TY - JOUR
T1 - Boson–fermion duality in a gravitational background
AU - Ferreiros, Yago
AU - Fradkin, Eduardo
N1 - Publisher Copyright:
© 2018 Elsevier Inc.
PY - 2018/12
Y1 - 2018/12
N2 - 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 [Formula presented] scalar field coupled to a [Formula presented] 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–Fisher–Chern–Simons complex scalar and a free Dirac fermion. We also show that, even away from the IR, keeping the [Formula presented] 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.
AB - 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 [Formula presented] scalar field coupled to a [Formula presented] 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–Fisher–Chern–Simons complex scalar and a free Dirac fermion. We also show that, even away from the IR, keeping the [Formula presented] 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.
KW - Composite boson
KW - Field theory duality
KW - Quantum Hall effect
KW - Topological field theory
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U2 - 10.1016/j.aop.2018.10.001
DO - 10.1016/j.aop.2018.10.001
M3 - Article
AN - SCOPUS:85055113539
SN - 0003-4916
VL - 399
SP - 1
EP - 25
JO - Annals of Physics
JF - Annals of Physics
ER -