# Self-adjoint local boundary problems on compact surfaces. I. Spectral flow

## TL;DR

This paper computes the spectral flow for self-adjoint elliptic operators on compact surfaces with boundary, showing it depends on boundary topology and acts as a universal invariant for such operator paths.

## Contribution

It provides a topological formula for spectral flow and establishes its universality as an additive invariant for boundary value problems.

## Key findings

- Spectral flow expressed via boundary topological data
- Spectral flow is a universal additive invariant
- Extension to parametrized families and K-theory index

## Abstract

The paper deals with first order self-adjoint elliptic differential operators on a smooth compact oriented surface with non-empty boundary. We consider such operators with self-adjoint local boundary conditions. The paper is focused on paths in the space of such operators connecting two operators conjugated by a unitary automorphism. The first result is the computation of the spectral flow for such paths in terms of the topological data over the boundary. The second result is the universality of the spectral flow: we show that the spectral flow is a universal additive invariant for such paths, if the vanishing on paths of invertible operators is required. In the next paper of the series we generalize these results to families of such operators parametrized by points of an arbitrary compact space instead of an interval. The integer-valued spectral flow is replaced then by the family index taking values in the $K^1$-group of the base space.