# SUSY QM and Atiyah-Singer index theorem

Consider maps $$t\mapsto x^i(t)$$ from circle to some Riemannian (spin) manifold and Lagrangian

$$\mathcal L = \frac12 g_{ij}(x) \partial_t x^i \partial_t x^j + \frac12 g_{ij} \psi^j \left(\delta^i_k \partial_t + \Gamma^i_{mk} \partial_t x^m\right)\psi^k, \tag{1}$$

where $$\psi^k$$ are real Grassmann variables. This is supersymmetric under

$$\delta x^i =\epsilon \psi^i, \qquad \delta\psi^i=\epsilon \partial_t x^i.\tag{2}$$

We want to compute

$$\operatorname{Tr}(-)^Fe^{-\beta H}=\int_\text{periodic}[dx][d\psi] \exp \left(-\int_0^\beta dt \mathcal L\right)\tag{3}$$

in the limit $$\beta \to 0$$.

My question is: to see that the lagrangian for quadratic fluctuations around constant configurations

$$\xi^i=x^i -x^i_0, \qquad\eta^i=\psi^i-\psi^i_0,\tag{4}$$

(namely the one surviving in $$\beta \to 0$$ limit) is

$$\mathcal L^{(2)}=\frac12 g_{ij}(x_0) \partial_t \xi^i \partial_t \xi^j - \frac14 R_{ijkl}\xi^i\partial_t\xi^j \psi_0^k\psi_0^l +\frac{i}2\eta^a\partial_t\eta^a.\tag{5}$$

What are the right substitutions to make, besides using Riemann normal coordinates and vielbein $$e_i^a e_j^b \eta_{ab}=g_{ij}$$?

References: Friedan and Windey or Alvarez-Gaume'.

1. OP wants to evaluate the index $${\rm Tr}(-)^F e^{-\beta H}=\int_{PBC}\![d\phi]\int_{PBC}[d\psi]~ e^{-S_E[\phi,\psi]},\tag{2.5}$$ in Ref. 1 with periodic boundary conditions (PBC) for both the boson $$\phi\equiv x$$ and the fermion $$\psi$$. Hence the corresponding Fourier components are labelled by integers$$^1$$ $$n\in\mathbb{Z}$$. One may argue that (2.5) does not depend on $$\beta$$, so that one may consider the $$\beta\to 0^+$$ limit.

2. Before discussing OP's model with 1 real fermion, we first discuss a model with 2 real fermions $$\psi_1^j$$ and $$\psi_2^j$$, or equivalently, 1 complex fermion $$\psi^j=(\psi_1^j+i\psi_2^j)/\sqrt{2}$$. Following Ref. 2, The Minkowski Lagrangian reads$$^2$$ $$L_M~=~ \frac{1}{2}g_{ij}(x)\dot{x}^i\dot{x}^j +\frac{i}{2}\sum_{\alpha\in\{1,2\}}\psi^i_{\alpha} g_{ij}(x) \left(\dot{\psi}^j_{\alpha}+\dot{x}^m\Gamma^j_{mk}(x)\psi^k_{\alpha}\right)$$ $$+\frac{1}{4}R_{ijk\ell}(x)\psi^i_1\psi^j_1\psi^k_2\psi^{\ell}_2\tag{74}$$ with real fermions in Ref. 2, or equivalently $$L_M~=~ \frac{1}{2}g_{ij}(\phi)\dot{\phi}^i\dot{\phi}^j +i\psi^{\ast i} g_{ij}(\phi) \frac{D\psi^j}{dt} -\frac{1}{4}R_{ijk\ell}(\phi)\psi^{\ast i}\psi^{\ast j}\psi^k\psi^{\ell}\tag{4.1}$$ with complex fermions in Ref. 1. We next Wick rotate $$\tau=it$$. The Euclidean action is $$S_E~=~\int_0^{\beta} d\tau ~ L_E.$$ Next split an arbitrary virtual path $$(x,\psi)$$ into zeromodes and non-constant modes: $$x^i(\tau)~=~x^i_0 +\xi^i(\tau), \qquad \psi^j_{\alpha}(\tau)~=~\psi^j_{\alpha 0}+\eta^j_{\alpha}(\tau).$$ The non-constant modes $$(\xi,\eta)$$ have Fourier components labelled by non-zero integers $$n\in\mathbb{Z}\backslash\{0\}$$. Let us rescale from old unprimed variables to new primed variables $$\tau~=~ \beta\tau^{\prime},\qquad L_E~=~ \frac{L^{\prime}_E}{\beta},\qquad S_E~=~ S^{\prime}_E, \qquad g_{ij}~=~ g^{\prime}_{ij},$$ $$x^i_0~=~ x^{i\prime}_0,\qquad \xi^i~=~ \sqrt{\beta}\xi^{i\prime},\qquad \psi^i_{\alpha 0} ~=~\frac{\psi^{i\prime}_{\alpha 0}}{\beta^{1/4}},\qquad \eta^i_{\alpha}~=~ \eta^{i\prime}_{\alpha}.$$ One may argue that the total path integral (super)Jacobian of this transformation is precisely $$1$$, e.g. via zeta-function regularization $$\prod_{n\in\mathbb{Z}} \sqrt{\beta} ~=~1.$$ We drop the primes from the notation from now on. Only quadratic terms in $$(\xi,\eta)$$ survive in the Euclidean Lagrangian $$L^{(2)}_E~=~ \frac{1}{2}g_{ij}(x_0)\dot{\xi}^i\dot{\xi}^j +\eta^{\ast i} g_{ij}(x_0)\dot{\eta}^j -\frac{1}{4}R_{ijk\ell}(x_0)\psi^{\ast i}_0\psi^{\ast j}_0\psi^k_0\psi^{\ell}_0 +{\cal O}(\beta).$$ The term $$\psi^{\ast i}_0 g_{ij}(x_0)\dot{\eta}^j$$ vanishes identically because of PBC.

The above scaling procedure $$\beta\to 0^+$$ is an example of localization to constant paths $$(x^i(\tau),\psi^j_{\alpha}(\tau))~\longrightarrow~ (x^i_0,\psi^j_{\alpha 0})$$ in a path integral. The full path integral is given by the Duistermaat-Heckman formula and the method of steepest descent for the Euclidean signature; or equivalently, the WKB stationary phase formula for Minkowski signature, cf. Refs. 4-5.

The fermionic d.o.f. mimic the de Rham complex. One may show that the Witten index (2.5) becomes the Euler characteristics, cf. the Chern-Gauss-Bonnet theorem, and eq. (4.3) in Ref. 1. $$\Box$$

3. We now return to OP's question. OP's Euclidean Lagrangian [which corresponds to eq. (73) in Ref. 2] is obtained by imposing the additional condition $$\psi_1^i~=~\psi_2^i~=~\psi^i/\sqrt{2}.$$ The quartic curvature term disappears by symmetry. The rescaling from old unprimed variables to new primed variables now reads $$\tau~=~ \beta\tau^{\prime},\qquad L_E~=~ \frac{L^{\prime}_E}{\beta},\qquad S_E~=~ S^{\prime}_E, \qquad g_{ij}~=~ g^{\prime}_{ij},$$ $$x^i_0~=~ x^{i\prime}_0,\qquad \xi^i~=~ \sqrt{\beta}\xi^{i\prime},\qquad \psi^i_0 ~=~\frac{\psi^{i\prime}_0}{\sqrt{\beta}},\qquad \eta^i~=~ \eta^{i\prime}.$$ One may argue that the total path integral (super)Jacobian of this transformation is still precisely $$1$$. Only quadratic terms in $$(\xi,\eta)$$ survive in the Euclidean Lagrangian $$L^{(2)}_E~=~ \frac{1}{2}g_{ij}(x_0)\dot{\xi}^i\dot{\xi}^j -\frac{1}{4}R_{ijab}(x_0)\psi^a_0\psi^b_0\xi^i\dot{\xi}^j+\frac{1}{2}\eta^a \dot{\eta}^a+{\cal O}(\beta).\tag{78}$$ cf. Ref. 2. Here $$a,b$$ are flat indices (aka. as vielbein indices). The calculation (78) is simplified by using Riemann normal coordinates, cf. $$\Gamma^{k}_{ij}(x)~\sim~ \frac{1}{2} R^k{}_{i\ell j}(x_0) \xi^{\ell} \tag{5.10}$$ in Ref. 3.

The fermions now form a Clifford algebra, so that the index (2.5) becomes the Dirac/A-roof genus, cf. eqs. (79)-(81) in Ref. 2. $$\Box$$

References:

1. L. Alvarez-Gaume, Supersymmetry and the Atiyah-Singer index theorem, Comm. Math. Phys. 90 (1983) 161.

2. L. Alvarez-Gaume and E. Witten, Gravitational Anomalies, Nucl.Phys. B234 (1984) 269.

3. D. Friedan and P. Windey, Supersymmetric Derivation of the Atiyah-Singer Index and the Chiral Anomaly, Nucl.Phys. B235 (1984) 395.

4. R.J. Szabo, Equivariant Localization of Path Integrals, hep-th/9608068.

5. S. Cordes, G. Moore and S. Ramgoolam, Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories, hep-th/9411210; Chapter 12.

6. H. Oogori, Lecture 8: Supersymmetry and index theorems.

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$$^1$$ Anti-periodic boundary conditions (ABC) corresponds to half-integer modes.

$$^2$$ We have corrected some index typos in eq. (74) of Ref. 2. Another typo in Ref. 2 is that $$\tau$$ in eq. (74) should be $$t$$. Later Ref. 2 forgets to remove an $$i$$ in front of the Wick-rotated kinetic term for the fermions in eq. (78).

• thanks for the nice answer; 1. is it also possible to see this using the lagrangian I write? 2. can you explain how you get 1 for the determinant of transformation: $\xi$ cancels $\psi_2$, but what happens for finite dimensional $\psi_0$? 3. I'm not sure I understand the rescalings: are you making a transformation, say, $\xi(\tau) = \sqrt{\beta}\xi'(\tau')$, and then explicitely rescaling the lagrangian?
– jj_p
Nov 11 '14 at 14:07
• 1. Yes. See update. 2. See update. 3. Yes. May 20 '19 at 11:02
• Additional references: 7. L.I. Nicolaescu, Notes on the Atiyah-Singer Index Theorem, 2013. Feb 18 at 13:59