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The $S^1$-Equivariant signature for semi-free actions as an index formula

Juan Camilo Orduz

TL;DR

The paper constructs an induced Dirac–Schrödinger-type operator on the orbit space $M_0/S^1$ of a semi-free $S^1$-action and proves that its index realizes the $S^1$-equivariant signature defined by Lott. By applying the Brüning–Heintze push-down framework, it pushes down a suitable operator from the original manifold, analyzes its mean curvature and cone-data near the fixed-point set, and obtains a self-adjoint, discrete operator whose chiral index matches the signature in the Witt case. The Witt condition ensures vanishing eta-terms and yields the index formula $\text{ind}(\mathscr{D}^+) = \sigma_{S^1}(M) = \int_{M_0/S^1} L(T(M_0/S^1), g^{T(M_0/S^1)})$, connecting analysis on the quotient with the intersection-signature of the stratified quotient. For the non-Witt case the operator remains essentially self-adjoint and its index is defined, with conjectured equivalence to $\sigma_{S^1}(M)$, extending the analytic realization beyond Witt spaces.

Abstract

John Lott defined an integer-valued signature $σ_{S^1}(M)$ for the orbit space of a compact orientable manifold with a semi-free $S^1$-action but he did not construct a Dirac-type operator which has this signature as its index. We construct such operator on the orbit space and we show that it is essentially unique and that its index coincides with Lott's signature, at least when the stratified space satisfies the so-called Witt condition. For the non-Witt case, this operator remains essentially self-adjoint (in contrast to the Hodge de-Rham operator) and it has a well defined index which we conjecture will also compute $σ_{S^1}(M)$. This article is a condensed version of the original author's PhD Thesis where the theory of induced Dirac-Schrödinger-type operators is developed in detail.

The $S^1$-Equivariant signature for semi-free actions as an index formula

TL;DR

The paper constructs an induced Dirac–Schrödinger-type operator on the orbit space of a semi-free -action and proves that its index realizes the -equivariant signature defined by Lott. By applying the Brüning–Heintze push-down framework, it pushes down a suitable operator from the original manifold, analyzes its mean curvature and cone-data near the fixed-point set, and obtains a self-adjoint, discrete operator whose chiral index matches the signature in the Witt case. The Witt condition ensures vanishing eta-terms and yields the index formula , connecting analysis on the quotient with the intersection-signature of the stratified quotient. For the non-Witt case the operator remains essentially self-adjoint and its index is defined, with conjectured equivalence to , extending the analytic realization beyond Witt spaces.

Abstract

John Lott defined an integer-valued signature for the orbit space of a compact orientable manifold with a semi-free -action but he did not construct a Dirac-type operator which has this signature as its index. We construct such operator on the orbit space and we show that it is essentially unique and that its index coincides with Lott's signature, at least when the stratified space satisfies the so-called Witt condition. For the non-Witt case, this operator remains essentially self-adjoint (in contrast to the Hodge de-Rham operator) and it has a well defined index which we conjecture will also compute . This article is a condensed version of the original author's PhD Thesis where the theory of induced Dirac-Schrödinger-type operators is developed in detail.

Paper Structure

This paper contains 28 sections, 56 theorems, 161 equations, 2 figures.

Table of Contents

  1. Brüning-Heintze contruction
  2. The isomorphism $\Phi$
  3. Induced operators on the principal orbit type
  4. Lott's $S^1$-equivariant signature formula
  5. Definition of the equivariant $S^1$-signature
  6. The equivariant $S^1$-signature formula for semi-free actions
  7. Sketch of the proof of the signature formula
  8. The Witt condition
  9. Induced Dirac-Schrödinger operator on $M_0/S^1$
  10. The mean curvature $1$-form
  11. The operator $T(D)$
  12. Decomposition of $S^1$-invariant differential forms
  13. Construction of the bundle $F$
  14. Description of the isomorphism $\Phi$
  15. Description of the operator $S(D)$
  16. ...and 13 more sections

Key Result

Theorem 1

The operator $\mathscr{D}':\Omega_c(M_0/S^1)\longrightarrow \Omega_c(M_0/S^1)$ is given explicitly by where $\bar{\kappa}$ is the mean curvature form and $\widehat{c}(\bar{\varphi}_0)$ is a bounded endomorphism. In addition $\mathscr{D}'$ satisfies:

Figures (2)

  • Figure 1: Mapping cylinder of the $\mathbb{C}P^N$-fibration $\pi_{\mathcal{F}}:\mathcal{F}\longrightarrow F.$
  • Figure 2: Decomposition of $M/S^1$ as a manifold with boundary and a $t$-neighborhood of the fixed point set.

Theorems & Definitions (97)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Example 1.2: Exterior Algebra
  • Lemma 1.3: JO17
  • Theorem 1.4: BH78
  • Lemma 1.5: BH78
  • Proposition 1.6: BH78
  • Proposition 1.7: BH78
  • Proposition 2.1: L00
  • ...and 87 more