T-dualization of type II superstring theory in double space

The European Physical Journal C, Mar 2017

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. T-dualization 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, \(y_a\), in double space coordinate \(Z^M=(x^\mu ,y_\mu )\). Requiring that the T-dual transformation law after the exchange \(x^a\leftrightarrow y_a\) 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 \(\hbox {II}^\star \) ones.

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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)


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B. Nikolić, B. Sazdović. T-dualization of type II superstring theory in double space, The European Physical Journal C, 2017, pp. 197, Volume 77, Issue 3, DOI: 10.1140/epjc/s10052-017-4758-0