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The T-dual coordinate in open bosonic string theory with one target space dimension compactified is:
X'(z,z*)= X(z) - X(z*)
right? Am I correct in thinking that this definition can change when the target space isn't Minkowski (i.e. with G_\mu\nu and B_\mu\nu fields turned on)?
Does one find that there are components of the string c.o.m. momentum, p^\mu (from the non-compact directions), in the definition of X'^25 for general backgrounds? (In particular where G_\mu\25 and B_\mu\25 are non-zero).
Lubos Motl
Sep3-04, 03:11 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, d70yxj wrote:\n\n> X\'(z,z*)= X(z) - X(z*)\n>\n> right? Am I correct in thinking that this definition can change when\n> the target space isn\'t Minkowski (i.e. with G_\\mu\\nu and B_\\mu\\nu\n> fields turned on), mixing in components of the string c.o.m. momentum\n> p^\\mu in the non-compact directions?\n\nDefinitely, you are correct. It is more difficult to define the T-dual\ncoordinate if the background is not flat and/or if it contains a B-field.\nYou must even think what T-duality you want to consider in this more\ngeneral case. The standard T-duality converting a circle of radius R to a\ncircle of radius 1/R can be generalized to more general backgrounds with\nroughly the same topology, but it is more subtle to decide which\ncoordinate is the T-dualized one (usually it is the coordinate along a\nU(1) isometry).\n\nIf you choose X^9 to be the coordinate along the U(1) isometry, it is\nstill true that the other components of X will be unchanged while X^9\nwould be transformed much like above. If it\'s impossible to set g^{9i} and\nB^{9i} to zero by a coordinate transformation, you will have troubles to\nfind the T-dual background. Recall that the U(1) isometry is essential to\nmake things simple. If the background has a broken U(1) isometry, it\ncorresponds to a condensate of particles with nonzero momentum around the\ncircle, which get mapped by T-duality to particles with nonzero winding.\n\nWell, the T-dual picture will have a condensate of winding states which\nmakes the background non-geometric. This restricts the backgrounds that\ncan be simply T-dualized to a subset, and the T-duality transformation of\nX^9 is never too much more difficult. Of course the rule "R goes to 1/R"\nbecomes more complicated, and if the radius of the circle changes with the\nother coordinates, the T-dual background will have a non-constant dilaton,\nand so forth.\n__________________________________________ ____________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 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, d70yxj wrote:
> X'(z,z*)= X(z) - X(z*)
>
> right? Am I correct in thinking that this definition can change when
> the target space isn't Minkowski (i.e. with G_\mu\nu and B_\mu\nu
> fields turned on), mixing in components of the string c.o.m. momentum
> p^\mu in the non-compact directions?
Definitely, you are correct. It is more difficult to define the T-dual
coordinate if the background is not flat and/or if it contains a B-field.
You must even think what T-duality you want to consider in this more
general case. The standard T-duality converting a circle of radius R to a
circle of radius 1/R can be generalized to more general backgrounds with
roughly the same topology, but it is more subtle to decide which
coordinate is the T-dualized one (usually it is the coordinate along a
U(1) isometry).
If you choose X^9 to be the coordinate along the U(1) isometry, it is
still true that the other components of X will be unchanged while X^9
would be transformed much like above. If it's impossible to set g^{9i} and
B^{9i} to zero by a coordinate transformation, you will have troubles to
find the T-dual background. Recall that the U(1) isometry is essential to
make things simple. If the background has a broken U(1) isometry, it
corresponds to a condensate of particles with nonzero momentum around the
circle, which get mapped by T-duality to particles with nonzero winding.
Well, the T-dual picture will have a condensate of winding states which
makes the background non-geometric. This restricts the backgrounds that
can be simply T-dualized to a subset, and the T-duality transformation of
X^9 is never too much more difficult. Of course the rule "R goes to 1/R"
becomes more complicated, and if the radius of the circle changes with the
other coordinates, the T-dual background will have a non-constant dilaton,
and so forth.
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Hi, thanks for the help. I'd like to say a bit more about what motivated the question above, to see if I'm on the right track. What I had in mind was following the D-branes book, Ch5. He takes a classical string sigma model action and assumes that bckgrd fields G, B and \phi can be turned on, but are independent of one of the target space coordinates, say X^9.
He then introduces a second, different action with no X^9 field, but with a lagrange multiplier called X`^9, and a new worldsheet vector, v_a, in appropriate places.
Then one of two things; either look at the constraint equation for X'^9 (which involves v_a), and upon substituting the solution for v_a back into the action one has the original string theory action, but with some arbitrary scalar field appearing in the place of X^9 (which may as well be called X^9). Alternatively, look at the equations of motion for v_a, which relates X^\mu, X'^9, v_a and some of the bckgrd fields. Using these substitute into the action for v_a, and one again has a string sigma model, but with X' in place of X, and G, B and the dilaton fields transformed in the usual T-dual way.
Sorry for ineffectually paraphrasing the calculation, but I presume it's fairly standard or can be easily looked up there or elsewhere. Anyway, am I correct in thinking this is saying that a string action with X, G, B and \phi is classically equivalent to the same action but with the T-dual fields X', G', B' and \phi' replacing the original worldsheet and background fields? I.e., is this a classical statement of the T-duality of the theory?
Finally, my original question above...substituting the equation of motion for X'^9 into that for v_a gives a relation between X^9, the T-dual coordinate X'^9 and some of the other fields; in the case of a flat background with everything turned off, this gives the usual relation between coordinate and T-dual coordinate, i.e. interchanges string centre of mass momentum and length, something like:
d_\sigma X^9= - id_\tau X'^9
and d_\tau X^9 = id_\sigma X'^9.
But with background fields turned on it looks like the string momenta in the other directions will be mixed in. Suppose the X^9 direction is on a circle of radius R, but also say I set G_9,1 = 1. Then I think that
X'^9(\sigma=pi) - X'^9(\sigma=0) = pi*(n/R + p^1)
So it seems like classical strings looked at in the T-dual picture to this non-trivial background do not all stretch between the same hypersurfaces, but rather there is a p^1 dependence for where the ends of the strings are in the X'^9 direction. Have I got this completely wrong? If not how is this interpreted, should I imagine the D-branes stretching depending on the p^1 momentum of the string attached to it?
(Apologies for the verbose post).
>(del X', delbar X') = (del X, -delbar X). Therefore, I would not
> say that it is a classical symmetry on the worldsheet: for X's,
> it is rather a form of S-duality on the worldsheet. On the other
> hand, T-duality is a "classical" symmetry in spacetime because
> it holds order by order in stringy perturbation theory. Consequently,
Well, I just mean that the actions of the two sigma models are classically equivalent, with X' replacing X, and appropriate redefinitions of the target space fields. Obviously the G, B and \phi fields aren't on the worldsheet, but they do come into the definition of the sigma model.
>>But with background fields turned on it looks like the string momenta
>>in the other directions will be mixed in. Suppose the X^9 direction is
>>on a circle of radius R, but also say I set G_9,1 = 1. Then I think
>>that
>[Moderator's note: nonzero G_{9,1} is T-dual to nonzero B_{9,1} -
> it's because G_{9,\mu} is the Kaluza-Klein gauge field A_\mu
> for the momentum, and the T-dual of the momentum is the winding
> whose gauge field is B_{9,\mu}. LM]
Yep, I'm completely happy with this. From the sigma model analysis outlined above B'_{\mu,9} = G_{\mu,9}/G_{9,9}, right?
>>X'^9(\sigma=pi) - X'^9(\sigma=0) = pi*(n/R + p^1)
>>So it seems like classical strings looked at in the T-dual picture to
>>this non-trivial background do not all stretch between the same
>>hypersurfaces, but rather there is a p^1 dependence for where the ends
>>of the strings are in the X'^9 direction. Have I got this completely
>>wrong? If not how is this interpreted, should I imagine the D-branes
>>stretching depending on the p^1 momentum of the string attached to it?
>[Moderator's note: Right. If you look in the previous note, you see that
> physics involves a nonzero B-field. While the momenta - the canonically
> dual variables to the positions - always live on a lattice that includes
> the point "zero", the lattice of velocities may be shifted if the B-field
> is nonzero - a sort of stringy Aharonov-Bohm effect. The B-field
> corresponds to a coupling of momentum and winding, and if you subtract
> p^1\sigma / \pi from your X'^9(\sigma), you will get a single valued
> variable. This subtraction is equivalent to going from momentum to
Sorry, I still don't quite get this...I'm thinking of open strings here, and expecting them to stretch between hypersurfaces. Why am I subtracting p^1\sigma from X'(\sigma)? Maybe just point me to the right page in Polchinski....
[Moderator's note: I thought you wanted closed strings - see page 249
(middle) and 250 of Polchinski I for the description of the shift
between velocities and momenta. The situation of open strings is
related, but you should keep in mind that the open strings can end
on various D-branes, and the distance between the D-branes is T-dual
to the Wilson line that breaks U(2) to U(1)^2, for example. One must
also realize that B+F is the only physical combination because the
electromagnetic potential A living on the brane transforms under the
B-field's gauge invariance, B\to B+d\lambda, A\to A-\lambda. LM]
Right, thanks, I think I understand what I did wrong now. Btw, I was always considering open strings with no chan-paton factors, and the equation I have in the earlier posts:
X'^9(\sigma=pi) - X'^9(\sigma=0) = pi*(n/R + p^1)
is supposed to be the set of possible lengths of an open string stretching between two copies of the same D-brane, but with an unwanted dependence on p^1. I've taken a step too far though, as it should be:
X'^9(\sigma=pi) - X'^9(\sigma=0) = pi*(G_{9,9}p^9 + G_{1,9}p^1)
I was then (foolishly) assuming p^9 = n/R, but I think really it should be (G_{9,9}p^9 + G_{1,9}p^1)=n/R that's quantized, I guess. Then you have the usual equation for the length of the string stretching between D-branes, with no p^1 dependence.
So it was a red herring, really - at least I think so, now. The classical sigma model calculation still seems like a neat way to see the transformation of all the background fields, though, and the more general relation between X and X' with background fields turned on.
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