Wall-crossings and a categorification of K-theory stable bases of the Springer resolution

We compare the $K$ -theory stable bases of the Springer resolution associated to different affine Weyl alcoves. We prove that (up to relabelling) the change of alcoves operators are given by the Demazure–Lusztig operators in the affine Hecke algebra. We then show that these bases are categorified by the Verma modules of the Lie algebra, under the localization of Lie algebras in positive characteristic of Bezrukavnikov, Mirković, and Rumynin. As an application, we prove that the wall-crossing matrices of the $K$ -theory stable bases coincide with the monodromy matrices of the quantum cohomology of the Springer resolution.

Residual categories for (co)adjoint Grassmannians in classical types

In our previous paper we suggested a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety of Picard number 1 to the structure of its derived category of coherent sheaves. Here we generalize this conjecture, make it more precise, and support it by the examples of (co)adjoint homogeneous varieties of simple algebraic groups of Dynkin types $\mathrm {A}_n$ and $\mathrm {D}_n$ , that is, flag varieties $\operatorname {Fl}(1,n;n+1)$ and isotropic orthogonal Grassmannians $\operatorname {OG}(2,2n)$ ; in particular, we construct on each of those an exceptional collection invariant with respect to the entire automorphism group. For $\operatorname {OG}(2,2n)$ this is the first exceptional collection proved to be full.

Left Demazure–Lusztig Operators on Equivariant (Quantum) Cohomology and K-Theory

We study the Demazure–Lusztig operators induced by the left multiplication on partial flag manifolds $G/P$. We prove that they generate the Chern–Schwartz–MacPherson classes of Schubert cells (in equivariant cohomology), respectively their motivic Chern classes (in equivariant K-theory), in any partial flag manifold. Along the way, we advertise many properties of the left and right divided difference operators in cohomology and K-theory and their actions on Schubert classes. We apply this to construct left divided difference operators in equivariant quantum cohomology, and equivariant quantum K-theory, generating Schubert classes and satisfying a Leibniz rule compatible with the quantum product.

Landau–Ginzburg/Calabi–Yau Correspondence for a Complete Intersection via Matrix Factorizations

By generalizing the Landau–Ginzburg/Calabi–Yau correspondence for hypersurfaces, we can relate a Calabi–Yau complete intersection to a hybrid Landau–Ginzburg model: a family of isolated singularities fibered over a projective line. In recent years Fan, Jarvis, and Ruan have defined quantum invariants for singularities of this type, and Clader and Clader–Ross have provided an equivalence between these invariants and Gromov–Witten invariants of complete intersections, in this way quantum cohomology yields an identification of the cohomology groups of the Calabi–Yau and of the hybrid Landau–Ginzburg model. It is not clear how to relate this to the known isomorphism descending from derived equivalences (due to Segal and Shipman, and Orlov and Isik). We answer this question for Calabi–Yau complete intersections of two cubics.

Wilson loop algebras and quantum K-theory for Grassmannians

We study the algebra of Wilson line operators in three-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric U(M ) gauge theories with a Higgs phase related to a complex Grassmannian Gr(M, N ), and its connection to K-theoretic Gromov-Witten invariants for Gr(M, N ). For different Chern-Simons levels, the Wilson loop algebra realizes either the quantum cohomology of Gr(M, N ), isomorphic to the Verlinde algebra for U(M ), or the quantum K-theoretic ring of Schubert structure sheaves studied by mathematicians, or closely related algebras.

GLSMs for exotic Grassmannians

In this paper we explore nonabelian gauged linear sigma models (GLSMs) for symplectic and orthogonal Grassmannians and flag manifolds, checking e.g. global symmetries, Witten indices, and Calabi-Yau conditions, following up a proposal in the math community. For symplectic Grassmannians, we check that Coulomb branch vacua of the GLSM are consistent with ordinary and equivariant quantum cohomology of the space.

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

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.

Coalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution of Prime Numbers

The occurrence and frequency of a phenomenon of resonance (namely the coalescence of some Dubrovin canonical coordinates) in the locus of small quantum cohomology of complex Grassmannians are studied. It is shown that surprisingly this frequency is strictly subordinate and highly influenced by the distribution of prime numbers. Two equivalent formulations of the Riemann hypothesis are given in terms of numbers of complex Grassmannians without coalescence: the former as a constraint on the disposition of singularities of the analytic continuation of the Dirichlet series associated to the sequence counting non-coalescing Grassmannians, and the latter as asymptotic estimate (whose error term cannot be improved) for their distribution function.