View Full Version : spectral triples and critical dimension
In math-ph/0306046, Martinetti ennounces the fundamental theorem of commutative spectral triples (theorem 2.11) making explicit a coefficient
(n-2)/24. I can not tell now if Gracia and Varilly follow the same convention,
but it seems obvious that the goal is to remark the existence of the same
term that bosonic string theory has.
Has this similitude been noticed by someone in the string community?
Lubos Motl
Sep3-04, 04:53 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>On Fri, 3 Sep 2004, arivero wrote:\n\n> Has this similitude been noticed by someone in the string community?\n\nThat\'s a really puzzling coefficient (n-2)/24 - exactly (minus) the energy\nof the bosonic theory first quantized ground state in "n" spacetime\ndimensions. I would guess that it is not well-known in the string\ncommunity.\n\nThis similarity has not started with the 2003 paper that you mentioned.\nSee for example\n\nhttp://www.arxiv.org/abs/hep-th/9805077\n\nwhich is a hep-th paper that you have cited yourself. This paper has a\ncute abstract: We are unable to formulate lattice gauge theory in terms of\nConnes\' spectral triples. ;-) The paper nevertheless refers to a couple of\nrelated sources - for example, some axioms by Connes that are "tailored\nsuch that there is a one-to-one correspondence between commutative\nspectral triples and Riemannian spin manifolds". This certainly sounds\nrelevant for your question because the (n-2)/24 is related to CFTs and\ntherefore to Riemannian manifolds.\n\nThis stuff sounds too mathematical to me, but potentially interesting. If\nsomeone understands "commutative spectral triples" and their isomorphism\nto Riemannian spin manifolds, I would be happy to hear a short\npresentation because it may include a nontrivial reinterpretation and/or\nabstract generalization of the stringy perturbative expansions.\n\nAll the best\nLubos\n_____________________________________ _________________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\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 Fri, 3 Sep 2004, arivero wrote:
> Has this similitude been noticed by someone in the string community?
That's a really puzzling coefficient (n-2)/24 - exactly (minus) the energy
of the bosonic theory first quantized ground state in "n" spacetime
dimensions. I would guess that it is not well-known in the string
community.
This similarity has not started with the 2003 paper that you mentioned.
See for example
http://www.arxiv.org/abs/http://www.arxiv.org/abs/hep-th/9805077
which is a hep-th paper that you have cited yourself. This paper has a
cute abstract: We are unable to formulate lattice gauge theory in terms of
Connes' spectral triples. ;-) The paper nevertheless refers to a couple of
related sources - for example, some axioms by Connes that are "tailored
such that there is a one-to-one correspondence between commutative
spectral triples and Riemannian spin manifolds". This certainly sounds
relevant for your question because the (n-2)/24 is related to CFTs and
therefore to Riemannian manifolds.
This stuff sounds too mathematical to me, but potentially interesting. If
someone understands "commutative spectral triples" and their isomorphism
to Riemannian spin manifolds, I would be happy to hear a short
presentation because it may include a nontrivial reinterpretation and/or
abstract generalization of the stringy perturbative expansions.
All the best
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Urs Schreiber
Sep6-04, 03:29 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>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag\nnews:Pine.LNX.4.31.0409031744140.1318-100000@feynman.harvard.edu...\n\n> This stuff sounds too mathematical to me, but potentially interesting. If\n> someone understands "commutative spectral triples" and their isomorphism\n> to Riemannian spin manifolds, I would be happy to hear a short\n> presentation because it may include a nontrivial reinterpretation and/or\n> abstract generalization of the stringy perturbative expansions.\n\nHi Lubos -\n\na commutative spectral triple of some Riemannian manifold M is a triple\nconsisting of\n\n- the set of smooth functions on that manifold. Under ordinary\nmultiplication they form an algebra, which is usually called A in this\ncontext\n\n- a Hilbert space on which this algebra is represented by self-adjoint\noperators. In this case this is just L^2(M) and the functions are\nrepresented as multiplication operators\n\n- a Dirac operator D on a graded Hilbert space which includes the above\nHilbert space. We can just choose the standard Hilbert space of a spin\nparticle on M and let D be the standard Dirac operator for that particle\n\n\nNow there is a theorem that given these three objects (A,H,D) you can from\nthem reconstruct the topological manifold M as well as a metric on it. The\ntopology is encoded in A and the metric in D, roughly.\n\nThe basic idea is to consider commutators of the form\n\n[D,f]\n\nwhere D is the Dirac operator and f an element of A as a generalized notion\nof gradient of f. Well, it is nothing but\n\n[D,f] = gamma^mu partial_mu f\n\nwhere gamma is the representation of the Clifford algebra on H.\n\nNow the point is that, [D,f] is itself an operator and we can consider it\'s\noperator norm. This operator norm can be interpreted as the norm of the\ngradient of f. This way a notion of distance on M is extracted from the\nabstract spectral triple, roughly as follows:\n\nGiven any two points on M (which can be characterized as ideals in the\nalgebra A) you can consider a function f which takes the value 0 at one\npoint and has gradient equal to 1 in the above sense. Then its value at the\nother point is nothing but the geodesic distance between the two points.\n\nThis is really a solution of the famous "Can you hear the shape of a\ndrum?"-question. The answer to that is, no, from the spectrum of the Laplace\noperator on some manifold M alone you cannot completely reconstructe the\nRiemannian manifold M, only some aspects of it. But Connes\' method shows\nthat when you replace the Laplace operator by its square root, the Dirac\noperator, and also include A and H, this is enough information to\nreconstruct M.\n\nMotivated by this theorem that Riemannian geometry can be recoved from a\ncommutative spectral triple, the idea of Connes\' non-commutative spectral\ngeometry is is to consider spectral triples where the algebra A is any\nabstract associative algebra and D any odd-graded operator, and to interpret\nthat as describing a generalized form of geometry.\n\n\nAnother point that Connes has popularized is the notion of "spectral\naction". He noted that the trace of the exponential of some power of the\nDirac operator D of a commutative spectral triple has an expansion whose\nfirst terms are nothing but the einstein Hilbert action on M for that\nparticular metric represented by D in the above way. There are higher order\ncorrection terms, too.\n\nEven more intersting, Connes and Lott showed that when one uses a slightly\nnon-commutative spectral triple, where the algebra A of smooth functions is\nreplaced by the algebra of smooth functions times a certain product of the\nalgebras of quaternions and complex numbers, the spectral action obtained\nfrom the associated spectral triple contains not only the Einstein-Hilbert\nterm, but also the action of Yang-Mills for the standard model gauge group.\nThis is called the Connes-Lott model.\n\n\nI was always intrigued by the question if this is related to string theory\nsomehow. Since every good idea is part of string theory, and since spectral\ntriples and the spectral action looks like a good idea, it might be one way\nto look at strings.\n\nThere have been attempts to make such a connection in the 90s, but, I was\ntold, the rise of the Matrix Model has washed away interest in the\n"spectral" aspects of noncommutative geometry in favor of the "algebraic"\naspects, where the non-coommutativity of the algebra itself plays the more\nprominent role, while the Dirac operator and its spectrum are ignored.\n\nThe most interesting approach to "spectral string theory" which I know of is\nthat by Ali Chamseddine\n\nAli H. Chamseddine,\nAn Effective Superstring Spectral Action\nhttp://arxiv.org/abs/hep-th/9705153 .\n\nHe had the obvious but very intriguing idea to use the worldsheet\nsupercharge as a Dirac operator D in a spectral triple, use the\nsuperstring\'s Hilbert space H as the Hilbert space and then try to compute\nthe associated spectral action.\n\nTo the lowest in which he works, and ignoring higher modes of the string, he\nclaims that the spectral action obtained from this spectral triple does\nindeed reproduce the string\'s effective action, i.e. the Einstein-Hilbert\nterm and the contribution from the B field.\n\nA while ago I enjoyed thinking about how to maybe improve on this result,\nbut I didn\'t make much progress. More discussion of the paper by Chamseddine\nas well as some half-baked random thoughts on how to proceed are given at\n\nhttp://golem.ph.utexas.edu/string/archives/000322.html .\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>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag
news:Pine.LNX.4.31.0409031744140.1318-100000@feynman.harvard.edu...
> This stuff sounds too mathematical to me, but potentially interesting. If
> someone understands "commutative spectral triples" and their isomorphism
> to Riemannian spin manifolds, I would be happy to hear a short
> presentation because it may include a nontrivial reinterpretation and/or
> abstract generalization of the stringy perturbative expansions.
Hi Lubos -
a commutative spectral triple of some Riemannian manifold M is a triple
consisting of
- the set of smooth functions on that manifold. Under ordinary
multiplication they form an algebra, which is usually called A in this
context
- a Hilbert space on which this algebra is represented by self-adjoint
operators. In this case this is just L^2(M) and the functions are
represented as multiplication operators
- a Dirac operator D on a graded Hilbert space which includes the above
Hilbert space. We can just choose the standard Hilbert space of a spin
particle on M and let D be the standard Dirac operator for that particle
Now there is a theorem that given these three objects (A,H,D) you can from
them reconstruct the topological manifold M as well as a metric on it. The
topology is encoded in A and the metric in D, roughly.
The basic idea is to consider commutators of the form
[D,f]
where D is the Dirac operator and f an element of A as a generalized notion
of gradient of f. Well, it is nothing but
[D,f] = \gamma^\mu partial_mu f
where \gamma is the representation of the Clifford algebra on H.
Now the point is that, [D,f] is itself an operator and we can consider it's
operator norm. This operator norm can be interpreted as the norm of the
gradient of f. This way a notion of distance on M is extracted from the
abstract spectral triple, roughly as follows:
Given any two points on M (which can be characterized as ideals in the
algebra A) you can consider a function f which takes the value at one
point and has gradient equal to 1 in the above sense. Then its value at the
other point is nothing but the geodesic distance between the two points.
This is really a solution of the famous "Can you hear the shape of a
drum?"-question. The answer to that is, no, from the spectrum of the Laplace
operator on some manifold M alone you cannot completely reconstructe the
Riemannian manifold M, only some aspects of it. But Connes' method shows
that when you replace the Laplace operator by its square root, the Dirac
operator, and also include A and H, this is enough information to
reconstruct M.
Motivated by this theorem that Riemannian geometry can be recoved from a
commutative spectral triple, the idea of Connes' non-commutative spectral
geometry is is to consider spectral triples where the algebra A is any
abstract associative algebra and D any odd-graded operator, and to interpret
that as describing a generalized form of geometry.
Another point that Connes has popularized is the notion of "spectral
action". He noted that the trace of the exponential of some power of the
Dirac operator D of a commutative spectral triple has an expansion whose
first terms are nothing but the einstein Hilbert action on M for that
particular metric represented by D in the above way. There are higher order
correction terms, too.
Even more intersting, Connes and Lott showed that when one uses a slightly
non-commutative spectral triple, where the algebra A of smooth functions is
replaced by the algebra of smooth functions times a certain product of the
algebras of quaternions and complex numbers, the spectral action obtained
from the associated spectral triple contains not only the Einstein-Hilbert
term, but also the action of Yang-Mills for the standard model gauge group.
This is called the Connes-Lott model.
I was always intrigued by the question if this is related to string theory
somehow. Since every good idea is part of string theory, and since spectral
triples and the spectral action looks like a good idea, it might be one way
to look at strings.
There have been attempts to make such a connection in the 90s, but, I was
told, the rise of the Matrix Model has washed away interest in the
"spectral" aspects of noncommutative geometry in favor of the "algebraic"
aspects, where the non-coommutativity of the algebra itself plays the more
prominent role, while the Dirac operator and its spectrum are ignored.
The most interesting approach to "spectral string theory" which I know of is
that by Ali Chamseddine
Ali H. Chamseddine,
An Effective Superstring Spectral Action
http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/9705153 .
He had the obvious but very intriguing idea to use the worldsheet
supercharge as a Dirac operator D in a spectral triple, use the
superstring's Hilbert space H as the Hilbert space and then try to compute
the associated spectral action.
To the lowest in which he works, and ignoring higher modes of the string, he
claims that the spectral action obtained from this spectral triple does
indeed reproduce the string's effective action, i.e. the Einstein-Hilbert
term and the contribution from the B field.
A while ago I enjoyed thinking about how to maybe improve on this result,
but I didn't make much progress. More discussion of the paper by Chamseddine
as well as some half-baked random thoughts on how to proceed are given at
http://golem.ph.utexas.edu/string/archives/000322.html .
Robert C. Helling
Sep6-04, 09:10 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 Fri, 3 Sep 2004 17:53:37 -0400, Lubos Motl <motl@feynman.harvard.edu> wrote:\n\n> This stuff sounds too mathematical to me, but potentially interesting. If\n> someone understands "commutative spectral triples" and their isomorphism\n> to Riemannian spin manifolds, I would be happy to hear a short\n> presentation because it may include a nontrivial reinterpretation and/or\n> abstract generalization of the stringy perturbative expansions.\n\nLet me write a response that is somewhat complementary to Urs\'. First,\nfor all the details consult my favourite review of non-commutative\ngeometry (written long before string people got interested in this\nstuff):\n\nAN INTRODUCTION TO NONCOMMUTATIVE SPACES AND THEIR GEOMETRY.\nBy Giovanni Landi (Trieste U. & INFN, Naples),. Jan 1997. 186pp.\ne-Print Archive: hep-th/9701078\n\nI assume you are comfortable with the usual string theory version of\nNCG, that is how a manifold is encoded in the algebra of continious\nfunctions on it. Actually, there are different flavours to this\nregarding which part of the structur of the maifold you want to encode\nin the algebra:\n\nFirst of all, there are the points in the manifold. They correspond to\nirreducible represenations (as all irreps are of the form\npi_x(f)=f(x)) or maximal ideals (they are all of the form\nI_x = {f|f(x)=0} ). This stuff is well known but only reconstructs the\nmanifold as a set (of points). Without throwing in any further\ninformation you can as well recover the topology (this is where the\nrequirement of the continuity of the functions comes in): This is less\nwell known among physicists but actually quite simple: In your first\nyear you learn that you specify the topology of a space by declaring\nsome of its subsets as open (up to some conditions on uniions and\nintersections). It is however equivalent to specify a "closure"\nfunction that maps all subsets to their closure (again up to some\nconditions like the closure doesn\'t add anything if acting on a\nclosure).\n\nIn terms of continuous functions that has a spimple expression: Start\nwith some subset S. Then consider all continuous functions vanishing\non S, call this ideal I_S. Then the closre C(S) is the set of points\non which all functions in I_S vanish. You should be able to\nreformulate this for yoursef in the algebraic language.\n\nOK, now we encoded the manifold as a topological set. This level of\nalgebraization is what is used in the string literature following\nDoulas and Hull and Connes Douglas Schwartz (or Seiberg and Witten if\nyou like). Then they express everything in terms of star products of\nordinary functions and \'borrow\' the usual integrals to write down the\naction. But that is not very intrinsic from the NCG point of view\nbecause the integral is thrown in not in terms of the non-commutative\nalgebra.\n\nBut you can define this extra structure also in the formal way: First,\nthere is an algebraic way to define differentiation, but that\'s quite\ninvolved and we won\'t need it here. The next bit of structure you\nwould like to reformulate in an algebraic way is a metric on your\nspace. This is where the spectral triple comes in: In short, the\nadditional structure you throw in is one operator that acts as a Dirac\noperator. There are some issues with the fine print as always in\nfunctional analysis as not all functions are differentiable and thus\nthe Dirac operator is only defined on a dense subset and therefore you\nhave to be careful with domains of definitions when you write down\nstuff like a commutator [D,A]=DA-AD, but let us igore those.\n\nThe nice thing about the Dirac operator is that it acts on spinors to\nproduce spinors (as opposed to the gradient that acts on scalar to\nproduce vectors), so it is easier to treat as it leaves you in the\nsame space (that is spin bundles) but again that is not really\nessential. What is important that in your C* algebra you have a norm\n(which is the supremum norm of your continious function): If your\nmanifold is compact it maps a function to the maximum of |f(x)| over\nthe manifold and if your space is not compact you have to be a bit\nmore careful in the beginning to make sure this always exists. Then if\nD is your Dirac operator, you can compute the norm of [D,f] and that\nis the maximum of the length of the gradient (as you see after two\nlines of gamma algebra). Now you can ask the question: Given two\npoints x and y, what is the largest difference |f(x)-f(y)| over all\nfunctions f such that the norm of [D,f] is <=1? Well, lets assume\nthere is a geodesic between x and y. Then the maximum will be achieved\nfor a function that has its gradient allways maximal and parallel to\nthe tanget of the geodesic. Thus, the maximum of |f(x)-f(y)| will be\nthe length of the geodesic.\n\nAgain, you can reformulate this using only algebraic properties (again\ntalking about values of irreps instead of f(x) etc) and from this\nobtain a definition of geodesic distance between points which directly\nleads to a metric.\n\nFinally we want to define an integral for a function f. Let us assume\nour Manifold is Riemannian of dimension d and compact. Then the Dirac\noperator is elliptic and has a spectrum that is positive and\ndiscrete. It is a fact of life, that the n-th eigenvalue of f |D|^(-d)\ngoes like 1/n, so the trace of that operator diverges\nlogarithmically. The surprising thing is that the divergence can be\ndefined properly: Let\'s call the n-th eigenvalue of the above operator\nl_n and consider the sums\n\n1/log(N) sum_{n=1}^N l_n\n\nThere is a sense in which the limit N->oo exists and that is called\nthe Dixmier trace (see section 5 of the review for details). It turns\nout that it is equal to the integral of f, only up to a number and\nthat numer is the one including (n-2)/24 that you were refering to (or\ntake the constant function f=1 to define it).\n\nThis log divergence is of course related to the log divergence of a\nfermion loop on that manifold and thus it can for example be given an\nexact meaning in terms of heat-kernels or any of your other\nregularization methods. But the upshot of all this is that this number\nyou cited is related to a one loop effect and there is probably its\nrelation to string theory as well. Again, for the details read the\nreview. It is really well written and understandable.\n\nBest\nRobert\n\n--\n..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO o.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO\nRobert C. Helling Department of Applied Mathematics and Theoretical Physics\nUniversity of Cambridge\nprint "Just another Phone: +44/1223/766870\nstupid .sig\\n"; http://www.aei-potsdam.mpg.de/~helling\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 Fri, 3 Sep 2004 17:53:37 -0400, Lubos Motl <motl@feynman.harvard.edu> wrote:
> This stuff sounds too mathematical to me, but potentially interesting. If
> someone understands "commutative spectral triples" and their isomorphism
> to Riemannian spin manifolds, I would be happy to hear a short
> presentation because it may include a nontrivial reinterpretation and/or
> abstract generalization of the stringy perturbative expansions.
Let me write a response that is somewhat complementary to Urs'. First,
for all the details consult my favourite review of non-commutative
geometry (written long before string people got interested in this
stuff):
AN INTRODUCTION TO NONCOMMUTATIVE SPACES AND THEIR GEOMETRY.
By Giovanni Landi (Trieste U. & INFN, Naples),. Jan 1997. 186pp.
e-Print Archive: http://www.arxiv.org/abs/hep-th/9701078
I assume you are comfortable with the usual string theory version of
NCG, that is how a manifold is encoded in the algebra of continious
functions on it. Actually, there are different flavours to this
regarding which part of the structur of the maifold you want to encode
in the algebra:
First of all, there are the points in the manifold. They correspond to
irreducible represenations (as all irreps are of the form
\pi_x(f)=f(x)) or maximal ideals (they are all of the form
I_x = {f|f(x)=0} ). This stuff is well known but only reconstructs the
manifold as a set (of points). Without throwing in any further
information you can as well recover the topology (this is where the
requirement of the continuity of the functions comes in): This is less
well known among physicists but actually quite simple: In your first
year you learn that you specify the topology of a space by declaring
some of its subsets as open (up to some conditions on uniions and
intersections). It is however equivalent to specify a "closure"
function that maps all subsets to their closure (again up to some
conditions like the closure doesn't add anything if acting on a
closure).
In terms of continuous functions that has a spimple expression: Start
with some subset S. Then consider all continuous functions vanishing
on S, call this ideal I_S. Then the closre C(S) is the set of points
on which all functions in I_S vanish. You should be able to
reformulate this for yoursef in the algebraic language.
OK, now we encoded the manifold as a topological set. This level of
algebraization is what is used in the string literature following
Doulas and Hull and Connes Douglas Schwartz (or Seiberg and Witten if
you like). Then they express everything in terms of star products of
ordinary functions and 'borrow' the usual integrals to write down the
action. But that is not very intrinsic from the NCG point of view
because the integral is thrown in not in terms of the non-commutative
algebra.
But you can define this extra structure also in the formal way: First,
there is an algebraic way to define differentiation, but that's quite
involved and we won't need it here. The next bit of structure you
would like to reformulate in an algebraic way is a metric on your
space. This is where the spectral triple comes in: In short, the
additional structure you throw in is one operator that acts as a Dirac
operator. There are some issues with the fine print as always in
functional analysis as not all functions are differentiable and thus
the Dirac operator is only defined on a dense subset and therefore you
have to be careful with domains of definitions when you write down
stuff like a commutator [D,A]=DA-AD, but let us igore those.
The nice thing about the Dirac operator is that it acts on spinors to
produce spinors (as opposed to the gradient that acts on scalar to
produce vectors), so it is easier to treat as it leaves you in the
same space (that is spin bundles) but again that is not really
essential. What is important that in your C* algebra you have a norm
(which is the supremum norm of your continious function): If your
manifold is compact it maps a function to the maximum of |f(x)| over
the manifold and if your space is not compact you have to be a bit
more careful in the beginning to make sure this always exists. Then if
D is your Dirac operator, you can compute the norm of [D,f] and that
is the maximum of the length of the gradient (as you see after two
lines of \gamma algebra). Now you can ask the question: Given two
points x and y, what is the largest difference |f(x)-f(y)| over all
functions f such that the norm of [D,f] is <=1? Well, lets assume
there is a geodesic between x and y. Then the maximum will be achieved
for a function that has its gradient allways maximal and parallel to
the tanget of the geodesic. Thus, the maximum of |f(x)-f(y)| will be
the length of the geodesic.
Again, you can reformulate this using only algebraic properties (again
talking about values of irreps instead of f(x) etc) and from this
obtain a definition of geodesic distance between points which directly
leads to a metric.
Finally we want to define an integral for a function f. Let us assume
our Manifold is Riemannian of dimension d and compact. Then the Dirac
operator is elliptic and has a spectrum that is positive and
discrete. It is a fact of life, that the n-th eigenvalue of f |D|^(-d)
goes like 1/n, so the trace of that operator diverges
logarithmically. The surprising thing is that the divergence can be
defined properly: Let's call the n-th eigenvalue of the above operator
l_n and consider the sums
1/log(N) sum_{n=1}^N l_n
There is a sense in which the limit N->oo exists and that is called
the Dixmier trace (see section 5 of the review for details). It turns
out that it is equal to the integral of f, only up to a number and
that numer is the one including (n-2)/24 that you were refering to (or
take the constant function f=1 to define it).
This log divergence is of course related to the log divergence of a
fermion loop on that manifold and thus it can for example be given an
exact meaning in terms of heat-kernels or any of your other
regularization methods. But the upshot of all this is that this number
you cited is related to a one loop effect and there is probably its
relation to string theory as well. Again, for the details read the
review. It is really well written and understandable.
Best
Robert
--
..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo. oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/766870
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
Robert C. Helling
Sep6-04, 09:10 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 Fri, 3 Sep 2004 17:53:37 -0400, Lubos Motl <motl@feynman.harvard.edu> wrote:\n\n> This stuff sounds too mathematical to me, but potentially interesting. If\n> someone understands "commutative spectral triples" and their isomorphism\n> to Riemannian spin manifolds, I would be happy to hear a short\n> presentation because it may include a nontrivial reinterpretation and/or\n> abstract generalization of the stringy perturbative expansions.\n\nLet me write a response that is somewhat complementary to Urs\'. First,\nfor all the details consult my favourite review of non-commutative\ngeometry (written long before string people got interested in this\nstuff):\n\nAN INTRODUCTION TO NONCOMMUTATIVE SPACES AND THEIR GEOMETRY.\nBy Giovanni Landi (Trieste U. & INFN, Naples),. Jan 1997. 186pp.\ne-Print Archive: hep-th/9701078\n\nI assume you are comfortable with the usual string theory version of\nNCG, that is how a manifold is encoded in the algebra of continious\nfunctions on it. Actually, there are different flavours to this\nregarding which part of the structur of the maifold you want to encode\nin the algebra:\n\nFirst of all, there are the points in the manifold. They correspond to\nirreducible represenations (as all irreps are of the form\npi_x(f)=f(x)) or maximal ideals (they are all of the form\nI_x = {f|f(x)=0} ). This stuff is well known but only reconstructs the\nmanifold as a set (of points). Without throwing in any further\ninformation you can as well recover the topology (this is where the\nrequirement of the continuity of the functions comes in): This is less\nwell known among physicists but actually quite simple: In your first\nyear you learn that you specify the topology of a space by declaring\nsome of its subsets as open (up to some conditions on uniions and\nintersections). It is however equivalent to specify a "closure"\nfunction that maps all subsets to their closure (again up to some\nconditions like the closure doesn\'t add anything if acting on a\nclosure).\n\nIn terms of continuous functions that has a spimple expression: Start\nwith some subset S. Then consider all continuous functions vanishing\non S, call this ideal I_S. Then the closre C(S) is the set of points\non which all functions in I_S vanish. You should be able to\nreformulate this for yoursef in the algebraic language.\n\nOK, now we encoded the manifold as a topological set. This level of\nalgebraization is what is used in the string literature following\nDoulas and Hull and Connes Douglas Schwartz (or Seiberg and Witten if\nyou like). Then they express everything in terms of star products of\nordinary functions and \'borrow\' the usual integrals to write down the\naction. But that is not very intrinsic from the NCG point of view\nbecause the integral is thrown in not in terms of the non-commutative\nalgebra.\n\nBut you can define this extra structure also in the formal way: First,\nthere is an algebraic way to define differentiation, but that\'s quite\ninvolved and we won\'t need it here. The next bit of structure you\nwould like to reformulate in an algebraic way is a metric on your\nspace. This is where the spectral triple comes in: In short, the\nadditional structure you throw in is one operator that acts as a Dirac\noperator. There are some issues with the fine print as always in\nfunctional analysis as not all functions are differentiable and thus\nthe Dirac operator is only defined on a dense subset and therefore you\nhave to be careful with domains of definitions when you write down\nstuff like a commutator [D,A]=DA-AD, but let us igore those.\n\nThe nice thing about the Dirac operator is that it acts on spinors to\nproduce spinors (as opposed to the gradient that acts on scalar to\nproduce vectors), so it is easier to treat as it leaves you in the\nsame space (that is spin bundles) but again that is not really\nessential. What is important that in your C* algebra you have a norm\n(which is the supremum norm of your continious function): If your\nmanifold is compact it maps a function to the maximum of |f(x)| over\nthe manifold and if your space is not compact you have to be a bit\nmore careful in the beginning to make sure this always exists. Then if\nD is your Dirac operator, you can compute the norm of [D,f] and that\nis the maximum of the length of the gradient (as you see after two\nlines of gamma algebra). Now you can ask the question: Given two\npoints x and y, what is the largest difference |f(x)-f(y)| over all\nfunctions f such that the norm of [D,f] is <=1? Well, lets assume\nthere is a geodesic between x and y. Then the maximum will be achieved\nfor a function that has its gradient allways maximal and parallel to\nthe tanget of the geodesic. Thus, the maximum of |f(x)-f(y)| will be\nthe length of the geodesic.\n\nAgain, you can reformulate this using only algebraic properties (again\ntalking about values of irreps instead of f(x) etc) and from this\nobtain a definition of geodesic distance between points which directly\nleads to a metric.\n\nFinally we want to define an integral for a function f. Let us assume\nour Manifold is Riemannian of dimension d and compact. Then the Dirac\noperator is elliptic and has a spectrum that is positive and\ndiscrete. It is a fact of life, that the n-th eigenvalue of f |D|^(-d)\ngoes like 1/n, so the trace of that operator diverges\nlogarithmically. The surprising thing is that the divergence can be\ndefined properly: Let\'s call the n-th eigenvalue of the above operator\nl_n and consider the sums\n\n1/log(N) sum_{n=1}^N l_n\n\nThere is a sense in which the limit N->oo exists and that is called\nthe Dixmier trace (see section 5 of the review for details). It turns\nout that it is equal to the integral of f, only up to a number and\nthat numer is the one including (n-2)/24 that you were refering to (or\ntake the constant function f=1 to define it).\n\nThis log divergence is of course related to the log divergence of a\nfermion loop on that manifold and thus it can for example be given an\nexact meaning in terms of heat-kernels or any of your other\nregularization methods. But the upshot of all this is that this number\nyou cited is related to a one loop effect and there is probably its\nrelation to string theory as well. Again, for the details read the\nreview. It is really well written and understandable.\n\nBest\nRobert\n\n--\n..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO o.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO\nRobert C. Helling Department of Applied Mathematics and Theoretical Physics\nUniversity of Cambridge\nprint "Just another Phone: +44/1223/766870\nstupid .sig\\n"; http://www.aei-potsdam.mpg.de/~helling\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 Fri, 3 Sep 2004 17:53:37 -0400, Lubos Motl <motl@feynman.harvard.edu> wrote:
> This stuff sounds too mathematical to me, but potentially interesting. If
> someone understands "commutative spectral triples" and their isomorphism
> to Riemannian spin manifolds, I would be happy to hear a short
> presentation because it may include a nontrivial reinterpretation and/or
> abstract generalization of the stringy perturbative expansions.
Let me write a response that is somewhat complementary to Urs'. First,
for all the details consult my favourite review of non-commutative
geometry (written long before string people got interested in this
stuff):
AN INTRODUCTION TO NONCOMMUTATIVE SPACES AND THEIR GEOMETRY.
By Giovanni Landi (Trieste U. & INFN, Naples),. Jan 1997. 186pp.
e-Print Archive: http://www.arxiv.org/abs/hep-th/9701078
I assume you are comfortable with the usual string theory version of
NCG, that is how a manifold is encoded in the algebra of continious
functions on it. Actually, there are different flavours to this
regarding which part of the structur of the maifold you want to encode
in the algebra:
First of all, there are the points in the manifold. They correspond to
irreducible represenations (as all irreps are of the form
\pi_x(f)=f(x)) or maximal ideals (they are all of the form
I_x = {f|f(x)=0} ). This stuff is well known but only reconstructs the
manifold as a set (of points). Without throwing in any further
information you can as well recover the topology (this is where the
requirement of the continuity of the functions comes in): This is less
well known among physicists but actually quite simple: In your first
year you learn that you specify the topology of a space by declaring
some of its subsets as open (up to some conditions on uniions and
intersections). It is however equivalent to specify a "closure"
function that maps all subsets to their closure (again up to some
conditions like the closure doesn't add anything if acting on a
closure).
In terms of continuous functions that has a spimple expression: Start
with some subset S. Then consider all continuous functions vanishing
on S, call this ideal I_S. Then the closre C(S) is the set of points
on which all functions in I_S vanish. You should be able to
reformulate this for yoursef in the algebraic language.
OK, now we encoded the manifold as a topological set. This level of
algebraization is what is used in the string literature following
Doulas and Hull and Connes Douglas Schwartz (or Seiberg and Witten if
you like). Then they express everything in terms of star products of
ordinary functions and 'borrow' the usual integrals to write down the
action. But that is not very intrinsic from the NCG point of view
because the integral is thrown in not in terms of the non-commutative
algebra.
But you can define this extra structure also in the formal way: First,
there is an algebraic way to define differentiation, but that's quite
involved and we won't need it here. The next bit of structure you
would like to reformulate in an algebraic way is a metric on your
space. This is where the spectral triple comes in: In short, the
additional structure you throw in is one operator that acts as a Dirac
operator. There are some issues with the fine print as always in
functional analysis as not all functions are differentiable and thus
the Dirac operator is only defined on a dense subset and therefore you
have to be careful with domains of definitions when you write down
stuff like a commutator [D,A]=DA-AD, but let us igore those.
The nice thing about the Dirac operator is that it acts on spinors to
produce spinors (as opposed to the gradient that acts on scalar to
produce vectors), so it is easier to treat as it leaves you in the
same space (that is spin bundles) but again that is not really
essential. What is important that in your C* algebra you have a norm
(which is the supremum norm of your continious function): If your
manifold is compact it maps a function to the maximum of |f(x)| over
the manifold and if your space is not compact you have to be a bit
more careful in the beginning to make sure this always exists. Then if
D is your Dirac operator, you can compute the norm of [D,f] and that
is the maximum of the length of the gradient (as you see after two
lines of \gamma algebra). Now you can ask the question: Given two
points x and y, what is the largest difference |f(x)-f(y)| over all
functions f such that the norm of [D,f] is <=1? Well, lets assume
there is a geodesic between x and y. Then the maximum will be achieved
for a function that has its gradient allways maximal and parallel to
the tanget of the geodesic. Thus, the maximum of |f(x)-f(y)| will be
the length of the geodesic.
Again, you can reformulate this using only algebraic properties (again
talking about values of irreps instead of f(x) etc) and from this
obtain a definition of geodesic distance between points which directly
leads to a metric.
Finally we want to define an integral for a function f. Let us assume
our Manifold is Riemannian of dimension d and compact. Then the Dirac
operator is elliptic and has a spectrum that is positive and
discrete. It is a fact of life, that the n-th eigenvalue of f |D|^(-d)
goes like 1/n, so the trace of that operator diverges
logarithmically. The surprising thing is that the divergence can be
defined properly: Let's call the n-th eigenvalue of the above operator
l_n and consider the sums
1/log(N) sum_{n=1}^N l_n
There is a sense in which the limit N->oo exists and that is called
the Dixmier trace (see section 5 of the review for details). It turns
out that it is equal to the integral of f, only up to a number and
that numer is the one including (n-2)/24 that you were refering to (or
take the constant function f=1 to define it).
This log divergence is of course related to the log divergence of a
fermion loop on that manifold and thus it can for example be given an
exact meaning in terms of heat-kernels or any of your other
regularization methods. But the upshot of all this is that this number
you cited is related to a one loop effect and there is probably its
relation to string theory as well. Again, for the details read the
review. It is really well written and understandable.
Best
Robert
--
..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo. oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/766870
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
Robert C. Helling
Sep6-04, 09:10 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 Fri, 3 Sep 2004 17:53:37 -0400, Lubos Motl <motl@feynman.harvard.edu> wrote:\n\n> This stuff sounds too mathematical to me, but potentially interesting. If\n> someone understands "commutative spectral triples" and their isomorphism\n> to Riemannian spin manifolds, I would be happy to hear a short\n> presentation because it may include a nontrivial reinterpretation and/or\n> abstract generalization of the stringy perturbative expansions.\n\nLet me write a response that is somewhat complementary to Urs\'. First,\nfor all the details consult my favourite review of non-commutative\ngeometry (written long before string people got interested in this\nstuff):\n\nAN INTRODUCTION TO NONCOMMUTATIVE SPACES AND THEIR GEOMETRY.\nBy Giovanni Landi (Trieste U. & INFN, Naples),. Jan 1997. 186pp.\ne-Print Archive: hep-th/9701078\n\nI assume you are comfortable with the usual string theory version of\nNCG, that is how a manifold is encoded in the algebra of continious\nfunctions on it. Actually, there are different flavours to this\nregarding which part of the structur of the maifold you want to encode\nin the algebra:\n\nFirst of all, there are the points in the manifold. They correspond to\nirreducible represenations (as all irreps are of the form\npi_x(f)=f(x)) or maximal ideals (they are all of the form\nI_x = {f|f(x)=0} ). This stuff is well known but only reconstructs the\nmanifold as a set (of points). Without throwing in any further\ninformation you can as well recover the topology (this is where the\nrequirement of the continuity of the functions comes in): This is less\nwell known among physicists but actually quite simple: In your first\nyear you learn that you specify the topology of a space by declaring\nsome of its subsets as open (up to some conditions on uniions and\nintersections). It is however equivalent to specify a "closure"\nfunction that maps all subsets to their closure (again up to some\nconditions like the closure doesn\'t add anything if acting on a\nclosure).\n\nIn terms of continuous functions that has a spimple expression: Start\nwith some subset S. Then consider all continuous functions vanishing\non S, call this ideal I_S. Then the closre C(S) is the set of points\non which all functions in I_S vanish. You should be able to\nreformulate this for yoursef in the algebraic language.\n\nOK, now we encoded the manifold as a topological set. This level of\nalgebraization is what is used in the string literature following\nDoulas and Hull and Connes Douglas Schwartz (or Seiberg and Witten if\nyou like). Then they express everything in terms of star products of\nordinary functions and \'borrow\' the usual integrals to write down the\naction. But that is not very intrinsic from the NCG point of view\nbecause the integral is thrown in not in terms of the non-commutative\nalgebra.\n\nBut you can define this extra structure also in the formal way: First,\nthere is an algebraic way to define differentiation, but that\'s quite\ninvolved and we won\'t need it here. The next bit of structure you\nwould like to reformulate in an algebraic way is a metric on your\nspace. This is where the spectral triple comes in: In short, the\nadditional structure you throw in is one operator that acts as a Dirac\noperator. There are some issues with the fine print as always in\nfunctional analysis as not all functions are differentiable and thus\nthe Dirac operator is only defined on a dense subset and therefore you\nhave to be careful with domains of definitions when you write down\nstuff like a commutator [D,A]=DA-AD, but let us igore those.\n\nThe nice thing about the Dirac operator is that it acts on spinors to\nproduce spinors (as opposed to the gradient that acts on scalar to\nproduce vectors), so it is easier to treat as it leaves you in the\nsame space (that is spin bundles) but again that is not really\nessential. What is important that in your C* algebra you have a norm\n(which is the supremum norm of your continious function): If your\nmanifold is compact it maps a function to the maximum of |f(x)| over\nthe manifold and if your space is not compact you have to be a bit\nmore careful in the beginning to make sure this always exists. Then if\nD is your Dirac operator, you can compute the norm of [D,f] and that\nis the maximum of the length of the gradient (as you see after two\nlines of gamma algebra). Now you can ask the question: Given two\npoints x and y, what is the largest difference |f(x)-f(y)| over all\nfunctions f such that the norm of [D,f] is <=1? Well, lets assume\nthere is a geodesic between x and y. Then the maximum will be achieved\nfor a function that has its gradient allways maximal and parallel to\nthe tanget of the geodesic. Thus, the maximum of |f(x)-f(y)| will be\nthe length of the geodesic.\n\nAgain, you can reformulate this using only algebraic properties (again\ntalking about values of irreps instead of f(x) etc) and from this\nobtain a definition of geodesic distance between points which directly\nleads to a metric.\n\nFinally we want to define an integral for a function f. Let us assume\nour Manifold is Riemannian of dimension d and compact. Then the Dirac\noperator is elliptic and has a spectrum that is positive and\ndiscrete. It is a fact of life, that the n-th eigenvalue of f |D|^(-d)\ngoes like 1/n, so the trace of that operator diverges\nlogarithmically. The surprising thing is that the divergence can be\ndefined properly: Let\'s call the n-th eigenvalue of the above operator\nl_n and consider the sums\n\n1/log(N) sum_{n=1}^N l_n\n\nThere is a sense in which the limit N->oo exists and that is called\nthe Dixmier trace (see section 5 of the review for details). It turns\nout that it is equal to the integral of f, only up to a number and\nthat numer is the one including (n-2)/24 that you were refering to (or\ntake the constant function f=1 to define it).\n\nThis log divergence is of course related to the log divergence of a\nfermion loop on that manifold and thus it can for example be given an\nexact meaning in terms of heat-kernels or any of your other\nregularization methods. But the upshot of all this is that this number\nyou cited is related to a one loop effect and there is probably its\nrelation to string theory as well. Again, for the details read the\nreview. It is really well written and understandable.\n\nBest\nRobert\n\n--\n..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO o.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO\nRobert C. Helling Department of Applied Mathematics and Theoretical Physics\nUniversity of Cambridge\nprint "Just another Phone: +44/1223/766870\nstupid .sig\\n"; http://www.aei-potsdam.mpg.de/~helling\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 Fri, 3 Sep 2004 17:53:37 -0400, Lubos Motl <motl@feynman.harvard.edu> wrote:
> This stuff sounds too mathematical to me, but potentially interesting. If
> someone understands "commutative spectral triples" and their isomorphism
> to Riemannian spin manifolds, I would be happy to hear a short
> presentation because it may include a nontrivial reinterpretation and/or
> abstract generalization of the stringy perturbative expansions.
Let me write a response that is somewhat complementary to Urs'. First,
for all the details consult my favourite review of non-commutative
geometry (written long before string people got interested in this
stuff):
AN INTRODUCTION TO NONCOMMUTATIVE SPACES AND THEIR GEOMETRY.
By Giovanni Landi (Trieste U. & INFN, Naples),. Jan 1997. 186pp.
e-Print Archive: http://www.arxiv.org/abs/hep-th/9701078
I assume you are comfortable with the usual string theory version of
NCG, that is how a manifold is encoded in the algebra of continious
functions on it. Actually, there are different flavours to this
regarding which part of the structur of the maifold you want to encode
in the algebra:
First of all, there are the points in the manifold. They correspond to
irreducible represenations (as all irreps are of the form
\pi_x(f)=f(x)) or maximal ideals (they are all of the form
I_x = {f|f(x)=0} ). This stuff is well known but only reconstructs the
manifold as a set (of points). Without throwing in any further
information you can as well recover the topology (this is where the
requirement of the continuity of the functions comes in): This is less
well known among physicists but actually quite simple: In your first
year you learn that you specify the topology of a space by declaring
some of its subsets as open (up to some conditions on uniions and
intersections). It is however equivalent to specify a "closure"
function that maps all subsets to their closure (again up to some
conditions like the closure doesn't add anything if acting on a
closure).
In terms of continuous functions that has a spimple expression: Start
with some subset S. Then consider all continuous functions vanishing
on S, call this ideal I_S. Then the closre C(S) is the set of points
on which all functions in I_S vanish. You should be able to
reformulate this for yoursef in the algebraic language.
OK, now we encoded the manifold as a topological set. This level of
algebraization is what is used in the string literature following
Doulas and Hull and Connes Douglas Schwartz (or Seiberg and Witten if
you like). Then they express everything in terms of star products of
ordinary functions and 'borrow' the usual integrals to write down the
action. But that is not very intrinsic from the NCG point of view
because the integral is thrown in not in terms of the non-commutative
algebra.
But you can define this extra structure also in the formal way: First,
there is an algebraic way to define differentiation, but that's quite
involved and we won't need it here. The next bit of structure you
would like to reformulate in an algebraic way is a metric on your
space. This is where the spectral triple comes in: In short, the
additional structure you throw in is one operator that acts as a Dirac
operator. There are some issues with the fine print as always in
functional analysis as not all functions are differentiable and thus
the Dirac operator is only defined on a dense subset and therefore you
have to be careful with domains of definitions when you write down
stuff like a commutator [D,A]=DA-AD, but let us igore those.
The nice thing about the Dirac operator is that it acts on spinors to
produce spinors (as opposed to the gradient that acts on scalar to
produce vectors), so it is easier to treat as it leaves you in the
same space (that is spin bundles) but again that is not really
essential. What is important that in your C* algebra you have a norm
(which is the supremum norm of your continious function): If your
manifold is compact it maps a function to the maximum of |f(x)| over
the manifold and if your space is not compact you have to be a bit
more careful in the beginning to make sure this always exists. Then if
D is your Dirac operator, you can compute the norm of [D,f] and that
is the maximum of the length of the gradient (as you see after two
lines of \gamma algebra). Now you can ask the question: Given two
points x and y, what is the largest difference |f(x)-f(y)| over all
functions f such that the norm of [D,f] is <=1? Well, lets assume
there is a geodesic between x and y. Then the maximum will be achieved
for a function that has its gradient allways maximal and parallel to
the tanget of the geodesic. Thus, the maximum of |f(x)-f(y)| will be
the length of the geodesic.
Again, you can reformulate this using only algebraic properties (again
talking about values of irreps instead of f(x) etc) and from this
obtain a definition of geodesic distance between points which directly
leads to a metric.
Finally we want to define an integral for a function f. Let us assume
our Manifold is Riemannian of dimension d and compact. Then the Dirac
operator is elliptic and has a spectrum that is positive and
discrete. It is a fact of life, that the n-th eigenvalue of f |D|^(-d)
goes like 1/n, so the trace of that operator diverges
logarithmically. The surprising thing is that the divergence can be
defined properly: Let's call the n-th eigenvalue of the above operator
l_n and consider the sums
1/log(N) sum_{n=1}^N l_n
There is a sense in which the limit N->oo exists and that is called
the Dixmier trace (see section 5 of the review for details). It turns
out that it is equal to the integral of f, only up to a number and
that numer is the one including (n-2)/24 that you were refering to (or
take the constant function f=1 to define it).
This log divergence is of course related to the log divergence of a
fermion loop on that manifold and thus it can for example be given an
exact meaning in terms of heat-kernels or any of your other
regularization methods. But the upshot of all this is that this number
you cited is related to a one loop effect and there is probably its
relation to string theory as well. Again, for the details read the
review. It is really well written and understandable.
Best
Robert
--
..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo. oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/766870
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
Lubos Motl
Sep6-04, 07:10 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>Dear Robert and Urs,\n\nthank you for your extensive answers about the spectral triples - many\npoints are very interesting.\n\nWell, let me admit that the very notion of "triples" in this context is\nnot too attractive for me. I thought that there would be related to\ntriples of things of the same type - like in the seven-author paper\n"Triples, fluxes, and something else".\n\nThese triples of three completely different things always look to me like\na type of mathematical, bureaucratic nonsense. There is nothing canonical\nabout (A,H,D) being a triple - it could also be a sextuplet (A,H,D,R,F,V)\nwhere A,H,D are essentially the same things as before, R is what you use\nto count instead of real numbers, F is the third basic operation that you\nuse for occasional functional integral calculations (the three basic\noperations are addition, subtraction, and Feynman\'s integral) and V is the\ntype of visa that you need to do this job legally. ;-)\n\nOK, more seriously, there seems to be something very interesting about the\ncoefficient (d-2)/24 that I have not understood yet. My understanding of\nRobert\'s last paragraph is that the trace of the (-d)-th power of the\nDirac operator on a d-dimensional manifold (I took f=1) equals, in some\nappropriate sense, the logarithm of a UV cutoff times the volume of the\nmanifold times (or over) (d-2)/24. Do I understand it well?\n\nThis is just very interesting for the kind of "worldsheets for\nworldsheets" ideas that I am still trying to study. It\'s because (the\npower of) the Dirac operator lives on the d-dimensional "target space",\nand its trace is, in some sense, target space physics (or mathematics),\nbut the relation of the operator\'s trace to its integral seems to involve\na constant that would naturally appear (as a ground state energy, central\ncharge, or something like that) in a light-cone gauge pre-string theory\ndescribing the d-dimensional target space. Does it have a simple CFT\nexplanation? I am sure that these relations are connected with CFT and the\nappearance of 1/24 is not a coincidence - for example see the paper\n\nhttp://www.math.univ-metz.fr/~benameur/Adv1864.pdf\n\ntalks about conformal invariant regulators in the context of the Dixmier\ntrace... If someone has a proof of the ratio (d-2)/24, I will be happy to\nsee it.\n\nAll the best\nLubos\n_____________________________________ _________________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\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>Dear Robert and Urs,
thank you for your extensive answers about the spectral triples - many
points are very interesting.
Well, let me admit that the very notion of "triples" in this context is
not too attractive for me. I thought that there would be related to
triples of things of the same type - like in the seven-author paper
"Triples, fluxes, and something else".
These triples of three completely different things always look to me like
a type of mathematical, bureaucratic nonsense. There is nothing canonical
about (A,H,D) being a triple - it could also be a sextuplet (A,H,D,R,F,V)
where A,H,D are essentially the same things as before, R is what you use
to count instead of real numbers, F is the third basic operation that you
use for occasional functional integral calculations (the three basic
operations are addition, subtraction, and Feynman's integral) and V is the
type of visa that you need to do this job legally. ;-)
OK, more seriously, there seems to be something very interesting about the
coefficient (d-2)/24 that I have not understood yet. My understanding of
Robert's last paragraph is that the trace of the (-d)-th power of the
Dirac operator on a d-dimensional manifold (I took f=1) equals, in some
appropriate sense, the logarithm of a UV cutoff times the volume of the
manifold times (or over) (d-2)/24. Do I understand it well?
This is just very interesting for the kind of "worldsheets for
worldsheets" ideas that I am still trying to study. It's because (the
power of) the Dirac operator lives on the d-dimensional "target space",
and its trace is, in some sense, target space physics (or mathematics),
but the relation of the operator's trace to its integral seems to involve
a constant that would naturally appear (as a ground state energy, central
charge, or something like that) in a light-cone gauge pre-string theory
describing the d-dimensional target space. Does it have a simple CFT
explanation? I am sure that these relations are connected with CFT and the
appearance of 1/24 is not a coincidence - for example see the paper
http://www.math.univ-metz.fr/~benameur/Adv1864.pdf
talks about conformal invariant regulators in the context of the Dixmier
trace... If someone has a proof of the ratio (d-2)/24, I will be happy to
see it.
All the best
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Urs Schreiber
Sep7-04, 03:42 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>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag\nnews:Pine.LNX.4.31.0409061954500.6816-100000@feynman.harvard.edu...\n\n> Well, let me admit that the very notion of "triples" in this context is\n> not too attractive for me. I thought that there would be related to\n> triples of things of the same type - like in the seven-author paper\n> "Triples, fluxes, and something else".\n>\n> These triples of three completely different things always look to me like\n> a type of mathematical, bureaucratic nonsense. There is nothing canonical\n> about (A,H,D) being a triple\n\nHi Lubos -\n\nI know what you mean, but would like to offer a very natural interpretation\nof these triples:\n\nThey are nothing but a mathematician\'s way to talk about supersymmetric\nquantum mechanics/field theory.\n\nAll the triple says is that there is a Hilbert space with operators on it,\nand that (at least) one of these operators, namely D, is a supersymmetry\ngenerator.\n\nYou could reformulate the idea behind Connes\' spectral triples as:\n\na) "Supersymmetric quantum mechanics encodes Riemannian geometry and its\ngeneralizations."\n\nand inded this is what has been done for instance in math-ph/9807006 and\nhep-th/9706132, giving lots of details.\n\nIn this sense it agrees pretty nicely with the string theory insight that\n\nb) "Superconformal field theories on the worldsheet encode spacetime and its\ngeneralizations."\n\nWhen SCFT is regarded as an infinite-dimensional quantum mechanics in loop\nspace statement b) becomes just a special case of statement a).\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>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag
news:Pine.LNX.4.31.0409061954500.6816-100000@feynman.harvard.edu...
> Well, let me admit that the very notion of "triples" in this context is
> not too attractive for me. I thought that there would be related to
> triples of things of the same type - like in the seven-author paper
> "Triples, fluxes, and something else".
>
> These triples of three completely different things always look to me like
> a type of mathematical, bureaucratic nonsense. There is nothing canonical
> about (A,H,D) being a triple
Hi Lubos -
I know what you mean, but would like to offer a very natural interpretation
of these triples:
They are nothing but a mathematician's way to talk about supersymmetric
quantum mechanics/field theory.
All the triple says is that there is a Hilbert space with operators on it,
and that (at least) one of these operators, namely D, is a supersymmetry
generator.
You could reformulate the idea behind Connes' spectral triples as:
a) "Supersymmetric quantum mechanics encodes Riemannian geometry and its
generalizations."
and inded this is what has been done for instance in http://www.arxiv.org/abs/math-ph/9807006 and
http://www.arxiv.org/abs/hep-th/9706132, giving lots of details.
In this sense it agrees pretty nicely with the string theory insight that
b) "Superconformal field theories on the worldsheet encode spacetime and its
generalizations."
When SCFT is regarded as an infinite-dimensional quantum mechanics in loop
space statement b) becomes just a special case of statement a).
Robert C. Helling
Sep7-04, 07:09 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>On Mon, 6 Sep 2004 20:10:59 -0400, Lubos Motl <motl@feynman.harvard.edu> wrote:\n\n> Well, let me admit that the very notion of "triples" in this context is\n> not too attractive for me. I thought that there would be related to\n> triples of things of the same type - like in the seven-author paper\n> "Triples, fluxes, and something else".\n\nOK agreed, you could also call a representation of a Lie algebra a\ntriplet (A,V,rho): It consists of an algebra, a vector space and a map\n(with properties) from the algebra to the endomorphisms of the vector\nspace.\n\nBTW I explicitly mention the Hilbert space in my post but I was\nrefering to it as "functional analysis fine print" or something along\nthe lines. Think of it as the L_2 space of the manifold and the\nalgebra as the continious functions. Those are different function\nspaces but they contain a common dense subset (if you allow me to talk\nabout elements of L_2 as it they were functions, strictly speaking\nthey are equivalence classes of functions that can differ on sets of\nmeasure zero).\n\n> OK, more seriously,\n\nGood.\n\n> there seems to be something very interesting about the\n> coefficient (d-2)/24 that I have not understood yet. My understanding of\n> Robert\'s last paragraph is that the trace of the (-d)-th power of the\n> Dirac operator on a d-dimensional manifold (I took f=1) equals, in some\n> appropriate sense, the logarithm of a UV cutoff times the volume of the\n> manifold times (or over) (d-2)/24. Do I understand it well?\n\nYes, correct, at least in the limit of large cut-off. Up to some\npowers of two and pi. The correct formula is (9.3) in Landi\'s\nreview. First of all, the above trace scales correctly under scale\ntransformations. The surprising thing is of course that the constant\nof proportionality is the same for all manifolds.\n\nMaybe I can say a bit more than yesterday. First of all, here is a\nshort cut thru the review: Assuming you already know the basic stuff,\nyou can start reading chapter 5 "The spectral calculus", here the he\nintroduces all the tricks you can play with the Dirac operator and\nwhat these Dixmier traces are (they are some regularization of the\nusual trace). The stuff about the omega-limits is interesting but we\ndon\'t need it as in our case the ordinary limits exist. Make sure you\nunderstand the examples. You only need to read until including section\n5.5.\n\nThen ignore section 6 about differential forms and 7 about vector\nbundles and 8 about YM theory and the NC standard model (which at tree\nlevel predicts a Higgs mass of 279GeV, see p611 of Connes\' book, to\nunderstand this was my original motivation in 1997 to learn this\nstuff) but then start reading again in section 9 where 9.1 and 9.2\nhint in the direction that relate the Einstein action to these\ntraces.\n\nThe trick is to use the heat-kernel\n\nh(s) = sum_i exp(-s lambda_i)\n\nwhere lambda_i is the i\'s eigenvalue. Then you can show (if your\nGerman is still in working order, read ch. 3 of my master thesis\nhttp://www.aei-potsdam.mpg.de/research/thesis/helling_dipl.ps.gz )\nthat (again up to computable constants) h(s) s^(d/2) is a power series\nin s^2 and the coefficicient of s^2 is proportional to the Einstein\nHilbert action. From what I say in that thesis (and I am to lazy to\nrepeat here), this is really the log divergence of the 1-loop\neffective action of a scalar field on that manifold.\n\nOh, I forgot, of course expressions of the lambda_i are encoded in the\nDixmier traces of the Dirac operator and the Laplacian.\n\nRobert\n\n\n--\n..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO o.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO\nRobert C. Helling Department of Applied Mathematics and Theoretical Physics\nUniversity of Cambridge\nprint "Just another Phone: +44/1223/766870\nstupid .sig\\n"; http://www.aei-potsdam.mpg.de/~helling\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, 6 Sep 2004 20:10:59 -0400, Lubos Motl <motl@feynman.harvard.edu> wrote:
> Well, let me admit that the very notion of "triples" in this context is
> not too attractive for me. I thought that there would be related to
> triples of things of the same type - like in the seven-author paper
> "Triples, fluxes, and something else".
OK agreed, you could also call a representation of a Lie algebra a
triplet (A,V,\rho): It consists of an algebra, a vector space and a map
(with properties) from the algebra to the endomorphisms of the vector
space.
BTW I explicitly mention the Hilbert space in my post but I was
refering to it as "functional analysis fine print" or something along
the lines. Think of it as the L_2 space of the manifold and the
algebra as the continious functions. Those are different function
spaces but they contain a common dense subset (if you allow me to talk
about elements of L_2 as it they were functions, strictly speaking
they are equivalence classes of functions that can differ on sets of
measure zero).
> OK, more seriously,
Good.
> there seems to be something very interesting about the
> coefficient (d-2)/24 that I have not understood yet. My understanding of
> Robert's last paragraph is that the trace of the (-d)-th power of the
> Dirac operator on a d-dimensional manifold (I took f=1) equals, in some
> appropriate sense, the logarithm of a UV cutoff times the volume of the
> manifold times (or over) (d-2)/24. Do I understand it well?
Yes, correct, at least in the limit of large cut-off. Up to some
powers of two and \pi. The correct formula is (9.3) in Landi's
review. First of all, the above trace scales correctly under scale
transformations. The surprising thing is of course that the constant
of proportionality is the same for all manifolds.
Maybe I can say a bit more than yesterday. First of all, here is a
short cut thru the review: Assuming you already know the basic stuff,
you can start reading chapter 5 "The spectral calculus", here the he
introduces all the tricks you can play with the Dirac operator and
what these Dixmier traces are (they are some regularization of the
usual trace). The stuff about the \omega-limits is interesting but we
don't need it as in our case the ordinary limits exist. Make sure you
understand the examples. You only need to read until including section
5.5.
Then ignore section 6 about differential forms and 7 about vector
bundles and 8 about YM theory and the NC standard model (which at tree
level predicts a Higgs mass of 279GeV, see p611 of Connes' book, to
understand this was my original motivation in 1997 to learn this
stuff) but then start reading again in section 9 where 9.1 and 9.2
hint in the direction that relate the Einstein action to these
traces.
The trick is to use the heat-kernel
h(s) = sum_i \exp(-s \lambda_i)
where \lambda_i is the i's eigenvalue. Then you can show (if your
German is still in working order, read ch. 3 of my master thesis
http://www.aei-potsdam.mpg.de/research/thesis/helling_dipl.ps.gz )
that (again up to computable constants) h(s) s^(d/2) is a power series
in s^2 and the coefficicient of s^2 is proportional to the Einstein
Hilbert action. From what I say in that thesis (and I am to lazy to
repeat here), this is really the log divergence of the 1-loop
effective action of a scalar field on that manifold.
Oh, I forgot, of course expressions of the \lambda_i are encoded in the
Dixmier traces of the Dirac operator and the Laplacian.
Robert
--
..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo. oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/766870
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
Urs Schreiber
Sep7-04, 10:35 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>"Robert C. Helling" <helling@ariel.physik.hu-berlin.de> schrieb im\nNewsbeitrag news:2q5ejlFrk5l3U1-100000@uni-berlin.de...\n> On Mon, 6 Sep 2004 20:10:59 -0400, Lubos Motl <motl@feynman.harvard.edu>\nwrote:\n\n> > there seems to be something very interesting about the\n> > coefficient (d-2)/24 that I have not understood yet. My understanding of\n> > Robert\'s last paragraph is that the trace of the (-d)-th power of the\n> > Dirac operator on a d-dimensional manifold (I took f=1) equals, in some\n> > appropriate sense, the logarithm of a UV cutoff times the volume of the\n> > manifold times (or over) (d-2)/24. Do I understand it well?\n>\n> Yes, correct, at least in the limit of large cut-off. Up to some\n> powers of two and pi. The correct formula is (9.3) in Landi\'s\n> review. First of all, the above trace scales correctly under scale\n> transformations. The surprising thing is of course that the constant\n> of proportionality is the same for all manifolds.\n\nOk, so this explains at the level of formulas why (d-2)/24 shows up. But\nwhat does it imply conceptually in the NCG context? Does it imply any\nspecial behaviour of the spectral action principle in d=26 or something? Can\nwe see any stringy physics anywhere?\n\nFor instance when I look at that theorem 2.11 in hep-th/0306046 which\nAlejandro mentioned at the beginning of this thread I see that the\nfunctional\n\nS(D) = int D^2-d\n\ntakes its minimum always for D=D_s, the standard Dirac operator of the given\nspin structure and that for this Dirac operator S(D_s) is proportionall to\nthe Einstein-Hilbert action (for the metric encoded in D_s) times\n\n- (d-2)/24 ,\n\nright?\n\nSo this means that in the critical dimension this functional becomes\n*independent* of the Einstein-Hilbert action, because it vanishes\nidentically.\n\nWhat does this imply? Does it mean that the spectral action principle\nbecomes void in the critical dimension?\n\nI must admit I have to recall some details of the spectral action stuff.\nWhat is the precise relation between S(D_s) above and the spectral action,\ne.g. equation (1.28) of\nhttp://ernie.ecs.soton.ac.uk/opcit/cgi-bin/pdf?id=oai%3AarXiv.org%3Ahep-th%2F9606001 ?\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>"Robert C. Helling" <helling@ariel.physik.hu-berlin.de> schrieb im
Newsbeitrag news:2q5ejlFrk5l3U1-100000@uni-berlin.de...
> On Mon, 6 Sep 2004 20:10:59 -0400, Lubos Motl <motl@feynman.harvard.edu>
wrote:
> > there seems to be something very interesting about the
> > coefficient (d-2)/24 that I have not understood yet. My understanding of
> > Robert's last paragraph is that the trace of the (-d)-th power of the
> > Dirac operator on a d-dimensional manifold (I took f=1) equals, in some
> > appropriate sense, the logarithm of a UV cutoff times the volume of the
> > manifold times (or over) (d-2)/24. Do I understand it well?
>
> Yes, correct, at least in the limit of large cut-off. Up to some
> powers of two and \pi. The correct formula is (9.3) in Landi's
> review. First of all, the above trace scales correctly under scale
> transformations. The surprising thing is of course that the constant
> of proportionality is the same for all manifolds.
Ok, so this explains at the level of formulas why (d-2)/24 shows up. But
what does it imply conceptually in the NCG context? Does it imply any
special behaviour of the spectral action principle in d=26 or something? Can
we see any stringy physics anywhere?
For instance when I look at that theorem 2.11 in http://www.arxiv.org/abs/hep-th/0306046 which
Alejandro mentioned at the beginning of this thread I see that the
functional
S(D) = \int D^2-d
takes its minimum always for D=D_s, the standard Dirac operator of the given
spin structure and that for this Dirac operator S(D_s) is proportionall to
the Einstein-Hilbert action (for the metric encoded in D_s) times
- (d-2)/24 ,
right?
So this means that in the critical dimension this functional becomes
*independent* of the Einstein-Hilbert action, because it vanishes
identically.
What does this imply? Does it mean that the spectral action principle
becomes void in the critical dimension?
I must admit I have to recall some details of the spectral action stuff.
What is the precise relation between S(D_s) above and the spectral action,
e.g. equation (1.28) of
http://ernie.ecs.soton.ac.uk/opcit/cgi-bin/pdf?id=oai%3AarXiv.org%3Ahep-th%2F9606001 ?
I have uploaded a more detailed description of the history coincidence to
http://arxiv.org/abs/physics/0409022
(forget and forgive the last parragraphs of the article, I like to speculate)
Note the signs in first parragraph are exchanged, it is D^-n times D^2
The guy that best has explained commutative spectral triples is an australian, Adam Rennie. Besides, the Italian teams and the textbooks.
I am going to do some mediterranean travelling, so I am not sure if I will
be able to follow closely on this. At least, let me to complete Urs's comments.
Besides the dualitiy between algebras and spaces, told also by the people on quantum groups, spectral triples need two basic properties:
A "first order calculus", which is defined by asking some commutation
properties between the algebra and the dirac operator, of the
type [D,[D,A]]=0 or similar.
A Poincare duality tool, which is based in Kasparov K-theory. In fact the
first use of KKtheory in theoretical physics started here (warning to
googlers: it is very easy to mistake Kasparov K and Kaluza-Klein).
If this sounds a ring, please tell !
Besides, the integration over a manifold depends on a special trace, called
Dixmier trace (all very frenchy), and its extension, called Wodzisky' residue.
This trace have some scale-invariant properties, so yep it could be some CFT idea around. In the first french edition Connes tryed to relate it to Kogut-Wilson renormalisation and fixed points; this approach was retired in the English translation years after. But it seems some scaling is involved. Modernly an alternate view, via Cesaro summation, is used.
The volume element of a manifold is |D|^{-n}, and in some sense the calculation Tr_w D^2 |D|^{-n} is the integration of a two dimensional object along *all* the n-dimensional manifold. A kind of trick NCG is able to perform, even with non integer dimensions.
> The most interesting approach to "spectral string theory" which I know of is
> that by Ali Chamseddine
> Ali H. Chamseddine,
> An Effective Superstring Spectral Action
> http://arxiv.org/abs/http://www.arx.../hep-th/9705153 .
Perhaps the more continous effort on the line of spectral triples
and strings is the collaboration between RJ Szabo and the napolitan
group (Landi and Lizzi, mainly). Two early papers on it are:
http://arxiv.org/abs/hep-th/9706107
Target Space Duality in Noncommutative Geometry
Fedele Lizzi, Richard J. Szabo
http://arxiv.org/abs/hep-th/9812235
Spectral Geometry of Heterotic Compactifications
David D. Song, Richard J. Szabo
Last week I wrote Fedele about the coincidence, but we did not
discuss about it (I was more interested on hotel addresses). I
do not know Szabo, and just now I am not sure if he is the same
having some introductory string theory lectures around.
As for other references, there are the two books
Connes' red book (say, dual to G-S-W' green book)
Gracia B.-Varilly-Figueroa (dual to Polchinski)
Then some general lectures (from Schucker, Connes, Varilly, Lizzi
and others) around the arxiv, scattered between physics, math, math-ph and
hep-th, and then some specific papers on commutative triples, most
about discrete ones, an a couple about continous ones, from the
australiam team I mentioned (Adam Rennie, Steven Lord,...)
I will keep thinking on this. A last point of NCG is that it
is used to quotient spaces, a operation slightly more drastic
than compactification. The tools here are plain K-theory, in
the algebra side, and Morita equivalence, in the representation
theory. The trick is that all compact operators are in the
same K-theory classes, so they can be seen as different representations
of a single manifold (a point, for finite matrices). But perhaps
the discrete part in Connes-Lott triple is really the result
of a quotient from a greater dimensional space. Hmm.
Well, enough for the whole week. Cheers,
Alejandro Rivero
Alejandro
Sep12-04, 04:26 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>Lubos Motl <motl@feynman.harvard.edu> wrote in message news:<Pine.LNX.4.31.0409061954500.6816-100000@feynman.harvard.edu>...\n\n> explanation? I am sure that these relations are connected with CFT and the\n> appearance of 1/24 is not a coincidence - for example see the paper\n>\n> http://www.math.univ-metz.fr/~benameur/Adv1864.pdf\n>\n> talks about conformal invariant regulators in the context of the Dixmier\n> trace...\n\nDuring my travel, I have remembered a related point: Dixmier trace has\na kind of scale invariance; this was remarked in the French version of the\nbook of Alain Connes, but de-emphatised in the english one. I will try to\nthink along this line.\n\nAlejandro\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>Lubos Motl <motl@feynman.harvard.edu> wrote in message news:<Pine.LNX.4.31.0409061954500.6816-100000@feynman.harvard.edu>...
> explanation? I am sure that these relations are connected with CFT and the
> appearance of 1/24 is not a coincidence - for example see the paper
>
> http://www.math.univ-metz.fr/~benameur/Adv1864.pdf
>
> talks about conformal invariant regulators in the context of the Dixmier
> trace...
During my travel, I have remembered a related point: Dixmier trace has
a kind of scale invariance; this was remarked in the French version of the
book of Alain Connes, but de-emphatised in the english one. I will try to
think along this line.
Alejandro
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