Contents

# Contents

## Definition

Given a Feynman propagator $\frac{1}{H}$, then the corresponding vacuum energy $Z$ is the logarithm of the functional determinant $det_{reg}$ of $H$

$Z \coloneqq -log\,det_{reg} H \,.$

## Properties

### In terms of special values of the zeta function

The vacuum energy is equivalently the special value of the zeta function $\zeta_H$ of $H$ given by the derivative at 0:

$Z = -\frac{1}{2}\zeta_H^\prime(0) \,.$

### In terms of the path integral and relation to generating function

Traditionally the vacuum energy is expressed in terms of a hypothetical path integral. (As opposed to the above zeta-function formalization this is not rigorous, but it serves to give the idea of why this is the vacuum energy and the the zeta-function expression may be taken to be the rigorous definition of the path integral heuristics.)

By analogy with finite-dimensional Gaussian integrals (see at Feynman diagram – For finitely many degrees of freedom) one expects that the Wick rotated vacuum amplitude version of the path integral (no field insertions, no boundary conditions) is

$\underset{\phi \in \mathbf{Fields}}{\int} \exp(- S_H(\phi)) D\phi = (det_reg H)^{-1/2} \,.$

Therefore

$log \underset{\phi \in \mathbf{Fields}}{\int} \exp(- S_H(\phi)) D\phi = -\frac{1}{2}log\, det_{reg} H$

is the generating functional for n-point functions. (…)

### As holomorphic potential for Determinant line bundle

Regard

$h \coloneqq \frac{1}{2} det_{reg}H$

as a hermitian structure on a holomorphic line bundle, hence, locally, as the absolute value-squared of the unit section $\phi_i$ of a holomorphic line bundle with respect to a local trivializing section (see at Chern connection).

$h|_{U_i} = {\Vert \phi_i \Vert}^2 \,.$

Then this line bundle is the determinant line bundle of $H$. (Quillen 85), review includes (Freed 87, p. 18, Qiu 12, section 2.8.1).

The Chern connection is

$A = \partial log(h) = \partial \frac{1}{2} det_{reg}H = \partial Z$

and the curvature differential 2-form is

$F = i \bar \partial\partial Z \,.$

eh? Something wrong with the factors of $1/2$ here…

## Applications

### The solution to the strong CP problem via axions

In the solution to the strong CP problem via axions it is analysis of the dependency of the vacuum energy of Yang-Mills theory on the theta-angle? $\theta$ which is argued to show that the axion expectation value $\theta = \langle a \rangle$ vanishes (Vafa-Witten 84).

For instance

• Claudio Scrucca, section 1.6 in Advanced quantum field theory pdf

• José Edelstein, page 2 of Lecture 8: 1-loop closed string vacuum amplitude, 2013 (pdf)

• Daniel Quillen, Determinants of Cauchy-Riemann Operators over a Riemann Surface, Functional Anal.

Appl. 19 (1985) 31.

• Daniel Freed, On determinant line bundles, Math. aspects of string theory, ed. S. T. Yau, World Sci. Publ. 1987, (revised pdf, dg-ga/9505002)

• Jia Qiu, section 2.8 of Lecture notes on topological field theory arXiv:1201.5550

The application to the axion solution to the strong CP problem is due to