Finite Matrix Model Research Articles

Exact solution of the [formula omitted] finite matrix model

We find the exact solutions of the Φ23 finite matrix model (Grosse-Wulkenhaar model). In the Φ23 finite matrix model, multipoint correlation functions are expressed as G|a11…aN11|…|a1B…aNBB|. The ∑i=1BNi-point function denoted by G|a11…aN11|…|a1B…aNBB| is given by the sum over all Feynman diagrams (ribbon graphs) on Riemann surfaces with B-boundaries, and each |a1i⋯aNii| corresponds to the Feynman diagrams having Ni-external lines from the i-th boundary. It is known that any G|a11…aN11|…|a1B…aNBB| can be expressed using G|a1|…|an| type n-point functions. Thus we focus on rigorous calculations of G|a1|…|an|. The formula for G|a1|…|an| is obtained, and it is achieved by using the partition function Z[J] calculated by the Harish-Chandra-Itzykson-Zuber integral. We give G|a|, G|ab|, G|a|b|, and G|a|b|c| as the specific simple examples. All of them are described by using the Airy functions.

Finite matrix model of quantum hall fluids on S2

Based on Haldane's spherical geometrical formalism of the two-dimensional quantum Hall fluids, the relation between the noncommutative geometry of S2 and the two-dimensional quantum Hall fluids is exhibited. A finite matrix model on the two-sphere is explicitly constucted as an effective description of the fractional quantum Hall fluids of finite extent, and the complete sets of physical quantum states of this matrix model are determined. We also describe how the low-lying excitations in the model are constructed in terms of the quasi-particle and quasi-hole excitations. It is shown that there exists a Haldane hierarchical structure in the two-dimensional quantum Hall fluid states of the matrix model. These hierarchical fluid states are generated by the parent fluid state by condensing the quasi-particle and quasi-hole excitations level by level.

Quasi-hole solutions in finite noncommutative Maxwell-Chern-Simons theory

We study Maxwell-Chern-Simons theory in 2 noncommutative spatial dimensions and 1 temporal dimension. We consider a finite matrix model obtained by adding a linear boundary field which takes into account boundary fluctuations. The pure Chern-Simons has been previously shown to be equivalent to the Laughlin description of the quantum Hall effect. With the addition of the Maxwell term, we find that there exists a rich spectrum of excitations including solitons with nontrivial "magnetic flux" and quasi-holes with nontrivial "charges", which we describe in this article. The magnetic flux corresponds to vorticity in the fluid fluctuations while the charges correspond to sources of fluid fluctuations. We find that the quasi-hole solutions exhibit a gap in the spectrum of allowed charge.

Matrix model and Ginsparg-Wilson relation

We discuss that the Ginsparg-Wilson relation, which has the key role in the recent development of constructing lattice chiral gauge theory, can play an important role to define chiral structures in finite matrix models and noncommutative geometries.

Super Yang-Mills theory as a random matrix model.

We generalize the Gervais-Neveu gauge to four-dimensional N=1 superspace. The model describes an N=2 super Yang-Mills theory. All chiral superfields (N=2 matter and ghost multiplets) exactly cancel to all loops. The remaining Hermitian scalar superfield (matrix) has a renormalizable massive propagator and simplified vertices. These properties are associated with N=1 supergraphs describing a superstring theory on a random lattice world sheet. We also consider all possible finite matrix models, and find they have a universal large-color limit. These could describe gravitational strings if the matrix-model coupling is fixed to unity, for exact electric-magnetic self-duality.

Finite matrix models with continuous local gauge invariance

We construct a hamiltonian lattice gauge theory which possesses local SU (2) gauge invariance and yet is defined on a Hilbert space of 5-dimensional real vectors for every link. This construction does not allow for generalization to arbitrary SU( N), but a small variation of it can be generalized to an SU( N) × U(1) local gauge invariant model. The latter is solvable in simple gauge sectors leading to trivial spectra. We display these by studying a U(1) local gauge invariant model with similar characteristics.