Abstract
Abstract We give a very simple derivation of the Atiyah-Patodi-Singer (APS) index theorem and its small generalization by using the path integral of massless Dirac fermions. It is based on the Fujikawa’s argument for the relation between the axial anomaly and the Atiyah-Singer index theorem, and only a minor modification of that argument is sufficient to show the APS index theorem. The key ingredient is the identification of the APS boundary condition and its generalization as physical state vectors in the Hilbert space of the massless fermion theory. The APS η-invariant appears as the axial charge of the physical states.
Highlights
We give a very simple derivation of the Atiyah-Patodi-Singer (APS) index theorem and its small generalization by using the path integral of massless Dirac fermions
For the purpose of this paper, the important fact shown in [26] is that the APS boundary condition corresponds to the vacuum state Ω| of massless fermions
We will use this fact in our derivation of the APS index theorem
Summary
Let us recall some facts about path integrals. They are very elementary, but are crucial for the purpose of the present paper. The relation between a physical state β| and the corresponding boundary condition β at τ = 0 can be worked out as in [26]. For the purpose of this paper, the important fact shown in [26] is that the APS boundary condition corresponds to the vacuum state Ω| of massless fermions. We will use this fact in our derivation of the APS index theorem. In the Euclidean path integral on the manifold X, we automatically get the vacuum |Ω at τ → +∞ This is the physical reason that the APS boundary condition is realized by the vacuum state
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