Complex (super)-matrix models with external sources and q-ensembles of Chern–Simons and ABJ(M) type

Leonardo Santilli,Miguel Tierz

doi:10.1088/1751-8121/abb6b0

Abstract

The Langmann–Szabo–Zarembo (LSZ) matrix model is a complex matrix model with a quartic interaction and two external matrices. The model appears in the study of a scalar field theory on the non-commutative plane. We prove that the LSZ matrix model computes the probability of atypically large fluctuations in the Stieltjes–Wigert matrix model, which is a q-ensemble describing U(N) Chern–Simons theory on the three-sphere. The correspondence holds in a generalized sense: depending on the spectra of the two external matrices, the LSZ matrix model either describes probabilities of large fluctuations in the Chern–Simons partition function, in the unknot invariant or in the two-unknot invariant. We extend the result to supermatrix models, and show that a generalized LSZ supermatrix model describes the probability of atypically large fluctuations in the ABJ(M) matrix model.

Highlights

We will consider a supersymmetric extension of the LSZ model, which we define as the supermatrix model (1.2)
We prove that the LSZ matrix model computes the probability of atypically large fluctuations in the Stieltjes–Wigert matrix model, which is a q-ensemble describing U (N ) Chern–Simons theory on the three-sphere
We extend the result to supermatrix models, and show that a generalized LSZ supermatrix model describes the probability of atypically large fluctuations in the ABJ(M) matrix model

Summary

Noncommutative scalar theory with background field

We review the LSZ model: in the first subsection, the geometric construction of the Moyal plane in the presence of a background magnetic field is sketched, while the second subsection is dedicated to the construction of a scalar field theory with quartic interaction. An example is given by a particle moving on a plane with a magnetic field of intensity B in the transverse direction; the momentum space becomes noncommutative R2, as the momentum operators modify according to: pj. In this case the covariant momenta Pj satisfy the commutation relation [Pj, Pk] = −iB jk. If the two frameworks are put together, that is, a transverse magnetic field is plugged in on a noncommutative plane, three possible harmonic oscillator pictures arise:.

Scalar theory on the fuzzy sphere

Solution of the matrix model