Residue Mirror Symmetry for Grassmannians

Self Cite

We study a correspondence between 3d $$ \mathcal{N} $$ N = 2 topologically twisted Chern-Simons-matter theories on S1× Σg and quantum K -theory of Grassmannians. Our starting point is a Frobenius algebra depending on a parameter β associated with an algebraic Bethe ansatz introduced by Gorbounov-Korff. They showed that the Frobenius algebra with β = −1 is isomorphic to the (equivariant) small quantum K -ring of the Grassmannian, and the Frobenius algebra with β = 0 is isomorphic to the equivariant small quantum cohomology of the Grassmannian. We apply supersymmetric localization formulas to the correlation functions of supersymmetric Wilson loops in the Chern-Simons-matter theory and show that the algebra of Wilson loops is isomorphic to the Frobenius algebra with β = −1. This allows us to identify the algebra of Wilson loops with the quantum K - ring of the Grassmannian. We also show that correlation functions of Wilson loops on S1× Σg satisfy the axiom of 2d TQFT. For β = 0, we show the correspondence between an A-twisted GLSM, the Frobenius algebra for β = 0, and the quantum cohomology of the Grassmannian. We also discuss deformations of Verlinde algebras, omega-deformations, and the K -theoretic I -functions of Grassmannians with level structures.

Section: Jhep08(2020)157mentioning

confidence: 99%

Section: Jhep08(2020)157mentioning

confidence: 99%

Section: Jhep08(2020)157mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

3d $$ \mathcal{N} $$ = 2 Chern-Simons-matter theory, Bethe ansatz, and quantum K -theory of Grassmannians

Ueda

Yoshida

2020

Self Cite

“…For the T [SU(2)] theory, the relevant geometry is the space of algebraic maps P 1 → P 1 -we found [50] particular useful for understanding the map space in this simple example. In coordinates [z 1 : z 2 ] for the base and [w 1 : w 2 ] for the target, algebraic maps P 1 → P 1 of degree d are specified by 2 homogeneous polynomials [w 1 (z 1 , z 2 ) : w 2 (z 1 , z 2 )] each of degree d. w and w define the same map if there exists α = 0 such that w = αw and so the coefficients of the polynomials can be compactified to projective space:…”

Section: Poincaré Polynomial Of Handsawmentioning

confidence: 99%

Factorisation of 3d $$ \mathcal{N} $$ = 4 twisted indices and the geometry of vortex moduli space

Samuel

Dorey

Zhang

2020

We study the twisted indices of N = 4 supersymmetric gauge theories in three dimensions on spatial S 2 with an angular momentum refinement. We demonstrate factorisation of the index into holomorphic blocks for the T [SU(N )] theory in the presence of generic fluxes and fugacities. We also investigate the relation between the twisted index, Hilbert series and the moduli space of vortices. In particular, we show that each holomorphic block coincides with a generating function for the χ t genera of the moduli spaces of "local" vortices. The twisted index itself coincides with a corresponding generating function for the χ t genera of moduli spaces of "global" vortices in agreement with a proposal of Bullimore et al. We generalise this geometric interpretation of the twisted index to include fluxes and Chern-Simons levels. For the T [SU(N )] theory, the relevant moduli spaces are the local and global versions of Laumon space respectively and we demonstrate the proposed agreements explicitly using results from the mathematical literature. Finally, we exhibit a precise relation between the Coulomb branch Hilbert series and the Poincaré polynomials of the corresponding vortex moduli spaces.

Grothendieck lines in 3d $$ \mathcal{N} $$ = 2 SQCD and the quantum K-theory of the Grassmannian

Closset,

Khlaif

2023

We revisit the 3d GLSM computation of the equivariant quantum K-theory ring of the complex Grassmannian from the perspective of line defects. The 3d GLSM onto X = Gr(Nc, nf) is a circle compactification of the 3d $$ \mathcal{N} $$ N = 2 supersymmetric gauge theory with gauge group $$ \textrm{U}{\left({N}_c\right)}_{k,k+l{N}_c} $$ U N c k , k + l N c and nf fundamental chiral multiplets, for any choice of the Chern-Simons levels (k, l) in the ‘geometric window’. For $$ k={N}_c-\frac{n_f}{2} $$ k = N c − n f 2 and l = −1, the twisted chiral ring generated by the half-BPS lines wrapping the circle has been previously identified with the quantum K-theory ring QKT(X). We identify new half-BPS line defects in the UV gauge theory, dubbed Grothendieck lines, which flow to the structure sheaves of the (equivariant) Schubert varieties of X. They are defined by coupling $$ \mathcal{N} $$ N = 2 supersymmetric gauged quantum mechanics of quiver type to the 3d GLSM. We explicitly show that the 1d Witten index of the defect worldline reproduces the Chern characters for the Schubert classes, which are written in terms of double Grothendieck polynomials. This gives us a physical realisation of the Schubert-class basis for QKT(X). We then use 3d A-model techniques to explicitly compute QKT(X) as well as other K-theoretic enumerative invariants such as the topological metric. We also consider the 2d/0d limit of our 3d/1d construction, which gives us local defects in the 2d GLSM, the Schubert defects, that realise equivariant quantum cohomology classes.