nLab fermionic path integral

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 $\phi : \Sigma \to X$

• fermions (gravitino): $\psi \in \Gamma(S(\Sigma) \otimes \phi^* T X)$, 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 $\phi$.

The action functional

$$S : (\phi, \psi) \mapsto S^{bos}(\phi) + S^{ferm}_{\phi}(\psi)$$

is the sum of the

• bosonic action

  $$S^{bos}(\phi) = \int_\Sigma \langle d \phi \wedge \star d \phi\rangle$$

• fermionic action

  $$S^{ferm}_\phi(\psi) = \int_\Sigma \langle \psi, D_\phi \psi\rangle$$

  where $D_\phi$ is a Dirac operator on $S \otimes \phi^* T M$ (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

$$\int [d \phi] [d \psi] \exp(S(\psi)(\phi,\psi))$$

can be computed in two steps

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

by first computing the integral over the fermions

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

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 $\phi$, but is instead a section $pfaff$ of a Pfaffian line bundle

$$\array{ && Pfaff \\ {}^{pfaff := Z_{eff}^{ferm}}\nearrow & \downarrow \\ C^{\infty}(\Sigma, X) &= & C^{\infty}(\Sigma, X) }$$

over the space of bosonic field configurations.

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

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

makes the fermionic path integral into a genuine function

$$Z_{eff}^{ferm} : = (t \circ pfaff) : C^\infty(\Sigma, X) \to \mathbb{C} \,.$$

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

$$\int [d \phi] \exp(S^{bos}(\phi)) Z_{eff}^{ferm}(\phi)$$

Pfaffian bundles

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

For $n \in \mathbb{N}$, there the square root of the determinant of a skew symmetric $(n\times n)$-matrix $D$ – the Pfaffian of the matrix – can be understood as the Berezinian integral

$$pfaff(D) = \int [d \vec \theta] \exp( \langle \theta , D \theta\rangle) \in det \mathbb{R}^n$$

over the Grassmann algebra elements $\theta_i$. Written this way this is an element of the determinant line of $\mathbb{R}^n$: its identification with a number depends on the choice of basis for $\mathbb{R}^n$. For this case this is unproblematic, since there is a canonical choice of basis for the single vector space $\mathbb{R}^n$, but when $D$ instead depends on a parameter $\phi$, 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

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

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

$$D_\phi^{\pm} : \mathcal{H}_\phi^{\pm} \to \mathcal{H}^{\mp}_\phi \,.$$

We also have a “quaternionic structure”

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

Define then an open cover of the space $C^\infty(\Sigma,X)$ of the space of bosonic fields with open sets $U_\mu$ for $(0 \leq \mu)$ given by

$$U_\mu := \{ \phi \in C^\infty(\Sigma,X) | \mu \nin Spec D_\phi^2\} \,,$$

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

$$\mathcal{H}_\phi^{\mu \pm} := \oplus_{0 \leq \epsilon \leq \mu} Eig(D_\phi^2, \epsilon)$$

of eigenspaces of $D_\phi^2$ for eigenvalues bounded by $\mu$.

The Dirac operator that we are interested in is

$$D_\phi^\mu := J_\phi^- \circ D_\phi^+ : \mathcal{H}_\phi^{\mu,+} \to \mathcal{H}_\phi^{\mu,+} \,.$$

This defines now a finite-dimensional matrix

$$\langle -, D_\phi^\mu -\rangle$$

whose Berezinian integral is the Pfaffian

$$\int [d \psi] \exp(\langle \psi , D^\mu_\phi \psi \rangle ) = pfaff(D^\mu_\phi) \in det \mathcal{H}^{\mu \pi}_\phi \,.$$

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

• a smooth line bundle $Pfaff \to C^\infty(\Sigma, X)$

• with a smooth section $pfaff(D)$.

Moreover, there is canonically a Hermitian 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.

Related concepts

• Nicolai map

• path integral

• spinors in Yang-Mills theory