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

Microscopic quantum superpotential in N=1 gauge theories

We consider the N=1 super Yang-Mills theory with gauge group U(N), adjoint chiral multiplet X and tree-level superpotential Tr W(X). We compute the quantum effective superpotential W_mic as a function of arbitrary off-shell boundary conditions at infinity for the scalar field X. This effective superpotential has a remarkable property: its critical points are in one-to-one correspondence with the full set of quantum vacua of the theory, providing in particular a unified picture of solutions with different ranks for the low energy gauge group. In this sense, W_mic is a good microscopic effective quantum superpotential for the theory. This property is not shared by other quantum effective superpotentials commonly used in the literature, like in the strong coupling approach or the glueball superpotentials. The result of this paper is a first step in extending Nekrasov's microscopic derivation of the Seiberg-Witten solution of N=2 super Yang-Mills theories to the realm of N=1 gauge theories.Comment: 23 pages, 1 figure; typos corrected, version to appear in JHE

The Proof of the Dijkgraaf-Vafa Conjecture and application to the mass gap and confinement problems

Using generalized Konishi anomaly equations, it is known that one can express, in a large class of supersymmetric gauge theories, all the chiral operators expectation values in terms of a finite number of a priori arbitrary constants. We show that these constants are fully determined by the requirement of gauge invariance and an additional anomaly equation. The constraints so obtained turn out to be equivalent to the extremization of the Dijkgraaf-Vafa quantum glueball superpotential, with all terms (including the Veneziano-Yankielowicz part) unambiguously fixed. As an application, we fill non-trivial gaps in existing derivations of the mass gap and confinement properties in super Yang-Mills theories.Comment: 31 pages, 1 figure; v2: typos corrected; references, a note on Kovner-Shifman vacua (section 4.3) and a few clarifying comments in Section 3 added; v3: cosmetic changes, JHEP versio

Glueball operators and the microscopic approach to N=1 gauge theories

We explain how to generalize Nekrasov's microscopic approach to N=2 gauge theories to the N=1 case, focusing on the typical example of the U(N) theory with one adjoint chiral multiplet X and an arbitrary polynomial tree-level superpotential Tr W(X). We provide a detailed analysis of the generalized glueball operators and a non-perturbative discussion of the Dijkgraaf-Vafa matrix model and of the generalized Konishi anomaly equations. We compute in particular the non-trivial quantum corrections to the Virasoro operators and algebra that generate these equations. We have performed explicit calculations up to two instantons, that involve the next-to-leading order corrections in Nekrasov's Omega-background.Comment: 38 pages, 1 figure and 1 appendix included; v2: typos and the list of references corrected, version to appear in JHE

Intermixture of extended edge and localized bulk energy levels in macroscopic Hall systems

We study the spectrum of a random Schroedinger operator for an electron submitted to a magnetic field in a finite but macroscopic two dimensional system of linear dimensions equal to L. The y direction is periodic and in the x direction the electron is confined by two smooth increasing boundary potentials. The eigenvalues of the Hamiltonian are classified according to their associated quantum mechanical current in the y direction. Here we look at an interval of energies inside the first Landau band of the random operator for the infinite plane. In this energy interval, with large probability, there exist O(L) eigenvalues with positive or negative currents of O(1). Between each of these there exist O(L^2) eigenvalues with infinitesimal current O(exp(-cB(log L)^2)). We explain what is the relevance of this analysis to the integer quantum Hall effect.Comment: 29 pages, no figure

Diffusive shock acceleration in extragalactic jets

We calculate the temporal evolution of distributions of relativistic electrons subject to synchrotron and adiabatic processes and Fermi-like acceleration in shocks. The shocks result from Kelvin-Helmholtz instabilities in the jet. Shock formation and particle acceleration are treated in a self-consistent way by means of a numerical hydrocode. We show that in our model the number of relativistic particles is conserved during the evolution, with no need of further injections of supra-thermal particles after the initial one. From our calculations, we derive predictions for values and trends of quantities like the spectral index and the cutoff frequency that can be compared with observations.Comment: 12 pages containing 7 postscript figures; uses A&A macros. Accepted for publication in Astronomy and Astrophysic

Long-lived Bloch oscillations with bosonic Sr atoms and application to gravity measurement at micrometer scale

We report on the observation of Bloch oscillations on the unprecedented time scale of severalseconds. The experiment is carried out with ultra-cold bosonic strontium-88 loaded into a vertical optical standing wave. The negligible atom-atom elastic cross section and the absence of spin makes $^{88}$Sr an almost ideal Bose gas insensitive to typical mechanisms of decoherence due to thermalization and to external stray fields. The small size enables precision measurements of forces at micrometer scale. This is a challenge in physics for studies of surfaces, Casimir effects, and searches for deviations from Newtonian gravity predicted by theories beyond the standard model