Instantons on the Blown-up Surface and the Affine Vertex Algebra
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| Format: | Preprint |
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2025
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| _version_ | 1866915635003392000 |
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| author | Li, Wei-Ping Jiang, Qingyuan Zhao, Yu |
| author_facet | Li, Wei-Ping Jiang, Qingyuan Zhao, Yu |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_18959 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Instantons on the Blown-up Surface and the Affine Vertex Algebra Li, Wei-Ping Jiang, Qingyuan Zhao, Yu Algebraic Geometry High Energy Physics - Theory Representation Theory 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. |
| title | Instantons on the Blown-up Surface and the Affine Vertex Algebra |
| topic | Algebraic Geometry High Energy Physics - Theory Representation Theory |
| url | https://arxiv.org/abs/2511.18959 |