From $S^1$-fixed points to $\mathcal{W}$-algebra representations

Abstract

We study a set ${\mathcal{M}}_{K,N}$ parameterizing filtered $SL(K)$-Higgs bundles over ${\mathbb{CP}}^1$ with an irregular singularity at $z=\infty$, such that the eigenvalues of the Higgs field grow like $|\lambda |\sim |z^{N/K}\mathrm{d}z|$, where $K$ and $N$ are coprime. ${\mathcal{M}}_{K,N}$ carries a ${\mathbb{C}}^{\times }$-action analogous to the famous ${\mathbb{C}}^{\times }$-action introduced by Hitchin on the moduli spaces of Higgs bundles over compact curves. The construction of this ${\mathbb{C}}^{\times }$-action on ${\mathcal{M}}_{K,N}$ involves the rotation automorphism of the base ${\mathbb{CP}}^1$. We classify the fixed points of this ${\mathbb{C}}^{\times }$-action, and exhibit a curious $1$-$1$ correspondence between these fixed points and certain representations of the vertex algebra ${\mathcal{W}}_K$; in particular we have the relation $\mu =\frac{1}{12}\left(K-1-c_{\mathrm{eff}}\right)$, where $\mu$ is a regulated version of the $L^2$ norm of the Higgs field, and $c_{\mathrm{eff}}$ is the effective Virasoro central charge of the corresponding $W$-algebra representation. We also discuss a Bialynicki-Birula-type stratification of ${\mathcal{M}}_{K,N}$, where the strata are labeled by isomorphism classes of the underlying filtered vector bundles.

36 pages

References in corpus (13)

• Solutions Of Four-Dimensional Field Theories Via M Theory
• Wall-crossing, Hitchin Systems, and the WKB Approximation
• R-Twisting and 4d/2d Correspondences
• Schur Indices, BPS Particles, and Argyres-Douglas Theories
• Surface Defects and Chiral Algebras
• Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT
• Nahm Transform For Periodic Monopoles And N=2 Super Yang-Mills Theory
• Ends of the moduli space of Higgs bundles
• Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters
• Twisted wild character varieties
• Wild nonabelian Hodge theory on curves
• Hitchin systems in N=2 field theory
• Betti numbers of the moduli space of rank 3 parabolic Higgs bundles

Cited by in corpus (7)

• Singular Geometry and Higgs Bundles in String Theory
• Planar minimal surfaces with polynomial growth in the $\mathrm{Sp}(4,\mathbb{R})$-symmetric space
• Asymptotic Geometry of the Moduli Space of Parabolic $SL(2,\mathbb{C})$-Higgs Bundles
• Perspectives on the Asymptotic Geometry of the Hitchin Moduli Space
• Generic Ends of the Moduli Space of $SL(n,\mathbb{C})$-Higgs Bundles
• Branes and DAHA Representations
• Localization for affine $W$-algebras