Notes on Partition Functions on the Riemann surfaces

Partition Function on the Riemann surfaces

In this note we follow Lecture 1, part 3 of volume 2 of (Deligne et al. 1999) to calculate Euclidean partition function of free scalar fields with periodic boundary conditions.

The field theory datum are the following

  • is a compact Riemannian manifold. (The metric is irrelevant here)

  • The space of fields is a function class (smooth, distributional, Sobolev, etc.) defined on .

  • The action is the one of free (bosonic) scalar fields, i.e. The normalization constant is merely a convention for now.

First, we see that the space of fields admits the decomposition where is the universal cover of , which is in this case, and consists of functions that are -equivariant. Explicitly, we have the map with We denote the above function classes by to indicate that such decomposition is valid for arbitrary classes.

Note that , then by Hodge decomposition, each can be uniquely decomposed by where is the harmonic representative of according to .1 is a (oriented) curve starts from a base point in and is a single-valued function on .2

The free field action now reads where is the Laplacian. This suggest the following formula

Now the remaining problem is to calculate the Gaussian integral Thus, we employ the zeta function regularization:

Zeta function regularization.

Recall that for finite-dimensional Gaussian integral, we have the result (up to normalization) for positive definite . We define an analogy of the term.

The motivation of such definition comes from the finite-dimensional analog: If is a positive-definite operator on a Euclidean space, then and . Then .

By such definition, we can mimic finite-dimensional Gaussian integral and compute

We conclude that our final expression with the theorem.

Let us specifically look at -dimensional cases, i.e. . We regard as a Riemann surface with genus .

Proof. Let be a symplectic basis of with the corresponding basis of holomorphic -forms . We have where is the periodic matrix. Note that the imaginary part is positive definite. The harmonic form is of the form Here gives the harmonic forms in with -periods and -periods . The -norm becomes Then by Poisson summation formula, we have Inserting this back to the expression we have the desired result. ◻

Reference

Deligne, P., P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, J. W. Morgan, D. R. Morrison, and Edward Witten, eds. 1999. Quantum fields and strings: A course for mathematicians. Vol. 1, 2.

  1. This means ↩︎

  2. We have for any loop . by definition have zero monodromy and hence is single-valued.↩︎

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