T-dualization of type II superstring theory in double space

In this article we offer the new interpretation of T-dualization procedure of type II superstring theory in 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 $Z^M=(x^\mu,y_\mu)$. Demanding that the T-dual transformation law after exchange $x^a\leftrightarrow y_a$ has the same form as 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 change of bar spinors and R-R field strength. If we dualize odd number of dimensions $x^a$, such change flips type IIA/B to type II B/A. If we T-dualize time-like direction, one imaginary unit $i$ maps type II superstring theories to type $II^\star$ ones.


Idea od double space
Double space = initial coordinates plus T-dual partners -Siegel, Duff, Tseytlin about 25 years ago. Interest for this subject emerged again (Hull, Berman, Zwiebach) in the context of T-duality as O(d , d ) transformation. The approach of Duff 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 (arXiv:1505.06044, 1503.05580). All calculations are made in full double space. In double space T-duality is a symmetry transformation.
Bosonic T-duality -assumptions and approximations Bosonic T-dualization -we assume that background fields are independent of x µ . In mentioned reference, expressions for background fields as well as action are obtained in an iterative procedure as an expansion in powers of θ α andθ α . Every step in iterative procedure depends on previuous one, so, for mathematical simplicity, we consider only basic (θ andθ independent) components.
Fermionic T-duality -assumptions and consistency check Fermionic T-dualization -we assume that θ α andθ α are Killing directions. Consequently, auxiliary superfirlds are zero according to arXiv: 0405072. If we assume that rest of background fields are constant then their curvatures are zero. Using space-time field equations we confirmed the consistency of the choice of constant P αβ .
In both cases, under introduced assumptions, action gets the form are NS-R fields and F αβ is R-R field strength. Momenta π α and π α are canonically conjugated to θ α andθ α . All spinors are Majorana-Weyl ones.
All background fields are constant. Busher bosonic T-duality Global shift symmetry exists x a → x a + b, where index a is subset of µ. We introduce gauge fields v a ± and make change in the action ∂ ± x a → ∂ ± x a + v a ± . Additional term in the action where y a is Lagrange multiplier. It makes v a ± to be unphysical degrees of freedom. On the equations of motion for y a we get initial action, while, fixing x a to zero, on th equations of motion for v a ± we get T-dual action. Transformation laws Solution of the equation of motion for y a is v a ± = ∂ ± x a . Combining this solution with equations of motion for gauge fields v a ± we obtain T-dual transformation laws Here J ±µ = ± 2 κ Ψ α ±µ π ±α and θ ac T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks Transformation laws in double space where is so called generalized metric, while Ω MN is constant symmetric matrix and it is known as Bojan Nikolić Bosonic and fermionic T-dualization of type II superstring theory in T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks T-duality as permutation in double space Demanding that a Z M has the transformation law as initial coordinates Z M , we find the T-dual generalized metric and T-dual current T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks NS-NS background fields From (9) we obtain the T-dual NS-NS background fields which are in full agreement with those obtained by Buscher procedure T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks NS-R background fields From (10) we obtain the form of the T-dual NS-R fields From transformation laws we see that two chiral sectors transform differently. Consequently, there are two sets of vielbeins in T-dual picture as well two sets of gamma matrices. This T-dual vielbeins are connected by Lorentz transformation, while spinorial representation of this Lorentz transformation, a Ω α β , relates two sets of gamma matrices. In order to have unique set of gamma matrices, we have to multiply one fermionic index by a Ω α β . Model Bosonic T-duality Fermionic T-duality Concluding remarks R-R field strength R-R field strength couples fermionic momenta and, consequently, its T-dual can not be read from transformation law.
From the demand that term in the action is T-dual invariant, we obtain the form of the T-dual R-R field stregth where c is an arbitrary constant. For the specific value of c, we get the same expression as in Buscher procedure.

Basic facts
In last years it was seen that tree level superstring theories on certain supersymmetric backgrounds admit a symmetry which is called fermionic T-duality. This is a redefinition of the fermionic worldsheet fields similar to the redefinition we perform on bosonic variables when we do an ordinary T-duality.
Technically, the procedure is the same as in the bosonic case up to the fact that dualization will be done along θ α andθ α directions.

T-duality
Model Bosonic T-duality Fermionic T-duality Concluding remarks Action On the equations of motion for π α andπ α action (2) becomes Bosonic T-duality Fermionic T-duality Concluding remarks Fixing the chiral gauge invariance In the above action θ α appears only in the form ∂ − θ α and θ α in the form ∂ +θ α .
Using the BRST formalism we fix theis chiral gauge invariance adding to the action where α αβ is arbitrary non singular matrix. Model Bosonic T-duality Fermionic T-duality Concluding remarks

Transformation laws
Applying the same mathematical procedure as in the case of the bosonic T-dualization, we have where ϑ α andθ α are T-dual fermionic coordinates. Model Bosonic T-duality Fermionic T-duality Concluding remarks Transformation laws in double space Let us introduce double fermionic coordinates Transformation laws in double space are of the form

Generalized metric and currents
The generalized metric and the matrix A AB are The currents are of the form Fermionic T-dualization as permutation T-dual coordinates are is permutation matrix.
Bojan Nikolić Bosonic and fermionic T-dualization of type II superstring theory in Fermionic T-dualization as permutation Demanding that T-dual coordinates transformation laws are of the same form as those for initial coordinates we get The matrix A AB transforms as Background fields From these relations we obtain the R-R and NS-R T-dual background fields in the same form as in the Buscher procedure Fermionic T-duality Concluding remarks NS-NS background fields Π +µν is coupled by x's and we can not read the T-dual field from transformation laws.
As in the case of bosonic T-dualization, assuming that this term is invariant under T-dualization, we get the appropriate fermionic T-dual where c is an arbitrary constant.
Bojan Nikolić Bosonic and fermionic T-dualization of type II superstring theory in T-duality Model Bosonic T-duality Fermionic T-duality Concluding remarks

Concluding remarks
We represented both kind of T-dualizations of type II superstring as permutation symmetry in double space.
The successive T-dualizations make a group called T-duality group. In the case of type II superstring fermionic T-duality transformations are performed by the same matrices T a as in the bosonic string case. Consequently, the corresponding T-duality group is the same.
In the bosonic case there is an advantage of this approach. In one equation all T-dual theories (for any subset x a ) are contained. We do not have to repeat procedure for each specific choice of x a . This kind of approach could be helpful in better understanding of M-theory.