Testing the Boson/Fermion Duality on the Three-Sphere

Abstract

We study the duality between theories of a fundamental scalar or fermion coupled to $U(N)$ Chern-Simons gauge theory at the level of the three-sphere partition function, or equivalently entanglement entropy across a circle. The duality relation between the sphere free energies of the large $N$ bosonic and fermionic conformal theories is sensitive to certain shifts in the Chern-Simons level already at order $N$. We first study a similar duality in a $\mathcal{N}=2$ supersymmetric Chern-Simons matter theory, where it can be checked exactly using localization. At large $N$, the free energies of supersymmetric and non-supersymmetric theories are related in a simple way, and we use this fact to obtain an explicit solution for the scalar and fermion free energies which obey the duality map and have the expected weak and strong coupling behaviors. We also suggest that a worldline representation of the free energy allows to relate its calculation to a sum of Wilson loops in pure Chern-Simons theory. In the supersymmetric case, we find that this approach precisely reproduces the localization prediction. In the non-supersymmetric case, we observe that a result consistent with the boson/fermion duality can be obtained using a certain framing prescription, which remains to be understood. We also briefly discuss the case of massive theories, and compare the small mass expansion of the free energy to previously known results for the correlation functions of bilinear operators in flat space.

33 pages

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Cited by in corpus (8)

• All tree level scattering amplitudes in Chern-Simons theories with fundamental matter
• Strong-Weak Chern-Simons-Matter Dualities from a Lattice Construction
• Duality and an exact Landau-Ginzburg potential for quasi-bosonic Chern-Simons-Matter theories
• Master 3d Bosonization Duality with Boundaries
• The fermion-boson map for large d
• Fermi seas from Bose condensates in Chern-Simons matter theories and a bosonic exclusion principle
• Crossing Symmetry in Matter Chern-Simons Theories at finite $N$ and $k$
• The Hilbert Space of large $N$ Chern-Simons matter theories