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Categorical wall-crossing formula for Donaldson-Thomas theory on the resolved conifold

Yukinobu Toda

Abstract

We prove wall-crossing formula for categorical Donaldson-Thomas invariants on the resolved conifold, which categorifies Nagao-Nakajima wall-crossing formula for numerical DT invariants on it. The categorified Hall products are used to describe the wall-crossing formula as semiorthogonal decompositions. A successive application of categorical wall-crossing formula yields semiorthogonal decompositions of categorical Pandharipande-Thomas stable pair invariants on the resolved conifold, which categorify the product expansion formula of the generating series of numerical PT invariants on it.

Categorical wall-crossing formula for Donaldson-Thomas theory on the resolved conifold

Abstract

We prove wall-crossing formula for categorical Donaldson-Thomas invariants on the resolved conifold, which categorifies Nagao-Nakajima wall-crossing formula for numerical DT invariants on it. The categorified Hall products are used to describe the wall-crossing formula as semiorthogonal decompositions. A successive application of categorical wall-crossing formula yields semiorthogonal decompositions of categorical Pandharipande-Thomas stable pair invariants on the resolved conifold, which categorify the product expansion formula of the generating series of numerical PT invariants on it.

Paper Structure

This paper contains 41 sections, 48 theorems, 363 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and summary of the paper
  3. Categorical PT stable pair theory on the resolved conifold
  4. Categorical wall-crossing formula
  5. Outline of the proof of Theorem \ref{['intro:thm2']}
  6. Related works
  7. Acknowledgements
  8. Notation and Convention
  9. Preliminary
  10. The category of factorizations
  11. Attracting loci
  12. Kempf-Ness stratification
  13. Knörrer periodicity
  14. Categorified Hall products for quivers with super-potentials
  15. Moduli stacks of representations of quivers
  16. ...and 26 more sections

Key Result

Theorem 1.1

(Corollary cor:sod2.5) There exists a semiorthogonal decomposition of the form Here $a_{n, \beta}$ is defined by

Figures (5)

  • Figure 1: Wall-chamber structures
  • Figure 2: $(3, 7, 7, 10, 15) \in \mathbb{B}_{c}(d), d=5, c\ge 20$
  • Figure 3: Algorithm for $\chi=(4, 2, 1)$, $d=4$, $c=b=7$
  • Figure 4: $\delta$ and $\delta'$ for $\chi=(2,2,3,4,5) \in \mathbb{B}_{10}(5)$
  • Figure 5: Wall-chamber structures

Theorems & Definitions (100)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 2.5
  • proof
  • Definition 3.1
  • Remark 4.1
  • ...and 90 more