Boson-fermion duality in a gravitational background

TL;DR

This paper explores the duality between a relativistic bosonic theory with gauge fields and a Dirac fermion in curved spacetime, demonstrating their equivalence at the IR fixed point and emphasizing the role of short-range interactions.

Contribution

It establishes the boson-fermion duality in a gravitational background, including the effects of curvature and gauge fields, and highlights the importance of short-range interactions for the duality's consistency.

Findings

Duality holds at the IR fixed point between scalar and fermion theories.

Gravitational coupling arises naturally from spin factors in the Chern-Simons theory.

Short-range interactions are crucial for preserving worldline topology and duality integrity.

Abstract

We study the $2+1$ dimensional boson-fermion duality in the presence of background curvature and electromagnetic fields. The main players are, on the one hand, a massive complex $|\phi|^4$ scalar field coupled to a $U(1)$ Maxwell-Chern-Simons gauge field at level $1$, representing a relativistic composite boson with one unit of attached flux, and on the other hand, a massive Dirac fermion. We show that, in a curved background and at the level of the partition function, the relativistic composite boson, in the infinite coupling limit, is dual to a short-range interacting Dirac fermion. The coupling to the gravitational spin connection arises naturally from the spin factors of the Wilson loop in the Chern-Simons theory. A non-minimal coupling to the scalar curvature is included on the bosonic side in order to obtain agreement between partition functions. Although an explicit Lagrangian expression for the fermionic interactions is not obtained, their short-range nature constrains them to be irrelevant, which protects the duality in its strong interpretation as an exact mapping at the IR fixed point between a Wilson-Fischer-Chern-Simons complex scalar and a free Dirac fermion. We also show that, even away from the IR, keeping the $|\phi|^4$ term is of key importance as it provides the short-range bosonic interactions necessary to prevent intersections of worldlines in the path integral, thus forbidding unknotting of knots and ensuring preservation of the worldline topologies.