Abstract
Details are given on the zeta function metric and connection on the determinant line bundle over the Grassmannian associated to boundary value problems over an interval.
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Notes
- 1.
Throughout, \(\mbox{tr}=\mbox{tr}_{V}\) denotes the trace on a finite-dimensional vector space \(V\), \({\rm Tr}\) an operator trace, and \(\text{Tr\,}_{{\mathcal{C}}},\text{Tr\,}_{\mathbf{z}}\) the canonical and zeta regularized traces.
- 2.
We differ from 3 by a sign since we use the form on the dual bundle.
- 3.
Note that for bundles \(\xi^{i}\), \(i=1,2,3\), with connection inducing connections \(\nabla^{i,j}\) on \(\text{Hom}(\xi^{i},\xi^{j})\) one has for respective sections \(A,B\) of \(\text{Hom}(\xi^{2},\xi^{3})\), \(\text{Hom}(\xi^{1},\xi^{2})\), \(\nabla^{1,3}(AB)=\nabla^{1,2}(A)B+A\nabla^{2,3}(B)\), for any choice of connection \(\nabla^{2}\) on \(\xi^{2}\).
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Scott, S. (2023). Curvature of the Determinant Line Bundle for Elliptic Boundary Problems over an Interval. In: Cintio, A., Michelangeli, A. (eds) Trails in Modern Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-031-44988-8_12
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