Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

On spin structures and orientations for gauge-theoretic moduli spaces (1908.03524v3)

Published 9 Aug 2019 in math.DG, math.AG, and math.AT

Abstract: Let $X$ be a compact manifold, $G$ a Lie group, $P \to X$ a principal $G$-bundle, and $\mathcal{B}P$ the infinite-dimensional moduli space of connections on $P$ modulo gauge. For a real elliptic operator $E\bullet$ we previously studied orientations on the real determinant line bundle over $\mathcal{B}P$. These are used to construct orientations in the usual sense on smooth gauge theory moduli spaces, and have been extensively studied since the work of Donaldson. Here we consider complex elliptic operators $F\bullet$ and introduce the idea of spin structures, square roots of the complex determinant line bundle of $F_\bullet$. These may be used to construct spin structures in the usual sense on smooth complex gauge theory moduli spaces. We study the existence and classification of such spin structures. Our main result identifies spin structures on $X$ with orientations on $X \times S1$. Thus, if $P \to X$ and $Q \to X \times S1$ are principal $G$-bundles with $Q|{X\times{1}} \cong P$, we relate spin structures on $(\mathcal{B}_P,F\bullet)$ to orientations on $(\mathcal{B}Q,E\bullet)$ for a certain class of operators $F_\bullet$ on $X$ and $E_\bullet$ on $X\times S1$. Combined with arXiv:1811.02405, we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups $G=U(m), SU(m)$. In a sequel arXiv:2001.00113 we apply this to define canonical orientation data for all Calabi-Yau 3-folds $X$ over the complex numbers, as in Kontsevich-Soibelman arXiv:0811.2435, solving a long-standing problem in Donaldson-Thomas theory.

Summary

We haven't generated a summary for this paper yet.