nLab fermionic path integral



Quantum field theory


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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.


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

The action functional

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

is the sum of the

  • bosonic action

    S bos)(ϕ)= Σdϕdϕ S^{bos})(\phi) = \int_\Sigma \langle d \phi \wedge \star d \phi\rangle
  • fermionic action

    S ϕ ferm(ψ)= Σψ,D ϕψ S^{ferm}_\phi(\psi) = \int_\Sigma \langle \psi, D_\phi \psi\rangle

    where D ϕD_\phi is a Dirac operator on Sϕ *TMS \otimes \phi^* T M (the Dirac operator on SS twisted by the pullback of the Levi-Civita connection on T *XT^* X ).

One imagines than that the hypothetical path integral symboilically written as

[dϕ][dψ]exp(S(ψ)(ϕ,ψ)) \int [d \phi] [d \psi] \exp(S(\psi)(\phi,\psi))

can be computed in two steps

=[dϕ]exp(S bos(ϕ))([dψ]exp(S ϕ ferm(ψ))) \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(ϕ):=[dψ]exp(S ϕ ferm(ψ)) 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 ϕ\phi, but is instead a section pfaffpfaff of a Pfaffian line bundle

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

over the space of bosonic field configurations.

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

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

makes the fermionic path integral into a genuine function

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

Any such choice of tt is called a choice of quantum integrand.

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

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

Pfaffian bundles

We are implicitly assuming that dimΣ=2dim \Sigma = 2 or maybe 8n+28 n + 2 in the following. Needs to be generalized.

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

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

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

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

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

D ϕ ±: ϕ ± ϕ . D_\phi^{\pm} : \mathcal{H}_\phi^{\pm} \to \mathcal{H}^{\mp}_\phi \,.

We also have a “quaternionic structure”

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

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

U μ:={ϕC (Σ,X)|μninSpecD ϕ 2}, U_\mu := \{ \phi \in C^\infty(\Sigma,X) | \mu \nin Spec D_\phi^2\} \,,

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

Over these open subsets we have the finite rank vector bundles

ϕ μ±:= 0ϵμEig(D ϕ 2,ϵ) \mathcal{H}_\phi^{\mu \pm} := \oplus_{0 \leq \epsilon \leq \mu} Eig(D_\phi^2, \epsilon)

of eigenspaces of D ϕ 2D_\phi^2 for eigenvalues bounded by μ\mu.

The Dirac operator that we are interested in is

D ϕ μ:=J ϕ D ϕ +: ϕ μ,+ ϕ μ,+. 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 ϕ μ \langle -, D_\phi^\mu -\rangle

whose Berezinian integral is the Pfaffian

[dψ]exp(ψ,D ϕ μψ)=pfaff(D ϕ μ)det ϕ μπ. \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 μ\mu glue together to define

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

  • with a smooth section pfaff(D)pfaff(D).

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


For the sigma model describing the heterotic superstring propagating on a pseudo-Riemannian manifold XX, 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 XX. See there for more details.

Last revised on November 27, 2023 at 00:49:31. See the history of this page for a list of all contributions to it.