everyone
except me was well aware of...
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nEspecially if you would like some explicit examples, you might find the\nlanguage of exceptional collections more helpful. Exceptional\ncollections in certain special cases generate the derived category of\ncoherent sheaves. Physically, they\'re just a nice D-brane basis.\nSeiberg duality is then a special sequence of mutations. (For a\nmutation, roughly speaking, take two neighboring D-branes, form a bound\nstate, then erase one of the D-branes you started with from the\ncollection -- in other words a mutation is a special kind of change of\nbasis.)\n\nI\'ve always been a bit uncomfortable with these PL monodromies. They\nare not the same as the monodromies you get from solving the\nPicard-Fuchs equations and moving around the singular points of the\nKaehler moduli space of the local Calabi-Yau. The monodromies Aaron\nreferred to earlier were of the second kind, I believe. If there is\nanyone out there who can point me to a paper where these two ideas are\ncarefully related to each other, I would be very interested.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Especially if you would like some explicit examples, you might find the
language of exceptional collections more helpful. Exceptional
collections in certain special cases generate the derived category of
coherent sheaves. Physically, they're just a nice D-brane basis.
Seiberg duality is then a special sequence of mutations. (For a
mutation, roughly speaking, take two neighboring D-branes, form a bound
state, then erase one of the D-branes you started with from the
collection -- in other words a mutation is a special kind of change of
basis.)
I've always been a bit uncomfortable with these PL monodromies. They
are not the same as the monodromies you get from solving the
Picard-Fuchs equations and moving around the singular points of the
Kaehler moduli space of the local Calabi-Yau. The monodromies Aaron
referred to earlier were of the second kind, I believe. If there is
anyone out there who can point me to a paper where these two ideas are
carefully related to each other, I would be very interested.
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Chris wrote:\n\n> I\'ve always been a bit uncomfortable with these PL monodromies. They\n> are not the same as the monodromies you get from solving the\n> Picard-Fuchs equations and moving around the singular points of the\n> Kaehler moduli space of the local Calabi-Yau. The monodromies Aaron\n> referred to earlier were of the second kind, I believe. If there is\n> anyone out there who can point me to a paper where these two ideas are\n> carefully related to each other, I would be very interested.\n\nI thought these two kinds of monodromies are the same.\nMore precisely, one moves around singular points in the complex\nstructure moduli space of the mirror CY; only the complex structure\nmoduli are globally well defined, so one wants to make the loops\nthere. The effect is then a mixing of all the even dimensional\nbranes on the original CY, and this may be viewed, more or less per\ndef, as automorphisms of the category.\n\nFor some lit, see hep-th/0102198, hep-th/0110071, hep-th/0209161.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Chris wrote:
> I've always been a bit uncomfortable with these PL monodromies. They
> are not the same as the monodromies you get from solving the
> Picard-Fuchs equations and moving around the singular points of the
> Kaehler moduli space of the local Calabi-Yau. The monodromies Aaron
> referred to earlier were of the second kind, I believe. If there is
> anyone out there who can point me to a paper where these two ideas are
> carefully related to each other, I would be very interested.
I thought these two kinds of monodromies are the same.
More precisely, one moves around singular points in the complex
structure moduli space of the mirror CY; only the complex structure
moduli are globally well defined, so one wants to make the loops
there. The effect is then a mixing of all the even dimensional
branes on the original CY, and this may be viewed, more or less per
def, as automorphisms of the category.
For some lit, see http://www.arxiv.org/abs/hep-th/0102198, http://www.arxiv.org/abs/hep-th/0110071, http://www.arxiv.org/abs/hep-th/0209161.
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n> I thought these two kinds of monodromies are the same.\n> More precisely, one moves around singular points in the complex\n> structure moduli space of the mirror CY; only the complex structure\n> moduli are globally well defined, so one wants to make the loops\n> there. The effect is then a mixing of all the even dimensional\n> branes on the original CY, and this may be viewed, more or less per\n> def, as automorphisms of the category.\n>\n> For some lit, see hep-th/0102198, hep-th/0110071, hep-th/0209161.\n\nThanks for the refs. I\'ll take a look. In the meanwhile, let me ask a\nmore precise question. Consider the anticanonical bundle over P^2.\nThe PF equations for the mirror are relatively simple, and many people\nhave carefully studied all the monodromies. These monodromies have a\nwell defined action on the charges of a D-brane. Now take the\nfollowing exceptional collection on P^2\n\nO, O(1), O(2)\n\nThere is a mutation such that\n\nL_{O} O(1), O, O(2)\n\nis again an exceptional collection which generates the derived category\nof coherent sheaves on P^2. This mutation can be understood according\nto Vafa and collaborators as a Picard-Lefschetz monodromy in the W\nplane. Can someone tell me how to get this action on an exceptional\ncollection from a monodromy of the solutions to the PF equations?\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I thought these two kinds of monodromies are the same.
> More precisely, one moves around singular points in the complex
> structure moduli space of the mirror CY; only the complex structure
> moduli are globally well defined, so one wants to make the loops
> there. The effect is then a mixing of all the even dimensional
> branes on the original CY, and this may be viewed, more or less per
> def, as automorphisms of the category.
>
> For some lit, see http://www.arxiv.org/abs/hep-th/0102198, http://www.arxiv.org/abs/hep-th/0110071, http://www.arxiv.org/abs/hep-th/0209161.
Thanks for the refs. I'll take a look. In the meanwhile, let me ask a
more precise question. Consider the anticanonical bundle over P^2.
The PF equations for the mirror are relatively simple, and many people
have carefully studied all the monodromies. These monodromies have a
well defined action on the charges of a D-brane. Now take the
following exceptional collection on P^2
O, O(1), O(2)
There is a mutation such that
L_{O} O(1), O, O(2)
is again an exceptional collection which generates the derived category
of coherent sheaves on P^2. This mutation can be understood according
to Vafa and collaborators as a Picard-Lefschetz monodromy in the W
plane. Can someone tell me how to get this action on an exceptional
collection from a monodromy of the solutions to the PF equations?
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Chris <herzog@kitp.ucsb.edu> wrote in message news: Now take the\n> following exceptional collection on P^2\n>\n> O, O(1), O(2)\n>\n> There is a mutation such that\n>\n> L_{O} O(1), O, O(2)\n>\n> is again an exceptional collection which generates the derived category\n> of coherent sheaves on P^2. This mutation can be understood according\n> to Vafa and collaborators as a Picard-Lefschetz monodromy in the W\n> plane. Can someone tell me how to get this action on an exceptional\n> collection from a monodromy of the solutions to the PF equations?\n\n\nI think I understand the source of your confusion. The Picard-Fuchs\ndifferential equations are obeyed by periods. However, not all solutions\ncorrespond to periods, as, for example, the average of two solutions\nof a linear equation is still a solution, but the average of two integral\nhomology cycles is not an integral cycle.\n\nFor your example, you need to find the linear combinations of (some basis\nof) solutions to the Picard-Fuchs equations which represent the "cycles"\nmirror to the objects in your exceptional collection (I use the quotation\nmarks because the noncompactness adds some subtlety).\n\nDoes this help at all? I hope so.\n\n-ez\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Chris <herzog@kitp.ucsb.edu> wrote in message news: Now take the
> following exceptional collection on P^2
>
> O, O(1), O(2)
>
> There is a mutation such that
>
> L_{O} O(1), O, O(2)
>
> is again an exceptional collection which generates the derived category
> of coherent sheaves on P^2. This mutation can be understood according
> to Vafa and collaborators as a Picard-Lefschetz monodromy in the W
> plane. Can someone tell me how to get this action on an exceptional
> collection from a monodromy of the solutions to the PF equations?
I think I understand the source of your confusion. The Picard-Fuchs
differential equations are obeyed by periods. However, not all solutions
correspond to periods, as, for example, the average of two solutions
of a linear equation is still a solution, but the average of two integral
homology cycles is not an integral cycle.
For your example, you need to find the linear combinations of (some basis
of) solutions to the Picard-Fuchs equations which represent the "cycles"
mirror to the objects in your exceptional collection (I use the quotation
marks because the noncompactness adds some subtlety).
Does this help at all? I hope so.
-ez
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nI should probably clarify that I am not asking this question because I\ndon\'t know where to begin. I\'m asking because I have a pretty good\nidea how to go about answering it, tried for a week or two, and failed.\n\n\nThere is in fact a nice paper by Mohri, Onjo, and Yang where these\nlinear combinations are worked out for P^2 (hep-th/0009072). They are\ncareful about issues of noncompactness, and I think their expressions\nfor the action of the monodromy on the D-brane charges is correct.\n\nLet M_1, M_2, and M_3 be 3x3 monodromy matrices which come from the\norbifold, large volume, and conifold points in the Kaehler moduli\nspace. These matrices act on the D-brane charges. One may consider\nsome arbitrary group element generated by M_1, M_2, and M_3.\nPresumably there is such an element that takes O(1) to L_O O(1)\n(although even here, I haven\'t checked recently, I\'m a little worried\nabout the general case, and one has to be careful about the gradings).\nAs far as I could tell, there is no such element which converts O,\nO(1), O(2) to L_O O(1), O, O(2).\n\nIn other words, I don\'t see a natural action of these PF monodromies on\nan exceptional collection which is equivalent to mutation. I only see\na very cloogy procedure where you stare at the collection,\ndecide which sheaves you want to mutate, construct some special element\nin this group of monodromies, and then act only on one sheaf.\n\nThis cloogy procedure seems rather different from Vafa et al.\'s W plane\npicture where all the D-branes are drawn and PL monodromies move one\nbrane past another rather than around special points in the Kaehler\nmoduli space.\n\nAhh, the difficulties of communicating through writing... But thanks\nfor your patience and your continued comments about my posts.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I should probably clarify that I am not asking this question because I
don't know where to begin. I'm asking because I have a pretty good
idea how to go about answering it, tried for a week or two, and failed.
There is in fact a nice paper by Mohri, Onjo, and Yang where these
linear combinations are worked out for P^2 (http://www.arxiv.org/abs/hep-th/0009072). They are
careful about issues of noncompactness, and I think their expressions
for the action of the monodromy on the D-brane charges is correct.
Let M_1, M_2, and M_3 be 3x3 monodromy matrices which come from the
orbifold, large volume, and conifold points in the Kaehler moduli
space. These matrices act on the D-brane charges. One may consider
some arbitrary group element generated by M_1, M_2, and M_3.
Presumably there is such an element that takes O(1) to L_O O(1)
(although even here, I haven't checked recently, I'm a little worried
about the general case, and one has to be careful about the gradings).
As far as I could tell, there is no such element which converts O,
O(1), O(2) to L_O O(1), O, O(2).
In other words, I don't see a natural action of these PF monodromies on
an exceptional collection which is equivalent to mutation. I only see
a very cloogy procedure where you stare at the collection,
decide which sheaves you want to mutate, construct some special element
in this group of monodromies, and then act only on one sheaf.
This cloogy procedure seems rather different from Vafa et al.'s W plane
picture where all the D-branes are drawn and PL monodromies move one
brane past another rather than around special points in the Kaehler
moduli space.
Ahh, the difficulties of communicating through writing... But thanks
for your patience and your continued comments about my posts.
Urs Schreiber
Apr1-05, 12:15 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Chris" <herzog@kitp.ucsb.edu> schrieb im Newsbeitrag\nnews:1112237688.579818.204810@z14g200 0cwz.googlegroups.com...\n>\n> Especially if you would like some explicit examples, you might find the\n> language of exceptional collections more helpful. Exceptional\n> collections in certain special cases generate the derived category of\n> coherent sheaves. Physically, they\'re just a nice D-brane basis.\n> Seiberg duality is then a special sequence of mutations. (For a\n> mutation, roughly speaking, take two neighboring D-branes, form a bound\n> state, then erase one of the D-branes you started with from the\n> collection -- in other words a mutation is a special kind of change of\n> basis.)\n\n\nI have had a look at your paper hep-th/0405118.\n\nI think I understand in principle what an exceptional collection of sheaves\nis, what a mutation of an exceptional collection is and that to each\nexceptional collection is associated a "Beilison quiver", which is not susy,\nand an "exceptional quiver", which is, where the latter can be obtained from\nthe former by promoting relations to generators.\n\nNow, you discuss a notion "SD" of Seiberg duality, acting on an exceptional\ncollection by means of mutations, and show that, via the susy quiver\nassociated with this collection, this operation indeed coincides with the\nordinary operation of Seiberg duality at the level of the gauge theory\ndefined by the quiver. Right?\n\nThe relation to tilting equivalences of this construction which one\nimmediately suspects to be there, is not clear in detail yet, you say,\nbecause, remarkably, Bondal has shown that the derived category of reps of\nthe _Beilison_ quiver is equivalent to the derived category of coherent\nsheaves, while for the relation to tilting equivalences it would have to be\nthe derived category of reps of the susy quiver instead. Right?\n\n\nI\'d have a more general question about these equivalent derived quiver rep\ncategories:\n\nThere is a basic fact that every category is equivalent to a skeletal one. A\nskeletal category is one where all isomorphic objects are actually equal,\ni.e. where the only isomorphisms are identity arrows.\n\nGiven any category we can obtain the skeletal one that it is equivalent to\nby picking one representative object from each class of isomorphic objects.\n\nIn general there will be more than one way to truncate a given category to\nan equivalent skeletal one by picking such representatives.\n\nNow, I might be mistaken, but I believe that a derived category of coherent\nsheaves D(X) in general has many more objects than a derived category\nD(Rep(Q)) of quiver reps that it is equivalent to.\n\nThis would suggest that we can think of D(Rep(Q)) as obtained from the full\nD(X) by eliminating objects from isomorphism classes of objects. It might\nnot actually yield a skeletal category, but one with smaller isomorphism\nclasses of objects.\n\nThis would naturally explain the "gauge freedom" (given by Seiberg\nduality/mutations/tilting equivalences/etc.) in going from D(X) to\nD(Rep(Q)).\n\nIs this what is going on? (It\'s maybe obvious, but I have not seen it stated\nexplicitly anywhere yet.)\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Chris" <herzog@kitp.ucsb.edu> schrieb im Newsbeitrag
news:1112237688.579818.204810@z14g2000cwz.googlegr oups.com...
>
> Especially if you would like some explicit examples, you might find the
> language of exceptional collections more helpful. Exceptional
> collections in certain special cases generate the derived category of
> coherent sheaves. Physically, they're just a nice D-brane basis.
> Seiberg duality is then a special sequence of mutations. (For a
> mutation, roughly speaking, take two neighboring D-branes, form a bound
> state, then erase one of the D-branes you started with from the
> collection -- in other words a mutation is a special kind of change of
> basis.)
I have had a look at your paper http://www.arxiv.org/abs/hep-th/0405118.
I think I understand in principle what an exceptional collection of sheaves
is, what a mutation of an exceptional collection is and that to each
exceptional collection is associated a "Beilison quiver", which is not susy,
and an "exceptional quiver", which is, where the latter can be obtained from
the former by promoting relations to generators.
Now, you discuss a notion "SD" of Seiberg duality, acting on an exceptional
collection by means of mutations, and show that, via the susy quiver
associated with this collection, this operation indeed coincides with the
ordinary operation of Seiberg duality at the level of the gauge theory
defined by the quiver. Right?
The relation to tilting equivalences of this construction which one
immediately suspects to be there, is not clear in detail yet, you say,
because, remarkably, Bondal has shown that the derived category of reps of
the _Beilison_ quiver is equivalent to the derived category of coherent
sheaves, while for the relation to tilting equivalences it would have to be
the derived category of reps of the susy quiver instead. Right?
I'd have a more general question about these equivalent derived quiver rep
categories:
There is a basic fact that every category is equivalent to a skeletal one. A
skeletal category is one where all isomorphic objects are actually equal,
i.e. where the only isomorphisms are identity arrows.
Given any category we can obtain the skeletal one that it is equivalent to
by picking one representative object from each class of isomorphic objects.
In general there will be more than one way to truncate a given category to
an equivalent skeletal one by picking such representatives.
Now, I might be mistaken, but I believe that a derived category of coherent
sheaves D(X) in general has many more objects than a derived category
D(Rep(Q)) of quiver reps that it is equivalent to.
This would suggest that we can think of D(Rep(Q)) as obtained from the full
D(X) by eliminating objects from isomorphism classes of objects. It might
not actually yield a skeletal category, but one with smaller isomorphism
classes of objects.
This would naturally explain the "gauge freedom" (given by Seiberg
duality/mutations/tilting equivalences/etc.) in going from D(X) to
D(Rep(Q)).
Is this what is going on? (It's maybe obvious, but I have not seen it stated
explicitly anywhere yet.)
Urs Schreiber
Apr1-05, 01:37 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"WL" <wolfgang.lerche@cern.ch> schrieb im Newsbeitrag\nnews:1112280189.430706.170220@l41g200 0cwc.googlegroups.com...\n\n> For some lit, see hep-th/0102198, hep-th/0110071, hep-th/0209161.\n\nFollowing references in these I came across\n\nSeidel & Thomas\nBraid Group actions on Derived Categories of Coherent Sheaves\nmath.AG/0001043\n\nI have only had time to read the first few pages. The authors discuss the\ngroup\n\nAuteq(D(X))\n\nof autoequivalences of the derived category D(X) of coherent sheaves.\n\nNow, as we have discussed here before, I know from the papers by Yekutieli\n\nhttp://www.math.bgu.ac.il/~amyekut/publications/publications.html\n\nthat the group of autoequivalence of the derived category of\n_quiver_representations_ is known and called the derived Picard group DPic\nassociated to the path algebra of the quiver. From what Seidel and Thomas\nwrite on their page 2 together with some facts stated by Yekutieli DPic must\nbe closely related to Auteq(D(X)).\n\nI am hoping it is actually the same group. But is it?\n\nDPic consists of isomorphism classes of all tilting complexes.\n\nNot sure about the symbols used on the very top of p. 3 in Seidel&Thomas.\nIs that the same as tilting complexes?\n\nThanks for any help.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>"WL" <wolfgang.lerche@cern.ch> schrieb im Newsbeitrag
news:1112280189.430706.170220@l41g2000cwc.googlegr oups.com...
> For some lit, see http://www.arxiv.org/abs/hep-th/0102198, http://www.arxiv.org/abs/hep-th/0110071, http://www.arxiv.org/abs/hep-th/0209161.
Following references in these I came across
Seidel & Thomas
Braid Group actions on Derived Categories of Coherent Sheaves
math.AG/0001043
I have only had time to read the first few pages. The authors discuss the
group
Auteq(D(X))
of autoequivalences of the derived category D(X) of coherent sheaves.
Now, as we have discussed here before, I know from the papers by Yekutieli
http://www.math.bgu.ac.il/~amyekut/publications/publications.html
that the group of autoequivalence of the derived category of
_quiver_representations_ is known and called the derived Picard group DPic
associated to the path algebra of the quiver. From what Seidel and Thomas
write on their page 2 together with some facts stated by Yekutieli DPic must
be closely related to Auteq(D(X)).
I am hoping it is actually the same group. But is it?
DPic consists of isomorphism classes of all tilting complexes.
Not sure about the symbols used on the very top of p. 3 in Seidel&Thomas.
Is that the same as tilting complexes?
Thanks for any help.
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Urs Schreiber wrote:\n\n> Now, you discuss a notion "SD" of Seiberg duality, acting on an exceptional\n> collection by means of mutations, and show that, via the susy quiver\n> associated with this collection, this operation indeed coincides with the\n> ordinary operation of Seiberg duality at the level of the gauge theory\n> defined by the quiver. Right?\n\nThat\'s right, but I suspect I was a little brief in section 6, and I\ndon\'t discuss superpotentials which ideally I should have. The\nsequence of mutations equivalent to Seiberg duality was proposed before\nme by Wijnholt (hep-th/0212021) and by Vafa and collaborators\n(hep-th/0110028). What I showed is that this sequence of mutations is\na well defined operation on a special subset of exceptional collections\n-- those which generate something I called a strong helix. The work\ncleared up a lot of confusion I had about these weird exceptional\ncollections which seemed to produce pathological gauge theories. The\nstrong helix guys never have these pathologies.\n\n> The relation to tilting equivalences of this construction which one\n> immediately suspects to be there, is not clear in detail yet, you say,\n> because, remarkably, Bondal has shown that the derived category of reps of\n> the _Beilison_ quiver is equivalent to the derived category of coherent\n> sheaves, while for the relation to tilting equivalences it would have to be\n> the derived category of reps of the susy quiver instead. Right?\n\nAt the time I wrote the paper, what you say is correct. Since then, I\nthink Bridgeland (math.AG/0502050) and Paul Aspinwall and Ilarion Melnikov\n(hep-th/0405134) have gone a long way, if not all the way, toward showing\nthis equivalence for the local Calabi-Yau. Aaron, or somebody else,\ncorrect me if I\'m wrong.\n\n> I\'d have a more general question about these equivalent derived quiver\n> rep categories:\n\nThis one I\'ll have to think about. I\'ll just say again, in case it\'s\nrelevant, that in general there appear to be far more exceptional\ncollections than there are reasonable quiver gauge theories for a given\ndel Pezzo. It\'s only the strong helix guys that give good quivers.\nSeiberg duality takes one strong helix guy to another.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Urs Schreiber wrote:
> Now, you discuss a notion "SD" of Seiberg duality, acting on an exceptional
> collection by means of mutations, and show that, via the susy quiver
> associated with this collection, this operation indeed coincides with the
> ordinary operation of Seiberg duality at the level of the gauge theory
> defined by the quiver. Right?
That's right, but I suspect I was a little brief in section 6, and I
don't discuss superpotentials which ideally I should have. The
sequence of mutations equivalent to Seiberg duality was proposed before
me by Wijnholt (http://www.arxiv.org/abs/hep-th/0212021) and by Vafa and collaborators
(http://www.arxiv.org/abs/hep-th/0110028). What I showed is that this sequence of mutations is
a well defined operation on a special subset of exceptional collections
-- those which generate something I called a strong helix. The work
cleared up a lot of confusion I had about these weird exceptional
collections which seemed to produce pathological gauge theories. The
strong helix guys never have these pathologies.
> The relation to tilting equivalences of this construction which one
> immediately suspects to be there, is not clear in detail yet, you say,
> because, remarkably, Bondal has shown that the derived category of reps of
> the _Beilison_ quiver is equivalent to the derived category of coherent
> sheaves, while for the relation to tilting equivalences it would have to be
> the derived category of reps of the susy quiver instead. Right?
At the time I wrote the paper, what you say is correct. Since then, I
think Bridgeland (math.AG/0502050) and Paul Aspinwall and Ilarion Melnikov
(http://www.arxiv.org/abs/hep-th/0405134) have gone a long way, if not all the way, toward showing
this equivalence for the local Calabi-Yau. Aaron, or somebody else,
correct me if I'm wrong.
> I'd have a more general question about these equivalent derived quiver
> rep categories:
This one I'll have to think about. I'll just say again, in case it's
relevant, that in general there appear to be far more exceptional
collections than there are reasonable quiver gauge theories for a given
del Pezzo. It's only the strong helix guys that give good quivers.
Seiberg duality takes one strong helix guy to another.
Aaron Bergman
Apr2-05, 05:33 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <3b5he2F6c8b90U1@news.dfncis.de>,\nUrs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n> The relation to tilting equivalences of this construction which one\n> immediately suspects to be there, is not clear in detail yet, you say,\n> because, remarkably, Bondal has shown that the derived category of reps of\n> the _Beilison_ quiver is equivalent to the derived category of coherent\n> sheaves, while for the relation to tilting equivalences it would have to be\n> the derived category of reps of the susy quiver instead. Right?\n\nAs Chris mentions, the equivalence for the completed quiver is\ndemonstrated by Bridgeland (math.AG/0502050).\n\n[...]\n\n> Now, I might be mistaken, but I believe that a derived category of coherent\n> sheaves D(X) in general has many more objects than a derived category\n> D(Rep(Q)) of quiver reps that it is equivalent to.\n\nWhy do you think that? Among other things, the objects in a category\nmight not even form a set (although I guess the do (?) in this case), so\nI\'m not sure what you mean by \'more objects\'. There are at least aleph\nobjects in each category. More than that, there are certainly aleph\nobjects in each equivalence class on the quiver rep side.\n\n> This would suggest that we can think of D(Rep(Q)) as obtained from the full\n> D(X) by eliminating objects from isomorphism classes of objects. It might\n> not actually yield a skeletal category, but one with smaller isomorphism\n> classes of objects.\n\nD(Rep(Q)) is definitely not skeletal.\n\n> This would naturally explain the "gauge freedom" (given by Seiberg\n> duality/mutations/tilting equivalences/etc.) in going from D(X) to\n> D(Rep(Q)).\n\nThe Seiberg dualities generally change Q. So, you have equivalences to\nthe derived category of representations of *different* quivers. I really\ndoubt that these derived equivalences lift to isomorphisms of the quiver\nalgebras (though it could be true). If you like, they are distinguished\nby the residual structure (by which I mean the t-structure) inherited\nfrom the fact that they are derived categories rather than just abstract\ntriangulated categories.\n\nAlthough it can\'t happen when the canonical class isn\'t trivial, it\'s\nworth pointing out that two different varieties can have equivalent\nderived categories of coherent sheaves, so this isn\'t at all unique to\nderived module categories.\n\nAaron\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <3b5he2F6c8b90U1@news.dfncis.de>,
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
> The relation to tilting equivalences of this construction which one
> immediately suspects to be there, is not clear in detail yet, you say,
> because, remarkably, Bondal has shown that the derived category of reps of
> the _Beilison_ quiver is equivalent to the derived category of coherent
> sheaves, while for the relation to tilting equivalences it would have to be
> the derived category of reps of the susy quiver instead. Right?
As Chris mentions, the equivalence for the completed quiver is
demonstrated by Bridgeland (math.AG/0502050).
[...]
> Now, I might be mistaken, but I believe that a derived category of coherent
> sheaves D(X) in general has many more objects than a derived category
> D(Rep(Q)) of quiver reps that it is equivalent to.
Why do you think that? Among other things, the objects in a category
might not even form a set (although I guess the do (?) in this case), so
I'm not sure what you mean by 'more objects'. There are at least \aleph
objects in each category. More than that, there are certainly \aleph
objects in each equivalence class on the quiver rep side.
> This would suggest that we can think of D(Rep(Q)) as obtained from the full
> D(X) by eliminating objects from isomorphism classes of objects. It might
> not actually yield a skeletal category, but one with smaller isomorphism
> classes of objects.
D(Rep(Q)) is definitely not skeletal.
> This would naturally explain the "gauge freedom" (given by Seiberg
> duality/mutations/tilting equivalences/etc.) in going from D(X) to
> D(Rep(Q)).
The Seiberg dualities generally change Q. So, you have equivalences to
the derived category of representations of *different* quivers. I really
doubt that these derived equivalences lift to isomorphisms of the quiver
algebras (though it could be true). If you like, they are distinguished
by the residual structure (by which I mean the t-structure) inherited
from the fact that they are derived categories rather than just abstract
triangulated categories.
Although it can't happen when the canonical class isn't trivial, it's
worth pointing out that two different varieties can have equivalent
derived categories of coherent sheaves, so this isn't at all unique to
derived module categories.
Aaron
Aaron Bergman
Apr2-05, 05:34 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <3b5m6sF6fihqmU1@news.dfncis.de>,\nUrs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n> Now, as we have discussed here before, I know from the papers by Yekutieli\n>\n> http://www.math.bgu.ac.il/~amyekut/publications/publications.html\n>\n> that the group of autoequivalence of the derived category of\n> _quiver_representations_ is known and called the derived Picard group DPic\n> associated to the path algebra of the quiver. From what Seidel and Thomas\n> write on their page 2 together with some facts stated by Yekutieli DPic must\n> be closely related to Auteq(D(X)).\n>\n> I am hoping it is actually the same group. But is it?\n\nGiven that the two categories are equivalent, it seems that the groups\nof autoequivalences ought to be isomorphic.\n>\n> DPic consists of isomorphism classes of all tilting complexes.\n>\n> Not sure about the symbols used on the very top of p. 3 in Seidel&Thomas.\n> Is that the same as tilting complexes?\n\nNo. Those symbols give a Fourier-Mukai transform. Any equivalence\nbetween derived categories of coherent sheaves can be written as one. It\nlooks a lot like a tilting equivalence, however. With a tilting\nequivalence, you have some bimodule complex, X, and get a functor\n\nF(-) = - \\otimes^L X\n\nRickard, I guess, proved that all derived equivalences of the module\ncategories are of this form.\n\nFor coherent sheaves, given two varieties A and B, let X be an object in\nD(A x B). Then the Fourier-Mukai transform is\n\n\\pi_B_*(pi_A^* - \\otimes^L X)\n\nIn other words, we take the object on A and pull it back to A x B. We\nthen tensor it with the object X and push forward to get an object in B.\n\nWhy these look so similar will have to be answered by someone more\nknowledgeable than I.\n\nAaron\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <3b5m6sF6fihqmU1@news.dfncis.de>,
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
> Now, as we have discussed here before, I know from the papers by Yekutieli
>
> http://www.math.bgu.ac.il/~amyekut/publications/publications.html
>
> that the group of autoequivalence of the derived category of
> _quiver_representations_ is known and called the derived Picard group DPic
> associated to the path algebra of the quiver. From what Seidel and Thomas
> write on their page 2 together with some facts stated by Yekutieli DPic must
> be closely related to Auteq(D(X)).
>
> I am hoping it is actually the same group. But is it?
Given that the two categories are equivalent, it seems that the groups
of autoequivalences ought to be isomorphic.
>
> DPic consists of isomorphism classes of all tilting complexes.
>
> Not sure about the symbols used on the very top of p. 3 in Seidel&Thomas.
> Is that the same as tilting complexes?
No. Those symbols give a Fourier-Mukai transform. Any equivalence
between derived categories of coherent sheaves can be written as one. It
looks a lot like a tilting equivalence, however. With a tilting
equivalence, you have some bimodule complex, X, and get a functor
F(-) = - \otimes^L X
Rickard, I guess, proved that all derived equivalences of the module
categories are of this form.
For coherent sheaves, given two varieties A and B, let X be an object in
D(A x B). Then the Fourier-Mukai transform is
\pi_B_*(\pi_A^* - \otimes^L X)
In other words, we take the object on A and pull it back to A x B. We
then tensor it with the object X and push forward to get an object in B.
Why these look so similar will have to be answered by someone more
knowledgeable than I.
Aaron
Urs Schreiber
Apr4-05, 10:15 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Aaron Bergman" <abergman@physics.utexas.edu> schrieb im Newsbeitrag\nnews:abergman-F25070.19522901042005@localhost...\n> In article <3b5he2F6c8b90U1@news.dfncis.de>,\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n>> Now, I might be mistaken, but I believe that a derived category of\n>> coherent sheaves D(X) in general has many more objects than a derived\n>> category D(Rep(Q)) of quiver reps that it is equivalent to.\n>\n> Why do you think that?\n\nYeah, right, why did I think that??\n\n> Among other things, the objects in a category might not even form a set\n> (although I guess the do (?) in this case), so I\'m not sure what you mean\n> by \'more objects\'. There are at least aleph objects in each category. More\n> than that, there are certainly aleph objects in each equivalence class on\n> the quiver rep side.\n\nTrue, certainly. The reason why I made that claim was that we always get the\nquiver from a distinguished subset of all possible branes. Taking a\ndistinguished subsets smells like choosing representatives from isomorphism\nclasses. But of course there is something more subtle going on here.\n\nHm, maybe I should ask the following, maybe seemingly unrelated question:\n\nIn the Douglas/Moore context a representation of a quiver defines a vacuum\nof the effective gauge theory. The dimension of the vector spaces at the\nvertices of the quiver give the number of 3+0 branes and the linear maps\nbetween them the \'noncommutative distance\' between these.\n\nNow, if an element of Rep(Q) gives a vacuum of the gauge theory, what does\nan element of D(Rep(Q)), i.e. a complex of quiver representations, tell us\nabout the gauge theory?\n\nI guess the problem is that since in D(Rep(Q)) anti-branes enter the picture\nwe will in general no longer have a susy theory in R^3,1. But can one say\nanything about the effective gauge theory?\n\n\n>> This would naturally explain the "gauge freedom" (given by Seiberg\n>> duality/mutations/tilting equivalences/etc.) in going from D(X) to\n>> D(Rep(Q)).\n>\n> The Seiberg dualities generally change Q. So, you have equivalences to the\n> derived category of representations of *different* quivers.\n\n> I really doubt that these derived equivalences lift to isomorphisms of the\n> quiver algebras\n\nYes (and I don\'t think I claimed otherwise).\n\n> (though it could be true).\n\nLet\'s see:\n\nTwo algebras A and B are (Morita) equivalent iff there are weakly invertible\nobjects T in the category A-Mod-B of A-B bimodules.\n\nIf so, they are also derived equivalent, since the complex T\' in D(A-Mod-B)\nwith T concentrated in degree 0 is a 2-sided tilting complex, i.e. derived\nmultiplication with T\' gives an equivalence.\n\nBut the converse need not hold, I guess. A and B can be derived equivalent\nwithout being equivalent. I think I have seen an example for this discussed\nsomewhere in Yekutieli\'s papers, but I forget where.\n\nThere is a quiver Q(A) associated to any A and, IIRC, if A is what is called\nhereditary, it is equivalent to the path algebra K(Q(A)) of its own quiver\nQ(A), i.e.\n\nK(Q(A)) ~ A .\n\nSo if we assume A and B to be hereditary the above statements apply also to\nquivers.\n\n\n> Although it can\'t happen when the canonical class isn\'t trivial, it\'s\n> worth pointing out that two different varieties can have equivalent\n> derived categories of coherent sheaves,\n\nI have seen that stated somewhere, recently. It seems related to a question\nthat I talked about with Jarah Evslin, a while ago. You know, he travels the\nworld talking about a hierarchy of descriptions "embedding", "homotopy",\n"homology", "K-homology", etc. where the items in the beginning of the list\nknow more about the configuration but less about conserved charges.\n\nSo I asked him where derived categories would fit into the picture, or if\nthey subsumed it all, or what. Since an object of the derived cat of\ncoherent sheaves knows about embedding information, but apparently not about\nall embedding information, it should presumeably be located at the beginning\nof the list but after "embedding". On the other hand, it might not properly\nfit in that type of list at all.\n\n\n> so this isn\'t at all unique to derived module categories.\n\nWhich string theoretic processes can distinguish between two different\nvarieties with the same derived category of coherent sheaves?\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Aaron Bergman" <abergman@physics.utexas.edu> schrieb im Newsbeitrag
news:abergman-F25070.19522901042005@localhost...
> In article <3b5he2F6c8b90U1@news.dfncis.de>,
> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
>> Now, I might be mistaken, but I believe that a derived category of
>> coherent sheaves D(X) in general has many more objects than a derived
>> category D(Rep(Q)) of quiver reps that it is equivalent to.
>
> Why do you think that?
Yeah, right, why did I think that??
> Among other things, the objects in a category might not even form a set
> (although I guess the do (?) in this case), so I'm not sure what you mean
> by 'more objects'. There are at least \aleph objects in each category. More
> than that, there are certainly \aleph objects in each equivalence class on
> the quiver rep side.
True, certainly. The reason why I made that claim was that we always get the
quiver from a distinguished subset of all possible branes. Taking a
distinguished subsets smells like choosing representatives from isomorphism
classes. But of course there is something more subtle going on here.
Hm, maybe I should ask the following, maybe seemingly unrelated question:
In the Douglas/Moore context a representation of a quiver defines a vacuum
of the effective gauge theory. The dimension of the vector spaces at the
vertices of the quiver give the number of 3+0 branes and the linear maps
between them the 'noncommutative distance' between these.
Now, if an element of Rep(Q) gives a vacuum of the gauge theory, what does
an element of D(Rep(Q)), i.e. a complex of quiver representations, tell us
about the gauge theory?
I guess the problem is that since in D(Rep(Q)) anti-branes enter the picture
we will in general no longer have a susy theory in R^3,1. But can one say
anything about the effective gauge theory?
>> This would naturally explain the "gauge freedom" (given by Seiberg
>> duality/mutations/tilting equivalences/etc.) in going from D(X) to
>> D(Rep(Q)).
>
> The Seiberg dualities generally change Q. So, you have equivalences to the
> derived category of representations of *different* quivers.
> I really doubt that these derived equivalences lift to isomorphisms of the
> quiver algebras
Yes (and I don't think I claimed otherwise).
> (though it could be true).
Let's see:
Two algebras A and B are (Morita) equivalent iff there are weakly invertible
objects T in the category A-Mod-B of A-B bimodules.
If so, they are also derived equivalent, since the complex T' in D(A-Mod-B)
with T concentrated in degree is a 2-sided tilting complex, i.e. derived
multiplication with T' gives an equivalence.
But the converse need not hold, I guess. A and B can be derived equivalent
without being equivalent. I think I have seen an example for this discussed
somewhere in Yekutieli's papers, but I forget where.
There is a quiver Q(A) associated to any A and, IIRC, if A is what is called
hereditary, it is equivalent to the path algebra K(Q(A)) of its own quiver
Q(A), i.e.
K(Q(A)) ~ A .
So if we assume A and B to be hereditary the above statements apply also to
quivers.
> Although it can't happen when the canonical class isn't trivial, it's
> worth pointing out that two different varieties can have equivalent
> derived categories of coherent sheaves,
I have seen that stated somewhere, recently. It seems related to a question
that I talked about with Jarah Evslin, a while ago. You know, he travels the
world talking about a hierarchy of descriptions "embedding", "homotopy",
"homology", "K-homology", etc. where the items in the beginning of the list
know more about the configuration but less about conserved charges.
So I asked him where derived categories would fit into the picture, or if
they subsumed it all, or what. Since an object of the derived cat of
coherent sheaves knows about embedding information, but apparently not about
all embedding information, it should presumeably be located at the beginning
of the list but after "embedding". On the other hand, it might not properly
fit in that type of list at all.
> so this isn't at all unique to derived module categories.
Which string theoretic processes can distinguish between two different
varieties with the same derived category of coherent sheaves?
Urs Schreiber
Apr4-05, 10:42 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nOn Sat, 2 Apr 2005, Aaron Bergman wrote:\n\n> In article <3b5m6sF6fihqmU1@news.dfncis.de>,\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n>\n>> Now, as we have discussed here before, I know from the papers by Yekutieli\n>>\n>> http://www.math.bgu.ac.il/~amyekut/publications/publications.html\n>>\n>> that the group of autoequivalence of the derived category of\n>> _quiver_representations_ is known and called the derived Picard group DPic\n>> associated to the path algebra of the quiver. From what Seidel and Thomas\n>> write on their page 2 together with some facts stated by Yekutieli DPic\n>> must be closely related to Auteq(D(X)).\n>>\n>> I am hoping it is actually the same group. But is it?\n>\n> Given that the two categories are equivalent, it seems that the groups of\n> autoequivalences ought to be isomorphic.\n>>\n>> DPic consists of isomorphism classes of all tilting complexes.\n>>\n>> Not sure about the symbols used on the very top of p. 3 in Seidel&Thomas.\n>> Is that the same as tilting complexes?\n>\n> No. Those symbols give a Fourier-Mukai transform. Any equivalence between\n> derived categories of coherent sheaves can be written as one. It looks a lot\n> like a tilting equivalence, however. With a tilting equivalence, you have\n> some bimodule complex, X, and get a functor\n>\n> F(-) = - \\otimes^L X\n>\n> Rickard, I guess, proved that all derived equivalences of the module\n> categories are of this form.\n>\n> For coherent sheaves, given two varieties A and B, let X be an object in D(A\n> x B). Then the Fourier-Mukai transform is\n>\n> \\pi_B_*(pi_A^* - \\otimes^L X)\n>\n> In other words, we take the object on A and pull it back to A x B. We then\n> tensor it with the object X and push forward to get an object in B.\n>\n> Why these look so similar will have to be answered by someone more\n> knowledgeable than I.\n\n\nInterestingly, maybe, both transformations again look similar to the one\nrelating them, namely the BKR transformation that Aspinwall reviews on the\ntop of p. 104 of hep-th/0403166.\n\nHm, probably "conjugating" a tilting equivalence by a BKR transformation\ngives a Fourier-Mukai transformation and vice versa?\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Sat, 2 Apr 2005, Aaron Bergman wrote:
> In article <3b5m6sF6fihqmU1@news.dfncis.de>,
> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
>
>> Now, as we have discussed here before, I know from the papers by Yekutieli
>>
>> http://www.math.bgu.ac.il/~amyekut/publications/publications.html
>>
>> that the group of autoequivalence of the derived category of
>> _quiver_representations_ is known and called the derived Picard group DPic
>> associated to the path algebra of the quiver. From what Seidel and Thomas
>> write on their page 2 together with some facts stated by Yekutieli DPic
>> must be closely related to Auteq(D(X)).
>>
>> I am hoping it is actually the same group. But is it?
>
> Given that the two categories are equivalent, it seems that the groups of
> autoequivalences ought to be isomorphic.
>>
>> DPic consists of isomorphism classes of all tilting complexes.
>>
>> Not sure about the symbols used on the very top of p. 3 in Seidel&Thomas.
>> Is that the same as tilting complexes?
>
> No. Those symbols give a Fourier-Mukai transform. Any equivalence between
> derived categories of coherent sheaves can be written as one. It looks a lot
> like a tilting equivalence, however. With a tilting equivalence, you have
> some bimodule complex, X, and get a functor
>
> F(-) = - \otimes^L X
>
> Rickard, I guess, proved that all derived equivalences of the module
> categories are of this form.
>
> For coherent sheaves, given two varieties A and B, let X be an object in D(A
> x B). Then the Fourier-Mukai transform is
>
> \pi_B_*(\pi_A^* - \otimes^L X)
>
> In other words, we take the object on A and pull it back to A x B. We then
> tensor it with the object X and push forward to get an object in B.
>
> Why these look so similar will have to be answered by someone more
> knowledgeable than I.
Interestingly, maybe, both transformations again look similar to the one
relating them, namely the BKR transformation that Aspinwall reviews on the
top of p. 104 of http://www.arxiv.org/abs/hep-th/0403166.
Hm, probably "conjugating" a tilting equivalence by a BKR transformation
gives a Fourier-Mukai transformation and vice versa?
Aaron Bergman
Apr4-05, 11:15 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <Pine.LNX.4.62.0504041133530.19977@feynman.harvard .edu>,\nUrs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n> Interestingly, maybe, both transformations again look similar to the one\n> relating them, namely the BKR transformation that Aspinwall reviews on the\n> top of p. 104 of hep-th/0403166.\n\nThe Bridgeland-King-Reid transformation is basically a Fourier-Mukai\ntransform. I\'m not sure what you mean by saying that it relates normaly\nFMTs and tilting.\n\n> Hm, probably "conjugating" a tilting equivalence by a BKR transformation\n> gives a Fourier-Mukai transformation and vice versa?\n\nBKR relates the derived categories of equivariant coherent sheaves on a\nspace with a finite group action with the derived category of coherent\nsheaves on a particular crepant resolution of the orbifold. No quivers\nin sight.\n\n(but maybe if you look carefully on the horizon, you might be able to\nmake them out....)\n\nAaron\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <Pine.LNX.4.62.0504041133530.19977@feynman.harvard. edu>,
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
> Interestingly, maybe, both transformations again look similar to the one
> relating them, namely the BKR transformation that Aspinwall reviews on the
> top of p. 104 of http://www.arxiv.org/abs/hep-th/0403166.
The Bridgeland-King-Reid transformation is basically a Fourier-Mukai
transform. I'm not sure what you mean by saying that it relates normaly
FMTs and tilting.
> Hm, probably "conjugating" a tilting equivalence by a BKR transformation
> gives a Fourier-Mukai transformation and vice versa?
BKR relates the derived categories of equivariant coherent sheaves on a
space with a finite group action with the derived category of coherent
sheaves on a particular crepant resolution of the orbifold. No quivers
in sight.
(but maybe if you look carefully on the horizon, you might be able to
make them out....)
Aaron
Urs Schreiber
Apr4-05, 11:21 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nOn Mon, 4 Apr 2005, Aaron Bergman wrote:\n\n> In article <Pine.LNX.4.62.0504041133530.19977@feynman.harvard .edu>,\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n>\n>> Interestingly, maybe, both transformations again look similar to the one\n>> relating them, namely the BKR transformation that Aspinwall reviews on the\n>> top of p. 104 of hep-th/0403166.\n>\n> The Bridgeland-King-Reid transformation is basically a Fourier-Mukai\n> transform. I\'m not sure what you mean by saying that it relates normaly FMTs\n> and tilting.\n\n\nI am following Aspinwall\'s review. Before he states BKR on p. 103 he makes\non pp. 101-102 the point that G-equivariant sheaves on \\C^d are just a\ndifferent interpretation of the MacKay quiver. This is, I believe, the\nreasoning by which he claims on p. 104 that\n\n"The BKR result gives a precise recipe for mapping between these two\nderived categories [...]"\n\nand I believe by "these two derived categories" he means that of coherent\nsheaves and that of quiver reps.\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Mon, 4 Apr 2005, Aaron Bergman wrote:
> In article <Pine.LNX.4.62.0504041133530.19977@feynman.harvard. edu>,
> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
>
>> Interestingly, maybe, both transformations again look similar to the one
>> relating them, namely the BKR transformation that Aspinwall reviews on the
>> top of p. 104 of http://www.arxiv.org/abs/hep-th/0403166.
>
> The Bridgeland-King-Reid transformation is basically a Fourier-Mukai
> transform. I'm not sure what you mean by saying that it relates normaly FMTs
> and tilting.
I am following Aspinwall's review. Before he states BKR on p. 103 he makes
on pp. 101-102 the point that G-equivariant sheaves on \C^d are just a
different interpretation of the MacKay quiver. This is, I believe, the
reasoning by which he claims on p. 104 that
"The BKR result gives a precise recipe for mapping between these two
derived categories [...]"
and I believe by "these two derived categories" he means that of coherent
sheaves and that of quiver reps.
Aaron Bergman
Apr4-05, 11:27 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <3bd3k6F5i1v0uU1@news.dfncis.de>,\nUrs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n> True, certainly. The reason why I made that claim was that we always get the\n> quiver from a distinguished subset of all possible branes. Taking a\n> distinguished subsets smells like choosing representatives from isomorphism\n> classes. But of course there is something more subtle going on here.\n\nThink of the choice of an exceptional collection as a choice of \'basis\'.\n\n> Hm, maybe I should ask the following, maybe seemingly unrelated question:\n>\n> In the Douglas/Moore context a representation of a quiver defines a vacuum\n> of the effective gauge theory. The dimension of the vector spaces at the\n> vertices of the quiver give the number of 3+0 branes and the linear maps\n> between them the \'noncommutative distance\' between these.\n>\n> Now, if an element of Rep(Q) gives a vacuum of the gauge theory, what does\n> an element of D(Rep(Q)), i.e. a complex of quiver representations, tell us\n> about the gauge theory?\n>\n> I guess the problem is that since in D(Rep(Q)) anti-branes enter the picture\n> we will in general no longer have a susy theory in R^3,1. But can one say\n> anything about the effective gauge theory?\n\nI have no idea how to interpret it on the gauge theory side, but from\nthe equivalence of categories, we know that an element in D(Rep(Q))\ndescribes a boundary state for the B-model.\n\n[...]\n\n> > Although it can\'t happen when the canonical class isn\'t trivial, it\'s\n> > worth pointing out that two different varieties can have equivalent\n> > derived categories of coherent sheaves,\n>\n> I have seen that stated somewhere, recently. It seems related to a question\n> that I talked about with Jarah Evslin, a while ago. You know, he travels the\n> world talking about a hierarchy of descriptions "embedding", "homotopy",\n> "homology", "K-homology", etc. where the items in the beginning of the list\n> know more about the configuration but less about conserved charges.\n\nJarah was talking about various generalized cohomology theories, I\nthink. The derived category isn\'t a cohomology theory in any sense that\nI know -- which isn\'t to say that there isn\'t some sense that I don\'t\nknow about.\n\n> So I asked him where derived categories would fit into the picture, or if\n> they subsumed it all, or what. Since an object of the derived cat of\n> coherent sheaves knows about embedding information, but apparently not about\n> all embedding information, it should presumeably be located at the beginning\n> of the list but after "embedding". On the other hand, it might not properly\n> fit in that type of list at all.\n\nWhat is \'embedding information\'?\n\n> > so this isn\'t at all unique to derived module categories.\n>\n> Which string theoretic processes can distinguish between two different\n> varieties with the same derived category of coherent sheaves?\n\nNone, if you believe in dualities, I think.\n\nAaron\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <3bd3k6F5i1v0uU1@news.dfncis.de>,
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
> True, certainly. The reason why I made that claim was that we always get the
> quiver from a distinguished subset of all possible branes. Taking a
> distinguished subsets smells like choosing representatives from isomorphism
> classes. But of course there is something more subtle going on here.
Think of the choice of an exceptional collection as a choice of 'basis'.
> Hm, maybe I should ask the following, maybe seemingly unrelated question:
>
> In the Douglas/Moore context a representation of a quiver defines a vacuum
> of the effective gauge theory. The dimension of the vector spaces at the
> vertices of the quiver give the number of 3+0 branes and the linear maps
> between them the 'noncommutative distance' between these.
>
> Now, if an element of Rep(Q) gives a vacuum of the gauge theory, what does
> an element of D(Rep(Q)), i.e. a complex of quiver representations, tell us
> about the gauge theory?
>
> I guess the problem is that since in D(Rep(Q)) anti-branes enter the picture
> we will in general no longer have a susy theory in R^3,1. But can one say
> anything about the effective gauge theory?
I have no idea how to interpret it on the gauge theory side, but from
the equivalence of categories, we know that an element in D(Rep(Q))
describes a boundary state for the B-model.
[...]
> > Although it can't happen when the canonical class isn't trivial, it's
> > worth pointing out that two different varieties can have equivalent
> > derived categories of coherent sheaves,
>
> I have seen that stated somewhere, recently. It seems related to a question
> that I talked about with Jarah Evslin, a while ago. You know, he travels the
> world talking about a hierarchy of descriptions "embedding", "homotopy",
> "homology", "K-homology", etc. where the items in the beginning of the list
> know more about the configuration but less about conserved charges.
Jarah was talking about various generalized cohomology theories, I
think. The derived category isn't a cohomology theory in any sense that
I know -- which isn't to say that there isn't some sense that I don't
know about.
> So I asked him where derived categories would fit into the picture, or if
> they subsumed it all, or what. Since an object of the derived cat of
> coherent sheaves knows about embedding information, but apparently not about
> all embedding information, it should presumeably be located at the beginning
> of the list but after "embedding". On the other hand, it might not properly
> fit in that type of list at all.
What is 'embedding information'?
> > so this isn't at all unique to derived module categories.
>
> Which string theoretic processes can distinguish between two different
> varieties with the same derived category of coherent sheaves?
None, if you believe in dualities, I think.
Aaron
Urs Schreiber
Apr4-05, 11:38 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nOn Mon, 4 Apr 2005, Aaron Bergman wrote:\n\n> In article <3bd3k6F5i1v0uU1@news.dfncis.de>,\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n>> I have seen that stated somewhere, recently. It seems related to a question\n>> that I talked about with Jarah Evslin, a while ago. You know, he travels\n>> the world talking about a hierarchy of descriptions "embedding",\n>> "homotopy", "homology", "K-homology", etc. where the items in the beginning\n>> of the list know more about the configuration but less about conserved\n>> charges.\n>\n> Jarah was talking about various generalized cohomology theories, I think. The\n> derived category isn\'t a cohomology theory in any sense that I know -- which\n> isn\'t to say that there isn\'t some sense that I don\'t know about.\n\n\nI just thought that since the derived category in fact contains much (all)\nof the information in the rest of the list (K-theory for one, but also\npositions of branes, i.e. "embedding information") it might be\npart of the list. But maybe that entire list is better thought of as\ndescribing a series of steps by which we forget information in D(X) until\nwe arrive at K-theory.\n\n\n>> So I asked him where derived categories would fit into the picture, or if\n>> they subsumed it all, or what. Since an object of the derived cat of\n>> coherent sheaves knows about embedding information, but apparently not\n>> about all embedding information, it should presumeably be located at the\n>> beginning of the list but after "embedding". On the other hand, it might\n>> not properly fit in that type of list at all.\n>\n> What is \'embedding information\'?\n\n\nI mean the position of the D-branes in target space.\n\n\n>> Which string theoretic processes can distinguish between two different\n>> varieties with the same derived category of coherent sheaves?\n>\n> None, if you believe in dualities, I think.\n\n\nCool. As you may have seen, I am wondering whether there should be a sort\nof categorified Gelfan-Naimark theorem which would tell us that we can\nalways reconstruct the space of string configurations from the\n"categorified function algebra" D(X) ~ D(A-Mod). For that to be true,\nstrings may not be able to distinguish things that D(X) cannot\ndistinguish.\n\nOn the other hand, it is really D(A-Mod-A) (the derived category of\nbimodules) which should be called a 2-algebra.\n\nDo you know if D(A-Mod-A) is triangulated when D(A-Mod) is?\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Mon, 4 Apr 2005, Aaron Bergman wrote:
> In article <3bd3k6F5i1v0uU1@news.dfncis.de>,
> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
>> I have seen that stated somewhere, recently. It seems related to a question
>> that I talked about with Jarah Evslin, a while ago. You know, he travels
>> the world talking about a hierarchy of descriptions "embedding",
>> "homotopy", "homology", "K-homology", etc. where the items in the beginning
>> of the list know more about the configuration but less about conserved
>> charges.
>
> Jarah was talking about various generalized cohomology theories, I think. The
> derived category isn't a cohomology theory in any sense that I know -- which
> isn't to say that there isn't some sense that I don't know about.
I just thought that since the derived category in fact contains much (all)
of the information in the rest of the list (K-theory for one, but also
positions of branes, i.e. "embedding information") it might be
part of the list. But maybe that entire list is better thought of as
describing a series of steps by which we forget information in D(X) until
we arrive at K-theory.
>> So I asked him where derived categories would fit into the picture, or if
>> they subsumed it all, or what. Since an object of the derived cat of
>> coherent sheaves knows about embedding information, but apparently not
>> about all embedding information, it should presumeably be located at the
>> beginning of the list but after "embedding". On the other hand, it might
>> not properly fit in that type of list at all.
>
> What is 'embedding information'?
I mean the position of the D-branes in target space.
>> Which string theoretic processes can distinguish between two different
>> varieties with the same derived category of coherent sheaves?
>
> None, if you believe in dualities, I think.
Cool. As you may have seen, I am wondering whether there should be a sort
of categorified Gelfan-Naimark theorem which would tell us that we can
always reconstruct the space of string configurations from the
"categorified function algebra" D(X) ~ D(A-Mod). For that to be true,
strings may not be able to distinguish things that D(X) cannot
distinguish.
On the other hand, it is really D(A-Mod-A) (the derived category of
bimodules) which should be called a 2-algebra.
Do you know if D(A-Mod-A) is triangulated when D(A-Mod) is?
Aaron Bergman
Apr4-05, 11:42 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <Pine.LNX.4.62.0504041216560.20114@feynman.harvard .edu>,\nUrs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n> I am following Aspinwall\'s review. Before he states BKR on p. 103 he makes\n> on pp. 101-102 the point that G-equivariant sheaves on \\C^d are just a\n> different interpretation of the MacKay quiver. This is, I believe, the\n> reasoning by which he claims on p. 104 that\n>\n> "The BKR result gives a precise recipe for mapping between these two\n> derived categories [...]"\n>\n> and I believe by "these two derived categories" he means that of coherent\n> sheaves and that of quiver reps.\n\nThe BKR result gives the map between the categories I state. The result\non the bottom of p. 102 relates the equivariant derived category to the\nderived category of quiver reps. The combination of the two results\ngives what you state.\n\nAaron\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <Pine.LNX.4.62.0504041216560.20114@feynman.harvard. edu>,
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
> I am following Aspinwall's review. Before he states BKR on p. 103 he makes
> on pp. 101-102 the point that G-equivariant sheaves on \C^d are just a
> different interpretation of the MacKay quiver. This is, I believe, the
> reasoning by which he claims on p. 104 that
>
> "The BKR result gives a precise recipe for mapping between these two
> derived categories [...]"
>
> and I believe by "these two derived categories" he means that of coherent
> sheaves and that of quiver reps.
The BKR result gives the map between the categories I state. The result
on the bottom of p. 102 relates the equivariant derived category to the
derived category of quiver reps. The combination of the two results
gives what you state.
Aaron
Urs Schreiber
Apr4-05, 11:45 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nOn Mon, 4 Apr 2005, Aaron Bergman wrote:\n\n> The BKR result gives the map between the categories I state. The result on\n> the bottom of p. 102 relates the equivariant derived category to the derived\n> category of quiver reps. The combination of the two results gives what you\n> state.\n\n\nGood. So I guess it is fair to say that it is that result on the bottom of\np.102 which in the end is responsible for the similarity of the form of\nthe equivalence of derived module categories and derived coh. sheave\ncategories.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Mon, 4 Apr 2005, Aaron Bergman wrote:
> The BKR result gives the map between the categories I state. The result on
> the bottom of p. 102 relates the equivariant derived category to the derived
> category of quiver reps. The combination of the two results gives what you
> state.
Good. So I guess it is fair to say that it is that result on the bottom of
p.102 which in the end is responsible for the similarity of the form of
the equivalence of derived module categories and derived coh. sheave
categories.
Aaron Bergman
Apr4-05, 12:12 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <Pine.LNX.4.62.0504041243560.20184@feynman.harvard .edu>,\nUrs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n> On Mon, 4 Apr 2005, Aaron Bergman wrote:\n>\n> > The BKR result gives the map between the categories I state. The result on\n> > the bottom of p. 102 relates the equivariant derived category to the\n> > derived\n> > category of quiver reps. The combination of the two results gives what you\n> > state.\n>\n> Good. So I guess it is fair to say that it is that result on the bottom of\n> p.102 which in the end is responsible for the similarity of the form of\n> the equivalence of derived module categories and derived coh. sheave\n> categories.\n\nThe result on the bottom of p. 102 is specific to orbifolds of C^n, I\nthink. The general equivalence between quiver categories and\ntriangulated categories with strong exceptional collections follows from\nderived Morita equivalence which is very closely related to tilting.\n\nAaron\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <Pine.LNX.4.62.0504041243560.20184@feynman.harvard. edu>,
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
> On Mon, 4 Apr 2005, Aaron Bergman wrote:
>
> > The BKR result gives the map between the categories I state. The result on
> > the bottom of p. 102 relates the equivariant derived category to the
> > derived
> > category of quiver reps. The combination of the two results gives what you
> > state.
>
> Good. So I guess it is fair to say that it is that result on the bottom of
> p.102 which in the end is responsible for the similarity of the form of
> the equivalence of derived module categories and derived coh. sheave
> categories.
The result on the bottom of p. 102 is specific to orbifolds of C^n, I
think. The general equivalence between quiver categories and
triangulated categories with strong exceptional collections follows from
derived Morita equivalence which is very closely related to tilting.
Aaron
Urs Schreiber
Apr4-05, 12:14 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nOn Mon, 4 Apr 2005, Aaron Bergman wrote:\n\n> The result on the bottom of p. 102 is specific to orbifolds of C^n, I think.\n> The general equivalence between quiver categories and triangulated categories\n> with strong exceptional collections follows from derived Morita equivalence\n> which is very closely related to tilting.\n\nI see. Given any quiver, by wich I mean any old graph, can we always get a\ncompactification that is described by that quiver?\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Mon, 4 Apr 2005, Aaron Bergman wrote:
> The result on the bottom of p. 102 is specific to orbifolds of C^n, I think.
> The general equivalence between quiver categories and triangulated categories
> with strong exceptional collections follows from derived Morita equivalence
> which is very closely related to tilting.
I see. Given any quiver, by wich I mean any old graph, can we always get a
compactification that is described by that quiver?
Aaron Bergman
Apr4-05, 12:33 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <Pine.LNX.4.62.0504041229160.20149@feynman.harvard .edu>,\nUrs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n> On Mon, 4 Apr 2005, Aaron Bergman wrote:\n>\n> > In article <3bd3k6F5i1v0uU1@news.dfncis.de>,\n> > Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n>\n> >> I have seen that stated somewhere, recently. It seems related to a\n> >> question\n> >> that I talked about with Jarah Evslin, a while ago. You know, he travels\n> >> the world talking about a hierarchy of descriptions "embedding",\n> >> "homotopy", "homology", "K-homology", etc. where the items in the\n> >> beginning\n> >> of the list know more about the configuration but less about conserved\n> >> charges.\n> >\n> > Jarah was talking about various generalized cohomology theories, I\n> > think. The derived category isn\'t a cohomology theory in any sense\n> > that I know -- which isn\'t to say that there isn\'t some sense that\n> > I don\'t know about.\n>\n> I just thought that since the derived category in fact contains much (all)\n> of the information in the rest of the list (K-theory for one, but also\n> positions of branes, i.e. "embedding information") it might be\n> part of the list. But maybe that entire list is better thought of as\n> describing a series of steps by which we forget information in D(X) until\n> we arrive at K-theory.\n\nI think the processes are very different. I don\'t know much about this\nhierarchy of generalized cohomology theories -- it\'s been described to\nme a few times, but it\'s never sunk it -- but they\'re all still\ncohomologies theories that obey most of the Eilenberg-Steenrod axioms.\nIn particular, I don\'t think there\'s any surjective map from one of\nthese theories to another in general. They\'re probably related by\nspectral sequences or some such, I\'d guess.\n\nThe derived category, on the other hand, really is a category. The map\nto the Grothendieck group (which is a lot like K-theory) is a type of\ndecategorification.\n\nAaron\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <Pine.LNX.4.62.0504041229160.20149@feynman.harvard. edu>,
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
> On Mon, 4 Apr 2005, Aaron Bergman wrote:
>
> > In article <3bd3k6F5i1v0uU1@news.dfncis.de>,
> > Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
>
> >> I have seen that stated somewhere, recently. It seems related to a
> >> question
> >> that I talked about with Jarah Evslin, a while ago. You know, he travels
> >> the world talking about a hierarchy of descriptions "embedding",
> >> "homotopy", "homology", "K-homology", etc. where the items in the
> >> beginning
> >> of the list know more about the configuration but less about conserved
> >> charges.
> >
> > Jarah was talking about various generalized cohomology theories, I
> > think. The derived category isn't a cohomology theory in any sense
> > that I know -- which isn't to say that there isn't some sense that
> > I don't know about.
>
> I just thought that since the derived category in fact contains much (all)
> of the information in the rest of the list (K-theory for one, but also
> positions of branes, i.e. "embedding information") it might be
> part of the list. But maybe that entire list is better thought of as
> describing a series of steps by which we forget information in D(X) until
> we arrive at K-theory.
I think the processes are very different. I don't know much about this
hierarchy of generalized cohomology theories -- it's been described to
me a few times, but it's never sunk it -- but they're all still
cohomologies theories that obey most of the Eilenberg-Steenrod axioms.
In particular, I don't think there's any surjective map from one of
these theories to another in general. They're probably related by
spectral sequences or some such, I'd guess.
The derived category, on the other hand, really is a category. The map
to the Grothendieck group (which is a lot like K-theory) is a type of
decategorification.
Aaron
Urs Schreiber
Apr4-05, 01:00 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nOn Mon, 4 Apr 2005, Aaron Bergman wrote:\n\n> The map to the\n> Grothendieck group (which is a lot like K-theory) is a type of\n> decategorification.\n\nCould it be that the precise version of this statement is that the\nGrothendiek group is precisely the decategorification (i.e. the set of\nisomorphism classes of) the *t-structure* of D(X) that comes with the\nPi-stability condition?\n\nI am guessing this on the basis of what I think to have understood from\nBridgeland\'s math.AG/0212237, which you pointed me to recently.\nUnfortunately I have not really studied that paper, just looked at it here\nand there.\n\nSo let\'s see:\n\nBridgeland defines a slicing on D(X) (his def. 3.3, of course he works\nwith arbitrary triangulated categories) as a collection of subcategories\n\\P(\\phi) which contain all the "stable" objects of phase phi. (Hm, is\nthat even the right way to put it?)\n\nThen in prop. 5.3 he relates that to t-structures. It seems that the\ncollection \\P(\\phi > 0) gives a t-structure. So apparently the\nt-structure is something like the subcategory of D(X) containing all\nstable brane configurations. That should be what gives the Grothendieck\ngroup by decategorification.\n\nHm, maybe not reall. Something like that might be true, though.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Mon, 4 Apr 2005, Aaron Bergman wrote:
> The map to the
> Grothendieck group (which is a lot like K-theory) is a type of
> decategorification.
Could it be that the precise version of this statement is that the
Grothendiek group is precisely the decategorification (i.e. the set of
isomorphism classes of) the *t-structure* of D(X) that comes with the
\Pi-stability condition?
I am guessing this on the basis of what I think to have understood from
Bridgeland's math.AG/0212237, which you pointed me to recently.
Unfortunately I have not really studied that paper, just looked at it here
and there.
So let's see:
Bridgeland defines a slicing on D(X) (his def. 3.3, of course he works
with arbitrary triangulated categories) as a collection of subcategories
\P(\phi) which contain all the "stable" objects of phase \phi. (Hm, is
that even the right way to put it?)
Then in prop. 5.3 he relates that to t-structures. It seems that the
collection \P(\phi > 0) gives a t-structure. So apparently the
t-structure is something like the subcategory of D(X) containing all
stable brane configurations. That should be what gives the Grothendieck
group by decategorification.
Hm, maybe not reall. Something like that might be true, though.
Aaron Bergman
Apr4-05, 01:00 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <Pine.LNX.4.62.0504041312540.20244@feynman.harvard .edu>,\nUrs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n> On Mon, 4 Apr 2005, Aaron Bergman wrote:\n>\n> > The result on the bottom of p. 102 is specific to orbifolds of C^n,\n> > I think. The general equivalence between quiver categories and\n> > triangulated categories with strong exceptional collections follows\n> > from derived Morita equivalence which is very closely related to\n> > tilting.\n>\n> I see. Given any quiver, by wich I mean any old graph, can we always get a\n> compactification that is described by that quiver?\n\nNo. At least if we require SUSY, this can be seen by the fact that the\ncompactification must be Calabi-Yau. This implies that the Serre functor\nin the derived category must just be equal to a shift. This is clearly\nnot true for a general quiver.\n\nI think Hanany\'s written some papers on the types of quivers that can\narise in string compactifications, but I don\'t remember the references.\n\nAaron\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <Pine.LNX.4.62.0504041312540.20244@feynman.harvard. edu>,
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
> On Mon, 4 Apr 2005, Aaron Bergman wrote:
>
> > The result on the bottom of p. 102 is specific to orbifolds of C^n,
> > I think. The general equivalence between quiver categories and
> > triangulated categories with strong exceptional collections follows
> > from derived Morita equivalence which is very closely related to
> > tilting.
>
> I see. Given any quiver, by wich I mean any old graph, can we always get a
> compactification that is described by that quiver?
No. At least if we require SUSY, this can be seen by the fact that the
compactification must be Calabi-Yau. This implies that the Serre functor
in the derived category must just be equal to a shift. This is clearly
not true for a general quiver.
I think Hanany's written some papers on the types of quivers that can
arise in string compactifications, but I don't remember the references.
Aaron
Aaron Bergman
Apr5-05, 02:07 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In article <Pine.LNX.4.62.0504041336190.20280@feynman.harvard .edu>,\nUrs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n\n> On Mon, 4 Apr 2005, Aaron Bergman wrote:\n>\n> > The map to the\n> > Grothendieck group (which is a lot like K-theory) is a type of\n> > decategorification.\n>\n> Could it be that the precise version of this statement is that the\n> Grothendiek group is precisely the decategorification (i.e. the set of\n> isomorphism classes of) the *t-structure* of D(X) that comes with the\n> Pi-stability condition?\n\nYou don\'t need t-structures to define the Grothendieck group. It\'s a\nnatural construction on any triangulated category:\n\nTake the free abelian group on all isomorphism classes of objects and\nquotient out by the relation\n\n[A] - [B] + [C]\n\nwhenever A -> B -> C -> A[1] is a triangle.\n\nAaron\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <Pine.LNX.4.62.0504041336190.20280@feynman.harvard. edu>,
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
> On Mon, 4 Apr 2005, Aaron Bergman wrote:
>
> > The map to the
> > Grothendieck group (which is a lot like K-theory) is a type of
> > decategorification.
>
> Could it be that the precise version of this statement is that the
> Grothendiek group is precisely the decategorification (i.e. the set of
> isomorphism classes of) the *t-structure* of D(X) that comes with the
> \Pi-stability condition?
You don't need t-structures to define the Grothendieck group. It's a
natural construction on any triangulated category:
Take the free abelian group on all isomorphism classes of objects and
quotient out by the relation
[A] - [B] + [C]
whenever A -> B -> C -> A[1] is a triangle.
Aaron
Urs Schreiber
Apr5-05, 03:40 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Aaron Bergman" <abergman@physics.utexas.edu> schrieb im Newsbeitrag\nnews:abergman-DEF3C3.19513404042005@localhost...\n> In article <Pine.LNX.4.62.0504041336190.20280@feynman.harvard .edu>,\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:\n>\n>> On Mon, 4 Apr 2005, Aaron Bergman wrote:\n>>\n>> > The map to the Grothendieck group (which is a lot like K-theory) is a\n>> > type of decategorification.\n>>\n>> Could it be that the precise version of this statement is that the\n>> Grothendiek group is precisely the decategorification (i.e. the set of\n>> isomorphism classes of) the *t-structure* of D(X) that comes with the\n>> Pi-stability condition?\n>\n> You don\'t need t-structures to define the Grothendieck group. It\'s a\n> natural construction on any triangulated category:\n>\n> Take the free abelian group on all isomorphism classes of objects and\n> quotient out by the relation\n>\n> [A] - [B] + [C]\n>\n> whenever A -> B -> C -> A[1] is a triangle.\n\n\nYes, that\'s the definition. Now you mentioned that this is "a type of\ndecategorification". Namely, it is decategorification but followed by\ndividing out by certain relations, which tell us which binding and decay\nprocesses can occur.\n\nSo I was wondering if we can realize the Grothendiek group as a proper\ndecategoification (not followed by anything). This would amount to first\ngoing to a subcategory T of D(X) and then taking isomorphism classes of T\nsuch that we get the Grothendieck group on the nose.\n\nSince the relations that we want to divide out by describe D-brane\nreactions, I was speculating that a subcategory which does the trick might\nbe a t-structure, since that should sort of only conists of the stable end\nproducts of any reaction chain, where "stable" is defined by some stability\ncondition.\n\nMaybe that idea is wrong, of course. But I would like to understand if\nsomething in this spirit could be correct. This is related to my old\nquestion about how to subtract in triangulated categories.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Aaron Bergman" <abergman@physics.utexas.edu> schrieb im Newsbeitrag
news:abergman-DEF3C3.19513404042005@localhost...
> In article <Pine.LNX.4.62.0504041336190.20280@feynman.harvard. edu>,
> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:
>
>> On Mon, 4 Apr 2005, Aaron Bergman wrote:
>>
>> > The map to the Grothendieck group (which is a lot like K-theory) is a
>> > type of decategorification.
>>
>> Could it be that the precise version of this statement is that the
>> Grothendiek group is precisely the decategorification (i.e. the set of
>> isomorphism classes of) the *t-structure* of D(X) that comes with the
>> \Pi-stability condition?
>
> You don't need t-structures to define the Grothendieck group. It's a
> natural construction on any triangulated category:
>
> Take the free abelian group on all isomorphism classes of objects and
> quotient out by the relation
>
> [A] - [B] + [C]
>
> whenever A -> B -> C -> A[1] is a triangle.
Yes, that's the definition. Now you mentioned that this is "a type of
decategorification". Namely, it is decategorification but followed by
dividing out by certain relations, which tell us which binding and decay
processes can occur.
So I was wondering if we can realize the Grothendiek group as a proper
decategoification (not followed by anything). This would amount to first
going to a subcategory T of D(X) and then taking isomorphism classes of T
such that we get the Grothendieck group on the nose.
Since the relations that we want to divide out by describe D-brane
reactions, I was speculating that a subcategory which does the trick might
be a t-structure, since that should sort of only conists of the stable end
products of any reaction chain, where "stable" is defined by some stability
condition.
Maybe that idea is wrong, of course. But I would like to understand if
something in this spirit could be correct. This is related to my old
question about how to subtract in triangulated categories.
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