# Elliptic genus for manifolds with boundary

Let M be a closed spin manifold of dimension $d$. One form of the elliptic genus of $M$ is $$F(q)=q^{-d/8} \hat A(M) {\rm ch} \otimes_{k=1/2,3/2,\cdots} \Lambda_{q^k}T \otimes_{\ell=1}^\infty S_{q^\ell}T [M]$$ where the notation follows that of E. Witten, The Index of the Dirac Operator in Loop Space." The coefficient of $q^{n/2-d/8}$ is the index of a Dirac operator $D_n$ which acts on sections of $S \otimes T_{R_n}$ where $S$ is the spinor bundle and $T_{R_n}$ is the bundle associated to a representation $R_n$ of $Spin(d)$ with the first few representations being $$R_0=1, \qquad R_1=T, \qquad R_2=\Lambda^2 T \oplus T$$ where $T$ is the fundamental (vector) representation.

I'm interested in the generalization of the elliptic genus to manifolds with boundary. In the actual application I'm interested in one eventually takes the boundary to infinity to obtain a noncompact manifold, but I'd be happy to understand the situation for a compact manifold with boundary first. The index of the Dirac operator in such a situation acquires boundary corrections of the form $$CS[ \partial M] - \frac{1}{2}(\eta(0)+h)$$ where $h$ is the number of zero modes of the Dirac operator on $\partial M$ and $\eta(0)$ is the $\eta$ invariant. In the examples I'm interested in I believe the Chern-Simons contributions $CS[\partial M]$ vanish.

Summing up these boundary contributions to the index of $D_n$ weighted by $q^{n/2-d/8}$ leads to a "boundary" contribution to the elliptic genus on manifolds with boundary with the "bulk" contribution given by $F(q)$. My questions are whether this variant of the elliptic genus has been studied and if so where, whether this leads to interesting invariants of manifolds with boundary, and whether the modular properties of the bulk and boundary contributions are known.

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## 2 Answers

This, or a similar variant has been studied in Secondary Invariants for String Bordism and tmf and The f-invariant and index theory. The deviation from modularity caused by the boundary gives the interesting invariant of the boundary.

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I studied such modular invariants in my thesis, Modular invariants for manifolds with boundary, http://www.tdx.cat/handle/10803/3071 , http://www.tdx.cat/bitstream/handle/10803/3071/migc1de2.pdf , http://www.tdx.cat/bitstream/handle/10803/3071/migc2de2.pdf , and never find the time to get back to them, they are very interesting.

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Thank you for the reference. –  Jeff Harvey Jan 19 '12 at 13:46
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