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 $\varphi :\Sigma \to X$

• fermions (gravitino): $\psi \in \Gamma (S(\Sigma )\otimes {\varphi }^{*}TM)$, 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 $\varphi$.

The action functional

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

is the sum of the

• bosonic action

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

• fermionic action

  $$S_{\varphi }^{\mathrm{ferm}}(\psi )={\int }_{\Sigma }⟨\psi ,D_{\varphi }\psi ⟩$$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

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

can be computed in two steps

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

$$\mathrm{pfaff}(\varphi ):=\int [d\psi ]\exp (S_{\varphi }^{\mathrm{ferm}}(\psi ))$$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

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

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

makes the fermionic path integral into a genuine function

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

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

Pfaffian bundles

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

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

$$\mathrm{pfaff}(D)=\int [d\stackrel{\rightharpoonup }{\theta }]\exp (⟨\theta ,D\theta ⟩)\in \det {ℝ}^n$$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

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

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

We also have a “quaternionic structure”

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

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

$$U_{\mu }:=\{\varphi \in C^{\mathrm{\infty }}(\Sigma ,X)\mid \mu nin\mathrm{Spec}{D}_{\varphi }^2\},$$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

$${\mathscr{H}}_{\varphi }^{\mu \pm }:={\oplus }_{0\le ϵ\le \mu }\mathrm{Eig}(D_{\varphi }^2,ϵ)$$\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

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

$$\int [d\psi ]\exp (⟨\psi ,D_{\varphi }^{\mu }\varphi ⟩)=\mathrm{pfaff}(D_{\varphi }^{\mu })\in \det {\mathscr{H}}_{\varphi }^{\mu \pi }.$$\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 $\mathrm{Pfaff}\to C^{\mathrm{\infty }}(\Sigma ,X)$

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

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.

Related entries

• path integral

• spinors in Yang-Mills theory