# nLab fermionic path integral

superalgebra

and

supergeometry

## Applications

#### Integration theory

integration

analytic integrationcohomological integration
measureorientation in generalized cohomology
Riemann/Lebesgue integration, of differential formspush-forward in generalized cohomology/in differential cohomology

# Contents

## Idea

The path integral over the fermionic variables of the standard kinetic action functional for fermions (see for instance spinors in Yang-Mills theory) has a well-defined meaning as a section of the Pfaffian line bundle of the corresponding Dirac operator.

## Setup

For definiteness, we consider a sigma model quantum field theory on a worldvolume $\Sigma$ and pseudo-Riemannian target spacetime $X$ with fields

• bosons: smooth functions $\varphi :\Sigma \to X$

• fermions (gravitino): $\psi \in \Gamma \left(S\left(\Sigma \right)\otimes {\varphi }^{*}TM\right)$, sections of the tensor product of a spinor bundle on $\Sigma$ and the pullback of the cotangent bundle of $X$ along the given bosonic field $\varphi$.

$S:\left(\varphi ,\psi \right)↦{S}^{\mathrm{bos}}\left(\varphi \right)+{S}_{\varphi }^{\mathrm{ferm}}\left(\psi \right)$S : (\phi, \psi) \mapsto S^{bos}(\phi) + S^{ferm}_{\phi}(\psi)

is the sum of the

• bosonic action

${S}^{\mathrm{bos}}\right)\left(\varphi \right)={\int }_{\Sigma }⟨d\varphi \wedge \star d\varphi ⟩$S^{bos})(\phi) = \int_\Sigma \langle d \phi \wedge \star d \phi\rangle
• fermionic action

${S}_{\varphi }^{\mathrm{ferm}}\left(\psi \right)={\int }_{\Sigma }⟨\psi ,{D}_{\varphi }\psi ⟩$S^{ferm}_\phi(\psi) = \int_\Sigma \langle \psi, D_\phi \psi\rangle

where ${D}_{\varphi }$ is a Dirac operator on $S\otimes {\varphi }^{*}TM$ (the Dirac operator on $S$ twisted by the pullback of the Levi-Civita connection on ${T}^{*}X$ ).

One imagines than that the hypothetical path integral symboilically written as

$\int \left[d\varphi \right]\left[d\psi \right]\mathrm{exp}\left(S\left(\psi \right)\left(\varphi ,\psi \right)\right)$\int [d \phi] [d \psi] \exp(S(\psi)(\phi,\psi))

can be computed in two steps

$\cdots =\int \left[d\varphi \right]\mathrm{exp}\left({S}^{\mathrm{bos}}\left(\varphi \right)\right)\left(\int \left[d\psi \right]\mathrm{exp}\left({S}_{\varphi }^{\mathrm{ferm}}\left(\psi \right)\right)\right)$\cdots = \int [d \phi] \exp(S^{bos}(\phi)) \left( \int [d \psi] \exp(S^{ferm}_\phi(\psi)) \right)

by first computing the integral over the fermions

$\mathrm{pfaff}\left(\varphi \right):=\int \left[d\psi \right]\mathrm{exp}\left({S}_{\varphi }^{\mathrm{ferm}}\left(\psi \right)\right)$pfaff(\phi) := \int [d \psi] \exp(S^{ferm}_\phi(\psi))

and then inserting this into the remaining bosonic integral. Now, as opposed to the bosonic integral, this fermionic integral can be given well-defined sense by interpreting it as an infinite-dimensional Berezinian integral.

However, while this makes the expression well defined, the result is not quite a function of $\varphi$, but is instead a section $\mathrm{pfaff}$ of a Pfaffian line bundle

$\begin{array}{ccc}& & \mathrm{Pfaff}\\ {}^{\mathrm{pfaff}:={Z}_{\mathrm{eff}}^{\mathrm{ferm}}}↗& ↓\\ {C}^{\infty }\left(\Sigma ,X\right)& =& {C}^{\infty }\left(\Sigma ,X\right)\end{array}$\array{ && Pfaff \\ {}^{pfaff := Z_{eff}^{ferm}}\nearrow & \downarrow \\ C^{\infty}(\Sigma, X) &= & C^{\infty}(\Sigma, X) }

over the space of bosonic field configurations.

If $\mathrm{Pfaff}$ is not isomorphic to the trivial line bundle, we say the system has a fermionic quantum anomaly. If instead $\mathrm{Pfaff}$ is trivializable, any choice of trivialization

$t:\mathrm{Paff}\stackrel{\simeq }{\to }{C}^{\infty }\left(\Sigma ,X\right)×ℂ$t : Paff \stackrel{\simeq}{\to} C^\infty(\Sigma, X) \times \mathbb{C}

makes the fermionic path integral into a genuine function

${Z}_{\mathrm{eff}}^{\mathrm{ferm}}:=\left(t\circ \mathrm{pfaff}\right):{C}^{\infty }\left(\Sigma ,X\right)\to ℂ\phantom{\rule{thinmathspace}{0ex}}.$Z_{eff}^{ferm} : = (t \circ pfaff) : C^\infty(\Sigma, X) \to \mathbb{C} \,.

Any such choice of $t$ is called a choice of quantum integrand.

With this one can then try to enter the remaining bosonic path integral

$\int \left[d\varphi \right]\mathrm{exp}\left({S}^{\mathrm{bos}}\left(\varphi \right)\right){Z}_{\mathrm{eff}}^{\mathrm{ferm}}\left(\varphi \right)$\int [d \phi] \exp(S^{bos}(\phi)) Z_{eff}^{ferm}(\phi)

## Pfaffian bundles

We are implicitly assuming that $\mathrm{dim}\Sigma =2$ or maybe $8n+2$ in the following. Needs to be generalized.

For $n\in ℕ$, there the square root of a skew symmetric $\left(n×n\right)$-matrix $D$ – the Pfaffian of the matric – can be understood as the Berezinian integral

$\mathrm{pfaff}\left(D\right)=\int \left[d\stackrel{⇀}{\theta }\right]\mathrm{exp}\left(⟨\theta ,D\theta ⟩\right)\in \mathrm{det}{ℝ}^{n}$pfaff(D) = \int [d \vec \theta] \exp( \langle \theta , D \theta\rangle) \in det \mathbb{R}^n

over the Grassmann algebra elements ${\theta }_{i}$. Written this way this is an element of the determinant line of ${ℝ}^{n}$: its identification with a number depends on the choice of basis for ${ℝ}^{n}$. For this case this is unproblematic, since there is a canonical choice of basis for the single vector space ${ℝ}^{n}$, but when $D$ instead depends on a parameter $\varphi$, then in general its Pfaffian can at best be a section of a determinant line bundle.

We now generalize this to the case that $D$ is not a finite-dimensional matrix, but a Dirac operator acting on spaces of sections of a spinor bundle. We discuss that we can reduce this “infinite-dimensional matrix” in a sense locally to a finite dimensional one in a consistent way, such that the above ordinary construction of Pfaffians applies.

In the above setup, write

${ℋ}_{\varphi }^{±}:=\Gamma \left({S}^{±}\otimes {\varphi }^{*}{T}^{*}X\right)$\mathcal{H}_\phi^{\pm} := \Gamma(S^{\pm} \otimes \phi^* T^* X)

for the space of spinor sections for given $\varphi :\Sigma \to X$. Then the choral Dirac operators a maps

${D}_{\varphi }^{±}:{ℋ}_{\varphi }^{±}\to {ℋ}_{\varphi }^{\mp }\phantom{\rule{thinmathspace}{0ex}}.$D_\phi^{\pm} : \mathcal{H}_\phi^{\pm} \to \mathcal{H}^{\mp}_\phi \,.

We also have a “quaternionic structure”

${J}_{\varphi }^{±}:{ℋ}_{\varphi }^{±}\to {ℋ}_{\varphi }^{\mp }$J_\phi^{\pm} : \mathcal{H}_\phi^{\pm} \to \mathcal{H}^{\mp}_\phi

Define then an open cover of the space ${C}^{\infty }\left(\Sigma ,X\right)$ of the space of bosonic fields with open sets ${U}_{\mu }$ for $\left(0\le \mu \right)$ given by

${U}_{\mu }:=\left\{\varphi \in {C}^{\infty }\left(\Sigma ,X\right)\mid \mu nin\mathrm{Spec}{D}_{\varphi }^{2}\right\}\phantom{\rule{thinmathspace}{0ex}},$U_\mu := \{ \phi \in C^\infty(\Sigma,X) | \mu \nin Spec D_\phi^2\} \,,

hence the collection of bosonic field configurations such that $\mu$ is not in the operator spectrum of the squared Dirac operator.

Over these open subsets we have the finite rank vector bundles

${ℋ}_{\varphi }^{\mu ±}:={\oplus }_{0\le ϵ\le \mu }\mathrm{Eig}\left({D}_{\varphi }^{2},ϵ\right)$\mathcal{H}_\phi^{\mu \pm} := \oplus_{0 \leq \epsilon \leq \mu} Eig(D_\phi^2, \epsilon)

of eigenspaces of ${D}_{\varphi }^{2}$ for eigenvalue?s bounded by $\mu$.

The Dirac operator that we are interested in is

${D}_{\varphi }^{\mu }:={J}_{\varphi }^{-}\circ {D}_{\varphi }^{+}:{ℋ}_{\varphi }^{\mu ,+}\to {ℋ}_{\varphi }^{\mu ,+}\phantom{\rule{thinmathspace}{0ex}}.$D_\phi^\mu := J_\phi^- \circ D_\phi^+ : \mathcal{H}_\phi^{\mu,+} \to \mathcal{H}_\phi^{\mu,+} \,.

This defines now a finite-dimensional matrix

$⟨-,{D}_{\varphi }^{\mu }-⟩$\langle -, D_\phi^\mu -\rangle

whose Berezinian integral is the Pfaffian

$\int \left[d\psi \right]\mathrm{exp}\left(⟨\psi ,{D}_{\varphi }^{\mu }\varphi ⟩\right)=\mathrm{pfaff}\left({D}_{\varphi }^{\mu }\right)\in \mathrm{det}{ℋ}_{\varphi }^{\mu \pi }\phantom{\rule{thinmathspace}{0ex}}.$\int [d \psi] \exp(\langle \psi , D^\mu_\phi \phi \rangle ) = pfaff(D^\mu_\phi) \in det \mathcal{H}^{\mu \pi}_\phi \,.

One shows that these constructions for each $\mu$ glue together to define

• a smooth line bundle $\mathrm{Pfaff}\to {C}^{\infty }\left(\Sigma ,X\right)$

• with a smooth section $\mathrm{pfaff}\left(D\right)$.

Moreover, there is canonically a hermitean metric? and a canonical unitary connection on a bundle (the Freed-Bismut connection?) on this bundle.

## Examples

For the sigma model describing the heterotic superstring propagating on a pseudo-Riemannian manifold $X$, the trivialization of the Pfaffian line bundle, hence the cancellation of its fermionic quantum anomaly is related to the existence of a (twisted) differential string structure on $X$. See there for more details.

Revised on November 7, 2012 22:53:44 by Urs Schreiber (82.169.65.155)