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Instantons on the Blown-up Surface and the Affine Vertex Algebra

Wei-Ping Li, Qingyuan Jiang, Yu Zhao

Abstract

Vafa-Witten observed that Yoshioka's blow-up formula for the Euler characteristics of rank $r$ instantons on an algebraic surface coincides with the character of the Wess-Zumino-Witten model for $\mathrm{SU}(r)$ at level $1$, and raised the question of finding a rational conformal field theory explanation for this striking coincidence. In this work, we provide an answer to this question by constructing and analyzing the affine $\mathrm{gl}_r$ action on various cohomology theories, including the Grothendieck group of coherent sheaves, Hochschild homology, Chow groups, and Hodge cohomology, of the moduli space of stable sheaves on a blown-up surface. A key ingredient in our proof is a representation-theoretic reformulation of the theory of Grassmannians of Tor-amplitude $[0,1]$-perfect complexes studied by the first-named author in terms of the spin representation of the finite-dimensional Clifford algebra. This may be viewed as a finite analog of the question of Vafa-Witten via the Boson-Fermion correspondence.

Instantons on the Blown-up Surface and the Affine Vertex Algebra

Abstract

Vafa-Witten observed that Yoshioka's blow-up formula for the Euler characteristics of rank instantons on an algebraic surface coincides with the character of the Wess-Zumino-Witten model for at level , and raised the question of finding a rational conformal field theory explanation for this striking coincidence. In this work, we provide an answer to this question by constructing and analyzing the affine action on various cohomology theories, including the Grothendieck group of coherent sheaves, Hochschild homology, Chow groups, and Hodge cohomology, of the moduli space of stable sheaves on a blown-up surface. A key ingredient in our proof is a representation-theoretic reformulation of the theory of Grassmannians of Tor-amplitude -perfect complexes studied by the first-named author in terms of the spin representation of the finite-dimensional Clifford algebra. This may be viewed as a finite analog of the question of Vafa-Witten via the Boson-Fermion correspondence.

Paper Structure

This paper contains 30 sections, 28 theorems, 175 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The main results of the paper
  3. Moduli of instantons, the conformal field theory and some related work
  4. Strategy of the proof of \ref{['thm:1.1']}
  5. The organization of the paper and some notations
  6. Acknowledgment
  7. Perverse coherent sheaves on the blow-ups
  8. Grassmannians of complexes and the Clifford algebra
  9. The orthogonalization and
  10. The dual representation and the grading
  11. The Chow correspondences on Grassmannians of complexes
  12. Geometric correspondences and the Clifford algebra action
  13. The semi-orthogonal relations
  14. Derived Categories of Derived Grassmannians
  15. Semiorthogonal decompositions and categorical equations
  16. ...and 15 more sections

Key Result

Theorem 1.1

If $\mathrm{gcd}(r,c_1\cdot H)=1$, then for any $l\in\mathbb{Z}$, we have an identification of $\hat{\mathrm{gl}}_r$-representations: Here, an extra smooth assumption assumptionS will be needed for the Hochschild homology group, the Chow group, and the Hodge cohomology group.

Figures (2)

  • Figure 1: The affine $(n,-l)$-plane. The red dashed curve indicates the parabolic trajectory passing through $\mathrm{M}^0(l,n)$.
  • Figure 2: Visualization for a bounded bigraded $\mathcal{E}(r)$-representation $V$ in the case $r=2$. Each $\mathbb{Z}$-module $V_{l,n}$ is placed at the lattice point $(n,-l)$. Points with $l\equiv 0 \pmod{2}$ are shown in red, and those with $l\equiv 1 \pmod{2}$ in blue. The region of potentially nonzero $V_{l,n}$ is foliated by parabolic trajectories $\mathbb{O}(l,n)$ with $l\in\{-1,0\}$ (and $n\ge n_0$ in the geometric situation $V_{l,n}=\mathbb{H}^*(\mathrm{M}^0(l,n))$). Parabolic trajectories $\mathbb{O}(l,n)$ with $l=0$ are drawn in red, and those with $l=-1$ in blue.

Theorems & Definitions (58)

  • Theorem 1.1: \ref{['main1']}
  • Theorem 2.1: Proposition 3.1 of kuhn2021blowup, Lemma 3.4 and Theorem 4.1 of 1050282810787470592
  • Theorem 2.2: Proposition 3.37 of 1050282810787470592
  • proof
  • Theorem 2.3: Lemma 3.22 of 1050282810787470592 and Lemma 3.3 of koseki2021categorical
  • Remark 2.4
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 48 more