Lectures on 𝐾-theoretic computations in enumerative geometry

Abstract

K-theoretic computations in enumerative geometry 6.4 Moduli of relative quasimaps 78 6.5 Degeneration formula and the glue operator 81 7 Nuts and bolts 86 7.1 The Tube 86 7.2 The Vertex 89 7.3 The index limit 92 7.4 Cap and capping 93 7.5 Large framing vanishing 94 8 Difference equations 97 8.1 Shifts of Kähler variables 97 8.2 Shifts of equivariant variables 99 8.3 Difference equations for vertices 103 9 Stable envelopes and quantum groups 106 9.1 K-theoretic stable envelopes 106 9.2 Triangle lemma and braid relations 110 9.3 Actions of quantum group 115 10 Quantum Knizhnik-Zamolodchikov equations 118 10.1 Commuting difference operators from R-matrices 118 10.2 Minuscule shifts and qKZ 120 10.3 Glue operator in the stable basis 123 10.

Twisted indices of 3d $$ \mathcal{N} $$ = 4 gauge theories and enumerative geometry of quasi-maps

“…In this limit, M m has an algebraic description as the moduli space of quasi-maps to the Higgs branch, Σ → M H , of degree m [7]. 1 More precisely, in the H-twist we recover the twisted quasi-maps to holomorphic symplectic quotients introduced in [14], while in the C-twist we find a generalisation to arbitrary genus of a construction of [15].…”

Quantum line defects and refined BPS spectra

“…A obvious enumerative problem is to compute χ(M , O M ). However this is not natural from the perspective of supersymmetry [35]. Supersymmetry is associated with virtual counts, or indices, which require a certain grading.…”

“…For example one can consider the Koszul resolution ( • E ∨ , d), where d is the contraction with the section s. Then O M arises as the cohomology in degree zero of this resolution. In homological algebra it is not natural to consider only the zeroth cohomology, but one would rather consider the virtual structure sheaf [4,35] O vir…”

“…In this note we will adapt the K-theoretic approach to the counting of BPS states developed by Nekrasov and Okounkov in [32,35] to study quantum line defects in class S supersymmetric quantum field theories. The formalism of framed BPS quivers associates to a line operator a certain enumerative problem of Donaldson-Thomas type, the counting of certain stable quiver representations [9,10,11,16].…”

“…We will show that such a problem can be refined to include the spin information of the BPS states. The main technical tool we will use is the K-theoretic version of the localization computation as introduced in [32,35]. Under certain conditions, the K-theoretic construction produces certain toric modules out of which we can compute ordinary framed BPS states by taking Euler characteristics.…”