A survey of some recent developments in GLSMs

1. Introduction

Gauged linear sigma models (GLSMs) were first described 30 years ago.¹ They quickly became vital tools in string compactifications, still used and developed today. The goal of this paper (and the corresponding talk at the workshop GLSMs@30) is to briefly survey some of the developments and current research areas in GLSMs. To be clear, there is not enough space to describe, much less give justice to, everything that has been developed or is being researched, but we do hope to outline many areas, and will reference related talks that took place at GLSMs@30.

2. Constructions of Geometries

Originally, GLSMs were used to give physical realizations of geometries of the form of complete intersections in symplectic quotients ${ℂ}^n∕∕G$. Briefly, the idea is to realize ${ℂ}^n∕∕G$ as a two-dimensional supersymmetric G-gauge theory with matter fields corresponding to ${ℂ}^n$, plus additional matter and a superpotential for which the complete intersection is the critical locus.

For example, to describe a hypersurface $\{G=0\}\subset {ℂ}^n∕∕G$, one starts with a gauge theory describing ${ℂ}^n∕∕G$, and adds a chiral superfield p and a superpotential $W=pG$, where p is chosen to transform under the action of G in such a way that W is gauge-invariant. If the hypersurface is smooth, then the critical locus reduces to

which is the desired hypersurface in ${ℂ}^n∕∕G$. We will refer to this as a “perturbative” description.

Nowadays we know of two alternative mechanisms that can be used to realize geometries :

•	Strong coupling effects in two-dimensional gauge theories can restrict the space of vacua. The prototype for this is the GLSM for the Grassmannian–Pfaffian system.2
•	Decomposition3,4 locally realizes a branched cover. Prototypes for this are GLSMs relating complete intersections of quadrics to branched covers.5

Let’s quickly walk through each of these in turn.

First, we consider nonperturbative constructions of Pfaffians.² The prototypical example is the GLSM for the complete intersection of seven hyperplanes in the Grassmannian $G(2,7)$, which is denoted by $G(2,7)[1^7]$. This GLSM is a $U(2)$ gauge theory with seven fundamentals ${\varphi }_i^a$ plus seven chiral superfields denoted by $p_{\alpha }$ which are charged under $detU(2)$, with a superpotential

For $r\gg 0$, this GLSM describes $G(2,7)[1^7]$, by the usual analysis. For $r\ll 0$, the analysis of this GLSM utilizes results from the strongly coupled gauge theory. Working locally in a Born–Oppenheimer approximation along the space of vevs of the $p_{\alpha }$ fields,

•	loci with one massless doublet (generic case) have no SUSY vacua,
•	loci with three massless doublets have one SUSY vacuum.

The resulting theory, the loci with three massless doublets, describes a Pfaffian variety inside the projective space ${\mathbb{P}}^6$ defined by the $p_{\alpha }$.

Next, we turn to nonperturbative constructions of branched covers.⁵ A simple example involves the GLSM for ${\mathbb{P}}^3[2,2]$. This is a $U(1)$ gauge theory, with four chiral multiplets ${\varphi }_i$ of charge $+1$, two chiral multiplets $p_{\alpha }$ of charge $-2$, and a superpotential

For $r\gg 0$, this describes ${\mathbb{P}}^3[2,2]=T^2$, by the usual analysis. For $r\ll 0$, working locally in a Born–Oppenheimer approximation on the space of vevs of the $p_{\alpha }$ fields, which is ${\mathbb{P}}^1$, the $S^{ij}$ acts as a mass matrix for the charge 1 fields ${\varphi }_i$. To correctly analyze this phase, we must use the fact that at low energies, the gauge theory (generically) has a trivially acting ${\mathbb{Z}}_2\subset U(1)$, hence a ${\mathbb{Z}}_2$ one-form symmetry, and so by decomposition,^3,4 is (generically) a double cover, away from the locus $\{detS=0\}$, where some of the ${\varphi }_i$ become massless. The resulting geometry is a double cover of ${\mathbb{P}}^1$ (the space of vevs of the $p_{\alpha }$), branched over a degree-four locus ($\{detS=0\}$), which is another $T^2$.

The GLSM for ${\mathbb{P}}^5[2,2,2]=K3$ can be analyzed very similarly. The $r\ll 0$ phase is a branched double cover of ${\mathbb{P}}^2$, branched over a degree 6 locus, which is another $K3$.

Starting in 3-folds, these examples become more interesting. The GLSM for ${\mathbb{P}}^7[2,2,2,2]$ describes a noncommutative resolution of a branched double cover, defined^6,7,8 in terms of derived categories. In particular, the GLSM gives a UV representation of a closed string CFT for a noncommutative resolution. The noncommutative structure is detected physically by studying matrix factorizations in (hybrid) Landau–Ginzburg phases — in other words, by examining D-branes.

These noncommutative resolutions were discussed elsewhere at this meeting, in talks of S. Katz, T. Schimannek, M. Romo, and J. Guo.

Another property of these 3-fold examples (both the Grassmannian/Pfaffian and the branched covers) is that the different GLSM phases are not birational to one another. This contradicted folklore of the time, which said that all (geometric) phases of a single GLSM should be birational. Instead, these phases are related by homological projective duality.^6,7,8 This has been studied in this context in mathematics, in variations of GIT quotients, see for example Refs. 9,10,11,12,13,14,15. Homological projective duality is beyond the scope of this overview, but was discussed elsewhere at this meeting, in talks of J. Guo and M. Romo.

Nowadays, we can also realize similar effects perturbatively. For example, Pfaffians can be described via the PAX and PAXY models.¹⁶ Perturbative and nonperturbative constructions can be exchanged by dualities, see e.g. Ref. 17.

3. Quantum Cohomology and Two-Dimensional Mirrors

One of the original applications of GLSMs was to make predictions for quantum cohomology rings of Fano toric varieties. For such spaces, we can use the GLSM to replace counting rational curves with an algebraic computation, on the Coulomb branch, that encodes the same result. In particular, quantum cohomology can be seen in a Coulomb branch computation. For example, under RG flow, the GLSM for ${\mathbb{P}}^n$ describes a space that shrinks to (classical) zero size, and then onto the Coulomb branch, where quantum cohomology is described as the classical critical locus of a twisted one-loop effective superpotential, instead of as a sum over rational curves.

For Fano symplectic quotients ${ℂ}^n∕∕G$ for $G=U{(1)}^k$, the twisted one-loop effective superpotential is of the form¹⁸

and the resulting critical locus $\{\partial \tilde{W}∕\partial {\sigma }_a=0\}$ is given by¹⁸ If the theory in the IR is a pure Coulomb branch, then these are the quantum cohomology relations.

To make this more concrete, let us specialize to ${\mathbb{P}}^n$. Under RG flow, the GLSM for ${\mathbb{P}}^n$ describes a space that shrinks to (classical) zero size, and then onto the Coulomb branch. The one-loop twisted effective superpotential is

which has critical locus given by the solution to namely This is precisely the well-known quantum cohomology ring relation for ${\mathbb{P}}^n$, identifying $\sigma$ with a generator of $H^2({\mathbb{P}}^n)$.

The same ideas also apply to nonabelian GLSMs, meaning, GLSMs describing spaces of the form ${ℂ}^n∕∕G$ for nonabelian G (and subvarieties thereof). For Fano ${ℂ}^n∕∕G$, RG flow again drives the GLSM out of a geometric phase and onto the Coulomb branch. Again the quantum cohomology ring arises as the critical locus of a superpotential, albeit with two subtleties :

•	The Coulomb branch is a Weyl-group orbifold of the $\sigma$’s.
•	The Coulomb branch is an open subset of the space of $\sigma$’s — an “excluded locus” is removed.

To make this discussion concrete, we turn to the example of the Grassmannian $G(k,n)$ of k-planes in ${ℂ}^n$. This can be described as the symplectic quotient ${ℂ}^{kn}∕∕U(k)$, where $U(k)$ acts as n copies of the fundamental representation. Here, the twisted one-loop effective superpotential is

using the fact that $Q_{ia}^b={\delta }_b^a$ for copies of the fundamental representation. In principle, the space of $\sigma$’s is orbifolded by the Weyl group of $U(k)$ (namely, the symmetric group $S_k$), which acts by interchanging the ${\sigma }_a$, and we also remove the “excluded locus” $\{{\sigma }_a={\sigma }_b,a\ne b\}$. The critical locus is computed from which implies It may not yet be manifest, but this defines the quantum cohomology ring relation for $G(k,n)$.

As a quick consistency check, we compute the number of vacua. The relation above is an order n polynomial, so for each value of a, there are k solutions, hence $kn$ possible values altogether. Taking into account the $S_k$ orbifold and the excluded locus, the number of admissible solutions to the critical locus equation is

as expected.

To make the relation to the quantum cohomology ring of the Grassmannian more clear, we can rewrite the critical locus equation (12) as follows. First, note that the ${\sigma }_a$ are k distinct roots of the nth order polynomial

Let ${\overline{\sigma }}_{a^{\prime }}$ denote the remaining $n-k$ roots. From Vieta’s theorem in algebra, the elementary symmetric polynomials $e_i$ in the ${\sigma }_a$ and ${\overline{\sigma }}_{a^{\prime }}$ obey Define and similarly for $\overline{\sigma }$, and then the result above from Vieta’s theorem can be written as which is a standard expression for the quantum cohomology ring of $G(k,n)$, see e.g. Eq. (3.16) in Ref. 19, where we interpret $c_t(\sigma )$ as the total Chern class of the universal subbundle S on $G(k,n)$, and $c_t(\overline{\sigma })$ as the total Chern class of the universal quotient bundle Q.

So far we have reviewed Coulomb-branch-based quantum cohomology computations in GLSMs. Another approach to these and related questions is to use mirror symmetry, which we will review next.

First, we will quickly review abelian mirrors.^20,21 Briefly, start with a $U{(1)}^r$ gauge theory with matter multiplets of charges ${\rho }_i^a$, corresponding to a quotient ${ℂ}^n∕∕U{(1)}^r$. The mirror is a Landa–Ginzburg model, defined by the chiral superfields

•	${\sigma }_a$, $a\in \{1,\dots ,r\}$, ${\sigma }_a={\overline{D}}_{+}{D}_{-}{V}_a$,
•	$Y^i$, mirror to the matter fields of the original theory, with periodicities $Y^i\sim Y^i+2\pi i$,

with superpotential

Next, we turn to mirrors to ${ℂ}^n∕∕G$ for G nonabelian.²² Here, we pick a Cartan torus $U{(1)}^r\subseteq G$, r the rank of G, and let $\rho$ defining the representation of G under which the matter multiplets transform. The mirror is a Weyl-group-orbifold of the Landau–Ginzburg model defined by the fields

•	${\sigma }_a$, $a\in \{1,\dots ,r\}$, ${\sigma }_a={\overline{D}}_{+}{D}_{-}{V}_a$,
•	$Y^i$, mirror to the matter fields of the original theory,
•	$X_{\tilde{\mu }}$, in one-to-one correspondence with the nonzero roots of $𝔤$,

and superpotential

where ${\rho }_i$ is a weight vector, and ${\alpha }_{\tilde{\mu }}$ is a root vector. In brief, the idea of the nonbelian mirror is that it is abelian mirror symmetry in the Cartan torus, at a generic point on the Coulomb branch.

In principle, both these mirrors have the property that correlation functions in the original A-twisted GLSM are the same as correlation functions in the B-twisted Landau–Ginzburg mirror. We can derive a mirror map for operators from the critical loci of the superpotential (19). From $\partial W∕\partial X_{\tilde{\mu }}=0$, we get

and from $\partial W∕\partial Y^i=0$, we get In both of these critical locus equations, the left-hand side can be interpreted in the B-twisted mirror, and the right-hand side can be interpreted in the original A-twisted GLSM.

Now, let us work through two examples. As before, we begin with the GLSM for ${\mathbb{P}}^n$. The mirror²⁰ is a Landau–Ginzburg model with superpotential

We can integrate out $\sigma$ and $Y^{n+1}$ to write where $q=exp(-t)$. The critical locus is computed from which implies so if we define $X=exp(-Y^i)$, then the ring relation in the quantum cohomology ring for ${\mathbb{P}}^n$.

Next, we turn to the Grassmannian $G(k,n)$. Here, the mirror²² is the $S_k$ orbifold of a Landau–Ginzburg model with superpotential

where We integrate out ${\sigma }_a$, $Y^{na}$ to obtain where

Next, we compute the critical locus. From

we find for all i. Similarly, from we find hence The operator mirror map is so the critical locus equation (35) recovers the expression for the ring relation in the quantum cohomology ring of $G(k,n)$ described earlier; in other words, becomes

Also, poles in the superpotential at $X_{\mu \nu }=0$ correspond to the excluded locus

for $\mu \ne \nu$.

On a related matter, there was a talk at the meeting on nonabelian T-duality by N. Cabo Bizet.

In passing, we would also like to mention two other important topics, which lack of space prevents us from describing in more detail :

•

Supersymmetric localization. Supersymmetric localization was first applied to two-dimensional GLSMs in Refs. 23 and 24, to our knowledge, and was quickly applied to give alternative physical computations of Gromov–Witten invariants,25 elliptic genera,26,27 and Gamma classes.28,29,30,31,32 These are important contributions, which we wanted to acknowledge, but lack of space prevents us from going into any detail.

•

D-branes in GLSMs. GLSMs on open strings were explored in detail in Ref. 33, which described e.g. the grade restriction rule. There is no space in this overview to explain any details, but this was discussed at the meeting in talks by I. Brunner, K. Hori, J. Guo, and K. Aleshkin.

4. Quantum Sheaf Cohomology

So far we have reviewed progress in GLSMs for two-dimensional theories with (2,2) supersymmetry. There also exist GLSMs for two-dimensional theories with (0,2) supersymmetry.^1,34,35 Briefly, in geometric phases, these describe a space X, along with a holomorphic vector bundle $\mathcal{ℰ}\to X$, obeying the constraint

These theories admit analogues^36,37 of the A, B model topological twists³⁸ :

•	The analogue of the A twist, known as the A/2 model, exists when $det{\mathcal{ℰ}}^{\ast }\cong K_X$, and has operators corresponding to elements of $H^{•}(X,{\wedge }^{•}{\mathcal{ℰ}}^{\ast })$.
•	The analogue of the B twist, known as the B/2 model, exists when $det\mathcal{ℰ}\cong K_X$, and has operators corresponding to elements of $H^{•}(X,{\wedge }^{•}\mathcal{ℰ})$.

These theories have (0,2) supersymmetry and reduce to the ordinary A, B models in the special case that $\mathcal{ℰ}=TX$.

The OPEs of local operators in these theories also describe generalizations of quantum cohomology, known as quantum sheaf cohomology, see e.g. Refs. 36, 37 and 39. We outline the details here.

First, recall that local operators in the ordinary A model with target space X correspond to elements of $H^{•,•}(X)=H^{•}(X,{\wedge }^{•}{T}^{\ast }X)$, and correlation functions are computed mathematically by intersection theory on a moduli space of curves.

Quantum sheaf cohomology^36,37,39 arises from an A/2-twisted theory, with target space X and bundle $\mathcal{ℰ}$. Local operators correspond to elements of $H^{•}(X,{\wedge }^{•}{\mathcal{ℰ}}^{\ast })$. These have a classical product

Correlation functions are computed by sheaf cohomology on a moduli space of curves, and the resulting local operator OPEs describe a deformation of the classical product structure above. This reduces to ordinary quantum cohomology in the special case that $\mathcal{ℰ}=TX$.

To be concrete, we outline a family of examples on ${\mathbb{P}}^1\times {\mathbb{P}}^1$. First, recall that the ordinary quantum cohomology ring is

Now, to define quantum sheaf cohomology, we must define a suitable bundle $\mathcal{ℰ}$. Take $\mathcal{ℰ}$ to be a deformation of the tangent bundle, described as the cokernel where for A, B, C, D constant $2\times 2$ matrices (subject to obvious nondegeneracy constraints) and w, z column vectors of homogeneous coordinates on either ${\mathbb{P}}^1$ factor. Then, the quantum sheaf cohomology ring of ${\mathbb{P}}^1\times {\mathbb{P}}^1,\mathcal{ℰ}$ is given by^40,41,42,43 When for example $A=D=I,B=C=0$, then $\mathcal{ℰ}=TX$ and the quantum sheaf cohomology ring (46) reduces to the ordinary quantum cohomology ring (43).

One way to compute quantum sheaf cohomology, for Fano spaces, is using GLSMs and Coulomb branches.^40,41 The basic idea is the same as in (2,2) supersymmetry: under RG flow, the GLSM flows onto a Coulomb branch where the OPE ring relations can be computed as the critical locus of a twisted one-loop effective superpotential.

In abelian cases, the resulting twisted superpotential is of the form

where $M_i({\sigma }_a)$ are matrices encoding tangent bundle deformations, and ${\mathrm{\Upsilon }}_a$ is a (0,2) Fermi superfield (part of the (2,2) vector multiplet). The critical locus equations are which imply

We have already discussed ${\mathbb{P}}^1\times {\mathbb{P}}^1$ examples, for which the quantum sheaf cohomology ring relations are

the same form as (49).

Another example is the Grassmannian $G(k,n)$. Let $\mathcal{ℰ}$ be a deformation of the tangent bundle, defined by the cokernel

where

Then, the quantum sheaf cohomology ring relations are^44,45

which for $\mathcal{ℰ}=TX$ reduce to which defines the ring relation of the ordinary quantum cohomology ring of $G(k,n)$, as discussed previously.

Quantum sheaf cohomology is now known for

•	Fano toric varieties,40,41,42,43
•	Grassmannians,44,45
•	flag manifolds,46

all with $\mathcal{ℰ}$ given by a deformation of the tangent bundle. (Sheaf cohomology on toric complete intersections has also been discussed.⁴⁷) More general cases are open questions.

There is also a notion of mirror symmetry for (0,2) supersymmetric theories, known as (0,2) mirror symmetry. Just as the original form of mirror symmetry relates pairs of Calabi–Yau’s X, Y, (0,2) mirror symmetry relates pairs $(X,\mathcal{ℰ})$, $(Y,\mathcal{ℱ})$, where X, Y are Calabi–Yau (not necessarily mirror in the ordinary sense) and $\mathcal{ℰ}\to X$, $\mathcal{ℱ}\to Y$ are holomorphic bundles such that

The twisted theories are closely related : which, for $\mathcal{ℰ}=TX$, $\mathcal{ℱ}=TY$, reduces to the standard relation between the ordinary A, B models on mirrors, and the standard relation between Hodge diamonds.

(0,2) mirror symmetry has been studied for many years. For example, numerical evidence was described in Ref. 48. There are (limited) proposals for mirror constructions, see e.g. Refs. 48,49,50,51,52.

For (0,2) GLSMs describing Fano spaces, (limited) proposals exist for (0,2) mirrors as (0,2) Landau–Ginzburg models. Consider for example the case of ${\mathbb{P}}^1\times {\mathbb{P}}^1$, with bundle $\mathcal{ℰ}$ given as the cokernel

where as before. If we restrict to diagonal matrices A, B, C, D, then a mirror (0,2) Landau-Ginzburg model is defined by where ${\mathrm{\Upsilon }}_i$, $F_i$, ${\tilde{\mathrm{\Upsilon }}}_j$, ${\tilde{F}}_j$ are (0,2) Fermi superfields, parts of (2,2) $\sigma$ and Y multiplets.

There were several talks at this meeting on various aspects of two-dimensional (0,2) theories, including talks of S. Gukov, M. Litvinov, and S. Franco.

In passing, we would also like to mention two other important topics, which lack of space prevents us from describing in more detail :

•

Triality. Triality is a property of (0,2) supersymmetric theories, first discussed in Ref. 53. This is an IR duality relating triples of theories. They have the following prototypical form. Briefly, a (0,2) theory describing the Grassmannian $G(k,n)$ with bundle $$S^{\oplus N}\oplus {(Q^{\ast })}^{2k+N-n}\oplus {(det{S}^{\ast })}^{\oplus 2}$$(63) (for S the universal subbundle and Q the universal quotient bundle) is IR equivalent to a (0,2) theory describing the Grassmannian $G(n-k,N)$ with bundle $$S^{\oplus 2k+N-n}\oplus {(Q^{\ast })}^n\oplus {(det{S}^{\ast })}^{\oplus 2},$$(64) and is also IR equivalent to a (0,2) theory describing the Grassmannian $G\ast N-n+k,2k+N-n$ with bundle $$S^{\oplus n}\oplus {(Q^{\ast })}^N\oplus {(det{S}^{\ast })}^{\oplus 2},$$(65) for k, n, N satisfying certain inequalities, which simultaneously guarantee both that the geometric description is sensible, and that supersymmetry is unbroken. Triality was discussed further in S. Franco’s talk.

•

GLSMs with H flux. These have a long history,54,55,56,57,58,59,60 and are often used to describe, for example, non-Kähler heterotic compactifications. The details are well beyond the scope of this short overview, but certainly deserve to be mentioned.

5. Quantum K Theory

Just as two-dimensional GLSMs can sometimes be used to compute quantum cohomology, it has been noted^61,62,63,64 that three-dimensional GLSMs can sometimes be used to compute quantum K theory. Furthermore, analogous to other examples in this survey, in many cases quantum K theory can be computed using Coulomb branch techniques.

The basic idea of the physical realization of quantum K theory is as follows (see for example Refs. 61,62,63,64). Consider a GLSM in three dimensions, on a three-manifold of the form $S^1\times {\mathrm{\Sigma }}_2$, where ${\mathrm{\Sigma }}_2$ is a Riemann surface. Quantum K theory arises as OPEs of Wilson lines wrapped on the $S^1$, moving parallel to one another along the base ${\mathrm{\Sigma }}_2$.

To compute those OPEs, one does a Kaluza–Klein reduction⁶⁵ along the $S^1$. One gets an effective low-energy two-dimensional theory (along ${\mathrm{\Sigma }}_2$), with an infinite tower of fields. Regularizing the sum of their contributions to the two-dimensional twisted one-loop effective superpotential has the effect of changing the ordinary log contributions to dilogarithms Li₂.

The Wilson line OPE relations are the critical loci of the two-dimensional twisted superpotential.^64,66,67,68

Let us work through a simple example. Consider a three-dimensional GLSM for ${\mathbb{P}}^n$, meaning a $U(1)$ gauge theory with $n+1$ chiral superfields of charge $+1$. The twisted one-loop effective superpotential for the two-dimensional theory, obtained after regularizing the sum of Kaluza–Klein states, and for the pertinent Chern-Simons level, is of the form

where $x=exp(2\pi iR\sigma )$ for R the radius of the $S^1$, and $\sigma$ the scalar of the two-dimensional vector multiplet. The critical locus of this superpotential is This is precisely the quantum K theory ring relation for ${\mathbb{P}}^n$, where we identify x with $S=\mathcal{𝒪}(-1)$, the tautological line bundle. (Classically, in K theory, $1-S={\mathcal{𝒪}}_D$ for D a hyperplane divisor, and the $(n+1)$-fold self-intersection of a divisor on an n-dimensional space vanishes.) (Superpotentials for more general cases have also been discussed.^65,66,69)

We can relate the quantum K theory ring relation to the quantum cohomology ring relation, in the limit that $R\to 0$. To that end, in that limit, expand

and it is straightforward to see that the ring relation (67) reduces to which is the standard quantum cohomology ring relation for ${\mathbb{P}}^n$.

For another example, we turn to the Grassmannian $G(k,n)$. For the pertinent Chern–Simons level, the twisted one-loop effective superpotential, after regularizing the sum over Kaluza–Klein modes, is given by

where $x_a=exp(2\pi iR{\sigma }_a)$, for R the radius of the $S^1$, and ${\sigma }_a$ the vev of the scalar in the two-dimensional vector multiplet on the Coulomb present. (Also present, though not written explicitly, are the Weyl-group ($S_k$) orbifold, and the excluded locus ${\sigma }_a\ne {\sigma }_b$.)

The critical locus of this superpotential is

This equation can be symmetrized as before using Vieta, to obtain

One can show⁷¹ that the symmetric polynomials in the $\overline{x}$ are interpreted as coupling to

so the ring relations (72) become or after simplification, where $\star$ denotes the quantum product, and This is a presentation^a of the quantum K theory ring of the Grassmannian $G(k,n)$.^69,70

There exists an analogous presentation of the quantum K theory ring of partial flag manifolds of the form^71,73

where $S_i$ is a universal subbundle of rank $k_i$. Weihong Xu’s talk at this meeting described this in greater detail.

In this discussion, we have mostly glossed over the role of Chern–Simons levels. The three-dimensional supersymmetric theory can certainly have Chern–Simons terms, and their levels modify the low-energy twisted one-loop effective superpotential $\tilde{W}$. We have chosen Chern–Simons levels in the expressions above to match quantum K theory results, corresponding to $U{(1)}_{-1∕2}$ quantization of the chirals (Ref. 68, Sec. 2.2), but one can also choose other values for the levels. It is believed that other choices correspond to the mathematical notion of levels discussed in Ref. 74, but a detailed dictionary is not known for all cases.

We have also glossed over Wilson line OPEs for more general cases, not necessarily associated with quantum K theory. These have been extensively studied in the literature, see e.g. Refs. 67, 68 and references therein.

Earlier we discussed the role of ordinary mirror symmetry and (0,2) mirror symmetry in computing e.g. quantum cohomology. Similarly, there is a notion of mirror symmetry in three-dimensional gauge theories, see for example Refs. 75,76,77,78,79,80. The details are, unfortunately, beyond the scope of this short survey.

Others at this meeting who spoke on various aspects of quantum K theory included P. Koroteev, Y. P. Lee, and W. Xu, and related work in three-dimensional GLSMs was discussed by C. Closset, H. Jockers, and M. Litvinov. There were also discussions of related notions in integrable systems in the talks of P. Koroteev and W. Gu.

6. Conclusions

In this overview we have surveyed a few relatively recent developments in the physics of GLSMs.

One question for the future is whether quantum K theory and quantum sheaf cohomology can be linked? The boundary of a three-dimensional $N=2$ theory is a two-dimensional (0,2) supersymmetric theory.^81,82,83,84 One could imagine moving bulk operators to the boundary and using the bulk/boundary correspondence to describe quantum sheaf cohomology (of the two-dimensional (0,2) boundary) as a module over quantum K theory (of the three-dimensional $N=2$ bulk). However, one issue is that the bulk operators are Wilson lines, not local operators, unlike the boundary; moving those bulk operators to the boundary would yield Wilson lines in the two-dimensional (0,2) supersymmetric boundary. To implement this program would require a mathematical interpretation of two-dimensional (0,2) Wilson lines in terms of (presumably descendants in) quantum sheaf cohomology.

One direction we have not surveyed is the newer mathematically rigorous approaches to GLSMs.^85,86,87,88 These are extremely interesting, but there is not enough space here to survey them. Those constructions were described in talks by H. Fan, E. Segal, C. C. Melissa Liu, and D. Favero.

Acknowledgments

E.S. would like to thank the Simons Center for hosting this conference, GLSMs@30. E.S. was partially supported by NSF grant PHY-2310588.

Notes

^a To be clear, the quantum K theory ring of $G(k,n)$ has been studied from a variety of perspectives in both the math and physics communities; see for example Ref. 72 for an early mathematics reference, and see for example Ref. 64 for an early physics reference.