# $\eta$-invariants of Riemann Surface

I am curious about a concrete computation of $\eta$-invariants for Riemann surface, e.g. Torus.

Is there any nice review or notes talking about the computation? Or is it possible to express it as some topological data, analogous to 3d case?

A related post to this subject: $\eta$ invariants of Pin+ manifolds $\mathbb{RP}^{8k}$

• Kirby-Taylor ( www3.nd.edu/~taylor/old/papers/PSKT.pdf, see Lemma 3.6) describes how to compute an invariant for surfaces with Pin$^-$ structures, as the Gauss sum of a quadratic refinement on the first homology group. I think it is basically the $\eta$ invariant, in light of the relation to fermion SPT phases. Jul 13 '15 at 15:55
• Recently I got some more understanding on $\eta$ invariant, it can be computed directly with the knowledge of the spectrum of, e.g., Dirac operator. e.g, the torus certainly provides a symmetric spectrum for Dirac operator, and therefore has vanishing $\eta$ invariants. Some less trivial examples are real projective space at even dimension, which are unorientable. See the post cited and comments there. Dec 11 '15 at 2:10
• Isn't it defined in terms of the spectrum of Dirac operators? So if we know the spectrum, we can compute the invariant. Dec 11 '15 at 2:32
• @MengCheng yes, that is one direct way to compute that, for example you can compute $\eta$ for $\mathbb{RP}^{2l}$. Other way to compute that is by applying APS index theorem, $e.g.,$ constructing a higher dimensional manifold bounding the manifold you'd like to compute. Dec 11 '15 at 5:31
• @MengCheng maybe I didn't understand your comment? I guess you might ask whether it has be defined for Dirac operator? if so, the answer is no, and it could be defined for more general operators. Dec 11 '15 at 5:35