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Equivariant Cerf theory and perturbative $SU(n)$ Casson invariants

Shaoyun Bai, Boyu Zhang

TL;DR

This work develops an equivariant Cerf theory for $G$-Morse functions and transfers it to gauge theory to define perturbative $SU(n)$ Casson invariants on integer homology spheres for all $n\ge 3$. It introduces an $H$-equivariant spectral flow and extends the representation framework to $ ilde{\mathcal{R}}_G$ to handle gauge-variant indexing via Chern–Simons corrections, while holonomy perturbations ensure transversality of perturbed moduli spaces. The main contributions include a complete description of perturbation-induced bifurcations, a finite-dimensional reduction argument (Kuranishi-type), and an explicit $n=4$ formula, with a robust program connecting these invariants to universal equivariant Euler characteristics. The results offer a structurally rich, perturbation-independent counting framework for $SU(n)$ gauge moduli, with potential links to generalized Seiberg–Witten theories and equivariant topology.

Abstract

We develop an equivariant Cerf theory for Morse functions on finite-dimensional manifolds with group actions, and adapt the technique to the infinite-dimensional setting to study the moduli space of perturbed flat $SU(n)$-connections. As a consequence, we prove the existence of perturbative $SU(n)$ Casson invariants on integer homology spheres for all $n\ge 3$, and write down an explicit formula when $n=4$. This generalizes the previous works of Boden and Herald.

Equivariant Cerf theory and perturbative $SU(n)$ Casson invariants

TL;DR

This work develops an equivariant Cerf theory for -Morse functions and transfers it to gauge theory to define perturbative Casson invariants on integer homology spheres for all . It introduces an -equivariant spectral flow and extends the representation framework to to handle gauge-variant indexing via Chern–Simons corrections, while holonomy perturbations ensure transversality of perturbed moduli spaces. The main contributions include a complete description of perturbation-induced bifurcations, a finite-dimensional reduction argument (Kuranishi-type), and an explicit formula, with a robust program connecting these invariants to universal equivariant Euler characteristics. The results offer a structurally rich, perturbation-independent counting framework for gauge moduli, with potential links to generalized Seiberg–Witten theories and equivariant topology.

Abstract

We develop an equivariant Cerf theory for Morse functions on finite-dimensional manifolds with group actions, and adapt the technique to the infinite-dimensional setting to study the moduli space of perturbed flat -connections. As a consequence, we prove the existence of perturbative Casson invariants on integer homology spheres for all , and write down an explicit formula when . This generalizes the previous works of Boden and Herald.

Paper Structure

This paper contains 28 sections, 56 theorems, 360 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $G$--Morse functions
  3. The equivariant topology of $M$
  4. $G$--Morse functions
  5. Equivariant indices of $G$--Morse functions
  6. Equivariant Cerf theory
  7. Local bifurcation models
  8. Statement of the equivariant Cerf theorem
  9. Preliminaries on linear algebra
  10. Transversality
  11. Proof Theorem \ref{['thm_equivairant_cerf_finite_dim']}
  12. Invariant counting of critical orbits
  13. Computations
  14. Extending $\mathcal{R}_G$ over $\mathbb{R}$
  15. Holonomy perturbations and transversality
  16. ...and 13 more sections

Key Result

Theorem 1.1

For every $n\ge 3$, there exists a function with the following property. Suppose $Y$ is an integer homology sphere, let be the trivial $\mathop{\mathrm{SU}}\nolimits(n)$--bundle over $Y$, let $\theta$ be the trivial connection of $P$. Then for a generic holonomy perturbation $\pi$, the critical set of the perturbed Chern-Simons functional consists of finitely many non-degenerate orbits. Let $\ma

Figures (2)

  • Figure 1: A possible bifurcation diagram for $\mathop{\mathrm{SU}}\nolimits(5)$
  • Figure 2: Critical orbits of $f_t$

Theorems & Definitions (175)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • ...and 165 more