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.


    Team Members

    Dr. Markus Upmeier
    Project leader
    Universität Augsburg

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