We study the anomalous dimensions for scalar operators for a three-dimensional Chern-Simons theory recently proposed in arXiv:0806.1218. We show that the mixing matrix at two-loop order is that for an integrable Hamiltonian of an SU(4) spin chain with sites alternating between the fundamental and the anti-fundamental representations. We find a set of Bethe equations from which the anomalous dimensions can be determined and give a proposal for the Bethe equations to the full superconformal group of OSp(2,2|6).

The Bethe ansatz for superconformal Chern-Simons

Introduction Hilbert scheme of points Framed moduli space of torsion free sheaves on $\mathbb{P}^2$ Hyper-Kahler metric on $(\mathbb{C}^2)^{[n]}$ Resolution of simple singularities Poincare polynomials of the Hilbert schemes (1) Poincare polynomials of Hilbert schemes (2) Hilbert scheme on the cotangent bundle of a Riemann surface Homology group of the Hilbert schemes and the Heisenberg algebra Symmetric products of an embedded curve, symmetric functions and vertex operators Bibliography Index.

Lectures on Hilbert schemes of points on surfaces

We propose a formalism inspired by matrix models to compute open and closed topological string amplitudes in the B-model on toric Calabi-Yau manifolds. We find closed expressions for various open string amplitudes beyond the disk, and in particular we write down the annulus amplitude in terms of theta functions on a Riemann surface. We test these ideas on local curves and local surfaces, providing in this way generating functionals for open Gromov-Witten invariants in the spirit of mirror symmetry. In the case of local curves, we study the open string sector near the critical point which leads to 2d gravity, and we show that toric D-branes become FZZT branes in a double-scaling limit. We use this connection to compute non-perturbative instanton effects due to D-branes that control the large order behavior of topological string theory on these backgrounds.

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Open string amplitudes and large order behavior in topological string theory

The macroscopic entropy and the attractor equations for BPS black holes in four-dimensional N = 2 supergravity theories follow from a variational principle for a certain `entropy function'. We present this function in the presence of R2-interactions and non-holomorphic corrections. The variational principle identifies the entropy as a Legendre transform and this motivates the definition of various partition functions corresponding to different ensembles and a hierarchy of corresponding duality invariant inverse Laplace integral representations for the microscopic degeneracies. Whenever the microscopic degeneracies are known the partition functions can be evaluated directly. This is the case for N = 4 heterotic CHL black holes, where we demonstrate that the partition functions are consistent with the results obtained on the macroscopic side for black holes that have a non-vanishing classical area. In this way we confirm the presence of a measure in the duality invariant inverse Laplace integrals. Most, but not all, of these results are obtained in the context of semiclassical approximations. For black holes whose area vanishes classically, there remain discrepancies at the semiclassical level and beyond, the nature of which is not fully understood at present.

https://iopscience.iop.org/article/10.1088/1126-6708/2006/03/074/pdf

Black hole partition functions and duality

The prepotential in N = 2 SUSY Yang-Mills theories enjoys remarkable properties One of the most interesting is its relation to the coordinate on the quantum moduli space u = tr 2 that results into recursion equations for the coefficients of the prepotential due to instantons In this work we show, with an explicit multi-instanton computation, that this relation holds true at arbitrary winding numbers Even more interestingly we show that its validity extends to the case in which gravitational corrections are taken into account if the correlators are suitably modified These results apply also to the cases in which matter in the fundamental and in the adjoint is included We also check that the expressions we find satisfy the chiral ring relations for the gauge case and compute the first gravitational correction

Matone's Relation in the Presence of Gravitational Couplings

We continue our study of the large N phase transition in q-deformed Yang-Mills theory on the sphere and its role in connecting topological strings to black hole entropy. We study in detail the chiral theory defined in terms of uncoupled single U(N) representations at large N and write down the resulting partition function by means of the topological vertex. The emergent toric geometry has three Kahler parameters, one of which corresponds to the expected fibration over 1. By taking a suitable double-scaling limit we recover the chiral Gross-Taylor string expansion. To analyse the phase transition we construct a matrix model which describes the chiral gauge theory. It has three distinct phases, one of which should be described by the closed topological string expansion. We verify this expectation by explicit comparison between the matrix model and the chiral topological string free energies. We also show that the critical point in the pertinent phase of the matrix model corresponds to a divergence of the topological string perturbation series.

Topological strings and large N phase transitions II: chiral expansion of q-deformed Yang-Mills theory

https://www.newton.ac.uk/files/preprints/ni08016.pdf

Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory

We use Ward identities derived from the generalized Konishi anomaly in order to compute effective superpotentials for SU(N), SO(N) and $Sp(N)$ supersymmetric gauge theories coupled to matter in various representations. In particular we focus on cubic and quartic tree level superpotentials. With this technique higher order corrections to the perturbative part of the effective superpotential can be easily evaluated.

Effective Superpotentials via Konishi Anomaly

We use Ward identities derived from the generalized Konishi anomaly in order to compute effective superpotentials for SU(N), SO(N) and Sp(N) supersymmetric gauge theories coupled to matter in various representations. In particular we focus on cubic and quartic tree level superpotentials. With this technique higher order corrections to the perturbative part of the effective superpotential can be easily evaluated.

Michele Cirafici

Papers

Topological strings and large N phase transitions II: chiral expansion of q-deformed Yang-Mills theory

Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory

Effective Superpotentials via Konishi Anomaly

Effective superpotentials via Konishi anomaly

Topological strings and largeNphase transitions I: Nonchiral expansion ofq-deformed Yang-Mills theory