Gerbes in Renormalization and Quantization of Infinite-Dimensional Moduli Spaces

The main difficulty in studying infinite-dimensional moduli spaces is the extraction of meaningful finite quantities by renormalization. We propose the application of new methods developed in string topology and higher category theory for an understanding from a different viewpoint. Recent developments established the mathematical machinery for more geometric approaches and our objective is to relate this approach to other techniques in string topology, particularly higher line bundles (also called gerbes). 

The main goals may be summarized as follows.

  1. Gerbes and Quillen metrics. Use gerbes to study Quillen metrics for families of \(\overline{\partial}\)-operators. Investigate whether in this context gerbes provide a geometric explanation for traditional \(\zeta\)-renormalization methods.
  2. Analytic and topological models for gerbes. Understand the equivalence between analytical and topological models for gerbes. In particular, describe their relationship without passing through a classification by Dixmier-Douady classes. Exhibit the role played by gerbes in geometric quantization. Study Hitchin's projective volume form in Yang-Mills theory from the perspective of gerbes. Consider also the moduli space of extremal metrics from this point of view.
  3. Regularized determinants and the Witten genus. Reconsider recent interpretations of the Witten genus, working with the derived space of holomorphic curves.


We develop a categorical index calculus for elliptic symbol families. The categorified index problems we consider are a secondary version of the traditional problem of expressing the index class in K-theory in terms of differential-topological data. They include orientation problems for moduli spaces as well as similar problems for skew-adjoint and self-adjoint operators. The main result of this paper is an excision principle which allows the comparison of categorified index problems on different manifolds. Excision is a powerful technique for actually solving the orientation problem; applications appear in the companion papers arXiv:1811.01096, arXiv:1811.02405, and arXiv:1811.09658.


Related project(s):
33Gerbes in renormalization and quantization of infinite-dimensional moduli spaces

 Let X be a compact manifold, D a real elliptic operator on X, G a Lie group, P a principal G-bundle on X, and B_P the infinite-dimensional moduli space of all connections on P modulo gauge, as a topological stack. For each connection \nabla_P, we can consider the twisted elliptic operator on X. This is a continuous family of elliptic operators over the base B_P, and so has an orientation bundle O^D_P over B_P, a principal Z_2-bundle parametrizing orientations of KerD^\nabla_Ad(P) + CokerD^\nabla_Ad(P) at each \nabla_P. An orientation on (B_P,D) is a trivialization of O^D_P.


In gauge theory one studies moduli spaces M of connections \nabla_P on P satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds (X, g). Under good conditions M is a smooth manifold, and orientations on (B_P,D) pull back to

orientations on M in the usual sense of differential geometry.


This is important in areas such as Donaldson theory, where one needs an orientation on M

to define enumerative invariants.


We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on (B_P,D), after fixing some algebro-topological information on X. We use these to construct canonical orientations on gauge theory moduli spaces, including new results for moduli spaces of flat connections on 2- and 3-manifolds,

instantons, the Kapustin-Witten equations, and the Vafa-Witten equations on 4-manifolds, and the Haydys-Witten equations on 5-manifolds.


Related project(s):
33Gerbes in renormalization and quantization of infinite-dimensional moduli spaces

 Suppose (X, g) is a compact, spin Riemannian 7-manifold, with Dirac operator D. Let G be SU(m) or U(m), and E be a rank m complex bundle with G-structure on X. Write  B_E for the infinite-dimensional moduli space of connections on E, modulo gauge. There is a natural principal Z_2-bundle O^D_E on B_E parametrizing orientations of det D_Ad A for twisted elliptic operators D_Ad A at each [A] in  B_E. A theorem of Walpuski shows O^D_E is trivializable.


We prove that if we choose an orientation for det D, and a flag structure on X in the sense of Joyce arXiv:1610.09836, then we can define canonical trivializations of O^D_E for all such bundles E on X, satisfying natural compatibilities.


Now let (X,\varphi,g) be a compact G_2-manifold, with d(*\varphi)=0. Then we can consider moduli spaces  M_E^G_2 of G_2-instantons on E over X, which are smooth manifolds under suitable transversality conditions, and derived manifolds in general. The restriction of O^D_E to M_E^G_2 is the Z_2-bundle of orientations on M_E^G_2. Thus, our theorem induces canonical orientations on all such G_2-instanton moduli spaces  M_E^G_2.


This contributes to the Donaldson-Segal programme arXiv:0902.3239, which proposes defining enumerative invariants of G_2-manifolds (X,\varphi,g) by counting moduli spaces  M_E^G_2, with signs depending on a choice of orientation. This paper is a sequel to Joyce-Tanaka-Upmeier arXiv:1811.01096, which develops the general theory of orientations on gauge-theoretic moduli spaces, and gives applications in dimensions 3,4,5 and 6.


Related project(s):
33Gerbes in renormalization and quantization of infinite-dimensional moduli spaces

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Team Members

Dr. Markus Upmeier
Project leader
Universität Augsburg

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