Quantum Integrability and Generalised Quantum Schubert Calculus

Abstract

We introduce and study a new mathematical structure in the generalised (quantum) cohomology theory for Grassmannians. Namely, we relate the Schubert calculus to a quantum integrable system known in the physics literature as the asymmetric six-vertex model. Our approach offers a new perspective on already established and well-studied special cases, for example equivariant K-theory, and in addition allows us to formulate a conjecture on the so-far unknown case of quantum equivariant K-theory.

57 pages, 10 figures; v2: some references added and some minor changes; v3: abstract shortened, some typos corrected and a discussion of the Bethe roots for the non-equivariant case added; v4: accepted version

References in corpus (3)

• Puzzles, positroid varieties, and equivariant K-theory of Grassmannians
• Equivariant quantum cohomology and Yang-Baxter algebras
• A Chevalley formula for the equivariant quantum K-theory of cominuscule varieties

Cited by in corpus (32)

• Baxter Q-operator from quantum K-theory
• Double Grothendieck polynomials and colored lattice models
• BPS Indices, Modularity and Perturbations in Quantum K-theory
• A Chevalley formula for the equivariant quantum K-theory of cominuscule varieties
• Symmetric functions and wavefunctions of the six-vertex model by Izergin-Korepin analysis
• 3d N=2 Chern-Simons-matter theory, Bethe ansatz, and quantum K-theory of Grassmannians
• Dual wavefunction of the Felderhof model
• Crystal structures for symmetric Grothendieck polynomials
• Landau-Ginzburg mirror, quantum differential equations and qKZ difference equations for a partial flag variety
• Combinatorial properties of symmetric polynomials from integrable vertex models in finite lattice
• Frozen Pipes: Lattice Models for Grothendieck Polynomials
• Integrability approach to Feher-Nemethi-Rimanyi-Guo-Sun type identities for factorial Grothendieck polynomials
• Yang-Baxter algebras as convolution algebras: The Grassmannian case
• Dual wavefunction of the symplectic ice
• Quantum K theory rings of partial flag manifolds
• Colored five-vertex models and Demazure atoms
• Neutral-fermionic presentation of the $K$-theoretic $Q$-function
• Type $\hat{\mathrm{C}}$ Brauer loop schemes and loop model with boundaries
• Quantum K theory of Grassmannians, Wilson line operators, and Schur bundles
• An affine deformation of the quantum cohomology ring of flag manifolds and periodic Toda lattice
• Schubert defects in Lagrangian Grassmannians
• Quantum inverse scattering method and generalizations of symplectic Schur functions and Whittaker functions
• Euler characteristics of cominuscule quantum K-theory
• A class of partition functions associated with $E_{\tau ,\eta }(g{l}_3)$ by Izergin-Korepin analysis
• Positivity determines the quantum cohomology of Grassmannians
• Loop Models and $K$-Theory
• Lattice Models, Hamiltonian Operators, and Symmetric Functions
• Dimers, crystals and quantum Kostka numbers
• Integrable systems and crystals for edge labeled tableaux
• Free fermionic probability theory and K-theoretic Schubert calculus
• Integrable models and $K$-theoretic pushforward of Grothendieck classes
• Factorial Schur and Grothendieck polynomials from Bott Samelson varieties