Non Abelian T-duality in Gauged Linear Sigma Models

Abelian T-duality in Gauged Linear Sigma Models (GLSM) forms the basis of the physical understanding of Mirror Symmetry as presented by Hori and Vafa. We consider an alternative formulation of Abelian T-duality on GLSM's as a gauging of a global $U(1)$ symmetry with the addition of appropriate Lagrange multipliers. For GLSMs with Abelian gauge groups and without superpotential we reproduce the dual models introduced by Hori and Vafa. We extend the construction to formulate non-Abelian T-duality on GLSMs with global non-Abelian symmetries. The equations of motion that lead to the dual model are obtained for a general group, they depend in general on semi-chiral superfields; for cases such as $SU(2)$ they depend on twisted chiral superfields. We solve the equations of motion for an $SU(2)$ gauged group with a choice of a particular Lie algebra direction of the vector superfield. This direction covers a non-Abelian sector that can be described by a family of Abelian dualities. The dual model Lagrangian depends on twisted chiral superfields and a twisted superpotential is generated. We explore some non-perturbative aspects by making an Ansatz for the instanton corrections in the dual theories. We verify that the effective potential for the $U(1)$ field strength in a fixed configuration on the original theory matches the one of the dual theory. Imposing restrictions on the vector superfield, more general non-Abelian dual models are obtained. We analyze the dual models via the geometry of their susy vacua.


Introduction
String theory is arguably the field of physics that has stimulated the most work in various fields of mathematics. One of the most prolific example is mirror symmetry. From the physical point view, mirror symmetry is an example of the power of symmetries and dualities in string theory. Indeed, the deepest physical explanations for mirror symmetry are rooted in the idea of duality, more specifically, T-duality. One of the first suggestions that mirror symmetry can be understood in the language of T-duality was put forward in [1]; the insightful work of Strominger, Yau and Zaslow further developed this idea in a particular context [2]. The state-of-the-art description of mirror symmetry presented in [3,4] ties together many works around the concept of T-duality, including non-perturbative effects.
A practical way to understand T-duality considers a system with a global U (1) symmetry, promotes this symmetry to a local symmetry by introducing a gauge field and a Lagrange multiplier enforcing flatness of the corresponding connection. Then, the T-dual models are obtained as the result of integration in different orders [5]. A similar proposal was put forward for the case of non-Abelian T-duality [6]. There has recently been a revival in implementations of non-Abelian T-duality (NATD) in the context of supergravity solutions fueled by an understanding of its implementation in the RR sector [7]. NATD has been widely applied as a solution generating technique for some supergravity backgrounds and has provided many new examples of pairs in the context of the AdS/CFT correspondence (see [8] and references therein). Given the mounting evidence in favor of NATD as a symmetry of supergravity with RR fields, and indirectly of a class of nonlinear sigma models in string theory, it is natural to explore the possibility that NATD could be implemented at the level of gauged linear sigma models (GLSM). Indeed, there is a well-established connection between GLSM and NLSM in the context of string theory.
In this manuscript we explore the implementation of NATD for GLSM. We have various motivations in mind. For example, we hope that further understanding of NATD in GLSM could lead to a clarification of mirror symmetry in more general Calabi-Yau varieties, in particular, for determinantal varieties [9], [10]. We follow the standard definition of T-duality stated above and implement it for a class of GLSM with a non-Abelian global symmetry. The process is carried out by taking a non Abelian global symmetry in a gauge theory and gauging it by the introduction of an extra non Abelian gauge field. Then a Lagrange multiplier term is added (as in the original formulation of the duality) to yield the original or the dual model depending on which field is integrated. We illustrate this procedure first in the case of a scalar field minimally coupled to a gauge field. Then we study several examples of Abelian T-duality in GLSMs. Integrating the vector superfield one obtains a set of equations of motion to express the original coordinates, chiral superfields, in terms of the dual coordinates, which are twisted chiral superfields.
Then we formulate non-Abelian T-duality for a GLSM U (1) theory with a generic global symmetry acting on chiral superfields. We gauge the symmetry by introducing a vector superfield and adding a Lagrangian term, with a non constrained superfield as Lagrange multiplier. Integrating the vector superfield one arrives to a set of equations of motion leading to the dual theory. The dual superfields satisfy a semichiral condition, but in special cases such as SU (2) they are twisted chiral. To explore further the duality we consider as a model a GLSM with two equally charged chiral superfields, with a global SU (2) symmetry. We study the model in detail, obtaining the dual Kähler potential and the twisted superpotential. First we focus in a family of Abelian dualities along an SU (2) vector superfield direction which turn to be more general because the dual theories describe a non Abelian set of the SU (2) vector superfield. Then we consider a restricted vector superfield that leads to a fully non-Abelian dual model.
A crucial ingredient that was missing in the original attempts to cast mirror symmetry as T-duality was the inclusion of nonperturbative effects such as instantons; this was addressed systematically in the work of Hori and Vafa [3]. Motivated by this approach we also take some steps in this direction and include corrections to the twisted super-potential and demonstrate that it satisfies some expected properties, as that leads to coinciding twisted effective superpotential for the gauge field strength in the original and the dual model. In addition, by dualizing an Abelian direction in SU (2) and considering the instanton twisted superpotential one obtains the same resulting theory as the one obtained via a U (1) dualization.
The manuscript is organized a follows: In Section 2 we start reviewing supersymmetric gauge theories in 2D with (2,2) supersymmetry. In Section 3 we present Abelian T-Duality in GLSMs. In 3.1 we start with an example of gauging a global symmetry in a gauge theory, which is given by two scalars fields minimally coupled to a U (1) gauge field. In 3.2 we present a (2,2) 2D gauged linear sigma model with two chiral superfields. In subsection 3.3 we describe Abelian T-duality in the mentioned GLSM by first dualizing a theory with a single U (1) along the phase of one chiral superfield. Then we consider the case of single a U (1) gauge group and multiple chiral superfields and their dualizations. We finish the section by considering Abelian T-duality of a theory with multiple U (1)s and multiple chiral superfields. In subsection 4.1 we formulate non-Abelian Tduality for GLSM with a U (1) gauge symmetry and a generic global symmetry group, obtaining the equations of motion that lead to the dual model. We discuss the generalities of the dual model obtained by dualizing an SU (2) symmetry in a U (1) gauge theory with two equally charged chiral superfields. Subsection 4.3 focuses on a particular case of dualization along an SU (2) single generator, and the following subsection 4.4 takes a dualization along a generic Abelian direction inside if the SU (2) group. Section 5 considers a more general dualization for a selection of the vector superfield, which is a truly non-Abelian direction. We analyze the vacuum manifold of the dual theory. Section 6 compares to the results of the vacua obtained with the standard Mirror symmetry by dualizing via Abelian T-duality in the scheme of Hori-Vafa. We present our conclusions and point to a number of interesting follow up directions and open problems in Section 7. In a series of appendices we present supporting calculations and adress some technical questions.

(2, 2) Supersymmetric gauge theory in dimensions
In this section we recall some aspects of superfield representations in 2D which are well understood from the 4D point of view; our goal is to setup the notation and describe the basic fields which are the content of the GLSMs. The treatment of this section follows closely the monograph "Superspace or one thousand and one lessons in supersymmetry" [11] and the work by Witten on GLSMs [12], to which the reader is referred for more details. We will describe chiral superfields, twisted chiral superfields, vector superfields and twisted chiral field strengths.
Let us consider the dimensional reduction to 2D of a supersymmetric N = 1 gauge theory in 4D. This 2D theory is supersymmetric with 2 supersymmetries: N = 2. In one way of realizing the dimensional reduction denoted (2,2) the are two left and two right supersymmetries in 2D [12]. This occurs because there are four SUSY generators in total: two right Q + ,Q + and two left Q − ,Q − . In another way one has only two right supersymmetries, denoted (0, 2). Here only the operators Q + ,Q + generate the 2D SUSY [12], this case will not be considered in the present work. We employ the N = 1 4D superfield language to describe (anti-)chiral, twisted (anti-)chiral and vector superfields. Twisted chiral superfields arise in the supersymmetric theory in 2D. In the context of non-linear sigma models there are as well semi-chiral superfields, which constitute representations of (2,2) SUSY [13,14]. In this manuscript we are interested on the class of GLSMs considered by Hori-Vafa in [3] where the representations of the original theory are given by chiral superfields. However in our description of non-Abelian T-dualities we will encounter semi-chiral conditions for superfields in Section 5.
Let us write the supersymmetric covariant derivatives in 2D: 5 (2.1) They can be denoted as D α andDα, with α = +, −. The superspace coordinates are denoted as (x µ , θ α ,θα), where the 2D space-time is given by x 0 , x 3 , and the extra coordinates are x 1 , x 2 . The covariant derivates (2.1) are relevant in the restrictions to a generic superfield that lead to the susy representations. In the expressions only the 2D coordinates x 0 and x 3 are relevant in the dimensional reduction from 4D, because the fields are considered independent of x 1 , x 2 . The chiral (Φ) and anti-chiral (Φ) superfields are defined as: They come from dimensional rduction of the 4d covariant derivativesDα = − ∂ ∂θα − iθ α σ µ αα ∂µ and Dα = ∂ ∂θ α + iθασ µ αα ∂µ [12].
Let us also define twisted chiral (Ψ) and twisted anti-chiral (Ψ) superfields as: A non-Abelian transformation for a chiral superfield Φ in the fundamental representation of certain non Abelian group is given by: where Λ represents the transformation parameter, and T A are the generators of the gauge group. The transformation parameter, Λ depend in general on superspace coordinates and takes values in the Lie algebra of the group spanned by the generators T A 's. The last condition in (2.4) on Λ makes it a chiral superfield and ensures that the transformed field Φ ′ is also chiral. Correspondingly, an antichiral superfieldΦ transforms as whereΛ represents an anti-chiral superfield depending on superspace coordinates. For local transformations, where Λ(x) depends on x µ , one has that Λ =Λ and a gauge field is introduced to make the action covariant. More precisely, the theory has a multiplet of real, Lie algebra valued, scalar superfields V = V A T A transforming under gauge transformations Λ andΛ as There are covariant derivatives with respect to Λ transformations (gauge chiral representation) and with respect toΛ transformations (gauge antichiral rep.). A set of derivatives covariant with respect to Λ transformations is given byDα =Dα and D α = e −V D α e V . A set of covariant derivatives with respect toΛ transformations is given byD ′α = e VDα e −V and D ′ α = D α . These two sets of covariant derivatives allow us to define a 2D gauge invariant field strength, as a twisted chiral superfield, as follows: where D α andDα are the gauge covariant derivatives. The field strength gauge transformation is given by Σ → e iΛ Σe −iΛ . Analogously, one can construct the field strengthΣ = 1 2 {D ′ − , D ′ + } which is aΛ covariant twisted antichiral superfield. One can write The adjoint conjugate of the field strength Σ is given bȳ , (2.9) and has gauge transformationΣ → e iΛΣ e −iΛ . For an Abelian gauge group one gets the following expression We denote these field strengths with a 0 subindex because they will represent the U (1) gauge group of the GLSM, while V and Σ will denote the gauged global symmetry.
We finish this section by presenting the non total derivatives expansions of twisted (anti-) chiral superfields and twisted field strengths of an Abelian vector field V 0 which are given by [15]: The ellipsis represents derivatives of fields contributing to the kinetic terms of the Lagrangian. Looking at the twisted chiral superfields expansions: x i is an scalar field, (χ + , χ − ) are spin 1 2 fermions and G i auxiliary fields. All the components depend on the space-time coordinates x 0 , x 3 . In the field strength expansion one has the scalar σ 0 , the spin 1 2 fermions (λ + , λ − ), the auxiliary field D, and the 2D gauge field strength v 03 . In the appendix E we write useful formulas for the superfield calculations performed in the work, in particular the conventions for integration in superspace are given.

Abelian T-Duality in GLSMs
In this section we formulate Abelian T-duality in gauged linear sigma models as a gauging of a global Abelian symmetry. First, in the bosonic context, we highlight the differences that this approach has with the more traditional T-duality in GLSM largely explained in [4]. We present the approach of gauging a global symmetry in a theory which already has local symmetries. We then consider the case of a GLSM with two chiral fields, a U (1) gauge group, and a global U (1) symmetry. Then we describe the procedure of dualizing all Abelian global symmetries for a GLSM U (1) theory with n chiral superfields Φ i with charges Q i . Next, we describe the dualization procedure for a theory with m U (1) gauge fields along the phase of n chiral superfields. This method applies for more general cases some of which we are going to consider in following sections of dualizing non Abelian global symmetries in GLSMs.

Dualization of a gauge theory with global symmetries
In this section we describe the T-dualization of a theory with scalar fields coupled to a gauge field with minimal coupling, which possesses an U (1) global symmetry. This example serves to illustrate the philosophy of the work of describing T duality as a gauging of a global symmetry in a gauge theory. The global symmetry U (1) acts as a translation on the scalar fields. The dualization of a scalar field minimally coupled to a gauge field is discussed in [3] via another procedure, which consists of constructing a Lagrangian that after integration of certain of its coordinates gives the original or the dual model. Our goal in this section is to show that the dualization of a gauge theory can be realized by gauging the global symmetry and adding a Lagrange multiplier term; this approach follows the prescription of de la Ossa and Quevedo [6].
Consider the theory of two scalar fields, φ 1 and φ 2 , minimally coupled to a U (1) gauge field A µ . This theory has the Lagrangian with Q 1 and Q 2 the charges under the U (1) gauge symmetry. The theory has gauge symmetry given by the transformations There is as well a global symmetry acting as translations on φ 1 and φ 2 which can be parametrized as This symmetry can be gauged by making the parameterΛ dependent on space-time coordinates. Thus a gauge fieldÃ µ needs to be added. Under this scheme one can construct the Lagrangian Integrating the lagrange multiplier field ψ, i.e; imposing its equation of motion one obtains the pure gauge restrictionF µν ǫ µν = 0, which applied on (3.2) leads to the original model (3.1). Instead, taking a variation of (3.2) with respect toÃ µ , that is, considering the equation of motion forÃ µ , we obtain: which upon substitution in (3.2) it leads to the dual Lagrangian: By making the redefinition ψ → 2ψ/Q 1 and fixingQ 1 vs. D to ensure (D − 1) 2 , one gets the normalization obtained by Hori-Vafa in their duality procedure [3]: This simple example shows that T-dualities can be described in gauge theories, as a gauging of remnant global symmetries. This procedure is equivalent to taking as a starting point a more general Lagrangian that encodes the model and the dual model [3]. It is important, however, to notice that in this case one requires the existence of an extra scalar field which we call spectator (because it is not charged under the gauged global symmetry) to allow for an extra global Abelian symmetry that can be gauged. We will encounter similar situations in more general cases.

Supersymmetric GLSM
Here we describe 2D (2,2) supersymetric GLSM's with a single U (1) and multiple chiral superfields. We give the Lagrangian containing the kinetic terms and gauge interactions, without a superpotential. Also we describe twisted superpotential with its instanton corrections. This theory is the starting point for most of the dualities that will be studied in this work.
The Lagrangian of a 2D (2,2) GLSM with gauge group U (1) and n chiral superfields Φ i with charges Q i can be written as [3] where t = r−iθ, with r the Fayet-Iliopoulos (FI) parameter and θ the theta angle; e denotes the U (1) gauge coupling. The classical theory is invariant under vector×axial-vector R-symmetries U (1) V × U (1) A , where Σ has charge (0, 2). The chiral superfields transform under the vector×axial-vector R-symmetries, the U (1) V transformation is given by Φ(x, θ ± ,θ ± ) → e iαq i V Φ(x, e −iα θ ± , e iαθ± ), and the U (1) A is given by Φ(x, θ ± ,θ ± ) → e iαq i A Φ(x, e ∓iα θ ± , e ±iαθ± ), with q V and q A charges with respect to the vector and axial-vector symmetries. The U (1) A symmetry has a chiral anomaly which is canceled for the charge relation: i Q i = 0. As we are considering simple examples meant as building blocks for more realistic cases we do not impose that condition.
In addition, the theory (3.6) has other global symmetries, which are at least (N − 1) U (1) symmetries, those are the phase rotations of the N chiral superfields modulo U (1) gauge transformations. Each charged chiral superfield Φ i can be dualized using the phase rotation symmetry. This gives rise to Abelian T-duality.
Considering one-loop effects, one can obtain an effective superpotential for the vector field, Σ 0 with a fix configuration [12]. This is done by computing corrections to the vacuum expectation value of the D term. The effect can be interpreted as an effective value for the FI term depending on the scalar component of Σ 0 , which is σ. The effective twisted superpotential reads [12,3] This effective superpotential will serve for comparison with the one obtained for Σ 0 in the T-dual model.

Abelian T-duality
Here we describe the T-Duality of a 2D supersymmetric (2,2) GLSM with Abelian gauge group U (1) and two chiral superfields with charges Q 1 and Q 2 . 6 This system has one global U (1) symmetry realized as the phase rotations of Φ 1 and Φ 2 modulo the U (1) gauge transformations. On the target space generated by the scalar components of Φ 1,2 it constitutes a system with one cyclical coordinate. The procedure we perform in this section is equivalent to the one presented in [3], but consists on gauging the global symmetry into the Lagrangian and adding a Lagrange multiplier in the procedure presented in [16].
Let us fix the gauge to remove the phase transformation of Φ 2 and then implement the T-Duality by gauging the phase rotation of the field Φ 1 . Namely under the Abelian global symmetry with parameter α the fields transform as (3.8) We promote the global symmetry to a local one by introducing a vector superfield V and the Lagrange multipliers Ψ andΨ superfields [17]. The Lagrangian reads where V 0 and Σ 0 = 1 2D + D − V 0 are the vector superfield and field strength of the U (1) gauge symmetry, while V and Σ = 1 2D + D − V are the vector superfield and field strength of the gauged symmetry. We have gauged the global U (1) symmetry via the vector superfield V and the use of the unconstrainted superfields Ψ andΨ which constitute Lagrange multipliers. Integrating those one gets the condition Σ = 0, which is a pure gauge field, leading to the original GLSM of two chiral superfields coupled to a U (1) vector superfield V 0 . The equation of motion obtained by varying the action w.r.t to V , δS δV = 0, is given bȳ with Λ = 1 2D + D − Ψ andΛ = 1 2D − D +Ψ . This determines Λ andΛ as twisted chiral and anti-twisted chiral superfields. From (3.10) one can write Plugging the expression for V in (3.9), the Lagrangian can be written as Let us work out the second term in the previous equation (3.12), it reads This gives the dual Lagrangian The second term in (3.14) has an extra piece with respect to the one obtained in equation (3.16) of [3]. However, this extra piece cancels due to the following property: which is fulfilled as a result of D +Φ1 = 0 and D −Φ1 = 0. A similar cancellation takes place with theΛ term. The final dual Lagrangian can be written as For Q = 1 it reproduces the dual Lagrangian obtained with the Hori-Vafa procedure, see 3.1 of [3]: There is, however, a difference: the presence of the spectator chiral superfield Φ 2 . From this Lagrangian and using the expansions superfields in (2.11) at the end of the previous section, one can compute explicitly the scalar potential. Next, we write the contributions to the scalar potential: where F 2 and x 2 are the auxiliary and scalar components of the superfield Φ 2 , y and G are the scalar and auxiliary components of the twisted chiral field Λ, and i, j = 1 are indices denoting the single twisted superfield with K 11 being the Kähler metric evaluated in the scalar y. In appendix C we compute generically the scalar potential and interactions for a given Kähler potential and twisted superpotential of twisted-chiral and twisted-antichiral superfields. The scalar potential can be written as: The contribution to the scalar potential of the auxiliary field F 2 field is zero since is not coupled to any field. Next, integrating out the auxiliary field D we find Substituting the auxiliary field in the potential and reducing terms we obtain the expression Moreover, we want to eliminate the auxiliary field G, then using the equation of motion K 11 G =σ 0 , the scalar potential reduces to This potential has three susy vacua at σ 0 = 0 and (Q 0 (y +ȳ) − (t +t)) + Q 0,2 x 2x2 = 0, x 2 = 0, (y +ȳ) = (t +t)/Q 0 or x 2 = (t +t)/Q 0,2 , (y +ȳ) = 0. Taking the Higgs branch with vacuum σ 0 = x 2 = 0 and (y +ȳ) = (t +t)/Q 0,2 the gauge field strength v 03 gets a large mass in the IR limit, thus it can be integrated out [3]. Let us integrate the field strength v 03 , the effective scalar potential is given by the sum of U and the terms proportional to v 03 , i.e.
integrating out v 03 we get the following expression Substituting this into (3.23) we obtain The supersymmetric vacuum lies at the locus Qy = t. The susy vacuum arises by taking the Higgs branch, i.e, σ 0 = 0 and integrating v 03 in the IR limit. This procedure is equivalent to integrating the twisted field strength Σ in the twisted superpotential (3.17) to obtain the relation ΛQ 0 − t = 0, which implies the previous relation for the scalar components y. The resulting theory comes from looking at the instanton contribution [3] to the total twisted superpotential, which can be written If we look at W tot and integrate out Λ this will give rise to the same effective potential for Σ 0 as in the original theory (3.7). If we integrate Σ 0 the condition ΛQ 0 − t = 0 leads to a constant twisted-superpotential W = e −t/Q 0 , having a theory with a zero scalar potential. We will compare the dual models constituting a family of Abelian T-dualities of section 4 with this result.
The Kähler metric of the dual theory reads .
Writing the scalar components of the superfields Φ as φ = ρe iθ and Λ as λ =ρ + iθ, the equation of motion (3.10) implies the relation ρ 2 =ρ/Q. One can describe distances in the dual system as The measure (3.27) implies the condition Re(λ) =ρ > 0, which after renormalization is corrected to Re(λ) ≥ − ln(Λ U V /µ), with Λ U V the UV cutoff and µ the energy scale [3]. The dual fields Λ and Λ in (3.17) are twisted chiral and twisted anti-chiral superfields. Therefore, the duality exchanges the chiral superfield Φ 1 by the twisted chiral superfield Λ.

Multiple chiral superfields
In this subsection we apply the method to describe T-duality by gauging multiple global symmetries in a gauge theory to the case of a GLSM 2D (2,2) with one U (1) gauge symmetry, with vector superfield and field strength V 0 and Σ 0 and N + 1 chiral superfields Φ i with charges Q i . This theory has U (1) N global symmetries that can be gauged, and from the resulting theory the gauge fields can be integrated out to lead to a dual model. This case has also been studied in the work by Hori-Vafa via their dualization method, here we show the equivalence to the T-duality procedure we are spousing.
We start with the Lagrangian (3.6) with n = N +1 chiral superfields. We consider a parametrization of the U (1) N global symmetries, such that each field Φ i , i = 1, ..., N is only charged under the U (1) i with chargeQ i , i.e. under the i symmetry chiral superfields transform as: (3.28) The gauging of these symmetries into the Lagrangian is given by where V i , Σ i and Ψ i denote respectively the vector superfield, field strength and Lagrange multiplier superfield associated to the gauged symmetry U (1) i . Every field Φ i with i = 1, ..., N has chargeQ i under the gauged symmetry. Φ N +1 is an spectator superfield, which is not charged under U (1) i , is not dualized, and serves to provide the original theory with an extra U (1) global symmetry.
The equation of motion obtained by making the variation of the action w.r.t to V i as δS δV i = 0 is given byΦ from it one can obtain an expression of the vector superfield of the gauged symmetry as Using (3.31) in (3.36) bellow, i.e., integrating each of the vector superfields V i one gets the dual lagrangian This constitutes a replicated version of (3.17). It gives an alternative procedure to performing the dualization made by Hori-Vafa [3] (see their subsection 3.2.1), without the need of deforming the D-term i.e. avoiding the procedure they called in that scheme localization. The equation fits their dual model by making ∀ iQi = 1. The total twisted superpotential, after adding the instanton contributions reads The procedure can be applied as well to more general GLSMs, with a larger gauge symmetry and larger global symmetries. In particular, it can be applied to the case of a GLSM with multiple U (1) gauge fields. We also expect that it can be applied to GLSMs with non Abelian gauge symmetries as the ones described in [10]; these extensions would be the subject of future work.

Multiple U(1)s
In this subsection we perform the dualization procedure along the global Abelian symmetries in the system of N chiral superfields and M U (1) gauge symmetries. One requires an extra set of M expectators fields in order to have U (1) N global symmetries not contained in the U (1) M gauge symmetry. One obtains the dual model that is encountered in Section 3.3 of [3] plus the neutral fields under the the gauged symmetries, which do not play a role in the duality, but act as spectator superfields which allow the existence of extra global Abelian symmetries.
The chiral superfields Φ i have chargeQ i under the gauged symmetry and charges Q i,a under the U (1) a GLSM gauge symmetries. The transformation parameters with respect to the gauge symmetry and the gauged symmetry read Λ i andΛ a . Both transformations are given by We start with a lagrangian where the global symmetries have been gauged, and Lagrange multipliers fields have been added: where V i and Σ i denote the vector superfield and field strength associated to the gauged U (1) i symmetry, V a and Σ a denote the vector superfield and field strength associated to the GLSM gauge symmetry U (1) a . There are M spectators chiral superfields with charges only under the original gauge group, and uncharged with respect to to the gauged symmetries. The equation of motion obtained by varying the action with respect to V i as δS δV i = 0 is given bȳ In the above equation We obtain as solution for the vector superfield of U (1) i : Plugging (3.37) into (3.35) the dual Lagrangian reads Let us compare now with the procedure of Hori-Vafa. Setting the charges under the gauged symmetries asQ i = 1, the total twisted superpotential, after adding the instanton contributions reads This coincides with the results of Section 3.3 in Hori-Vafa work but we employ the approach of gauging the remnant global symmetries. We have tested in this section the method of gauging a global symmetry in GLSMs, in order to describe Abelian T-dualities. Next, we are going to see the advantage of this method in implementing non-Abelian dualities in GLSMs.

Non Abelian T-Duality in GLSMs
In this section we implement NATD in a 2D (2,2) supersymmetric GLSM. We start with a GLSM with Abelian gauge group U (1), N chiral superfields Φ k,i with charges Q k , k n k = N , i = 1, ..., n k where n k is the number of chiral superfields with charge Q k . Due to the coincidence of charges there is a group of global non-Abelian symmetry which acts on the chiral superfields. The non-Abelian global symmetry, denoted by G, can be gauged incorporating to the action the vector superfield V and adding a Lagrange multiplier Ψ, that when integrated out gives back the initial action. After obtaining the equation of motion for a general gauge group we specialize to SU (2). We consider a simple model with two chiral superfields in an SU (2) doublet, and perform the dualization selecting an Abelian direction inside of SU (2); the results obtained for a family of Abelian directions apply as well to a set of non-Abelian directions. In this dual theory where the Lagrange multipliers are the new coordinates, we obtain the supersymmetric vacuum which, together with the instanton corrections, leads to the dual theory.

Non Abelian T-duality for a GLSM with global symmetry G
In this subsection we start with the general action of a (2,2) GLSM with Abelian gauge group U (1), N chiral superfields Φ k,i with charges Q k , and a global symmetry G. We gauge the global symmetry by introducing a vector superfield V and Lagrange multipliers. The vector superfield is integrated to obtain the equations of motion that allow to express the new action in terms of the Lagrange multipliers coordinates.
We start writing the Lagrangian where the Lagrange multipliers Ψ andΨ are unconstrained superfields in the adjoint representation of the non Abelian group G. When they are integrated out give the conditions Σ = 0 andΣ = 0. This condition is a pure gauge condition that leads to what we call the original model, describing a U (1) gauge theory with chiral superfields coupled to a U (1) vector superfield, the theory has lagrangian: . Let us now integrate by parts the Lagrange multipliers terms in (4.1). We first expand them using the definition of the field strength in (2.8) to get: Note that they are complex conjugate of each other. The variation of the exponential of the vector field δe V can be used to write the variation of the inverse e −V i.e. δe −V as: The variation δe V is not a gauge invariant quantity. A gauge invariant variation for the vector field V can be obtained as [11] ∆V We can write ∆V = ∆V a T a , where T a are the gauge group generators. The variations with respect to V of the different terms in the Lagrangian (4.1) are given next. First the variation of the kinetic term for chiral superfields is given by: Then we compute the variation of the lagrange multiplier terms: and the conjugate: Using the previous formulae the equation of motion obtained after integrating out V in (4.1) is given by: Now we would like to express V in terms of the dual fields. The gauge symmetry can be used to fix Φ andΦ. The vector superfield V is Hermitian, with V † = V , (e ±V ) † = e ±V and (∆V ) † = δe V e −V = e V ∆V e −V . This fact helps to simplify the equation of motion. First we notice that χ † = −e VD − e −V which helps us to write (4.6) as We make the definitionτ = D +Ψ and τ =D + Ψ. The lagrange multipliers terms in the action (4.1) can be written as Tr (ΨΣ) = − 1 2 Tr (τ χ) and Tr (ΨΣ) = 1 2 Tr (τχ). The equation of motion (4.7) reduces to the eom of the Abelian case given in (3.10).
In the Wess-Zumino(WZ) gauge there is a strong simplification of the equations, we have: In this gauge the object χ = e −V D − e V can be expressed as χ = χ a T a , this may be possible in other gauges was well. For SU (2) we have χ = χ a σ a at any gauge. Then (4.7) simplifies to The previous equation has the variation ∆V a and its conjugate. Let us argue that they coincide. As we noticed previously (∆V ) † = e V ∆V e −V . This implies (∆V a ) † T a = (e V T a e −V )∆V a , which for Tr T a = 0 implies ∆V a = (∆V a ) † . However if Tr T a = 0 the relation gives Re(∆V a )(T a − e V T a e −V ) + i Im(∆V a )(T a + e V T a e −V ) = 0. The mentioned set of equations has solutions Im(∆V a ) = 0 and ∆V a T a = e V ∆V a T a e −V . This last condition implies [V, ∆V ] = 0 which is a condition that arises also from the Wess Zumino gauge relations (4.8) if ∆V † = e V ∆V e −V holds. Therefore we will consider that the relation ∆V a = (∆V a ) † holds, and look at the equation of motion (4.7) under this consideration.

GLSM with an SU(2) symmetry
We now specify to a gauge linear sigma model with global SU (2) symmetry that can be gauged. This is the explicit example that we will explore in detail in this work. We consider the case of a U (1) gauge symmetry and two chiral superfields equally charged under it. We apply our procedure of gauging the global symmetry of the theory which in this case is an SU (2) symmetry. After integrating the vector superfield via its eom this leads us to the dual model.
The Lagrangian which includes as limits the two dual theories is given by where by V 0 and Σ 0 we denote the vector superfield and field strength the U (1) gauge group of the GLSM, Φ i andΦ i with indices i, j = 1, 2 denote the two chiral and anti-chiral superfields , V is the gauge field of SU (2) gauged symmetry with field strength Σ and Ψ is the lagrange multiplier unconstrained superfield. The superfields Φ andΦ are doublets under the SU (2) global symmetry. The original model is obtained by integrating out the lagrange multiplier Ψ obtaining a pure gauge configuration Σ = 0. The scalar potential in this model is given by The vacuum manifold for r > 0 is given by CP 1 ≡ {|φ 1 | 2 + |φ 2 | 2 = 2r}/U (1) and σ 0 = 0. The mirror of CP 1 obtained by dualizing with Abelian T-duality along two directions i.e. employing the two rotation symmetries of the two fields Φ 1 and Φ 2 is the A 1 Toda space [3] which will be compared with our dual models in Section 6. Now we want to use the global SU (2) symmetry to implement non Abelian T-duality in this model.
Employing this last equation it is possible to express (Φe 2QV 0 e V Φ) in terms of V , thus eliminating 3 of the 4 termsΦ i Φ j , and leaving one. Applying the property det e V = 1 one obtains The superfields X a gets simplified 7 in SU (2). It is possible to use the gauge freedom in the gauged model to setΦ 1 Φ 2 = 0, by setting one of those components to zero. Substituting (4.14) in (4.1) we arrive at the dual model, which we analyze in what follows. Let us look at the reduction to the Abelian case considering χ = e −V D − e V = D − V and only one generator σ a = 1, one gets: This equation coincides with the result for the eom of the Abelian duality given in (3.10).
Let us work the V equation of motion (4.13) to solve for V vs. the other fields. Simplifiying (4.13) with the condition Tr (χτ + τ χ) = 0 one gets (4.16) These constitute three equations of motion, one for each group algebra element. Note than on it the fundamental fields in the duality become X a andX a which are twisted chiral and twisted anti-chiral superfields. This is the case for SU (2), but in more general cases the fundamental fields appear to be the semi-chiral superfieldsD + Ψ a , recall the eom (4.7).
The three equations of motion, one for each group generator can be employed to express two components of e V in terms of the other two. 8 We take the eom (4.16) for the generators a = 1, 2.
Employing these relations it is possible to eliminate the V dependence in the kinetic term to get: The third constraint coming from (4.16) evaluated for σ 3 reads: which is an equation that allows to express the chiral superfields of the model vs. the twisted superfields of the dual model. One can divide (4.19) by Φ 2 1 to obtain The solution reads: Let us choose the solution with +. This solution can be employed to obtain the kinetic term of the lagrangian in terms of the dual coordinates X 1 , X 2 , X 3 : Let us take the solution with a plus on (4.21), using the gauge freedom is possible to fix F to 1 i.e. to set Φ 1 = Φ 2 . This implies a constraint between the real parts of the twisted chiral superfields One could also fix Φ 2 = iΦ 1 , then the gauge implies X 1 +X 1 = X 3 +X 3 = 0. This will be particulary useful at the time of analyzing the dual model geometry.

Dualization along σ
In this subsection we would like to proceed with a simple case, that is to take an Abelian direction inside the duality group along the generator σ 1 . In this scheme we perform the duality and analyze the dual theory obtained.
We start writing the vector superfield V aligned with the generator σ 1 , i.e. V = V 1 σ 1 , for the exponential of V that is: In Appendix B we present relevant formulas for SU (2) vector superfields and its equations of motion, in particular we write the expression that leads to (4.24). Using the equations of motion from the generators σ 1 and σ 2 one obtains from the equations (4.16) Those equations can be solved to obtain From it we get The fact that V 1 is a real vector superfield implies relations among the twisted chiral and the twisted anti-chiral superfields such that the argument of the logarithm is real. Let us use the relations: with F = Φ 2 /Φ 1 given by (4.21). One can show that V 1 satisfiying (4.27) fulfills (e V 1 ) † = e V 1 . One also has with (e −V 1 ) † = e −V 1 . As a check, it is possible to verify that e −V 1 e V 1 = 1 using equations (4.29) and (4.32). There are three contributions to dθ 4 (Tr ΨΣ + TrΨΣ) coming from the three terms in (4.28), those are: from the first term, also: coming from the second term, and: coming from the third term. This last contribution is a duality generated twisted superpotential. Thus, in this gauge we are able to eliminate the original fields and the gauged field in terms of the Lagrange multipliers which become dynamical fields.
The dual Lagrangian can be obtained starting with (4.10) and using all previous relations (4.22)(4.33)(4.35) to get: The last term is a twisted superpotential generated by the duality. One can make an Ansatz for the instanton corrections on the dual theory, compute the effective superpotential for Σ 0 and compare it with the one loop calculation done for that quantity in the original theory [12]. Since we are dualizing an Abelian direction inside the gauged group SU (2) we will take the Ansatz for the instanton corrections to be the same as the Hori and Vafa [3] ∆W = 2µe −X 1 giving the total twisted superpotential (4.37) By integrating out X 1 one gets X 1 = − ln( QΣ 0 µ ) and by plugging this into (4.37) we obtain This last expression coincides with (4.12), the effective superpotential for Σ 0 on the original theory. This gives evidence for the equivalence of both theories. Next, we analyze the supersymmetric vacua of the dual theory.

Scalar potential of the dual model
Let us look at the scalar potential for the twisted field strength of the GLSM and the twisted chiral superfields arising in the dual theory. The vanishing of the new potential is what will give the susy locus, i.e., the target space geometry of the dual theory.
Using the expansions for the twisted superfields given in (2.11) one can express the twisted superpotential in terms of the scalar components of the twisted superfields. All the terms contributing to (4.36) read The kinetic part for the scalar components of the twisted chiral superfields is given by [15,18] with i, j = 1, 2, 3. In appendix C we write generic expressions to compute the Lagrangian contribution of twisted chiral superfields for a given Kähler potential and twisted superpotential. In particular, we derive the second term in (4.40). Collecting all the contributions to the scalar potential depending on the dynamical fields σ 0 , x a and the auxiliary fields D and G i one gets: It is necessary to compute the equations of motion for the auxiliary fields. Solving for the auxiliary field D, 9 yields D = −2e 2 Q(x 1 +x 1 − (t +t)/(2Q)) and one gets the scalar potential The above expression for the scalar potential suggests that we use the equation fo motion for G 1 in order to eliminate the auxiliary fields completely. One obtains K 2jḠj = K 3jḠj = 0 and K 1jḠj = 2Qσ 0 . The previous system is solved by the condition σ 0 = 0. Thus we obtain the potential: whose locus is Re x 1 = Re t/(2Q) = r/(2Q) which represents the susy vacuum in the dual theory.
However, if one looks at the Higgs branch in the NLSM limit the field Σ can be integrated out [3], which in this description means to integrate v 03 . To obtain an effective scalar potential one starts from the scalar potential plus the interaction terms with v 03 : Integrating with respect to v 03 one obtains v 03 = −ie 2 Q(t −t − 2Qn 1 (x 1 −x 1 )) which leads to the scalar potential whose locus |x 1 − t/(2Q)| 2 = 0 represents the dual manifold. Adding the instanton contributions leads us to the dual model in the IR limit. It is a very simple model given by X 1 = t/(2Q) and constant twisted superpotential coming from (4.37): W = 2µe −t/(2Q) . Next, we are going to generalize the study for a family of Abelian directions inside the SU (2) gauged group.

Dualization along n a σ a ∈ SU(2), V = |V |n a σ a
In this subsection we perform a family of Abelian dualities along the direction of a lineal combination of the generators, leading to V = |V |n a σ a . We obtain the dual sigma model, from it the scalar potential and analyze the susy vacua. The dual theory scalar potential also describes truly non-Abelian models. There is only a Higgs branch which leads to the integration of the original U (1) field strength and to a dual theory. We make an ansatz for the instanton contributions to the twisted superpotential which leads to a correct effective potential for Σ 0 , and after the integration on Σ 0 leads to a dual Abelian theory.
In this case the lagrange multiplier term is given by d 4 θTr (ΨΣ) = d 4 θ 1 2 X ana |V |. From (4.16) we have: (4.45) As before the fact that |V | is a real vector superfield implies that K(X a ,X a ,n a ) is as well real, this gives a relation among the twisted chiral and the twisted anti-chiral superfields. The expression for K is given in the appendix B in (B.20). The gauge fixing condition (4.23) X 2 +X 2 = X 3 +X 3 = 0, guarantees Im(K(X a ,X a ,n a )) = 0. In appendix B we gathered useful formulae for SU (2) algebra and the vector superfields taking values in this algebra.
Note that, in principle, then a depend on the superspace coordinates with restrictionn † a =n a . Later in Section 5 we will see that ifn a is a real superfield with the restriction D −na = 0 one can also integrate the equations to have the dual Lagrangian in terms of twisted chiral superfields. The dual Lagrangian forn a constant may be written as The action (4.46) is invariant under the transformation X a → X a + 2πika na2Q , k a ∈ Z, taking into account that the θ angle has quantum symmetry t → t + 2πi.
(4.49) 10 Modifications to this condition should give a truly non Abelian direction inside the non-Abelian group that will be studied in Section 5. 11 As discussed in [3] in the Higgs branch one can also integrate Σ; in this scheme it is related to fixing σ0 at its vanishing vev and to integrate v03 which leads to the scalar potential.
Integrating also the auxiliary field G a we obtain K abḠb = 2Qσ 0 n a . This only lets a contribution to the scalar potential given by 2Qσ 0Ḡa n a ⊂ U . Now the constraints are solved to give U = −2Qσ 0Ḡa n a + 2Q 2 e 2 a (x a +x a )n a − (t +t)/(2Q) 2 , (4.50) The object O(x i ,x i , n j ) depends on the components of the Kähler metric. A susy vacuum appears at σ 0 = 0 and Re( a (x a +x a )n a ) = Re t/(2Q) = r/(2Q). Recall that the condition n a n a = 1 holds, see appendix B. The escalar potential does not show the U (1) gauge symmetry of the original model.

(4.51)
This is solved as n 2 = n 3 = 0 or σ 0 = 0. This makes O(x i ,x i , n j ) = −n 2 1 /K 11 = −n 1 (X 1 +X 1 ) and the potential is given by A vacuum appears at Re(x 1 ) = Re(t) 2Q , |σ 0 | = 0 which is located at U = 0 and is a stable minimum at Re(t) < 0. There is no minimum in the Coulomb branch.
Let us now make a similar analysis as the one performed in the previous section. Taking the Higgs branch one can go to the IR limit by integrating out the U (1) field strength Σ 0 . As σ 0 = 0 this is to integrate v 03 considering it as the fundamental field. We start with the effective potential including (4.52) and the interactions with v 03 to have a n a (x a +x a ))) + (4.53) In the last step we have integrated v 03 to obtain an effective potential in IR theory: x a n a − t/(2Q)| 2 . (4.54) The dual manifold of this theory is at the locus a x a n a − t 2Q = 0. This is a hyperplane in C 3 . But there are additional restrictions as gauge fixing and the twisted superpotential coming from instanton constributions.
We have obtained a family of dual models by gauging a U (1) symmetry inside the global SU (2) of the original model (obtained from (4.10) after integrating the Lagrange multiplier). Making an Ansatz for the expected instanton contributions the twisted superpotential of the dual models is given by W = 2QX a n a Σ 0 + 2µe −Xana − tΣ 0 . (4.55) By integrating out X a one gets X a n a = − ln( QΣ 0 µ ) and by plugging this into (4.55) we obtain again W ef f (Σ 0 ) ′ = −2Q ln QΣ 0 µ + 2QΣ 0 − tΣ 0 which is exactly the effective superpotential for the U (1) field strength of the GLSM(4.38).
Going to the Higgs branch, i.e., σ 0 = 0 in the IR limit one can integrate the gauge U (1) field Σ 0 [3] in the twisted superpotential to obtain the condition X a n a = t 2Q . This gives a constant twisted superpotential W = e −t/(2Q) , with scalar potential U = 0. Thus the vacuum space comes from the restrictions in the coordinates. Recall that one can fix the gauge symmetry to obtain Φ 1 = Φ 2 and this will imply the relation X 2 +X 2 = X 3 +X 3 = 0, as was seen in (4.23). Therefore the target space of the dual sigma model is the 1-complex dimensional space given by X a ∈ C, a n 2 a = 1. (4.56) a X a n a = t 2Q , X 2 +X 2 = X 3 +X 3 = 0, Quantum symmetry: X a n a → X a n a + 2πik a 2Q , k a ∈ Z.
This determines a family of dual manifolds given by the set {n a }. 12 If we fix the gauge symmetry differently to obtain Φ 1 = iΦ 2 and this will imply the relation X 1 +X 1 = X 3 +X 3 = 0. Therefore the target space of the dual sigma model is the 1-complex dimensional space given by X a ∈ C, a n 2 a = 1. (4.57) X a n a = t 2Q , X 1 +X 1 = X 3 +X 3 = 0, Quantum symmetry: X a n a → X a n a + 2πik a 2Q , k a ∈ Z. Both families of dual spaces (4.56) and (4.57) are equivalent under the permutation 1 ↔ 2. Next we are going to see how in this example the inclusion of a truly Abelian part of the duality can modify this supersymmetric vacuum. However there is a non Abelian subset of the SU (2) duality group that gives rise to vacua coinciding with the ones of this Abelian family. This will be done by relaxing the condition that the expansion coefficient in terms of SU (2) generators is constant.
5 Dualization in SU (2) direction n a σ a , with semichiral n a In this section we go beyond a family of Abelian T-dualities inside the non-Abelian T-duality, i.e., we consider a direction inside an SU (2) gauged group with some restrictions on the non Abelian vector superfield. We implement these conditions by writing the vector superfield as V = |V |(x µ , θ α ,θα)σ ana (x µ , θ α ,θα) with the vectorn a (x µ , θ α ,θα) depending on the superspace coordinates. The vector superfield of the gauged group will have the more general form V = σ a V a (x µ , θ α ,θα). We impose the condition D −na = 0 in order to obtain the dual theory in terms of twisted chiral superfields; we take this condition for convenience but it would be interesting to study the choices more systematically.
When D −na = 0 is fulfilled one can obtain for the Lagrange multiplier terms d 4 θTr (ΨΣ) = d 4 θ 1 2 X ana |V |. The fieldn a is a vector field and in addition satisfies D −na = 0. This is a semichiral condition in addition to the reality condition. In this section we will first search for a vector superfield satisfying this semichiral condition. Under this choice ofn a , we can further evaluate the Lagrange multiplier term in the action (4.10) With these ingredients, the dual action is composed by the action valid for a constantn a (4.36) with the addition of an extra term: The last contribution in (5.2) is a truly non-Abelian part of the duality. It gives an interaction term between the U (1) gauge field and the dual twisted superfields. To evaluate this contribution, we need to compute also D − V 0 , we will do so in the WZ gauge. We will explore the part of the extra contributions that reflect in an effective scalar potential.
Let us discuss the symmetries of this action. It is invariant under X a → X a + 2πika 2naQ , k a ∈ Z, taking into account that the θ angle has periodicity t → t + 2πi. The first two lines of (5.2) are invariant as the real part of X a is invariant and K is real. The last two are invariant, because they arise both from −2Q dθ 4 X a n a V 0 → − 2πika 2 dθ 4 V 0 = 2πika 2 dθ − dθ + Σ 0 . This corresponds to a displacement t → t − 2πik a . Is it possible to make an Ansatz for the instanton contributions in the dual theory as e −Xana however the effective twisted superpotential for Σ 0 fails to reproduce the effective superpotential in the original theory. This is a question that will be explored in future work.
One needs to impose alson a (ȳ) † =n a (ȳ). Performing the expansion of (5.7) one gets Recalling the reality condition the totaln a (ȳ) has the expression Let us compute nowD +na for further usē Notice that even forn a non constant, i.e., a non Abelian direction in the vector superfield but w a 03 = 0 the terms contributing to the scalar potential (not to the kinetic term) coincide with the ones forn a constant discussed in 4.4. This is simplified by recalling the reality condition ofn a (ȳ): Let us now write the vector superfield of the GLSM gauge group V 0 in the Wess-Zumino gauge [18] Furthermore the quantity D − V 0 appearing in (5.2) and the field strength read: In order to see if there are extra contributions to the scalar potential, we obtain Comparing with (4.47), there is an extra contribution to the scalar potential, only dependent on the gauged field boson w a 03 , the U (1) sum of vector potentials v 3 + v 0 and the twisted chiral scalar x a . The extra contribution is given by U = Q 2 w a 03 (v 3 + v 0 )(x a +x a ). This matches the observation that This renders a total scalar potential, in which we have included interaction terms with the U (1) gauge field component v 0 + v 3 . Via a gauge transformation this field combination can be taken to a constant 14 , therefore we can look at the effective potential obtained after taking the Higgs branch and integrating v 03 : The previous vacua found from (4.50) remains the same by taking ∀ a w a 03 = 0. This is a restriction on the semichiral real superfieldn a ton a = n a (ȳ) + iθ + γ + (ȳ) + iθ +γ + (ȳ), which restricts the gauged field V . This is very interesting, because implies that the results of previous section for a family of Abelian T-dualities inside of SU (2) is much more general than that. Becausen a is not constant one has really a vector superfield with algebra direction V = |V |(x, θ,θ)n a (x, θ)σ a , and even if n a has the discussed restrictions the components of V along the different generators V a = |V |(x, θ,θ)n a (x, θ) are different from each other. Making the vacua discussed in Subsection 4.4 vacua of a non Abelian dual theory. Furthermore let us allow w a 03 and (v 0 + v 3 ) to be different from zero, and let us look at how the Higgs branch vacuum with σ 0 = 0 gets modified. Let us see this in the gauge X 2 +X 2 = X 3 +X 3 = 0. The vacuum of (5.15) is given by )ω a 03 and A = 2Q 2 e 2 such that the potential (5.15) reads U = A|x a n a − t/(2Q)| 2 + B a (x a +x a ).
Without fixing the gauge one obtains for the vacuum equations The addition of the non Abelian contribution (5.14) changes the vacuum by changing the hyperplane to be x a n a = t 2Q − B 1 2An 1 . The condition ω 1 03 n 1 = ω 3 03 n 3 needs to be satisfied. In this vacuum there are two positive eigenvalues and four zero eigenvalues (0, 0, 0, 0, A, A). The positive eigenvalues represent the growing directions of the potential. The zero eigenvalues of the Hessian represents the real dimension 4 of the vacuum space.
One direction corresponding to 0 eigenvalue, the first in Table 5.1, gives a transformation that preserves the plane condition is Fixing the gauge as in (4.23) the new vacuum is given by the one complex dimensional curve: At the quantum level, i.e. the level of the partition function there is in addition the symmetry x a → x a + 2πika 2naQ , k a ∈ Z coming from the periodicity of t. The value of the potential in the minimum is given by U min = (B 1 (−2B 1 + An 1 Re(t 1 )))/(8An 2 1 ).
To obtain a supersymmetric minimum is required a zero scalar potential U = 0, this can be achieved by setting B 1 = An 1 Re(t 1 ) 2 , i.e. (v 0 + v 3 )ω 1 03 = 2Qe 2 n 1 Re(t 1 ). 15 We found in this section that the family of Abelian dualities inside of SU (2) describes really a subset of the full non-Abelian duality by giving a lagrangian where the coefficientn a determining the vector superfield can variate in the space-time, giving for the vector superfield components the dependence V a = |V |(x µ , θ α ,θα)n a (x µ ). This means that the scalar potential coincides in both cases. Nevertheless there are kinetic terms which make the lagrangians distinct. When we look at the vacuum we considered then a as constant coefficients, but they appear in the action making it invariant under local transformations that are not Abelian. Therefore further investigation should me made to clarify the connection. If one considers a more general expansion, then an additional interaction term between U (1) vector superfields of the GLSM and twisted chiral scalars arise (5.14), that deform this vacuum. The general conclusion is that that in this model a family of Abelian dualities inside the gauged group can cast relevant features of the non-Abelian T-duality. It would be interesting to check if this is a conclusion that holds for other duality groups.

Comparison with Abelian T-duality
Our aim in this section is to compare with previous models the non-Abelian T-dual model obtained starting from a U (1) GLSM with SU (2) global symmetry, i.e., two chiral superfields with equal charges. This original model has as target space CP 1 . What we would like to compare is the model obtained by selecting an Abelian family inside SU (2) group presented in Section 4.4. We have already shown that these dual models are more than just Abelian since they display some properties of the full non Abelian duality. First, we compare it with the mirror of the model and later to a model with a single U (1) direction dualized. With this aim first we review the T-dualization of two Abelian directions of the GLSM model of two chiral superfields with equal charges under U (1); this is the Toda A 1 model [3]. Then, we compare it with the model obtained by dualizing the phase rotations of one chiral superfield. Intuitively, this last model is the relevant case to compare with since the family of Abelian dualities embedded in SU (2) should match the dualization of a single Abelian symmetry.
Let us present the Hori-Vafa dual model obtained by dualizing along the two directions of the chiral superfields Φ 1 , Φ 2 to obtain the twisted chiral superfields Y 1 and Y 2 . The supersymmetric vacuum of the original theory is CP 1 . Performing two Abelian dualities one obtains a dual model with a twisted superpotential W (Θ) ∼ Θ + 1/Θ [3]. This gives rise to Toda A 1 space {z ∈ C * , W (z) := z + 1/z}. It is appropriate to integrate the U (1) gauge field strength in the IR limit since the mass of σ 0 , the scalar component of Σ 0 , is high due to the relation of the coupling and the energy scale e ≫ Λ. The dual model obtained by Hori-Vafa is given by the Lagrangian First, we consider the vacua obtained in the Higgs branch without taking into account the instanton corrections. Taking into account that the Kähler metric is given by K 11 = − 1 2(Y 1 +Ȳ 1 ) and K 22 = − 1 2(Y 2 +Ȳ 2 ) and that from the kinetic term one gets a term for the auxiliary fields given by K ij G iḠj . This Lagrangian leads to the scalar potential Let us integrate the auxiliary fields G 1 , G 2 and the field combination D − iv 03 to obtain: U ef f = 2e 2 |y 1 + y 2 − t| 2 + |σ 0 | 2 2(y 1 + y 2 +ȳ 1 +ȳ 2 ). (6. 3) The solution is given by y 1 + y 2 − t = 0 and σ 0 = 0. This is equivalent to considering the classical solution and to take the Higgs branch and then to integrate the field Σ 0 which in the IR limit develops a large mass. In the case when we only consider the potential for the dynamical fields in the GLSM, integrating out G 1 , G 2 and the real auxiliary field D one obtains the potential: Leading to a Higgs branch at σ 0 = 0 and Re(y 1 + y 2 ) = Re(t) = r. A Coulomb branch appears at Re(y 1 + y 2 ) = 0, σ 0 = 2e √ r, which is supersymmetric only at r = 0, therefore there is no supersymmetric Coulomb branch.
The interaction terms of twisted superfield scalars with v 03 add to the potential: In the Higgs branch the Σ field gets a large mass, and integrating it out is equivalent to integrating out v 03 from (6.5) to obtain (6.3).
Therefore the condition Y 1 +Y 2 = t comes from integrating out Σ 0 along the Higgs branch of the theory. The relevant twisted superpotential of the effective theory is given by W = e −Y 1 + e −t+Y 1 , or, after making the change of variables Θ = t/2 − Y 1 , one obtains W = e −t/2 (e −Θ + e Θ ). This model is different from the case considered here, of a family of Abelian dualities inside of SU (2), due to the fact that the dualization is done along two U (1) directions. Therefore it is natural to expect a different result.
Finally, we compare the dual model obtained dualizing the U (1) GLSM with two equal charged chiral superfields (3.9 along an U (1) direction of one of them, with the model of Abelian duality inside SU (2) presented in Section 4.4 (4.1). The kinetic part of (4.1) can be written in matrix form asΦ T M Φ, where M = e |V |naσa and Φ T = Φ, then it is possible to rotate the element M into an element generated by σ 1 to obtain This can be seen from formula (B.7). Then the kinetic term in (3.9) can be written as where the prime fields are obtained by diagonalizing the exponential of the vector superfield matrix to get Φ ′ Since the GLSM has an U (1) symmetry, we can make a gauge transformation V 0 → V 0 + V 1 2Q 0 which is substituted in (6.7) to obtain setting the charge Q to 1 this is exactly the kinetic term of the two chiral superfields GLSM (3.9). Also the Lagrange multiplier term is given by d 4 θTr (ΛΣ) = d 4 θX 1 V 1 . These observations establish that the gauged theories are equivalent. Now, let us compare the vacuum of both models.
In the model of duality along the direction σ 1 in SU (2) the vacuum is given at x 1 = t 2Q with constant twisted superpotential W = 2µe −t/(2Q) . On the dualization of the Abelian symmetry along one chiral superfield the susy vacuum is given at y = t/Q and the twisted superpotential reads W = e −t/Q . These results are equivalent by considering the appropriated normalization factors.
The family of Abelian dualities inside of SU (2) leads to a dual model with scalar potential, that coincides with the one of the models with more general vector superfields (5.7). The extension to more general vector superfields gives rise to extra terms in the potential (5.2), in this case one has to search for the appropriate instanton correction. It is likely that they would not have the very simple form of the Abelian cases. They will be needed in the exploration of determinantal CY varieties and their duals. In this Section we have checked that the non Abelian T-duality performed coincides in a limiting case with the Abelian T-dual models. This is an important sanity check and a necessary step toward testing the duality. With the methods developed here we plan to consider in the future the study of non-Abelian T-dualities in realistic examples.

Conclusions
We have used the method of gauging a global symmetry in field theories with subsequent addition of a Lagrange multiplier enforcing flatness condition and alternative elimination of fields to describe Tduality in GLSMs. Tackling the description of T-Duality in GLSMs by this gauging procedure allows us to obtain a Lagrangian with added Lagrange multipliers and a vector superfield. Integration of the latter leads to the dual model. The method applies to GLSMs with multiple U(1)'s and multiple chiral superfields. More importantly for our goals is that the method admits a generalization to the gauging of non Abelian global symmetries, which lead to non Abelian-T dualities in GLSMs. We have thus implemented non-Abelian T-duality at the level of GLSM. Considering a generic gauge group we gauged the non Abelian global symmetry. The equations of motion were obtained in general. The non-Abelian duality leads to twisted superfield dual coordinates only for certain cases constituting an important difference with the Abelian duality. In more generic cases the elementary fields in the equations of motion are semi-chiral.
We have analyzed in detail the concrete example of a GLSM with a single U (1) gauge field and two chiral superfields of equal charges. This theory has a global SU (2) symmetry which can be gauged. In the SU (2) case the fundamental fields of the duality are twisted chiral superfields. The terms of the Kähler potential of the dual theory are obtained in terms of the dual twisted superfields. First, assuming an Abelian direction inside of the gauged group the twisted superpotential can be obtained as well. This direction means that we consider a vector superfield proportional to a linear combination of SU (2) generators with constant coefficients. This constitutes a family of Abelian dualities which displays characteristics of the full non Abelian duality, as for example the Kähler potential. Furthermore, for coefficients depending only on a scalar component the scalar potential reduces to the one of a family of Abelian dualities. This observation implies that the afore mentioned family of Abelian dualities is more than it seems, it describes a truly non Abelian duality inside of SU (2), even if not the more general one. In a next step we generalized it to truly non-Abelian T-Duality by imposing a semi-chiralilty condition on the coefficients of the generator expansion. The supersymmetric vacuum of this non Abelian dual model differs only by constants from the supersymmetric vacuum of the family of Abelian T-dualities previously mentioned. The difference comes from an interaction term with the U (1) gauge fields. In the IR limit, i.e., taking the Higgs branch when the U (1) field strength is integrated out this contributes to the scalar potential, and therefore to the supersymmetric vacuum.
We took some steps toward the inclusion of nonperturbative effects. Namely, we postulated an Ansatz for the twisted-superpotential instanton corrections. In the dual theory along an Abelian direction of the group (which is a linear combination of the generators) we verified that the effective action for the U (1) gauge field coincides for the original and the dual model. These instanton corrections allow to compare the Abelian dual theory family inside the non-Abelian dual theory with the case of an Abelian T-dual model with a single U (1) dualization. The result is that they match, i.e., their effective potentials have the same vacuum and the twisted superpotential can be mapped. For the fully non Abelian case however, we do not know how to write the instanton corrections; this is an important direction that we leave for future investigations.
It is worth pointing out a number of interesting problems that our investigation naturally highlights. With the methods studied here it would be interesting to tackle some explicit examples of dualities for determinantal CY varieties such as those presented in [19]. In particular, it seems possible to arrive at a more systematic description of some Pfaffian CY presented in [20]. Note that explicit computations of Gromov-Witten invariants for the CY of [19] were given in [21], providing a concrete framework for testing the non Abelian T-duality proposed in this manuscript. Given the advances in explicit computation in GLSMs, it would be interesting to apply the technique of supersymmetric localization to our models in order to better understand the effects of non-Abelian T-duality.
Finally, we have seen glimpses of new representations, such as the semi-chiral superfield representation appearing in our construction. It would be useful to achieve a complete understanding of the possible representations, possibly opening a window into a formulation of non-Abelian T-duality for generalized geometry.

A Equations of GLSM non Abelian T-dualization
Here we present details of the calculations in Subsection 4.1 where we compute the equations of motion of the integrated vector superfield of the gauged symmetry.
In the following we expand the variations of the Lagrange multiplier terms in the Lagrangian. The variation wit respect to the gauged field V of the lagrangian term for the Lagrange multiplier Ψ is given by: The conjugate term reads: These expressions are valid for any group, and we write them for further reference.

B Formulae for SU (2) group
In this appendix we present relevant formulae for superfields calculations in the case of the SU (2) group. We start presenting relations among the generators, formulae for the exponentials, and expansions for the vector superfields. As well we integrate the equations of motion for the vector superfield (4.16).
For the generators T a of SU (n) gauge the following identity applies: For the case SU (2) lets us give the specification of (B.1) and another formula: Tr (σ a σ b σ c ) = 2iǫ abc .
The real vector superfield can be written as where σ a are the Pauli matrices,σ = (σ 1 , σ 2 , σ 3 ),n a = V a /|V | is an unitary vector and |V | = √ V a V a .
Looking at the exponential series expansion one gets Previous expansions combine to give There is a particular case easy to treat, this is when V = V ana σ a withn a independent of the superspace coordinates or satisfiying the restriction D −na = 0. Then lets us consider V = |V |n a σ a giving e −V D − e V = D − |V |n a σ a . For this case Tr ΨΣ =D + Ψ a D − |V |n a . The explicit form of the exponential e V in this case is One can then use the relations (4.17) to obtain Setting n 1 = n 2 = 0 the only possible solution implies n 3 = 0 also. So for the gauge V 1 = V 2 = 0 also V 3 = 0 is required. On the other hand n 3 = 0 implies that Let us study first the case n 3 = 0, in this case by summing up the equations (B.7) and (B.8) one gets: and by subtracting them: For convenience let us rewrite (4.21) as (B.11) The expression for F is given by Thus obtaining from (B.9) This is solved by As well from (B.10) one gets This is solved by One can equate (B.14) and (B.17) to obtain an expression for (e 2QV 0 |Φ 1 | 2 ). This will serve to finally eliminate the chiral and antichiral superfields Φ 1 andΦ 1 from the action. The result reads sinh |V |((n 2 + in 3 /2(F +F ))(X 1 +X 1 ) + ( This relation implies either n 1 = n 2 = 0, n 1 = X 1 = 0, n 2 = X 2 = 0 or X 1 = X 2 = 0. The most interesting possibilities seem to be the 2nd and the 3rd. The current situation with V 2 = V 3 = X 2 = 0, V = V 1 and |Φ 1 | 2 = |Φ 2 | 2 implies 2(X 1 +X 1 )(X 3 +X 3 ) = 0 which is an additional restriction on X 1 and X 3 . As well V 1 = V 3 = X 1 = 0, V = V 2 and |Φ 1 | 2 = |Φ 2 | 2 implies 2(X 2 +X 2 )(X 3 +X 3 ) = 0 which is an additional restriction on X 2 and X 3 .
A solution for |V | can also be obtained for generic values of n 1 , n 2 and n 3 using (B.7) and (B.8). This is given by The expression K(X i ,X i , n j ) in the argument of the logarithm is given by: . (B.20)

C Twisted Chiral Expansion
In this section we compute explicitly some useful expression in components of the (anti-) twisted chiral superfields used along the paper, that can become handy for working with twisted chiral superfields. These derivations are based on the procedure presented in the book of Wess and Bagger [18] for chiral superfields.
First, let us recall the expansions of twisted and anti-twisted chiral superfields In previous formulae y i , χ − ,χ + , G i represent the scalar, fermionic and auxiliary components of the twisted chiral superfield Y i . Their conjugates are the components of the (anti-) chiral su-perfieldȲ i . Consider a function K representing the Kähler potential that depends on twisted chiral and anti-twisted chiral fields Y i andȲ i , where i runs from 1, . . . , n. Now, assuming that K(Y 1 , . . . Y n ,Ȳ 1 , . . . ,Ȳ n ) can be Taylor expanded, we compute a generic term of the expansion namely, Since this generic can be regarded as the product of the monomial Y i 1 · · · Y i N andȲ j 1 · · ·Ȳ j M , thus let us Taylor expand the functions P (Y i 1 · · · Y i N ) andP (Ȳ j 1 · · ·Ȳ j M ) explicitly are given by where y = (y i 1 , . . . , y i N ). We can use this expressions since K can be regarded as products Y 's and Y 's The contribution to the Lagrangian is the term proportional to θθθθ neglecting fermionic terms, which is easily computed in (C.3) with the aid of the previous expression setting P = Y i 1 · · · Y i N andP =Ȳ j 1 · · ·Ȳ j M , we get (neglecting fermionic terms) hence the θθθθ term of K(Y i 1 , . . . , Y in ,Ȳ j 1 , . . . ,Ȳ jn ) is the sum over all the monomials (with their respective constants of the power expansion) of the form (C.6), thus this term reads as follows K(Y i 1 , . . . , Y in ,Ȳ j 1 , . . . ,Ȳ jn ) = · · · + 4θ +θ− θ −θ+ G iḠj ∂ 2 ∂y i ∂ȳ j K(y 1 , . . . , y i 1 ,ȳ1, . . . ,ȳī n ) In general if P = y i 1 · · · y i N andP =ȳ j 1 · · ·ȳ j M , summing over all the terms of the expansion of the Kähler potential together with the following results of Kähler geometry we express (C.6) as: D Dualization along σ 3 , vector superfield: |V | = V 3 In this appendix we discuss the dualization along a particular direction of the SU (2) global symmetry, this is the direction along the σ 3 generator. This case was not considered in the main text because it has the particularity that solving the equations of motion for the vector superfield in terms of the twisted chiral superfields one encounters the condition V 3 = 0. Nevertheless, one can write the dual model Lagrangian, and compare the suspersymmetric vacuum with the one obtained in the case of Abelian duality.
Let us consider that the e V components fullfill e V 12 = e V 21 = 0. This is to choose a direction along σ 3 , which constitutes a reduction to an Abelian model. For this case e V = e V 3 σ 3 = e −V 3 0 0 e V 3 , |V | = V 3 , n 3 = 1. (D.1) The two following relations coming from the equation of motion for V (4.16) hold e 2QV 0Φ 1 e V 11 Φ 1 = (X 1 +X 1 − i(X 2 +X 2 )) These imply which is a bit problematic because in this case e V 11 = 1/e V 22 . Let us look at the other term in the Lagrangian from this term a twisted superpotential is generated.
16 This observation can be connected with the fact that in the WZ gauge one has for the lagrange multiplier term: Let us briefly discuss that the dualization along σ 3 can be mapped to an Abelian dualization along the phase of one chiral superfield as the one discussed in Subsection 3.3. in the main text this discussion is done for the direction σ 1 which is the one analyzed systematically. The kinetic part of (4.1) can be written in matrix form asΦ T M Φ, where M = e |V |naσa and Φ T = Φ. It is possible to rotate the element M into an element generated by σ 3 to obtain e |V |σ 3 = e |V | 0 0 e −|V | . (D.7) Then the kinetic term in (4.1) can be written as Since the GLSM has an U (1) symmetry, we make a gauge transformation V 0 → V 0 + |V | 2Q 0 which is substituted in (D.8) to obtain setting the charge Q to 1 this is exactly the kinetic term of the two chiral superfields GLSM (3.9).

E Notation and identities
Here we give a short resume of many of the conventions for the supersymmetry calculations performed in the paper.
F An SU (2) scalar example, dualizing along σ 3 direction The aim of this appendix is to work out explicitly another example of gauging a global symmetry on a gauge theory. We consider two complex scalar fields φ 1 and φ 2 charged with identical charge Let us make a dualization of this symmetry only under the σ 3 generator direction. This is equivalent to having the two complex scalar fields of the doublet charged with opposite charges under another U (1), and to dualize in that direction. The corresponding covariant derivatives act on the fields as (F.1) Both fields have the same charge under the initial U (1) gauge so thatD µ = ∂ µ + iqÂ µ . The lagrangian once the extra U (1) symmetry has been gauged is given by To obtain the dual model we integrate (F.2) with respect to A µ to obtain Substituting the equation (F.3) in (F.2) one obtains The term −q 2 A 2 0 |φ| 2 will give −q 2 A 2 0 |φ| 2 = − (∂ 1 λ) 2 4q 2 |φ| 2 + The term q 2 A 2 1 |φ| 2 will give Such that: The two gauge U (1) transformations can be used to fix the components ρ = 0, 1 ofD ρ φ 1 φ * 1 to be real leading to the lagrangian Let us write explicitly φ 1 = r 1 e iθ 1 to see that (D µ φ 1 ) †Dµ φ 1 = (∂ µ r 1 ) 2 + r 2 1 (∂ µ θ 1 +Â µ ) 2 .
2. When we keep the two scalar fields φ 1 and no gauge field, the gauged symmetry can be used to eliminate one phase. Choosing the one of φ 1 one obtains the lagrangian with φ 1 = φ * 1 and λ the dual component. In this appendix we have presented the main elements of a non Abelian SU (2) duality which renders the scalar part of a GLSM dualization along σ 3 direction. Further analysis of this connection will be presented elsewhere.