Macroscopoic Three-Loop Amplitudes and the Fusion Rules from the Two-Matrix Model

From the computation of three-point singlet correlators in the two-matrix model, we obtain an explicit expression for the macroscopic three-loop amplitudes having boundary lengths $\ell_{i}$ $(i = 1\sim 3)$ in the case of the unitary series $(p,q)= (m+1,m)$ coupled to two-dimensional gravity. The sum appearing in this expression is found to conform to the structure of the CFT fusion rules while the summand factorizes through a product of three modified Bessel functions. We briefly discuss a possible generalization of these features to macroscopic $n$-loop amplitudes.Comment: 9 pages, no figure, late

Darboux Transformation for the Manin-Radul Supersymmetric KdV equation

In this paper we present a vectorial Darboux transformation, in terms of ordinary determinants, for the supersymmetric extension of the Korteweg-de Vries equation proposed by Manin and Radul. It is shown how this transformation reduces to the Korteweg-de Vries equation. Soliton type solutions are constructed by dressing the vacuum and we present some relevant plots.Comment: 14 pp, 2 figures, AMS-LaTeX, to appear in Phys. Lett.

Crum Transformations and Wronskian Type Solutions for Supersymmetric KdV equation

Darboux transformation is reconsidered for the supersymmetric KdV system. By iterating the Darboux transformation, a supersymmetric extension of the Crum transformation is obtained for the Manin-Radul SKdV equation, in doing so one gets Wronskian superdeterminant representations for the solutions. Particular examples provide us explicit supersymmetric extensions, super solitons, of the standard soliton of the KdV equation. The KdV soliton appears as the body of the super soliton.Comment: 13 pp, AMS-LaTe

Massive Hyper-Kahler Sigma Models and BPS Domain Walls

With the non-Abelian Hyper-Kahler quotient by U(M) and SU(M) gauge groups, we give the massive Hyper-Kahler sigma models that are not toric in the N=1 superfield formalism. The U(M) quotient gives N!/[M! (N-M)!] (N is a number of flavors) discrete vacua that may allow various types of domain walls, whereas the SU(M) quotient gives no discrete vacua. We derive BPS domain wall solution in the case of N=2 and M=1 in the U(M) quotient model.Comment: 16 pages, 1 figure, contribution to the Proceedings of the International Conference on "Symmetry Methods in Physics (SYM-PHYS10)" held at Yerevan, Armenia, 13-19 Aug. 200

qq-Virasoro Algebra and the Point-Splitting

It is shown that a particular $q$-deformation of the Virasoro algebra can be interpreted in terms of the $q$-local field $\Phi (x)$ and the Schwinger-like point-splitted Virasoro currents, quadratic in $\Phi (x)$. The $q$-deformed Virasoro algebra possesses an additional index $\alpha$, which is directly related to point-splitting of the currents. The generators in the $q$-deformed case are found to exactly reproduce the results obtained by probing the fields $X(z)$ (string coordinate) and $\Phi (z)$ (string momentum) with the non-splitted Virasoro generators and lead to a particular representation of the $SU_q (1,1)$ algebra characterized by the standard conformal dimension $J$ of the field. Some remarks concerning the $q$-vertex operator for the interacting $q$-string theory are made.Comment: 17 pages, Late

Large N and double scaling limits in two dimensions

Recently, the author has constructed a series of four dimensional non-critical string theories with eight supercharges, dual to theories of light electric and magnetic charges, for which exact formulas for the central charge of the space-time supersymmetry algebra as a function of the world-sheet couplings were obtained. The basic idea was to generalize the old matrix model approach, replacing the simple matrix integrals by the four dimensional matrix path integrals of N=2 supersymmetric Yang-Mills theory, and the Kazakov critical points by the Argyres-Douglas critical points. In the present paper, we study qualitatively similar toy path integrals corresponding to the two dimensional N=2 supersymmetric non-linear sigma model with target space CP^n and twisted mass terms. This theory has some very strong similarities with N=2 super Yang-Mills, including the presence of critical points in the vicinity of which the large n expansion is IR divergent. The model being exactly solvable at large n, we can study non-BPS observables and give full proofs that double scaling limits exist and correspond to universal continuum limits. A complete characterization of the double scaled theories is given. We find evidence for dimensional transmutation of the string coupling in some non-critical string theories. We also identify en passant some non-BPS particles that become massless at the singularities in addition to the usual BPS states.Comment: 38 pages, including an introductory section that makes the paper self-contained, two figures and one appendix; v2: typos correcte

Towards N=1 Super-Yang-Mills on the Lattice

We consider the lattice regularization of N=1 supersymmetric Yang--Mills theory with Wilson fermions. This formulation breaks supersymmetry at any finite lattice spacing; we discuss how Ward identities can be used to define a supersymmetric continuum limit, which coincides with the point where the gluino becomes massless. As a first step towards the understanding of the zero gluino-mass limit, we present results on the quenched low-lying spectrum of SU(2) N=1 Super-Yang--Mills, at $\beta=2.6$ on a $V=16^3 \times 32$ lattice, in the OZI approximation. Our results, in spite of the quenched and OZI approximations, are in remarkable agreement with theoretical predictions in the supersymmetric theory, for the states with masses which are not expected to get a large contribution from fermion loops.Comment: 25 Latex pages, 5 figure

Monopole and Dyon Bound States in N=2 Supersymmetric Yang-Mills Theories

We study the existence of monopole bound states saturating the BPS bound in N=2 supersymmetric Yang-Mills theories. We describe how the existence of such bound states relates to the topology of index bundles over the moduli space of BPS solutions. Using an $L^2$ index theorem, we prove the existence of certain BPS states predicted by Seiberg and Witten based on their study of the vacuum structure of N=2 Yang-Mills theories.Comment: 34 pages, harvma

On a Lorentz-Invariant Interpretation of Noncommutative Space-Time and Its Implications on Noncommutative QFT

By invoking the concept of twisted Poincar\' e symmetry of the algebra of functions on a Minkowski space-time, we demonstrate that the noncommutative space-time with the commutation relations $[x_\mu,x_\nu]=i\theta_{\mu\nu}$, where $\theta_{\mu\nu}$ is a {\it constant} real antisymmetric matrix, can be interpreted in a Lorentz-invariant way. The implications of the twisted Poincar\'e symmetry on QFT on such a space-time is briefly discussed. The presence of the twisted symmetry gives justification to all the previous treatments within NC QFT using Lorentz invariant quantities and the representations of the usual Poincar\'e symmetry.Comment: 12 pages, one reference adde

Gravitational shocks as a key ingredient of Gamma-Ray Bursts

We identify a novel physical mechanism that may be responsible for energy release in $\gamma$-ray bursts. Radial perturbations in the neutron core, induced by its collision with collapsing outer layers during the early stages of supernova explosions, can trigger a gravitational shock, which can readily eject a small but significant fraction of the collapsing material at ultra-relativistic speeds. The development of such shocks is a strong-field effect arising in near-critical collapse in General Relativity and has been observed in numerical simulations in various contexts, including in particular radially perturbed neutron star collapse, albeit for a tiny range of initial conditions. Therefore, this effect can be easily missed in numerical simulations if the relevant parameter space is not exhaustively investigated. In the proposed picture, the observed rarity of $\gamma$-ray bursts would be explained if the relevant conditions for this mechanism appear in only about one in every $10^4-10^5$ core collapse supernovae. We also mention the possibility that near-critical collapse could play a role in powering the central engines of Active Galactic Nuclei.Comment: 9 pages, 3 figure