International Journal of Modern Physics A

In this paper, we briefly survey some developments in gauged linear sigma models (GLSMs). Specifically, we give an overview of progress on constructions of GLSMs for various geometries, GLSM-based computations of quantum cohomology, quantum sheaf cohomology, and quantum K theory rings, as well as two-dimensional abelian and nonabelian mirror constructions.

Brane transport provides a way to realize functors between the categories of B-branes of different phases in gauged linear sigma model (GLSM). When appropriately designed, these functors induce derived equivalence between Calabi–Yau manifolds. In some cases, brane transport can also be used to extract the homological projective dual (HPD) of certain projective embeddings. We describe the GLSMs realizing derived equivalence between different geometries via brane transport with emphasis on the nonabelian models, and how to construct the GLSMs where brane transport gives rise to the embedding of HPD category into the derived category of universal hyperplane section. The latter gives rise to an explanation of the relationship between GLSM and HPD, and also a noncommutative geometric interpretation to HPD.

Our goal in this paper is to give a brief survey of recent developments in the study of M-theory on singular Calabi–Yau three-folds, topological strings in the presence of a flat but topologically nontrivial B-field and their relationship to hybrid phases of certain Gauged Linear Sigma Models (GLSM).

In this paper, we discuss a simple model for D-brane transport in nonabelian GLSMs. The model is the elliptic curve version of a nonabelian GLSM introduced by Hori and Tong and has gauge group $U(2)$. It has two geometric phases, both of which describe the same elliptic curve, once realized as a codimension five complete intersection in $G(2,5)$ and once as a determinantal variety. The determinantal phase is strongly coupled with unbroken $\text{SU}(2)$. There are two singular points in the moduli space where the theory has a Coulomb branch. Using grade restriction rules, we show how to transport B-branes between the two phases along paths avoiding the singular points. With the help of the GLSM hemisphere partition function, we compute analytic continuation matrices and monodromy matrices, confirming results obtained by different methods.

We discuss the realization of $2d$ $(0,2)$ gauge theories in terms of branes focusing on Brane Brick Models, which are T-dual to D1-branes probing toric Calabi–Yau 4-folds. These brane setups fully encode the infinite class of $2d$ $(0,2)$ quiver gauge theories on the worldvolume of the D1-branes and substantially streamline their connection to the probed geometries. We review various methods for efficiently generating Brane Brick Models. These algorithms are then used to construct $2d$ $(0,2)$ gauge theories for the cones over all the smooth Fano 3-folds and two infinite families of Sasaki–Einstein 7-manifolds with known metrics.

In this paper, we describe non-Abelian T-dualities for a two-dimensional (2D) gauged linear sigma model (GLSM) with $\mathcal{𝒩}=2$ supersymmetry (SUSY). We start with the case of left and right SUSY $(2,2)$, $U(1)$ gauge group, and global non-Abelian symmetries. Our analysis applies to the specialization of the GLSM with the global group $SU(2)\times SU(2)$, whose original model is the resolved conifold. We analyze the dual model and compute the periods on the dual geometry, matching the effective potential for the $U(1)$ gauge field, and discussing the coincident singularity at the conifold point. We further present the non-Abelian T-dualization of models with $(0,2)$ SUSY, analyzing the case of global symmetry $SU(2)$. In this case, the full non-Abelian duality can be solved. The dual geometries with and without nonperturbative corrections to the superpotential coincide. The structure of the dual nonperturbative corrections is determined based on symmetry arguments. The non-Abelian T-dualities reviewed here lead to potential new symmetries between physical theories.

Heterotic toriodal ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ orbifolds may possess discrete torsion between the two defining orbifold twists in the form of additional cocycle factors in their one-loop partition functions. Using Gauged Linear Sigma Models (GLSMs) the consequences of discrete torsion can be uncovered when the orbifold is smoothed out by switching on appropriate blowup modes. Here blowup modes with twisted oscillator excitations are chosen to reproduce bundles that are close to the standard embedding without torsion. The orbifold resolutions with discrete torsion are distinguished from resolutions without torsion, since they require NS5-branes at their exceptional cycles.

In this paper, we study supersymmetric Berry connections of 2d $\mathcal{𝒩}=(2,2)$ gauged linear sigma models (GLSMs) quantized on a circle, which are periodic monopoles, with the aim to provide a fruitful physical arena for recent mathematical constructions related to the latter. These are difference modules encoding monopole solutions via a Hitchin–Kobayashi correspondence established by Mochizuki. We demonstrate how the difference modules arise naturally by studying the ground states as the cohomology of a one-parameter family of supercharges. In particular, we show how they are related to one kind of monopole spectral data, a quantization of the Cherkis–Kapustin spectral curve, and relate them to the physics of the GLSM. By considering states generated by D-branes and leveraging the difference modules, we derive novel difference equations for brane amplitudes. We then show that in the conformal limit, these degenerate into novel difference equations for hemisphere partition functions, which are exactly calculable. When the GLSM flows to a nonlinear sigma model with Kähler target X, we show that the difference modules are related to the equivariant quantum cohomology of X.

In this paper, we apply two-dimensional supersymmetric gauge theories to directly construct a new Bethe ansatz for the wave functions of the q-boson hopping model, and then derive the q-boson algebras from this ansatz.

In this paper, we explore some recent advancements in enumerative algebraic geometry, focusing particularly on the role of quantum K-theory of quiver varieties as viewed through the lens of integrable systems. We highlight a number of conjectures and open questions.

In this paper, we review some new results we recently obtained about the infrared physics of 3d $\mathcal{𝒩}=2$ SQCD with a unitary gauge group, in particular in the presence of a nonzero Fayet–Iliopoulos parameter and with generic values of the Chern–Simons levels. We review the 3d GLSM (also known as 3d A-model) approach to the computation of the 3d $\mathcal{𝒩}=2$ twisted chiral ring of half-BPS lines. For particular values of the Chern–Simons levels, this twisted chiral ring has a neat interpretation in terms of the quantum K-theory (QK) of the Grassmannian manifold. We propose a new set of line defects of the 3d gauge theory, dubbed Grothendieck lines, which represent equivariant Schubert classes in the QK ring. In particular, we show that double Grothendieck polynomials, which represent the equivariant Chern characters of the Schubert classes, arise physically as Witten indices of certain quiver supersymmetric quantum mechanics. We also explain two distinct ways how to compute K-theoretic enumerative invariants using the 3d GLSM approach.

Based on the observed behavior of the superconformal index in three-dimensional ${𝒩}=2$ theories, we propose a quantity that can be considered as an analogue of the “effective central charge.” We discuss the general properties of this quantity and ways of computing it in a variety of different theories, including simple Lagrangian theories as well as more interesting strongly coupled examples that come from 3d-3d correspondence.

A homology class $d\in {\mathrm{\text{H}}}_2(X,\mathbb{Z})$ of a complex flag variety $X=G∕P$ is called a line degree if the moduli space ${\overline{\mathcal{ℳ}}}_{0,0}(X,d)$ of 0-pointed stable maps to X of degree d is also a flag variety $G∕P^{\prime }$. We prove a quantum equals classical formula stating that any n-pointed (equivariant, $\mathrm{\text{K}}$-theoretic, genus zero) Gromov–Witten invariant of line degree on X is equal to a classical intersection number computed on the flag variety $G∕P^{\prime }$. We also prove an n-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov–Witten invariants of the variety of complete flags $G∕B$. Our formulas make it straightforward to compute the big quantum $\mathrm{\text{K}}$-theory ring ${\mathrm{\text{QK}}}^{\mathrm{\text{big}}}(X)$ modulo the ideal $〈Q^d〉$ generated by degrees d larger than line degrees.

We give an alternate proof of the integrality conjecture of genus zero Gopakumar–Vafa type invariants on semi-positive varieties using algebraic geometry. The main technique is to relate Gopakumar–Vafa type invariants to quantum K-invariants and to utilize the integrality of the latter.

Sphere partition function integral for abelian Calabi–Yau GLSM is equal to the pairing of the I-function with its complex conjugate, where the real structure is essentially constructed from Iritani’s integral structure on the cohomology of the target.

We study the problem of computing Gopakumar–Vafa (GV) invariants for multiparameter families of symmetric Calabi–Yau threefolds admitting flops to diffeomorphic manifolds. There are infinite Coxeter groups, generated by permutations and flops, that act as symmetries on the GV-invariants of these manifolds. We describe how these groups are related to symmetries in GLSMs, and the existence of multiple mirrors. Some representation theory of these Coxeter groups is also discussed. The symmetries provide an infinite number of relations between the GV-invariants for each fixed genus. This remarkable fact is of assistance in obtaining higher-genus invariants via the BCOV recursion. This paper is based on joint work with Philip Candelas and Xenia de la Ossa.

In this paper, we survey the developments of the symplectic aspects about the gauged linear sigma model, invented by Witten 30 years ago.