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

Yalong Cao, Yukinobu Toda, Gufang Zhao

TL;DR

This work builds a robust framework to define and push forward $K$-theoretic invariants along $(-1)$-shifted Lagrangian correspondences on derived critical loci via critical pullbacks. The core construction uses a Koszul/spinor–type mechanism combined with specialization to the normal cone, yielding a map $f_{\pi_X}^!: K_0(X,\phi_X)\to K_0(M,\mathbb{Z}[1/2])$ with natural functorial, base-change, and bivariance properties. The authors develop a linearization of the Lagrangian category and prove a functorial calculus that underpins gluing formulas for quantum critical $K$-theory and DT$_4$ degeneration, with concrete applications to GLSMs, Nakajima-quiver varieties with potentials, and Joyce–Safronov-type conjectures. This framework generalizes and unifies several perspectives on categorified DT theory and shifted symplectic geometry, enabling new invariants on singular critical loci and robust tools for geometric representation theory. Overall, the paper provides foundational methods for quantum critical $K$-theory, degeneration formulas in DT$_4$ theory, and a $K$-theoretic realization of Joyce–Safronov ideas, with broad potential for further extensions in derived algebraic geometry and enumerative geometry.

Abstract

Given a regular function $φ$ on a smooth stack, and a $(-1)$-shifted Lagrangian $M$ on the derived critical locus of $φ$, under fairly general hypotheses, we construct a pullback map from the Grothendieck group of coherent matrix factorizations of $φ$ 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.

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

TL;DR

This work builds a robust framework to define and push forward -theoretic invariants along -shifted Lagrangian correspondences on derived critical loci via critical pullbacks. The core construction uses a Koszul/spinor–type mechanism combined with specialization to the normal cone, yielding a map with natural functorial, base-change, and bivariance properties. The authors develop a linearization of the Lagrangian category and prove a functorial calculus that underpins gluing formulas for quantum critical -theory and DT degeneration, with concrete applications to GLSMs, Nakajima-quiver varieties with potentials, and Joyce–Safronov-type conjectures. This framework generalizes and unifies several perspectives on categorified DT theory and shifted symplectic geometry, enabling new invariants on singular critical loci and robust tools for geometric representation theory. Overall, the paper provides foundational methods for quantum critical -theory, degeneration formulas in DT theory, and a -theoretic realization of Joyce–Safronov ideas, with broad potential for further extensions in derived algebraic geometry and enumerative geometry.

Abstract

Given a regular function on a smooth stack, and a -shifted Lagrangian on the derived critical locus of , under fairly general hypotheses, we construct a pullback map from the Grothendieck group of coherent matrix factorizations of to that of coherent sheaves on . 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 -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 -theoretic Donaldson-Thomas theory of local Calabi-Yau 4-folds, the other in confirming a -theoretic version of Joyce-Safronov conjecture.

Paper Structure

This paper contains 45 sections, 38 theorems, 302 equations.

Table of Contents

  1. Introduction
  2. Definition of critical pullbacks
  3. Properties of critical pullbacks
  4. Applications
  5. $K$-theoretic Joyce-Safronov conjecture
  6. Exact Lagrangians on derived critical loci
  7. Exact Lagrangians and relative shifted symplectic structures
  8. Setting of the paper
  9. Factorization categories and their $K$-theories
  10. Abelian categories of factorizations
  11. Derived categories of factorizations
  12. Critical $K$-theories
  13. Specialization to the normal cone in critical $K$-theory
  14. Some categorical preparations
  15. Potential function on deformation space
  16. ...and 30 more sections

Key Result

Theorem 1.1

There is a well-defined map (called critical pullback): from the Grothendieck group $K_0(X,\phi_X)$ of the category of coherent matrix factorizations of $(X,\phi_X)$ to the Grothendieck group of coherent sheaves on $M$.

Theorems & Definitions (116)

  • Theorem 1.1
  • Proposition 1.2: Proposition \ref{['prop:compare_OT']}
  • Theorem 1.3: Theorem \ref{['thm on funct']}
  • Theorem 1.4: Quantum critical $K$-theory, Theorem \ref{['thm on glue in GLSM']}
  • Remark 1.5
  • Theorem 1.6: Degeneration formula of $K$-theoretic $\mathop{\rm DT}\nolimits_4$ invariants, Theorem \ref{['thm on glue in DT4']}
  • Remark 1.7
  • Conjecture 1.8
  • Theorem 1.9
  • Definition 2.1
  • ...and 106 more