Simultaneous T-dualization of type II pure spinor superstring

In this article we consider simultaneous T-dualization of type II superstring action in pure spinor formulation. Simultaneous T-dualization means that we make T-dualization at the same time along some subset of initial coordinates marked by $x^a$. The only imposed assumption stems from the applicability of the Buscher T-dualization procedure - background fields do not depend on dualized directions $x^a$. In this way we obtain the full form of the T-dual background fields and T-dual transformation laws. Because two chiral sectors transform differently, there are two sets of vielbeins and gamma matrices connected by the local Lorentz transformation. Its spinorial representation is the same as in the constant background case. We also found the full expression for T-dual dilaton field.


Introduction
The importance of T-duality rose after M-theory was discovered. Five consistent superstring theories are connected by web of T and S dualities and make M-theory [1]. For example, T-duality connects type IIA and type IIB superstring theories in the sense that after odd number of T-dualizations type IIA/B turns into IIB/A, while after even number of T-dualizations type IIA/B stays unchanged [2,3].
T-dualization of type II superstrings was a subject of the papers [4,5,6,3]. In some papers T-dualization along single direction of the full action is considered [5,6]. Two chirality sectors under T-duality transform differentley and, consequently, in T-dual picture there are two sets of vielbeins and gamma matrices. But there is a local Lorentz appeared, pure spinor formalism [16,17,18,19,8]. It is pretty similar to GS one in the sense that space-time supersymmetry is manifest but it contains pure spinors λ α andλ α satisfying so called pure spinor constraints, λ α (Γ µ ) αβ λ β =λ α (Γ µ ) αβλ β = 0. Pure spinor formalism uses advantages of the previous two formalisms and avoids some disadvatages. In this article we will use pure spinor action of type II superstring from Ref. [8], where detailed derivation of the action is presented. The action is given in the form of an expansion in powers of θ α andθ α using (anti)holomorphicity and nilpotency conditions. In this article we want to analyze simultaneous T-dualization of the pure spinor superstring type II theory with only one assumption -background fields are independent of the coordinates x a along which we make T-dualization. This assumption stems from the applicability of the Buscher procedure. Our main goal is to find the full form of the T-dual background fields and T-dual transformation laws. We will compare these results with the constant background case [3].
We start with the action (2.1) and decompose the variables X M andX M (3.1) extracting directions x a along which we make T-dualization. Then we perform Buscher T-dualization procedure along x a obtaining the T-dual transformation laws and T-dual action. As in the constant background case, two chirality sectors transform differently under T-dualization. Consequently, there are two sets of vielbeins and gamma matrices, which are connected by local Lorentz transformation with the same form of the matrix a Ω as in the Ref. [3]. We keep unbar fermionic variables unchanged while bar variables are corrected by matrix a Ω, which is spinorial representation of local Lorentz transformation (see [3] for more details). After introducing proper fermionic variables, we read the form of the T-dual background fields.
We have written explicitly the expressions for T-dual physical superfields. For constant background they turn to the result of Ref. [3]. Expressions for the auxilary superfields and field strengths are completely new in the sense that terms containing them missed in the constant background case. In order to skip a large number of long expressions, these expressions are in superspace notation and they are not written in components.
Dilaton field is treated within quantum formalism. Using the result of Ref. [3] for the T-dual dilaton and the background fieldΠ +ab , coupling with ∂ + x a and ∂ − x b , we obtained the generalized expression for T-dual dilaton field.
2 Type II pure spinor superstring theory In this section we will introduce the type II pure spinor superstring action and, for completeness, calculate canonical Hamiltonian treating x µ , θ α andθ α as coordinates.

Pure spinor type II superstring action
The sigma model action for type II superstring of Ref. [8] is of the form where vectors X M andX N are left and right chiral supersymetric variables which components are defined as In the analysis we will use the action in the form (2.1). Just for completeness, the expanded form of the action is where we used matrix A M N in explicit form Matrix A M N containing type II superfields generally depend on x µ , θ α andθ α . The superfields A µν ,Ē µ α , E α µ and P αβ are physical superfields, because their first components are supergravity fields. The fields in the first column and first row are auxiliary superfirlds because they can be expressed in terms of the physical ones [8]. The rest ones, Ω µ,νρ (Ω µν,ρ ), C α µν (C µν α ) and S µν,ρσ , are curvatures (field strengths) for physical superfields.

Canonical Hamiltonian
Let us write the supervectors in the following form and currents (2.13) In order to find momenta canonically conjugated to the coordinates x µ , θ α andθ α , first we have to integrate out the fermionic momenta, π α andπ α , occuring in the currents J M andJ M . From the equations of motion for fermionic momenta it follows, respectively, Inserting these expressions back into the currents J M andJ M and summing the appropriate components we get Here we have (2.20) Using the expressions (2.17) we get the action where we introduced definitions Let us define some generalized metric g µν and Kalb-Ramond field b µν as Now we can find canonical momenta and then calculate canonical Hamiltonian. The expressions for canonical momenta are Let us introduce superspace coordinate and the corresponding canonically conjugated momenta All canonical momenta could be rewritten in more useful form as and It is useful to introduce current Using the expression for canonical momenta we have Then the action (2.21) takes a form From the definition of canonical Hamiltonian density we get The subscript st denotes supertransposition.

T-dualization along arbitrary number of coordinates
In this section we will make T-dualization along arbitrary subset of the coordinates x a . First we will make mathematical preparation extracting the desired directions from supervectors X M andX M . Then we will apply standard Buscher procedure assuming that background fields do not depend on x a . We will explicitly write the form of the physical superfields in order to make a comparison with constant background case [3].

Mathematical preparation
In order to make T-dualization along arbitrary bosonic directions x a , let us split the space time index µ in a and the undualized ones, i, and write the coordinates X M andX N in the form We note that vectors X M andX M have five block components and, consequently, A M N is 5 × 5 block matrix. In comparison with (2.2) we split Π µ into Π a and Π i as well as Γ µ into Γ a and Γ i . Let us introduce the notatioñ Applying this decomposition to the action (2.1), it gets a form

Buscher procedure
Let us perform T-dualization of the action (3.9) along x a directions. We assume that x a directions are Killing ones, so, background fields do not depend on them. Applyng standard procedure of Buscher T-dualization we replace ordinary world-sheet derivatives ∂ ± x a by covariant ones (3.10) In order to make the fields v a ± unphysical we add the term where y a are Lagrange multipliers. Taking into account that x a are isometry directions we choose the gauge, x a = 0, so that the gauge fixed action takes a form On the equations of motion for y a we obtain that field stregth is equal to zero which solution is v a ± = ∂ ± x a . In this way the action S f ix turns to the initial action S. On the equations of motion for gauge fields v a ± we have 14) into S f ix we get where we used the expressions for currents (3.8) and introduced the definitions

Transformation laws -two sets of vielbeins and gamma matrices
On the equation of motion for Lagrange multipliers y a we have, v a ± = ∂ ± x a . Substituting this into (3.14) and (3.15) we obtain transformation laws relating the initial and T-dual coordinates. For the following analysis, it is useful to use the expressions for a j M ± currents given in Eq.(3.1). Introducing notatioñ and currentsJ we rewrite the transformation laws (3.14) and (3.15) in the form while the inverse one is Here we introduced the fieldθ ab ± aŝ Note that the form of the transformation laws are the same as in the case of constant background fields [3]. But here all background fields depend on the undualized coordinates (θ α ,θ α , x i ). Transformation laws (3.22) and (3.23) can be rewritten in the same form as in [3] ∂ where we introduced the T-dual variables a Xμ = {y a , x i }. The matrices and theirs inverse which are connected by local Lorentz transformation Here a Ω is spinorial representation of the Lorentz transformation The underlined indices are Lorentz ones (denoted by a, b). The matrix Λ a b is a matrix of Lorentz transformation and it is given by the expression In T-dual theory, as a consequence of two types of Γ matrices, there are two types supersymmetry invariant variables In order to work with one set of gamma Γ matrices we have to introduce proper variables. We can rewrite bar expression as Let us preserve expressions for unbar variables, a θ α = θ α and a π α = π α , and change bar variables Now the forms of transformation of the supersymmetric invariants are the same. In short, fermionic index without bar is unchanged, while bar fermionic index is multiplied by a Ω. The further story, finding the spinorial representation of local Lorentz symmetry a Ω connectiong two kinds of vielbeins, is the same as in [6,5,3] and we will not repeat it. We will just write the final expression for matrix a Ω in spinorial representation Matrix Γ 11 has normalization constant to satisfy the condition (Γ 11 ) 2 = 1. Also we have

T-dual action
In this section we will find the form of the T-dual background fields in terms of the initial ones. The explicit form of the T-dual physical superfields we will compare with those obtained in the constant background case.

Relations between initial and T-dual background fields
We assume that T-dual action has the form of initial action but expressed in terms of the T-dual variables and background fields where a XM = aPM a ∂ + y a + ωM N a j N + , aXM = aPM a ∂ − y a +ωM N a j N − .

(4.2)
We did not write free field actions for pure spinors because they carry fermionic indices while we T-dualize along some subset of bosonic indices. Following the form of the initial theory we introduced for T-dual case The matrices ω andω are of the form We also need inverse matrices, ω −1 andω −1 , (4.7) During the calculation of above matrices we used the expressions for T-dual gamma matrices with upper and lower indices Let us note that we can put sign − in the definition of the T-dual matrices because that does not affect the Clifford algebra. The explicit form of the action given in (4.1) is Comparing this action with that from Eq.(3.18) we obtain where a A aN ≡P T aM a AMN , aĀM a ≡ a AMNPN a . Let us note that we expressed T-dual background fields in terms of the initial ones. The next step is to express components of the T-dual fields in terms of the components of the initial background fields. Also in order to find transfomation law for physicall superfields components in A M N we need the explicit expressions (4.21) We will find the explicit expressions for the following T-dual background fields: a A ab , a A a i ,Ē aα , a A i a , a A ij ,Ē i α , a E αa , a E α i and P αβ . Finding these expressions we can make comparison with constant background case and provide one check of validity of these expressions. The rest ones are background fileds which are not present in the constant background case and we will not write the explicit expressions for them.

Comparison with constant background case
In order to find T-dual field a A ab we take into considration the second component of the equation (4.14) or (4.15). The second component of (4.14) produces the equation We treat left-hand side and right-hand side of this equation as an expansion in powers of θ α . Equation apropriate coefficients we obtain the T-duals fields a A ab and a E αa Using the redefinitions A ab = κΠ +ab and a A ab = κ a Π ab + as well as the relationθ abΠ +bc = 1 2κ δ a c , we get In the case of the constant background fields (∆ ab = 0), the relation (4.24) transforms into because in that caseΠ +ab = Π +ab . The quantity ∆ ab is defined as Our result in general case gives the right limit for the constant background fields. The equation (4.25) in the limit of the constant background fields is in accordance with appropriate result in the constant background case [3]. Here we have in mind that field Ψ α µ is zero order term in the expansion of E α µ [8]. From the second component of (4.15) a A ab + κ 2 aĒ aβθbc we can get again the expression for a A ab , but we additionally have In the constant background caseĒ a α →Ψ a α andθ ab − keeps the form but now in terms Π +ab insteadΠ +ab . Consequently, here we also have good constant background limit.
Let us consider the equation which follows from equationg of the third components of the Eq.(4.14) Equating the zero components in the expansion we get Taking into account the redefinitions a A a i = κ a Π a +i , A ai = κΠ +ai , (4.33) we obtain, using the relation (Π −1 The constant background limit in this case means thatθ ab is defined via Π +ab instead Π +ab because ∆ ab = 0. Equating the third components of the Eq.(4.15), in the same way as in the previous case, we have The last two expressions in the limit of the constant background fields are in full correpsondence with the result obtained in the constant background case [3]. Considering appropriate component in the Eq.(4.16) we obtain This relation can be rewritten in the form In the constant background case explicit θ α andθ α dependence disappears andΠ +ab = Π +ab . Consequently, we get which is exactly the relation obtained in the constant background case [3]. Also we read from Eq.(4.16) In the constant background limit which effectively means that we put θ α =θ α = 0, A ia = κΠ +ia , A bi = κΠ +bi , (Π −1 + ) ab = 2κθ ab − and P αβ = 1 2κ e Φ 2 F αβ , we obtain relations from [3] aΨi

Dilaton field in T-dual picture
Dilaton field transformation under T-dualization is considered within path formalism [10,5,6,3,4]. Here we will use the result of these articles and combine it with the expression ofΠ +ab and obtain the generalized form of T-dual dilaton field. Also, just for completness, we will shortly address to the T-dual transformation of pure spinors. Taking in consideration the result for T-dual dilaton from Refs. [10,5,6,3,4] and the action (3.9) we get the form of the T-dual dilaton field (4.45) Using the expression forΠ +ab (3.7) in the form where ∆ ab is given in Eq.(4.27) and using some basic features of determinant and logarithm, we obtain In the constant background case ∆ ab = 0, dilaton, its T-dual and Π +ab become constant and we get agreement with the constant background result. From this expression we see that ignoring θ α andθ α dependent terms in [3] prevent us to get complete solution for the T-dual dilaton.

Concluding remarks
In this paper we have investigated simultaneous T-dualization of the pure spinor type II superstring described by the action of Ref. [8]. We assumed that background fields do not depend on the coordinates along which we make T-dualization. Our goal was to find the form of the T-dual background fields, especially physical superfields, and compare them with the constant background case of Ref. [3]. In relation to the articles [5,6], where single direction T-dualization is performed, here we demonstrated simultaneous T-dualization along some subset, x a (a = 1, 2, . . . d), of space-time directions. Also following Refs. [5,6,3], we found the form of the spinorial representation of local Lorentz transformation a Ω occuring in the T-dual picture. The action we used in this article is type II superstring action in pure spinor formulation of Ref. [8]. It is derived using nilpotrncy and (anti)holomorphicity conditions as an expansion in powers of θ α andθ α . In Ref. [3] we considered constant beckground version of this action obtained under certain assumptions -background fields are independent of all x µ coordinates and we take just first components in the expansions of background fields. In this way we lost information about the form of the all T-dual background fields as well as the form of T-dual physical superfields (their first components in the expansion are supergravity fields) is incomplete.
It is difficult to work with the expanded form of action (2.6) because it has large number of terms. We used condensed form of the action (2.1) and extracted in variables X M andX M terms containing derivatives of the directions along we T-dualize, ∂ ± x a . The rest part of these variables is denoted by current a j M ± . We inserted that expression into action and made T-dualization along x a direction. On the equation of motion for gauge fields v a ± we obtained T-dual action expressed in terms of T-dual coordinates y a and currents a j M ± . Under T-dualization the form of the action is preserved and consequetly, expressing the T-dual action in terms of the T-dual variables and fields, we finally got all T-dual background fields in the considered general case. In order to compare them with the constant background case of Ref. [3] we explicitlly wrote the expressions for physical superfields. In the limit of constant background fields obtained expressions turn into the expressions of Ref. [3].
Combining the equations of motion for Lagrange multipliers y a and gauge fields v a ± we obtain T-dual transformation laws in most general case of type II pure spinor superstring. Introducing appropriate notation we obtained the T-dual transformationn laws in the same form as in the constant background case. But let us stress that we consider the general case and that all background fields now depend on the undualized directions x i , θ α andθ α . Also we noticed that two chiral sectors transform differently and, as in constant background case [3], there are two sets of vielbeins and gamma matrices. But there is local Lorentz transformation connecting them. Using the results of [5,6,3] we obtained the general form of the local Lorentz transformation in spinorial representation a Ω which is the same as in the constant background case. In order to work with properly defined variables and background fields, fermions with bar index are multiplied by matrix a Ω.
The T-dual transformation of dilaton field Φ(x i , θ α ,θ α ) is treated within quantum formalism. The general form of the T-dual dilaton field in the case of simultaneous Tdualization is given in Ref. [3]. In this paper we found the general form of the fileds coupling T-dualized directions,Π +ab and aΠ ab + , and consequently, we obtained the generalized expression for T-dual dilaton field, where the additional term depends on the undualized directions x i , θ α andθ α . Let us note that dilaton field in initial theory also depends on these undualized directions.