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arXiv · 2503.06025

$K$-theoretic pullbacks for Lagrangians on derived critical loci

Yalong Cao·Yukinobu Toda·Gufang Zhao
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Abstract

Given a regular function $\phi$ on a smooth stack, and a $(-1)$-shifted Lagrangian $M$ on the derived critical locus of $\phi$, under fairly general hypotheses, we construct a pullback map from the Grothendieck group of coherent matrix factorizations of $\phi$ to that of coherent sheaves on $M$. This map satisfies a functoriality property with respect to the composition of Lagrangian correspondences, as well as the usual bivariance and base-change properties. We provide three applications of the construction, one in the definition of quantum $K$-theory of critical loci (Landau-Ginzburg models), paving the way to generalize works of Okounkov school from Nakajima quiver varieties to quivers with potentials, one in establishing a degeneration formula for $K$-theoretic Donaldson-Thomas theory of local Calabi-Yau 4-folds, the other in confirming a $K$-theoretic version of Joyce-Safronov conjecture.

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Yalong Cao, Yukinobu Toda, Gufang Zhao. 2025-03-08. $K$-theoretic pullbacks for Lagrangians on derived critical loci. https://arxiv.org/abs/2503.06025

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