Triple Point of a Scalar Field Theory on a Fuzzy Sphere

The model of a scalar field with quartic self-interaction on the fuzzy sphere has three known phases: a uniformly ordered phase, a disordered phase and a non-uniformly ordered phase, the last of which has no classical counterpart. These three phases are expected to meet at a triple point. By studying the infinite matrix size limit, we locate the position of this triple point to within a small triangle in terms of the parameters of the model. We find the triple point is closer to the coordinate origin of the phase diagram than previous estimates but broadly consistent with recent analytic predictions.Comment: 12 pages, 5 figure

A Multitrace Matrix Model from Fuzzy Scalar Field Theory

We present the analytical approach to scalar field theory on the fuzzy sphere which has been developed in arXiv:0706.2493 [hep-th]. This approach is based on considering a perturbative expansion of the kinetic term in the partition function. After truncating this expansion at second order, one arrives at a multitrace matrix model, which allows for an application of the saddle-point method. The results are in agreement with the numerical findings in the literature.Comment: 8 pages, talk given by CS at the International Workshop "Supersymmetries and Quantum Symmetries" (SQS'07), Dubna, July 30 - August 4 2007; to appear in the proceeding

Dimer geometry, amoebae and a vortex dimer model

We present a geometrical approach for studying dimers. We introduce a connection for dimer problems on bipartite and non-bipartite graphs. In the bipartite case the connection is flat but has non-trivial ${\bf Z}_2$ holonomy round certain curves. This holonomy has the universality property that it does not change as the number of vertices in the fundamental domain of the graph is increased. It is argued that the K-theory of the torus, with or without punctures, is the appropriate underlying invariant. In the non-bipartite case the connection has non-zero curvature as well as non-zero Chern number. The curvature does not require the introduction of a magnetic field. The phase diagram of these models is captured by what is known as an amoeba. We introduce a dimer model with negative edge weights that give rise to vortices. The amoebae for various models are studied with particular emphasis on the case of negative edge weights which corresponds to the presence of vortices. Vortices gives rise to new kinds of amoebae with certain singular structures which we investigate. On the amoeba of the vortex full hexagonal lattice we find the partition function corresponds to that of a massless Dirac doublet.Comment: 25 pages, 9 figures Latest version: some references added and typos remove

Commuting Quantum Matrix Models

We study a quantum system of $p$ commuting matrices and find that such a quantum system requires an explicit curvature dependent potential in its Lagrangian for the system to have a finite energy ground state. In contrast it is possible to avoid such curvature dependence in the Hamiltonian. We study the eigenvalue distribution for such systems in the large matrix size limit. A critical r\^ole is played by $p=4$. For $p\ge4$ the competition between eigenvalue repulsion and the attractive potential forces the eigenvalues to form a sharp spherical shell.Comment: 17 page

Quantised relativistic membranes and non-perturbative checks of gauge/gravity duality

We test the background geometry of the BFSS model using a D4-brane probe. This proves a sensitive test of the geometry and we find excellent agreement with the D4-brane predictions based on the solution of a membrane corresponding to the D4-brane propagating on this background.Comment: 7 pages, 2 figures, based on a talk, presented by D. O'C. at ISQS 25, 6-10 June, 2017, Prague, Czech Republic; to be published in Journal of Physics: Conference Serie

On the Phase Structure of Commuting Matrix Models

We perform a systematic study of commutative $SO(p)$ invariant matrix models with quadratic and quartic potentials in the large $N$ limit. We find that the physics of these systems depends crucially on the number of matrices with a critical r\^ole played by $p=4$. For $p\leq4$ the system undergoes a phase transition accompanied by a topology change transition. For $p> 4$ the system is always in the topologically non-trivial phase and the eigenvalue distribution is a Dirac delta function spherical shell. We verify our analytic work with Monte Carlo simulations.Comment: 37 pages, 13 figures, minor corrections, updated to match the published versio

The BFSS model on the lattice

We study the maximally supersymmetric BFSS model at finite temperature and its bosonic relative. For the bosonic model in $p+1$ dimensions, we find that it effectively reduces to a system of gauged Gaussian matrix models. The effective model captures the low temperature regime of the model including one of its two phase transitions. The mass becomes $p^{1/3}\lambda^{1/3}$ for large $p$, with $\lambda$ the 'tHooft coupling. Simulations of the bosonic-BFSS model with $p=9$ give $m=(1.965\pm .007)\lambda^{1/3}$, which is also the mass gap of the Hamiltonian. We argue that there is no `sign' problem in the maximally supersymmetric BFSS model and perform detailed simulations of several observables finding excellent agreement with AdS/CFT predictions when $1/\alpha'$ corrections are included.Comment: 23 pages, 11 figure

Environmentally Friendly Renormalization

We analyze the renormalization of systems whose effective degrees of freedom are described in terms of fluctuations which are ``environment'' dependent. Relevant environmental parameters considered are: temperature, system size, boundary conditions, and external fields. The points in the space of \lq\lq coupling constants'' at which such systems exhibit scale invariance coincide only with the fixed points of a global renormalization group which is necessarily environment dependent. Using such a renormalization group we give formal expressions to two loops for effective critical exponents for a generic crossover induced by a relevant mass scale $g$. These effective exponents are seen to obey scaling laws across the entire crossover, including hyperscaling, but in terms of an effective dimensionality, d\ef=4-\gl, which represents the effects of the leading irrelevant operator. We analyze the crossover of an $O(N)$ model on a $d$ dimensional layered geometry with periodic, antiperiodic and Dirichlet boundary conditions. Explicit results to two loops for effective exponents are obtained using a [2,1] Pad\'e resummed coupling, for: the ``Gaussian model'' ($N=-2$), spherical model ($N=\infty$), Ising Model ($N=1$), polymers ($N=0$), XY-model ($N=2$) and Heisenberg ($N=3$) models in four dimensions. We also give two loop Pad\'e resummed results for a three dimensional Ising ferromagnet in a transverse magnetic field and corresponding one loop results for the two dimensional model. One loop results are also presented for a three dimensional layered Ising model with Dirichlet and antiperiodic boundary conditions. Asymptotically the effective exponents are in excellent agreement with known results.Comment: 76 pages of Plain Tex, Postscript figures available upon request from [email protected], preprint numbers THU-93/14, DIAS-STP-93-1

Near commuting multi-matrix models

We investigate the radial extent of the eigenvalue distribution for Yang-Mills type matrix models. We show that, a three matrix Gaussian model with complex Myers coupling, to leading order in strong coupling is described by commuting matrices whose joint eigenvalue distribution is uniform and confined to a ball of radius R=(3Pi/2g)^(1/3). We then study, perturbatively, a 3-component model with a pure commutator action and find a radial extent broadly consistent with numerical simulations.Comment: 25 pages, appendix expanded, presentation improved, updated to match the published versio

Membrane Matrix models and non-perturbative checks of gauge/gravity duality

We compare the bosonic and maximally supersymmetric membrane models. We find that in Hoppe regulated form the bosonic membrane is well approximated by massive Gaussian quantum matrix models. In contrast the similarly regulated supersymmetric membrane, which is equivalent to the BFSS model, has a gravity dual description. We sketch recent progress in checking gauge/gravity duality in this context.Comment: 11 pages and 4 figures. To appear in the Proceedings of the Corfu Summer Institute 2015 "School and Workshops on Elementary Particle Physics and Gravity" 1-27 September 2015 Corfu, Greec