On the Coulomb Branch of a Marginal Deformation of N=4 SUSY Yang-Mills

We determine the exact vacuum structure of a marginal deformation of N=4 SUSY Yang-Mills with gauge group U(N). The Coulomb branch of the theory consists of several sub-branches which are governed by complex curves of the form Sigma_{n_{1}} U Sigma_{n_{2}} U Sigma_{n_{3}} of genus N=n_{1}+n_{2}+n_{3}. Each sub-branch intersects with a family of Higgs and Confining branches permuted by SL(2,Z) transformations. We determine the curve by solving a related matrix model in the planar limit according to the prescription of Dijkgraaf and Vafa, and also by explicit instanton calculations using a form of localization on the instanton moduli space. We find that Sigma_{n} coincides with the spectral curve of the n-body Ruijsenaars-Schneider system. Our results imply that the theory on each sub-branch is holomorphically equivalent to certain five-dimensional gauge theory with eight supercharges. This equivalence also implies the existence of novel confining branches in five dimensions.Comment: LaTeX file. 48 page

N=1* vacua, Fuzzy Spheres and Integrable Systems

We calculate the exact eigenvalues of the adjoint scalar fields in the massive vacua of N=1* SUSY Yang-Mills with gauge group SU(N). This provides a field theory prediction for the distribution of D3 brane charge in the AdS dual. We verify the proposal of Polchinski and Strassler that the D3-brane's lie on a fuzzy sphere in the supergravity limit and determine the corrections to this distribution due to worldsheet and quantum effects. The calculation also provides several new results concerning the equilibrium configurations of the N-body Calogero-Moser Hamiltonian.Comment: 20 page

Quantum counterpart of spontaneously broken classical PT symmetry

The classical trajectories of a particle governed by the PT-symmetric Hamiltonian $H=p^2+x^2(ix)^\epsilon$ ($\epsilon\geq0$) have been studied in depth. It is known that almost all trajectories that begin at a classical turning point oscillate periodically between this turning point and the corresponding PT-symmetric turning point. It is also known that there are regions in $\epsilon$ for which the periods of these orbits vary rapidly as functions of $\epsilon$ and that in these regions there are isolated values of $\epsilon$ for which the classical trajectories exhibit spontaneously broken PT symmetry. The current paper examines the corresponding quantum-mechanical systems. The eigenvalues of these quantum systems exhibit characteristic behaviors that are correlated with those of the associated classical system.Comment: 11 pages, 7 figure

Another Leigh-Strassler deformation through the Matrix model

In here the matrix model approach, by Dijkgraaf and Vafa, is used in order to obtain the effective superpotential for a certain deformation of N=4 SYM discovered by Leigh and Strassler. An exact solution to the matrix model Lagrangian is found and is expressed in terms of elliptic functions.Comment: 15 pages,2 figure

Wall Crossing and Instantons in Compactified Gauge Theory

We calculate the leading weak-coupling instanton contribution to the moduli-space metric of N=2 supersymmetric Yang-Mills theory with gauge group SU(2) compactified on R^3 x S^1. The results are in precise agreement with the semiclassical expansion of the exact metric recently conjectured by Gaiotto, Moore and Neitzke based on considerations related to wall-crossing in the corresponding four-dimensional theory.Comment: 24 pages, no figure

On Pseudo-Hermitian Hamiltonians and Their Hermitian Counterparts

In the context of two particularly interesting non-Hermitian models in quantum mechanics we explore the relationship between the original Hamiltonian H and its Hermitian counterpart h, obtained from H by a similarity transformation, as pointed out by Mostafazadeh. In the first model, due to Swanson, h turns out to be just a scaled harmonic oscillator, which explains the form of its spectrum. However, the transformation is not unique, which also means that the observables of the original theory are not uniquely determined by H alone. The second model we consider is the original PT-invariant Hamiltonian, with potential V=igx^3. In this case the corresponding h, which we are only able to construct in perturbation theory, corresponds to a complicated velocity-dependent potential. We again explore the relationship between the canonical variables x and p and the observables X and P.Comment: 9 pages, no figure

A New 2d/4d Duality via Integrability

We prove a duality, recently conjectured in arXiv:1103.5726, which relates the F-terms of supersymmetric gauge theories defined in two and four dimensions respectively. The proof proceeds by a saddle point analysis of the four-dimensional partition function in the Nekrasov-Shatashvili limit. At special quantized values of the Coulomb branch moduli, the saddle point condition becomes the Bethe Ansatz Equation of the SL(2) Heisenberg spin chain which coincides with the F-term equation of the dual two-dimensional theory. The on-shell values of the superpotential in the two theories are shown to coincide in corresponding vacua. We also identify two-dimensional duals for a large set of quiver gauge theories in four dimensions and generalize our proof to these cases.Comment: 19 pages, 2 figures, minor corrections and references adde

On O(1) contributions to the free energy in Bethe Ansatz systems: the exact g-function

We investigate the sub-leading contributions to the free energy of Bethe Ansatz solvable (continuum) models with different boundary conditions. We show that the Thermodynamic Bethe Ansatz approach is capable of providing the O(1) pieces if both the density of states in rapidity space and the quadratic fluctuations around the saddle point solution to the TBA are properly taken into account. In relativistic boundary QFT the O(1) contributions are directly related to the exact g-function. In this paper we provide an all-orders proof of the previous results of P. Dorey et al. on the g-function in both massive and massless models. In addition, we derive a new result for the g-function which applies to massless theories with arbitrary diagonal scattering in the bulk.Comment: 28 pages, 2 figures, v2: minor corrections, v3: minor corrections and references adde

The Coulomb branch of the Leigh-Strassler deformation and matrix models

The Dijkgraaf-Vafa approach is used in order to study the Coulomb branch of the Leigh-Strassler massive deformation of N=4 SYM with gauge group U(N). The theory has N=1 SUSY and an N-dimensional Coulomb branch of vacua, which can be described by a family of ``generalized'' Seiberg-Witten curves. The matrix model analysis is performed by adding a tree level potential that selects particular vacua. The family of curves is found: it consists of order N branched coverings of a base torus, and it is described by multi-valued functions on the latter. The relation between the potential and the vacuum is made explicit. The gauge group SU(N) is also considered. Finally the resolvents from which expectation values of chiral operators can be extracted are presented.Comment: 19 pages, 2 figures, late