# QLunch: Marie Fialová

**Speaker:** Marie Fialová from QMATH

**Title:** AHARONOV–CASHER THEOREM FOR A PLANE DOMAIN WITH HOLES WITH THE APS BOUNDARY CONDITION

**Abstract:** Consider the Dirac operator on a plane with a compactly supported smooth magnetic field perpendicular to the plane with total flux Φ. The Aharonov–Casher theorem tells us that the dimension of the kernel, i.e., the number of zero modes, of this operator is the largest integer strictly less than |Φ|/2π. The talk is focused on a similar result on the number of zero modes in an alternative setting. In particular we are interested in the Dirac operator on the complex plane outside a finite number of balls with a magnetic field supported inside each ball, i.e., a AharonovBohm setting. We consider the domain of the operator with the famous Atiyah–Patodi–Singer boundary condition on the boundaries of the balls. The number of zero modes depends only on the flux Φ_k through each ball mod 2π. If we assume that Φ_k ∈ [−π/2, π/2) the number of zero modes is again the largest integer strictly less than |Φ|/2π, where Φ = ∑ _k Φ_k . I will discuss the case of one ball where the theorem says that there cannot be any zero modes.