# susy QM and Atiyah-Singer index theorem

+ 3 like - 0 dislike
1786 views

Consider maps $t\mapsto x^i(t)$ from circle to some Riemannian (spin) manifold and lagrangian
$$\mathcal L = \frac12 g_{ij}(x) \partial_t x^i \partial_t x^j + \frac12 g_{ij} \psi^j \left(\delta^i_k \partial_t + \Gamma^i_{mk} \partial_t x^m\right)\psi^k,$$
where $\psi^k$ are real Grassmann variables. This is supersymmetric under
$$\delta x^i =\epsilon \psi^i, \qquad \delta\psi^i=\epsilon \partial_t x^i.$$
We want to compute
$$\operatorname{Tr}(-)^Fe^{-\beta H}=\int_\text{periodic}[dx][d\psi] \exp \left(-\int_0^\beta dt \mathcal L\right),$$
in the limit $\beta \to 0$.

My question is: to see that the lagrangian for quadratic fluctuations around constant configurations $\xi^i=x^i -x^i_0$, $\eta^i=\psi^i-\psi^i_0$ (namely the one surviving in $\beta \to 0$ limit) is
$$\mathcal L^{(2)}=\frac12 g_{ij}(x_0) \partial_t \xi^i \partial_t \xi^j - \frac14 R_{ijkl}\xi^i\partial_t\xi^j \psi_0^k\psi_0^l +\frac{i}2\eta^a\partial_t\eta^a,$$
what are the right substitutions to make, besides using Riemann normal coordinates and vielbein $e_i^a e_j^b \eta_{ab}=g_{ij}$?

I'm a bit confused. There are infinite number of constant configurations, so what exactly are $x_0$ and $\psi_0$?
Right, just fix one of them, and expand around it; the path integral measure splits as $[dx][d\psi]=[dx_0][d\psi_0][d\xi][d\eta]$.
To be honest I can't see how $[dx][d\psi]=[dx_0][d\psi_0][d\xi][d\eta]$. Let's take lattice regularization then the integral is just a finite dimensional multiple integral, then clearly for each fixed $x_0$ and $\psi_0$, we have $[dx][d\psi]=[d\xi][d\eta]$, and then shouldn't $[dx_0][d\psi_0][d\xi][d\eta]=\text{some volume}\cdot[dx][d\psi]$?
What I meant is: expand $x=x_0 + \xi$ around constant configuration $x_0(t)=x_0$, and the same for fermions. Then the measure splits as $[dx]=\mathcal N [dx_0][d\xi]$, where $\mathcal N$ is unknown normalization; my point is that, whatever $\beta$-dependent substitution you make, it should leave that measure invariant. Do you agree?
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverflo$\varnothing$Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.