William D. Linch, Brenno Carlini Vallilo
Recently, a gauged linear sigma model was proposed by Berkovits and Vafa which can be used to describe the AdS_5xS^5 superstring at finite and zero radius. In this paper we show that the model is classically integrable by constructing its first non-local conserved charge and a superspace Lax "quartet". Quantum conservation of the non-local charge follows easily from superspace rules.
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We study an N=(4,4) supersymmetric gauged linear sigma model which gives rise to the nonlinear sigma model for multi-centered KK-monopoles. We find a new T-duality transformation of the model even in the presence of F-terms. Performing T-duality, we find the gauged linear sigma model whose IR limit describes the exotic 522-brane with B-field.
Soliton modes in a gauged sigma model with interpolating potential have been investigated. Numerical solutions using a fourth order Runge -- Kutta method are discussed. By tuning the interpolation parameter the transition from symmetrybreaking to the symmetric phase is highlighted.
We discuss the N=2 supersymmetric extension of the gauged O(3) sigma model in (2+1) dimensions with an abelian Chern-Simons term. It is shown that the self-dual potential and the Bogomolny relations naturally appear as consequences of extended supersymmetry.
We formulate a gauged linear sigma model on a supermanifold. The structure of classical vacua is investigated and one-loop corrections are calculated for an auxiliary D-field. We find out a constraint for the one-loop divergence to vanish, which is consistent with a Ricci flatness condition for the supermanifold. Two types of D-branes are obtained by analysing supersymmetric boundary conditions. We also provide extensions to a non-abelian gauged linear sigma model and a (0,2) supersymmetric model.
We consider a (2+1) dimensional nonlinear O(3) sigma model with its U(1) subgroup gauged along with the inclusion of a self-interaction having symmetry breaking minima.The gauge field dynamics is governed by the Maxwell term.The model is shown to support topologically stable purely magnetic self-dual vortices.
We consider Chern-Simons gauged nonlinear sigma model with boundary which has a manifest bulk diffeomorphism invariance. We find that the Gauss's law can be solved explicitly when the nonlinear sigma model is defined on the Hermitian symmetric space, and the original bulk theory completely reduces to a boundary nonlinear sigma model with the target space of Hermitian symmetric space. We also study the symplectic structure, compute the diffeomorphism algebra on the boundary, and find an (enlarged) Virasoro algebra with classical central term.
We show that the degeneracy of topological solitons in the gauged O(3) non-linear sigma model with Chern-Simons term may be removed by chosing a self-interaction potential with symmetry - breaking minima. The topological solitons in the model has energy,charge,flux and angular momentum quantised in each topological sector.
The gauging of isometries in general sigma-models which include fermionic terms which represent the interaction of strings with background Yang-Mills fields is considered. Gauging is possible only if certain obstructions are absent. The quantum gauge anomaly is discussed, and the (1,0) supersymmetric generalisation of the gauged action given.
We study some aspects of Gauged Linear Sigma Models corresponding to orbifold singularities of the form $\BC^r/Γ$, for $r=2,3$ and $Γ= \BZ_n$ and $\BZ_n\times \BZ_m$. These orbifolds might be tachyonic in general. We compute expressions for the multi parameter sigma model Lagrangians for these orbifolds, in terms of their toric geometry data. Using this, we analyze some aspects of the phases of generic orbifolds of $\BC^r$.
We construct supersymmetric deformations of general, locally supersymmetric, nonlinear sigma models in three spacetime dimensions, by extending the pure supergravity theory with a Chern-Simons term and gauging a subgroup of the sigma model isometries, possibly augmented with R-symmetry transformations. This class of models is shown to include theories with standard Yang-Mills Lagrangians, with optional moment interactions and topological mass terms. The results constitute a general classification of three-dimensional gauged supergravities.
We study the supersymmetric extension of the gauged $ O(3) $ sigma model in $ 2+1 $ dimensions and find the supersymmetry algebra. We also discuss soliton solutions in case the Maxwell term is replaced by the Born-Infeld term. We show that by appropriate choice of the potential, the self-dual equations in the Born-Infeld case coincide with those of the Maxwell's case.
Worldsheet (0,2) gauged linear sigma models are often used to study supersymmetric heterotic string compactifications with non-trivial vector bundles. We make use of supersymmetric localization techniques to determine their one-loop partition functions. In particular we derive conditions which ensure that the full partition function is modular invariant and we propose a method to determine the massless and massive target space matter spectrum.
The geometry of (2,1) supersymmetric sigma-models with isometry symmetries is discussed. The gauging of such symmetries in superspace is then studied. We find that the coupling to the (2,1) Yang-Mills supermultiplet can be achieved provided certain geometric conditions are satisfied. We construct the general gauged action, using an auxiliary vector to generate the full non-polynomial structure.
We study systems of multiple localized closed string tachyons and the phenomena associated with their condensation, in C3/ZN nonsupersymmetric noncompact orbifold singularities using gauged linear sigma model constructions, following hep-th/0406039. Our study reveals close connections between the combinatorics of nonsupersymmetric flip transitions (between topologically distinct resolutions of the original singularity), the physics of tachyons of different degrees of relevance and the singularity structure of the corresponding residual endpoint geometries. This in turn can be used to study the…
The $O(3)$ nonlinear sigma model with its $U(1)$ subgroup gauged, where the gauge field dynamics is solely governed by a Chern-Simons term, admits both topological as well as nontopological self-dual soliton solutions for a specific choice of the potential. It turns out that the topological solitons are infinitely degenerate in any given sector.
We introduce the self-dual abelian gauged $O(3)$ sigma models where the Maxwell and Chern-Simons terms constitute the kinetic terms for the gauge field. These models have quite rich structures and various limits. Our models are found to exhibit both symmetric and broken phases of the gauge group. We discuss the pure Chern-Simons limit in some detail and study rotationally symmetric solitons.
Recently, we have studied the general Virasoro construction at one loop in the background of the general non-linear sigma model. Here, we find the action formulation of these new conformal field theories when the background sigma model is itself conformal. In this case, the new conformal field theories are described by a large class of new spin-two gauged sigma models. As examples of the new actions, we discuss the spin-two gauged WZW actions, which describe the conformal field theories of the generic affine-Virasoro construction, and the spin-two gauged g/h coset constructions. We are able to…
We study soliton solutions in 1+1 dimensional gauged sigma models, obtained by dimensional reduction from its 2+1 dimensional counterparts. We show that the Bogomol'nyi bound of these models can be expressed in terms of two conserved charges in a similar way to that of the BPS dyons in 3+1 dimensions. Purely magnetic vortices of the 2+1 dimensional completely gauged sigma model appear as charged solitons in the corresponding 1+1 dimensional theory. The scale invariance of these solitons is also broken because of the dimensional reduction. We obtain exact static soliton solutions of these model…
We propose an effective action for the eleven-dimensional (bosonic) Kaluza-Klein monopole solution. The construction of the action requires that the background fields admit an Abelian isometry group. The corresponding sigma-model is gauged with respect to this isometry. The gauged sigma-model is the source for the monopole solution. A direct (double) dimensional reduction of the action leads to the effective action of a 10-dimensional D-6-brane (IIA Kaluza-Klein monopole). We also show that the effective action of the 10-dimensional heterotic Kaluza-Klein monopole (which is a truncation of the…
Nonlinear sigma models arise in supergravity theories with or without matter couplings in various dimensions and they are important in understanding the duality symmetries of M-theory. With this motivation in mind, we review the salient features of gauged and ungauged nonlinear sigma models with or without Wess-Zumino terms for general target spaces in a minimal as well as lifted formulation. Relevant to the question of finding interesting vacua of gauged supergravity theories is the highly constrained potential which arises naturally in these theories. Motivated by this fact, we derive a gene…
The gauged sigma-model argument that string backgrounds related by T-dual give equivalent quantum theories is revisited, taking careful account of global considerations. The topological obstructions to gauging sigma-models give rise to obstructions to T-duality, but these are milder than those for gauging: it is possible to T-dualise a large class of sigma-models that cannot be gauged. For backgrounds that are torus fibrations, it is expected that T-duality can be applied fibrewise in the general case in which there are no globally-defined Killing vector fields, so that there is no isometry sy…
Vector $SO(3)$ gauged $O(4)$ sigma models on $\R_3$ are presented. The topological charge supplying the lower bound on the energy and rendering the soliton stable coincides with the Baryon number of the Skyrmion. These solitons have vanishing magnetic monopole flux. To exhibit the existence of such solitons, the equations of motion of one of these models is integrated numerically. The structure of the conserved Baryon current is briefly discussed.
We find the conditions under which a Riemannian manifold equipped with a closed three-form and a vector field define an on--shell N=(2,2) supersymmetric gauged sigma model. The conditions are that the manifold admits a twisted generalized Kaehler structure, that the vector field preserves this structure, and that a so--called generalized moment map exists for it. By a theorem in generalized complex geometry, these conditions imply that the quotient is again a twisted generalized Kaehler manifold; this is in perfect agreement with expectations from the renormalization group flow. This method ca…
We study both A-type and B-type D-branes in the gauged linear sigma model by considering worldsheets with boundary. The boundary conditions on the matter and vector multiplet fields are first considered in the large-volume phase/non-linear sigma model limit of the corresponding Calabi-Yau manifold, where we also find that we need to add a contact term on the boundary. These considerations enable to us to derive the boundary conditions in the full gauged linear sigma model, including the addition of the appropriate boundary contact terms, such that these boundary conditions have the correct non…
Gauging the M-2-brane effective action with respect to an Abelian isometry in such a way that the invariance under gauge transformations of the 3-form potential is maintained (slightly modified) we obtain a fully covariant action with 11-dimensional target space that gives the massive D-2-brane effective action upon dimensional reduction in the direction of the gauged isometry.
Existence of a supersymmetric extension of the gauged non-linear $Σ$-model and the transition from integrability to chaos
该研究发现3+1维规范非线性$\Sigma$模型在特定量子化条件下存在隐藏的(1+1)维$\mathcal{N}=2$超对称性，并探究了该超对称性在含时区域对从可积性到混沌跃迁的影响。
The (2+1) dimensional gauged O(3) nonlinear sigma model with Chern-Simons term is canonically quantized. Furthermore, we study a nonminimal coupling in this model implemented by means of a Pauli-type term. It is shown that the set of constraints of the model is modified by the introduction of the Pauli coupling. Moreover, we found that the quantum commutator relations in the nominimal case is independent of the Chern-Simons coefficient, in contrast to the minimal one.
We have performed a complete study of self-dual configurations in a $CPT$-odd and Lorentz-violating gauged $O(3)$ nonlinear sigma model. We have consistently implemented the Bogomol'nyi-Prasad-Sommerfield (BPS) formalism and obtained the correspondent differential first-order equations describing electrically charged self-dual configurations. The total energy and magnetic flux of the vortices, besides being proportional to the winding number, also depend explicitly on the Lorentz-violating coefficients belonging to the sigma sector. The total electrical charge is proportional to the magnetic f…
We develop the duality transformation rules in two-dimensional theories in the superfield formalism. Even if the chiral superfield which we dualize involves an F-term, we can dualize it by virtue of the property of chiral superfields. We apply the duality transformation rule of the neutral chiral superfield to the ${\cal N}=(4,4)$ gauged linear sigma model for five-branes. We also investigate the duality transformation rule of the charged chiral superfield in the ${\cal N} = (4,4)$ gauged linear sigma model for the $A_1$-type ALE space. In both cases we obtain the dual Lagrangians in the super…
We gauge the (2,2) supersymmetric non-linear sigma model whose target space has bihermitian structure (g, B, J_{\pm}) with noncommuting complex structures. The bihermitian geometry is realized by a sigma model which is written in terms of (2,2) semi-chiral superfields. We discuss the moment map, from the perspective of the gauged sigma model action and from the integrability condition for a Hamiltonian vector field. We show that for a concrete example, the SU(2) x U(1) WZNW model, as well as for the sigma models with almost product structure, the moment map can be used together with the corres…
We determine all the terms that are gauge-invariant up to a total spacetime derivative ("semi-invariant terms") for gauged non-linear sigma models. Assuming that the isotropy subgroup $H$ of the gauge group is compact or semi-simple, we show that (non-trivial) such terms exist only in odd dimensions and are equivalent to the familiar Chern-Simons terms for the subgroup $H$. Various applications are mentioned, including one to the gauging of the Wess-Zumino-Witten terms in even spacetime dimensions. Our approach is based on the analysis of the descent equation associated with semi-invariant ter…
It is shown that a gauged nonlinear $O(3)$ sigma model with anomalous magnetic moment interaction in $2+1$ dimensions is exactly integrable for static, self-dual field configurations. The matter fields are exactly equivalent to those of the usual ungauged nonlinear $O(3)$ sigma model. These static soliton solutions can be mapped into an Abelian purely magnetic vortex solutions through a suitable reduction of the non-Abelian gauge group. A relativistic Abelian model in $2+1$ dimensions is also presented where these purely magnetic vortices can be realized.
We study quantum aspects of the target space of the non-linear sigma model which is a low-energy effective theory of the gauged linear sigma model (GLSM). As such, we especially compute the exact sphere partition function of GLSM for KK$5$-branes whose background geometry is a Taub-NUT space, using the supersymmetric localization technique on the Coulomb branch. From the sphere partition function, we distill the world-sheet instanton effects. In particular, we show that concerning the single-centered Taub-NUT space, the instanton contributions exist only if the asymptotic radius of the $S^1$ f…
We first study a free particle on an $(n-1)$-sphere in an extended phase space, where the originally second-class Hamiltonian and constraints are now in strong involution. This allows for a Schrödinger representation and a Hamilton-Jacobi formulation of the model. We thereby obtain the free particle energy spectrum corresponding to that of a rigid rotator. We extend these considerations to a modified version of the field theoretical O(3) nonlinear sigma model, and obtain the corresponding energy spectrum as well as BRST Lagrangian.
Gauged Double Field Theory, Current Algebras and Heterotic Sigma Models
本文从流代数和σ模型角度研究规范双场论中的$O(D,D+n)$广义度规与规范对称性，证明其与杂化弦σ模型在$α'$领头阶一致，并揭示杂化弦非阿贝尔规范对称性及流代数的正确结构。
$U(1)$ gauged non-topological solitons in the 3+1 dimensional $O(3)$ sigma-model
该研究在3+1维平直时空的$U(1)$规范非线性$O(3)$ sigma模型中，发现并分析了带电磁场、非零角动量密度的新型非拓扑孤子，明确了其性质与存在范围。
We study a broad class of two dimensional gauged linear sigma models (GLSMs) with off-shell N=(2,2) supersymmetry that flow to nonlinear sigma models (NLSMs) on noncompact geometries with torsion. These models arise from coupling chiral, twisted chiral, and semichiral multiplets to known as well as to a new N=(2,2) vector multiplet, the constrained semichiral vector multiplet (CSVM). We discuss three kinds of models, corresponding to torsionful deformations of standard GLSMs realizing Kahler, hyperkahler, and Calabi-Yau manifolds. The (2,2) supersymmetry guarantees that these spaces are genera…
We investigate the existence of self-dual solitons with internal structure in a gauged $O(3)$ nonlinear sigma model immersed in a dielectric medium generated by a real scalar field (dubbed the source field). We consider rotationally symmetric configurations and applying the {Bogomol'nyi-Prasad-Sommerfield} formalism to obtain the energy lower bound and the respective {first-order differential equations (or self-dual equations).} By solving such a system of equations for three different dielectric media, we find the internal structure generates relevant changes in the soliton profiles when comp…
We discuss special perturbations of the gauged level $k$ WZNW model inspired by the $σ$-model perturbation of the nonunitary WZNW model. In the large $k$ limit there is a second conformal point in the vicinity of the ultarviolet fixed point. At the second critical point the conformal model has a rational Virasoro central charge which no longer corresponds to a coset construction. In spite of this fact the perturbative conformal model appears to be a unitary system as long as the underlying coset construction at the ultraviolet critical point is unitary.
We study gauged linear sigma models for noncompact Calabi-Yau manifolds described as a line bundle on a hypersurface in a projective space. This gauge theory has a unique phase if the Fayet-Iliopoulos parameter is positive, while there exist two distinct phases if the parameter is negative. We find four massless effective theories in the infrared limit, which are related to each other under the Calabi-Yau/Landau-Ginzburg correspondence and the topology change. In the T-dual theory, on the other hand, we obtain two types of exact massless effective theories: One is the sigma model on a newly ob…
(0,2) gauged linear sigma models with torsion, corresponding to principal torus bundles over warped CY bases, provide a useful framework for getting exact statements about perturbative dualities in the presence of fluxes. In this context we first study dualities mapping the torus fiber onto itself, implying the existence of quantization constraints on the torus moduli for consistency. Second, we investigate dualities mixing the principal torus bundle with the gauge bundle, relating the torsional GLSMs to ordinary ones corresponding to CY compactifications with non-standard embeddings, namely t…
The local BRST cohomology of the gauged non-linear sigma model on a group manifold is worked out for any Lie group G. We consider both, the case where the gauge field is dynamical and the case where it has no kinetic term (G/G topological theory). Our results shed a novel light on the problem of gauging the WZW term as well as on the nature of the topological terms introduced a few years ago by De Wit, Hull and Rocek. We also consider the BRST cohomology of the rigid symmetries of the ungauged model and recover the results of D'Hoker and Weinberg on the most general effective actions compatibl…
We propose two kinds of gauged linear sigma models whose moduli spaces are real eight-dimensional hyperKahler and Calabi-Yau manifolds, respectively. Here, hyperKahler manifolds have sp(2) holonomy in general and are dual to Type IIB (p,q)5-brane configurations. On the other hand, Calabi-Yau fourfolds are toric varieties expressed as quotient spaces. Our model involving fourfolds is different from the usual one which is directly related to a symplectic quotient procedure. Remarkable features in newly-found three-dimensional Chern-Simons-matter theories appear here as well, such as dynamical Fa…
The conditions under which a general two-dimensional non-linear sigma model is classically integrable are given. These requirements are found by demanding that the equations of motion of the theory are expressible as a zero curvature relation. Some new integrable two-dimensional sigma models are then presented.
This is a write-up of lectures on integrable sigma-models, which covers the following topics: (1) Homogeneous spaces, (2) Classical integrability of sigma-models in two dimensions, (3) Topological terms, (4) Background-field method and beta-function, (5) S-matrix bootstrap in the O(N) model, (6) Supersymmetric cosets and strings on AdS(d)xX.
The classical integrability the O(N) nonlinear sigma model on a half-line is examined, and the existence of an infinity of conserved charges in involution is established for the free boundary condition. For the case N=3 other possible boundary conditions are considered briefly.
We show that the massive Yang--Mills action having as a mass term the non-local operator introduced by Gubarov, Stodolsky, and Zakharov is classically equivalent to a principal gauged sigma model. The non-local mass corresponds to the topological term of the sigma model. The latter is obtained once the degrees of freedom implicitly generated in the non-local action are explicitly implemented as group elements. The non-local action is recovered by integrating out these group elements. In contrast to the usual gauge-fixed treatment, the sigma model point of view provides a safe framework in whic…
We investigate toric GLSMs as models for tachyon condensation in type II strings on space-time non-supersymmetric orbifold singularities. The A-model correlators in these theories satisfy a set of relations related to the topology of the resolved orbifold. Using these relations we compute the correlators and find a non-trivial chiral ring in the IR, which we interpret as supported on isolated Coulomb vacua of the theory.
We have studied the existence of topological Bogomol'nyi-Prasad-Sommerfield or self-dual configurations in a Lorentz-violating gauged $O(3)$ nonlinear sigma model, where $CPT$-even Lorentz-violating (LV) terms were introduced in both the gauge and $σ$% -field sectors. As happens in the usual gauged $σ$ model, purely magnetic self-dual configurations are allowed, maintaining some qualitative features of the standard ones. In a more involved configuration, Lorentz violation provides new self-dual magnetic solutions carrying an electric field but a null total electric charge. In both cases, the t…
We study two-dimensional ${\cal N}=(4,4)$ gauged linear sigma model (GLSM). Its low energy effective theory is a nonlinear sigma model whose target space gives rise to a configuration of five-branes in string theory. In this article we focus on sigma models for NS5-branes, KK5-branes and an exotic $5^2_2$-brane. In particular, we carefully analyze the GLSM for an exotic $5^2_2$-brane whose background configuration is multi-valued. The exotic $5^2_2$-brane is a concrete example of nongeometric configuration in string theory. We find that the exotic feature originates from the string winding coo…
Stress stability criterion of $U(1)$ gauged non-topological solitons in the 3+1 dimensional O(3) sigma-model
本文研究了3+1维O(3)西格玛模型中非拓扑孤立子的应力稳定性准则，分析了能量密度分布和稳定性条件。
The scale invariance of the $O(3)$ sigma model can be broken by gauging a $U(1)$ subgroup of the $O(3)$ symmetry and including a Maxwell term for the gauge field in the Lagrangian. Adding also a suitable potential one obtains a field theory of Bogomol'nyi type with topological solitons. These solitons are stable against rescaling and carry magnetic flux which can take arbitrary values in some finite interval. The soliton mass is independent of the flux, but the soliton size depends on it. However, dynamically changing the flux requires infinite energy, so the flux, and hence the soliton size, …
We construct and study a two parameter gauged linear sigma model with gauge group $(U(1)^2\times O(2))/{\mathbb Z}_2$ that has a dual model with gauge group $(U(1)^2\times SO(4))/{\mathbb Z}_2$. The model has two geometric phases, three hybrid phases and one phase whose character is unknown. One of the geometric phases is strongly coupled and the other is weakly coupled, where strong versus weak is exchanged under the duality. They correspond to two Calabi-Yau manifolds with $(h^{1,1},h^{2,1})=(2,24)$ which are birationally inequivalent but are expected to be derived equivalent. A region of th…
The purpose of this Letter is to present a computation of the interaction energy of gauged O(3) Chern-Simons vortices which are infinitely separated. The results will show the behaviour of the interaction energy as a function of the constant coupling the potential, which measures the relative strength of the matter self-coupling and the electromagnetic coupling. We find that vortices attract each other for $λ> 1 $ and repel when $λ< 1 $. When $λ=1 $ there is a topological lower bound on the energy. It is possible to saturate the bound if the fields satisfy a set of first order partial differen…
We construct yet another ${\mathcal N}=(4,4)$ gauged linear sigma model for the $A_N$-type ALE space. In our construction the toric data of the ALE space are manifest. Due to the $SU(2)_R$ symmetry, the F-term is automatically determined. The toric data, which govern the Kähler structures of the ALE space, are embedded into $U(1)$ charges of charged hypermultiplets. The F-term is also inevitable to determine the complex structures of the ALE space. In the IR limit, we obtain the Kähler potential of the $A_N$-type ALE space. We also find the origin of the ${\mathbb Z}_{N+1}$ orbifold symmetry i…
We study two-dimensional $\mathcal{N}=(2,2)$ supersymmetric gauged linear sigma models (GLSM) on the $Ω$-deformed sphere, $S^2_Ω$, which is a one-parameter deformation of the $A$-twisted sphere. We provide an exact formula for the $S^2_Ω$ supersymmetric correlation functions using supersymmetric localization. The contribution of each instanton sector is given in terms of a Jeffrey-Kirwan residue on the Coulomb branch. In the limit of vanishing $Ω$-deformation, the localization formula greatly simplifies the computation of $A$-twisted correlation functions, and leads to new results for non-abel…
We construct integrability preserving boundary conditions for Green-Schwarz sigma-models on semi-symmetric spaces. The boundary conditions are expressed as gluing conditions of the flat-connection, using an involutive metric preserving automorphism. We show that the boundary conditions preserve half of the space-time supersymmetry and an infinite set of conserved charges. We find integrable D-brane configurations for AdS_5 x S^5 and AdS_4 x CP^3 backgrounds.
Local symmetries is one of the most successful themes in modern theoretical physics. Although they are usually associated to Lie algebras, a gradual increase of interest in more general situations where local symmetries are associated to groupoids and algebroids has taken place in recent years. On the other hand, dualities is another persistently interesting theme in modern physics. One of the most prominent examples is provided by target space duality in string theory. The latter, Abelian or not, is usually associated to the presence of isometries, which is however a very restrictive assumpti…
Boson stars in the $U(1)$ gauged 3+1 dimensional $O(3)$ sigma-model
本文研究3+1维$U(1)$规范$O(3)$sigma模型中的自引力玻色子星，分析带电与旋转解、类螺旋频率依赖及质量阈值行为，并确定其存在域与物理性质。
We construct (0,2), D=2 gauged linear sigma model on a supermanifold in both the Abelian gauge group and the non-Abelian gauge group. The $\hat{U}$ operator provides consistency conditions for satisfying the SUSY invariance. Contrary to the Abelian gauge group, it is not essential to introduce the new operator in order to check the exact SUSY invariance of the Lagrangian density. However, in order to introduce the (0,2) chiral superfields, we need the $\hat{U}$ operator, because we can not define the (0,2) chirality conditions of the (0,2) chiral superfields without introducing the new operato…
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