Fermionic Matrix Model Research Articles

W-representations of the fermionic matrix and Aristotelian tensor models

We show that the fermionic matrix model can be realized by W-representation. We construct the Virasoro constraints with higher algebraic structures, where the constraint operators obey the Witt algebra and null 3-algebra. The remarkable feature is that the character expansion of the partition function can be easily derived from such Virasoro constraints. It is a τ-function of the KP hierarchy. We construct the fermionic Aristotelian tensor model and give its W-representation. Moreover, we analyze the fermionic red tensor model and present the W-representation and character expansion of the partition function.

Gauged fermionic matrix quantum mechanics

We consider the gauged free fermionic matrix model, for a single fermionic matrix. In the large N limit this system describes a c = 1/2 chiral fermion in 1 + 1 dimensions. The Gauss’ law constraint implies that to obtain a physical state, indices of the fermionic matrices must be fully contracted, to form a singlet. There are two ways in which this can be achieved: one can consider a trace basis formed from products of traces of fermionic matrices or one can consider a Schur function basis, labeled by Young diagrams. The Schur polynomials for the fermions involve a twisted character, as a consequence of Fermi statistics. The main result of this paper is a proof that the trace and Schur bases coincide up to a simple normalization coefficient that we have computed.

Asymptotics of partition functions in a fermionic matrix model and of relatedq-polynomials

In this paper, we study asymptotics of the thermal partition function of a model of quantum mechanical fermions with matrix-like index structure and quartic interactions. This partition function is given explicitly by a Wronskian of the Stieltjes-Wigert polynomials. Our asymptotic results involve the theta function and its derivatives. We also develop a new asymptotic method for general $q$-polynomials.

Spectra of eigenstates in fermionic tensor quantum mechanics

We study the $O(N_1)\times O(N_2)\times O(N_3)$ symmetric quantum mechanics of 3-index Majorana fermions. When the ranks $N_i$ are all equal, this model has a large $N$ limit which is dominated by the melonic Feynman diagrams. We derive an integral formula which computes the number of $SO(N_1)\times SO(N_2)\times SO(N_3)$ invariant states for any set of $N_i$. For equal ranks the number of singlets is non-vanishing only when $N$ is even, and it exhibits rapid growth: it jumps from $36$ in the $O(4)^3$ model to $595354780$ in the $O(6)^3$ model. We derive bounds on the values of energy, which show that they scale at most as $N^3$ in the large $N$ limit, in agreement with expectations. We also show that the splitting between the lowest singlet and non-singlet states is of order $1/N$. For $N_3=1$ the tensor model reduces to $O(N_1)\times O(N_2)$ fermionic matrix quantum mechanics, and we find a simple expression for the Hamiltonian in terms of the quadratic Casimir operators of the symmetry group. A similar expression is derived for the complex matrix model with $SU(N_1)\times SU(N_2)\times U(1)$ symmetry. Finally, we study the $N_3=2$ case of the tensor model, which gives a more intricate complex matrix model whose symmetry is only $O(N_1)\times O(N_2)\times U(1)$. All energies are again integers in appropriate units, and we derive a concise formula for the spectrum. The fermionic matrix models we studied possess standard 't Hooft large $N$ limits where the ground state energies are of order $N^2$, while the energy gaps are of order $1$.

Fermionic Matrix Models

We review a class of matrix models whose degrees of freedom are matrices with anti-commuting elements. We discuss the properties of the adjoint fermion one-matrix, two-matrix and gauge-invariant D-dimensional matrix models at large N and compare them with their bosonic counterparts, which are the more familiar Hermitian matrix models. We derive and solve the complete sets of loop equations for the correlators of these models and use these equations to examine critical behavior. The topological large N expansions are also constructed and their relation to other aspects of modern string theory, such as integrable hierarchies, is discussed. We use these connections to discuss the applications of these matrix models to string theory and induced gauge theories. We argue that as such the fermionic matrix models may provide a novel generalization of the discretized random surface representation of quantum gravity in which the genus sum alternates and the sums over genera for correlators have better convergence properties than their Hermitian counterparts. We discuss the use of adjoint fermions instead of adjoint scalars to study induced gauge theories. We also discuss two classes of dimensionally reduced models, a fermionic vector model and a supersymmetric matrix model, and discuss their applications to the branched polymer phase of string theories in target space dimensions D > 1 and also to the meander problem.

Observables and critical behaviour in fermionic matrix models

We review the properties of adjoint fermion one-, two-, and generic D-dimensional matrix models at large N. We derive and solve the complete sets of loop equations for the correlators of these models and examine the ensuing critical behavior. The topological 1/N-expansions are also constructed, and we discuss the applications of these matrix models to string theory and induced gauge theories.

Generalized Penner models to all genera.

We give a complete description of the genus expansion of the one-cut solution to the generalized Penner model. The solution is presented in a form which allows us in a very straightforward manner to localize critical points and to investigate the scaling behavior of the model in the vicinity of these points. We carry out an analysis of the critical behavior to all genera addressing all types of multicritical points. In certain regions of the coupling constant space the model must be defined via analytical continuation. We show in detail how this works for the Penner model. Using analytical continuation it is possible to reach the fermionic one-matrix model. We show that the critical points of the fermionic one-matrix model can be indexed by an integer m, as was the case for the ordinary Hermitian one-matrix model. Furthermore the mth multicritical fermionic model has to all genera the same value of ${\ensuremath{\gamma}}_{\mathrm{str}}$ as the mth multicritical Hermitian model. However, the coefficients of the topological expansion need not be the same in the two cases. We show explicitly how it is possible with a fermionic matrix model to reach a m=2 multicritical point for which the topological expansion has alternating signs, but otherwise coincides with the usual Painlev\'e expansion.