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Authors
V. A. Kazakov
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We review some old and new methods of reduction of the number of degrees of freedom from ~N^2 to ~N in the multi-matrix integrals.
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We find a family of (half) self-dual impurity models such that the self-dual (BPS) sector is exactly solvable, for any spatial distribution of the impurity, both in the topologically trivial case and for kink (or antikink) configurations. This allows us to derive the metric on the corresponding one-dimensional moduli space in an analytical form. Also the generalized translational symmetry is found in an exact form. This symmetry provides a motion on moduli space which transforms one BPS solution into another. Finally, we analyse exactly how vibrational properties (spectral modes) of the BPS solutions depend on the actual position on moduli space. These results are obtained both for the nontrivial topological sector (kinks or antikinks) as well as for the topologically trivial sector, where the motion on moduli space represents a kink-antikink annihilation process.
Some results for two distinct but complementary exactly solvable algebraic models for pairing in atomic nuclei are presented: 1) binding energy predictions for isotopic chains of nuclei based on an extended pairing model that includes multi-pair excitations; and 2) fine structure effects among excited $0^+$ states in $N approx Z$ nuclei that track with the proton-neutron ($pn$) and like-particle isovector pairing interactions as realized within an algebraic $sp(4)$ shell model. The results show that these models can be used to reproduce significant ranges of known experimental data, and in so doing, confirm their power to predict pairing-dominated phenomena in domains where data is unavailable.
In this short note we construct the DLCQ description of the flux seven-branes in type IIA string theory and discuss its basic properties. The matrix model involves dipole fields. We explain the relation of this nonlocal matrix model to various orbifolds. We also give a spacetime interpretation of the Seiberg-Witten-like map, proposed in a different context first by Bergman and Ganor, that converts this matrix model to a local, highly nonlinear theory.
In this paper a review is given of a class of sub-models of both approaches, characterized by the fact that they can be solved exactly, highlighting in the process a number of generic results related to both the nature of pair-correlated systems as well as collective modes of motion in the atomic nucleus.
Permutation invariant Gaussian matrix models were recently developed for applications in computational linguistics. A 5-parameter family of models was solved. In this paper, we use a representation theoretic approach to solve the general 13-parameter Gaussian model, which can be viewed as a zero-dimensional quantum field theory. We express the two linear and eleven quadratic terms in the action in terms of representation theoretic parameters. These parameters are coefficients of simple quadratic expressions in terms of appropriate linear combinations of the matrix variables transforming in specific irreducible representations of the symmetric group $S_D$ where $D$ is the size of the matrices. They allow the identification of constraints which ensure a convergent Gaussian measure and well-defined expectation values for polynomial functions of the random matrix at all orders. A graph-theoretic interpretation is known to allow the enumeration of permutation invariants of matrices at linear, quadratic and higher orders. We express the expectation values of all the quadratic graph-basis invariants and a selection of cubic and quartic invariants in terms of the representation theoretic parameters of the model.
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