T-dualization of type II superstring theory in double space
Eur. Phys. J. C
T-dualization of type II superstring theory in double space
B. Nikolic´ 0
B. Sazdovic´ 0
0 Institute of Physics Belgrade, University of Belgrade , Pregrevica 118, Belgrade , Serbia
In this article we offer a new interpretation of the T-dualization procedure of type II superstring theory in the double space framework. We use the ghost free action of type II superstring in pure spinor formulation in approximation of constant background fields up to the quadratic terms. Tdualization along any subset of the initial coordinates, x a , is equivalent to the permutation of this subset with subset of the corresponding T-dual coordinates, ya , in double space coordinate Z M = (x μ, yμ). Requiring that the T-dual transformation law after the exchange x a ↔ ya has the same form as the initial one, we obtain the T-dual NS-NS and NS-R background fields. The T-dual R-R field strength is determined up to one arbitrary constant under some assumptions. The compatibility between supersymmetry and T-duality produces a change of bar spinors and R-R field strength. If we dualize an odd number of dimensions x a , such a change flips type IIA/B to type II B/A. If we T-dualize the time-like direction, one imaginary unit i maps type II superstring theories to type II ones.
1 Introduction
T-duality is a fundamental feature of string theory [
1–8
]. As
a consequence of T-duality there is no physical difference
between string theory compactified on a circle of radius R
and circle of radius 1/R. This conclusion can be generalized
to tori of various dimensions.
The mathematical realization of T-duality is given by
Buscher T-dualization procedure [
4,5
]. If the background
fields have global isometries along some directions then we
can localize that symmetry introducing gauge fields. The
next step is to add the new term in the action with Lagrange
This work is supported in part by the Serbian Ministry of Education,
Science and Technological Development, under contract No. 171031.
multipliers which forces these gauge fields to be unphysical.
Finally, we can use gauge freedom to fix initial coordinates.
Varying this gauge fixed action with respect to the Lagrange
multipliers one gets the initial action and varying with respect
to the gauge fields one gets the T-dual action.
Buscher T-dualization can be applied along directions on
which background fields do not depend [
4–10
]. Such a
procedure was used in Refs. [
11–18
] in the context of closed string
noncommutativity. There is a generalized Buscher procedure
which deals with background fields depending on all
coordinates. The generalized procedure was applied to the case
of bosonic string moving in the weakly curved background
[
19,20
]. It leads directly to closed string noncommutativity
[
21
].
The Buscher procedure can be considered as the
definition of T-dualization. But there are also other frameworks
in which we can represent T-dualization which must be in
accordance with the Buscher procedure. Here we talk about
the double space formalism which was the subject of the
articles about 20 years ago [
22–26
]. Double space is spanned by
coordinates Z M = (x μ, yμ) (μ = 0, 1, 2, . . . , D−1), where
x μ and yμ are the coordinates of the D-dimensional initial
and T-dual space-time, respectively. Interest for this subject
emerged again with Refs. [
27–34
], where T-duality is related
with O(d, d) transformations. The approach of Ref. [22] has
been recently improved when the T-dualization along some
subset of the initial and corresponding subset of the T-dual
coordinates has been interpreted as permutation of these
subsets in the double space coordinates [
35,36
].
Let us motivate our interest in this subject. It is well known
that T-duality is important feature in understanding M-theory.
In fact, five consistent superstring theories are connected by
a web of T and S dualities. In the beginning we are going
to pay attention to the T-duality. To obtain formulation of
M-theory it is not enough to find all corresponding T-dual
theories. We must construct one theory which contains the
initial theory and all corresponding T-dual ones.
We have succeeded to realize such program in the bosonic
case, for both constant and weakly curved background. In
Refs. [
35,36
] we doubled all bosonic coordinates and obtain
the theory which contains the initial and all corresponding
Tdual theories. In such theory T-dualization along an arbitrary
set of coordinates x a is equivalent to replacement of these
coordinates with the corresponding T-dual ones, ya .
Therefore, T-duality in double space becomes symmetry
transformation with respect to permutation group.
Performing T-duality in supersymmetric case generates
new problems. In the present paper we are going to extend
such an approach to the type II theories. In fact, doubling all
bosonic coordinates we have unified types IIA, IIB as well
as type II [
37
] (obtained by T-dualization along time-like
direction) theor (...truncated)