η‐invariants and determinant lines
Xianzhe Dai,Daniel S. Freed +1 more
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In this article, a variational formula and a gluing law for the η−invariant of an odd dimensional manifold with boundary is investigated. But the dependence of the δ-invariance on the trivialization of the kernel of the Dirac operator on the boundary is best encoded by the statement that the exponential of the Δ-invarisant lives in the determinant line of the boundary.Abstract:
The η‐invariant of an odd dimensional manifold with boundary is investigated. The natural boundary condition for this problem requires a trivialization of the kernel of the Dirac operator on the boundary. The dependence of the η‐invariant on this trivialization is best encoded by the statement that the exponential of the η‐invariant lives in the determinant line of the boundary. Our main results are a variational formula and a gluing law for this invariant. These results are applied to reprove the formula for the holonomy of the natural connection on the determinant line bundle of a family of Dirac operators, also known as the ‘‘global anomaly formula.’’ The ideas developed here fit naturally with recent work in topological quantum field theory, in which gluing (which is a characteristic formal property of the path integral and the classical action) is used to compute global invariants on closed manifolds from local invariants on manifolds with boundary.read more
Citations
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Anomalies in string theory with D-branes
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References
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