International Journal of Modern Physics A

In the paper Int. J. Mod. Phys. A19, 4127 (2004), we started a program of creating a reference-book on matrix-model τ-functions, the new generation of special functions, which are going to play an important role in string theory calculations. The main focus of that paper was on the one-matrix Hermitian model τ-functions. The present paper is devoted to a direct counterpart for the Kontsevich and Generalized Kontsevich Model (GKM) τ-functions. We mostly focus on calculating resolvents (= loop operator averages) in the Kontsevich model, with a special emphasis on its simplest (Gaussian) phase, where exists a surprising integral formula, and the expressions for the resolvents in the genus zero and one are especially simple (in particular, we generalize the known genus zero result to genus one). We also discuss various features of generic phases of the Kontsevich model, in particular, a counterpart of the unambiguous Gaussian solution in the generic case, the solution called Dijkgraaf–Vafa (DV) solution. Further, we extend the results to the GKM and, in particular, discuss the p–q duality in terms of resolvents and corresponding Riemann surfaces in the example of dualities between (2, 3) and (3, 2) models.

We obtain Schrödinger quantum mechanics from Perelman's functional and from the Ricci-flow equations of a conformally flat Riemannian metric on a closed two-dimensional configuration space. We explore links with the recently discussed emergent quantum mechanics.

We provide a new model of higher-dimensional wormholes supported by phantom energy derived from a scalar field with negative kinetic term. We have shown that the Averaged Null Energy Condition (ANEC) violating phantom energy can be reduced as desired. We have also noticed that dimension of the space–time plays a crucial role for measuring the ANEC violating matter needed.

We propose the bosonic part of an action that defines M-theory. It possesses manifest SO(1, 10) symmetry and constructed based on the Lorentzian 3-algebra associated with U(N) Lie algebra. From our action, we derive the bosonic sector of BFSS matrix theory and IIB matrix model in the naive large N limit by taking appropriate vacua. We also discuss an interaction with fermions.

Raychaudhuri equation is generalized in the parametrized absolute parallelism geometry. This version of absolute parallelism is more general than the conventional one. The generalization takes into account the suggested interaction between the quantum spin of the moving particle and the torsion of the background gravitational field. The generalized Raychaudhuri equation obtained contains some extra terms, depending on the torsion of space–time, that would have some effects on the singularity theorems of general relativity. Under a certain condition, this equation could be reduced to the original Raychaudhuri equation without any need for a vanishing torsion.

In this paper we analyze the structure of the BRST charge of nonlinear superalgebras. We consider quadratic nonlinear superalgebras where a commutator (in terms of (super-)Poisson brackets) of the generators is a quadratic polynomial of the generators. We find the explicit form of the BRST charge up to cubic order in Faddeev–Popov ghost fields for arbitrary quadratic nonlinear superalgebras. We point out the existence of constraints on structure constants of the superalgebra when the nilpotent BRST charge is quadratic in Faddeev–Popov ghost fields. The general results are illustrated by simple examples of superalgebras.

Starting from an supersymmetric electric gauge theory with the multiple product gauge group and the bifundamentals, we apply Seiberg dual to each gauge group, obtain the supersymmetric dual magnetic gauge theories with dual matters including the gauge singlets. Then we describe the intersecting brane configurations, where there are NS-branes and D4-branes (and anti-D4-branes), of type IIA string theory corresponding to the meta-stable nonsupersymmetric vacua of this gauge theory.

We also discuss the case where the orientifold 4-planes are added into the above brane configuration. Next, by adding an orientifold 6-plane, we apply to an supersymmetric electric gauge theory with the multiple product gauge group (where a single symplectic or orthogonal gauge group is present) and the bifundamentals. Finally, we describe the other cases where the orientifold 6-plane intersects with NS-brane.

We construct the type IIA nonsupersymmetric meta-stable brane configurations corresponding to the various supersymmetric gauge theories. The D6-branes are both displaced and rotated where these deformations are described as the mass term and quartic term for the fundamental flavors, respectively. The multiplicity of the NS5-branes occurs in the superpotential order for adjoint, symmetric, or bifundamental matters. A rich pattern of nonsupersymmetric meta-stable states as well as the supersymmetric stable states is found.