[Paper Review] A remark on the space of metrics having non-trivial harmonic spinors

Nils Waterstraat|arXiv (Cornell University)|Jun 3, 2012

Advanced Operator Algebra Research8 references4 citations

TL;DR

This paper provides a concise proof that for a closed spin manifold $M$ of dimension $n \equiv 3 \mod 4$, the space of Riemannian metrics with invertible Dirac operator is either empty or has infinitely many path components. Using spectral flow and elementary homotopy theory, the author rederives and simplifies earlier results by Bär, showing that the index-theoretic obstruction to invertibility leads to a non-trivial topological structure in the space of metrics.

ABSTRACT

Let M be a closed spin manifold of dimension congruent to 3 modulo 4. We give a simple proof of the fact that the space of metrics on M with invertible Dirac operator is either empty or it has infinitely many path components.

Motivation & Objective

• To provide a simple, self-contained proof of the infinite path component structure of the space of metrics with invertible Dirac operator on closed spin manifolds with $n \equiv 3 \mod 4$.
• To re-derive and streamline earlier results by Bär (1996) and Dahl (2008) using spectral flow and homotopy-theoretic arguments.
• To establish that the set $R^{\textup{inv}}(M)$ of metrics with invertible Dirac operator is either empty or has infinitely many path components, using elementary topological tools.
• To extend the argument to twisted Dirac operators and conjecture that the corresponding space of invertible metrics also has infinitely many components.

Proposed method

• The proof uses spectral flow to track the change in the number of eigenvalues of the Dirac operator crossing zero as metrics are varied.
• It applies a homotopy-theoretic invariant $\Gamma(\gamma)$ that counts the net number of eigenvalues crossing zero in a given interval along a path $\gamma$ in the space of metrics.
• By constructing a sequence of paths $\gamma_k$ starting at a fixed metric $g_0 \in R^{\textup{inv}}(M)$, the method ensures $\Gamma(\gamma_k)$ are all distinct, implying the endpoints lie in different path components.
• The construction relies on the conformal invariance of the Dirac operator and the existence of a family of metrics on $S^n$ with eigenvalues that vary linearly across $[-2,2]$.
• The argument uses the Atiyah-Singer index theorem implicitly through the spectral behavior of the Dirac operator under metric deformation.
• A contradiction is derived by assuming two such metrics are in the same path component, which would force $\Gamma(\gamma_i) = \Gamma(\gamma_j)$, contradicting the construction.

Experimental results

Research questions

• RQ1 Does the space of Riemannian metrics with invertible Dirac operator on a closed spin manifold of dimension $n \equiv 3 \mod 4$ have infinitely many path components?
• RQ2 Can the infinite path component structure of $R^{\textup{inv}}(M)$ be established using spectral flow and elementary homotopy theory rather than advanced surgery theory?
• RQ3 Is the topological complexity of $R^{\textup{inv}}(M)$ preserved under twisting by a fixed connection on a vector bundle?
• RQ4 Can the spectral flow invariant $\Gamma(\gamma)$ be used to distinguish path components in the space of metrics with invertible Dirac operator?

Key findings

• The space $R^{\textup{inv}}(M)$ of Riemannian metrics with invertible Dirac operator on a closed spin manifold $M$ with $n \equiv 3 \mod 4$ is either empty or has infinitely many path components.
• The proof uses spectral flow to define a homotopy invariant $\Gamma(\gamma)$ that takes integer values and is additive under concatenation of paths.
• By constructing a sequence of paths $\gamma_k$ with strictly increasing $\Gamma(\gamma_k)$, the endpoints $\gamma_k(1)$ are shown to lie in distinct path components.
• The construction relies on the existence of a continuous family of metrics on $S^n$ whose Dirac operator has a single eigenvalue varying linearly from $-1$ to $1$ in $[-2,2]$.
• The argument shows that any two such metrics with different $\Gamma$-values cannot be connected by a path in $R^{\textup{inv}}(M)$, proving the infinite component structure.
• The result is extended to twisted Dirac operators via Bär’s 1997 results, leading to the conjecture that $R^{\textup{inv}}_{(F,\nabla)}(M)$ is either empty or has infinitely many components.