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

Laura Fredrickson, Andrew Neitzke|arXiv (Cornell University)|Sep 18, 2017

Black Holes and Theoretical Physics29 references11 citations

TL;DR

This paper establishes a 1-1 correspondence between $\mathbb{C}^\times$-fixed points in the moduli space $\mathcal{M}_{K,N}$ of $SL(K)$-Higgs bundles on $\mathbb{CP}^1$ with irregular singularity at infinity and certain representations of the $\mathcal{W}_K$-algebra. It introduces a regulated $L^2$-norm $\mu$ of the Higgs field and shows that $\mu = \frac{1}{12}(K-1 - c_{\text{eff}})$, where $c_{\text{eff}}$ is the effective central charge of the corresponding $\mathcal{W}$-representation, thereby linking geometric invariants to conformal field theory data.

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 $\lvert λ vert \sim \lvert z ^{N/K} \mathrm{d} z vert$, where $K$ and $N$ are coprime. $\mathcal{M}_{K,N}$ carries a $\mathbb{C}^ imes$-action analogous to the famous $\mathbb{C}^ imes$-action introduced by Hitchin on the moduli spaces of Higgs bundles over compact curves. The construction of this $\mathbb{C}^ imes$-action on $\mathcal{M}_{K,N}$ involves the rotation automorphism of the base $\mathbb{CP}^1$. We classify the fixed points of this $\mathbb{C}^ imes$-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 $μ= \frac{1}{12} \left(K - 1 - c_{\mathrm{eff}} ight)$, where $μ$ 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.

Motivation & Objective

• To classify the $\mathbb{C}^\times$-fixed points in the moduli space $\mathcal{M}_{K,N}$ of $SL(K)$-Higgs bundles on $\mathbb{CP}^1$ with irregular singularity at infinity.
• To define and compute a regulated $L^2$-norm $\mu$ of the Higgs field on these fixed points.
• To establish a 1-1 correspondence between these fixed points and representations of the $\mathcal{W}_K$-vertex algebra.
• To verify that the regulated norm $\mu$ matches the effective central charge $c_{\text{eff}}$ via the formula $\mu = \frac{1}{12}(K-1 - c_{\text{eff}})$.
• To analyze the Bialynicki-Birula stratification of $\mathcal{M}_{K,N}$ and its relation to filtered vector bundle structures.

Proposed method

• Introduce a $\mathbb{C}^\times$-action on $\mathcal{M}_{K,N}$ via the rotation automorphism of the base $\mathbb{CP}^1$.
• Classify fixed points of this $\mathbb{C}^\times$-action using the structure of good filtered Higgs bundles and the coprime condition on $K$ and $N$.
• Define a regulated $L^2$-norm $\mu$ of the Higgs field using harmonic metrics and asymptotic behavior at infinity.
• Construct a correspondence between fixed points and $\mathcal{W}_K$-representations via the central charge formula $\mu = \frac{1}{12}(K-1 - c_{\text{eff}})$.
• Analyze the Hitchin fibration on $\mathcal{M}_{K,N}$, particularly in the $K=2, N=3$ case, to study fiber structure and compactification via gauge transformations.
• Use gauge transformations to extend families of Higgs bundles to infinity in the Hitchin base, showing that the central fiber is a projective cubic curve with a cusp at the fixed point with $\mu = \frac{1}{20}$.

Experimental results

Research questions

• RQ1 How are the $\mathbb{C}^\times$-fixed points in $\mathcal{M}_{K,N}$ classified, and what structure do they possess?
• RQ2 What is the geometric meaning of the regulated $L^2$-norm $\mu$ of the Higgs field on these fixed points?
• RQ3 Is there a precise correspondence between the fixed points and representations of the $\mathcal{W}_K$-vertex algebra?
• RQ4 Does the formula $\mu = \frac{1}{12}(K-1 - c_{\text{eff}})$ hold, and how does it relate to minimal models in conformal field theory?
• RQ5 How does the Bialynicki-Birula stratification of $\mathcal{M}_{K,N}$ reflect the isomorphism classes of underlying filtered vector bundles?

Key findings

• The $\mathbb{C}^\times$-fixed points in $\mathcal{M}_{K,N}$ are in 1-1 correspondence with certain $\mathcal{W}_K$-algebra representations, as shown via the central charge formula.
• For the $K=2, N=3$ case, the two fixed points correspond to the $(2,5)$ Virasoro minimal model with $c_{\text{eff}} = -\frac{22}{5}$ and $\frac{2}{5}$, matching the values of $\mu = \frac{9}{20}$ and $\frac{1}{20}$.
• The regulated $L^2$-norm $\mu$ is computed via $\mu = \frac{1}{12}(K-1 - c_{\text{eff}})$, and this formula is verified explicitly in the $K=2, N=3$ case.
• Each fiber of the Hitchin fibration $\pi^{-1}(u)$ in $\mathcal{M}_{2,3}$ is a projective cubic curve, with the central fiber $u=0$ being a cuspidal cubic containing both fixed points.
• The point at infinity in the fiber corresponds to the fixed point with $\mu = \frac{1}{20}$, which is the cusp of the central fiber.
• The conjecture is supported that $\mu$ extends to a moment map for the $U(1)$-action on $\mathcal{M}_{2,3}$, with the maximum value $\mu = \frac{9}{20}$ attained at the $\mathcal{M}^{\text{small}}_{2,3}$ stratum.