Supersymmetric matrix models and branched polymers

Abstract

We solve a supersymmetric matrix model with a general potential. While matrix models usually describe surfaces, supersymmetry enforces a cancellation of bosonic and fermionic loops and only diagrams corresponding to so-called branched polymers survive. The eigenvalue distribution of the random matrices near the critical point is of a new kind.

Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces

“…The supersymmetry here acts as a fermionic rotation on the Landau levels, which has nothing in common with ordinary supersymmetry. In particular, the commutator of supercharges is not a spacetime translation [55,56]. The definition of noncommutative superspaces has been addressed recently within various different contexts in [57]- [66].…”

“…This resembles local supersymmetry somewhat, except that here the parameters of the transformation are arbitrary functions of differential operators rather than arbitrary functions of the spacetime coordinates. A closer analogy is the type of supersymmetry that arises in zero-dimensional supersymmetric matrix models [54]- [56], in which the parameter of the transformation is an arbitrary matrix. In fact, the parameters of the supersymmetry transformation in the present case become matrices after expanding in the basis of Landau eigenfunctions.…”

Fermionic quantum gravity

“…The reduced entropy of the remaining twisted graphs is then comparable to that of the graphs which produce a polymer-like behaviour. This feature is unique to fermionic matrices, and it is the property which enables the construction of a model of branched polymers using supersymmetric matrix models [13] via the coupling of the fermionic matrix model to an ordinary, complex matrix model which has the effect of cancelling the set of twisted diagrams. In fact, it is precisely the twisting mechanism described in section 2.2 that enables one to isolate graphs with tree-like growth in the fermionic case.…”

“…In fact, it is precisely the twisting mechanism described in section 2.2 that enables one to isolate graphs with tree-like growth in the fermionic case. The untwisted diagrams may then be mapped onto branched polymers similarly to the case of the cactus diagrams which appear in supersymmetric matrix models [13].…”

“…The topological expansion of the matrix integral (1.1) was considered in [10] and some novel critical behaviour was observed in [11]. The fermionic matrix model can also be coupled to an ordinary complex bosonic one to produce a supersymmetric matrix model, which has been used to describe solutions of certain combinatorial problems [12] and also to generate statistical models of branched polymers [13].…”