Mon, 06 Nov 2023
Laurence Mayther

There are two main methods of constructing compact manifolds with holonomy $G_2$, viz. resolution of singularities (first applied by Joyce) and twisted connect sum (first applied by Kovalev).  In the second case, there is a known invariant (the $\overline{\nu}$-invariant, introduced by Crowley–Goette–Nordström) which can, in many cases, be used to distinguish between different examples.  This invariant, however, has limitations; in particular, it cannot be computed on the $G_2$-manifolds constructed by resolution of singularities.


In this talk, I shall begin by discussing the notion of a $G_2$-manifold and the $\overline{\nu}$-invariant and its limitations.  In the context of this, I shall then introduce two new invariants of $G_2$-manifolds, termed $\mu$-invariants, and explain why these promise to overcome these limitations, in particular being well-suited to, and computable on, Joyce's examples of $G_2$-manifolds.  These invariants are related to $\eta$- and $\zeta$-invariants and should be regarded as the Morse indices of a $G_2$-manifold when it is viewed as a critical point of certain Hitchin functionals.  Time permitting, I shall explain how to prove a closed formula for the invariants on the orbifolds used in Joyce's construction, using Epstein $\zeta$-functions.

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