28 problems

Let $Z_{S^2}$ denote the partition function on the two-sphere of a two-dimensional ${\cal N}=(2,2)$ gauged linear sigma model (GLSM) whose infrared description is a non-linear sigm…

A torus manifold is a real $2n$-dimensional manifold with an effective action of $(S^1)^n$; a toral hypersurface is an at least $(S^1)^n$-equivariant hypersurface, and $Z$ denotes…

Negative-phase quotient conjecture. The Higgs branch of the negative phase is described by a nonlinear sigma model with target space

Window-brane conjecture. Branes with charges in the window correspond to the exceptional collection of the derived category of the geometry in the infrared Higgs branch.

Let $\mathcal{J}$ be the collection of subsets indexing the branes $\mathfrak{B}_{J}$, let $\hat{Z}_{\mathfrak{B}_{J}}(\mathbf{x})$ denote their A-periods, and let…

For GLSM data $(\mathsf{G},W,\rho_m,t,R)$ and a phase containing $\zeta=\operatorname{Re}(t)$, let $\pi_{\zeta}\colon \operatorname{MF}_{\mathsf{G}}(W)\to\mathcal{D}_{\zeta}$ be th…

Let a smooth Calabi–Yau variety be described by a GLSM, and let the hemisphere partition function be associated with an object in its brane category. Smooth Calabi–Yau A-period con…

Let a gauged linear sigma model (GLSM) describe a Calabi–Yau manifold, and let the hemisphere partition function be associated with an object in the bounded derived category of coh…

Let $(V,G,W,\mu)$ be a GLSM space satisfying the geometric phase condition, and let its classical vacuum … be a compact Kähler manifold. The GLSM correlation functions are maps … w…

Consider an elliptically fibered Calabi–Yau geometry with a geometric phase and a hybrid phase in which the fiber collapses. Let the relative conifold transformation denote the tra…

GLSM–Gromov–Witten correspondence conjecture. There is an element $\mathfrak c\in H_{\it CR}^*(\bar X_W;\Lambda_+)$ such that

A gauged linear sigma model (GLSM) may have distinct Kähler phases, corresponding to different geometric phases of the model. Two phases are twisted derived equivalent when the ass…

Let $X_{\beta}$ be the phase associated with a VGIT parameter $\beta \in L^{\vee}_{\mathbb{R}}$, and let FIPS denote the Fayet–Iliopoulos parameter space of the corresponding gauge…

Squashed toric models conjecture. The infrared fixed points of these GLSMs are the same as those of the theories studied by Hori, namely the infrared SCFTs obtained from NS5-branes…

Squashed orbifold flow conjecture. The GLSM describes an RG flow from this metric to the cigar SCFT.

Resolved $A_1$ flow conjecture. This GLSM flows from its tree-level metric to the resolved $A_1$ space, namely the Eguchi–Hanson theory.

Let $\bar{X}_W$ be the symplectic reduction in the geometric phase, let $H_{\rm CR}^*(\bar{X}_W;\Lambda)$ denote its Chen–Ruan cohomology, and let…

Let $\mathsf{Br}(\widetilde{\mathcal{Y}})$ be the proposed category of B-branes in the GLSM $\widetilde{\mathcal{Y}}$, viewed as a subcategory of $D^b(\widetilde{\mathcal{Y}})$. It…

Let $G$ act on a $G$-space $V$, and let $W\subset V$ be the zero locus of $G$-semi-invariant polynomials $f_i$, for $i=1,\ldots,r$. For $P\in\mathbb{C}[\mathfrak{t}]^{\mathcal{W}}$…

Let a phase of a gauged linear sigma model have hemisphere partition function $Z_{D^2}$, and let $Z_{D^2}$ determine the central charge of a D-brane in that phase. Exact-central-ch…

Let $X_{\theta}=[V\mathbin{/\mkern-6mu/\mkern-2mu}_{\theta}G]$ be a GIT quotient, where $V$ is a complex vector space, $G\subset\operatorname{GL}(V)$, and $\theta$ is a character o…

Smoothness conjecture. If a gauged linear $\sigma$-model is defined by pointsets ${\Scr A}$ and ${\Scr B}$ which are both ${\Scr A}$-complete and ${\Scr B}$-complete, then this mod…

Homological projective duality (HPD) is a framework relating derived categories of projective varieties and their dual data; an abelian gauged linear sigma model (GLSM) is a physic…

Let $Y$ be the Calabi–Yau threefold in $\mathbb{P}^6$ defined by the rank-$4$ locus of a $7\times 7$ antisymmetric matrix whose entries are linear in the homogeneous coordinates, a…

Many two-dimensional $\mathcal{N}=(2,2)$ nonlinear sigma models with Calabi–Yau target spaces admit ultraviolet descriptions as gauged linear sigma models. Two-sphere partition-fun…