Nick Dorey

The calculus of many instantons

We describe the modern formalism, ideas and applications of the instanton calculus for gauge theories with, and without, supersymmetry. Particular emphasis is put on developing a formalism that can deal with any number of instantons. This necessitates a thorough review of the ADHM construction of instantons with arbitrary charge and an in-depth analysis of the resulting moduli space of solutions. We review the construction of the ADHM moduli space as a hyper-Kahler quotient. We show how the functional integral in the semi-classical approximation reduces to an integral over the instanton moduli space in each instanton sector and how the resulting matrix partition function involves various geometrical quantities on the instanton moduli space: volume form, connection, curvature, isometries, etc. One important conclusion is that this partition function is the dimensional reduction of a higher- dimensional gauged linear sigma model which naturally leads us to describe the relation of the instanton calculus to D-branes in string theory. Along the way we describe powerful applications of the calculus of many instantons to supersymmetric gauge theories including (i) the gluino condensate puzzle in N = 1 theories (ii) Seiberg-Witten theory in N = 2 theories; and (iii) the AdS/CFT correspondence in N = 2 and N = 4 theories. Finally, we brielfy review the modifications of the instanton calculus for a gauge theory defined on a non-commutative spacetime and we also describe a new method for calculating instanton processes using a form of localization on the instanton moduli space.

Dyonic giant magnons

We study the classical spectrum of string theory on AdS5 ? S5 in the Hofman-Maldacena limit. We find a family of classical solutions corresponding to Giant Magnons with two independent angular momenta on S5. These solutions are related via Pohlmeyers reduction procedure to the charged solitons of the Complex sine-Gordon equation. The corresponding string states are dual to BPS boundstates of many magnons in the spin-chain description of planar = 4 SUSY Yang-Mills. The exact dispersion relation for these states is obtained from a purely classical calculation in string theory.

On the Dynamics of finite-gap solutions in classical string theory

We study the dynamics of finite-gap solutions in classical string theory on R × S 3 . Each solution is characterised by a spectral curve, �, of genus g and a divisor, γ, of degree g on the curve. We present a complete reconstruction of the general solution and identify the corresponding moduli-space, M (2g) R , as a real symplectic manifold of dimension 2g. The dynamics of the general solution is shown to be equivalent to a specific Hamiltonian integrable system with phase-space M (2g) R . The resulting description resembles the free motion of a rigid string on the Jacobian torus J(�). Interestingly, the canonically-normalised action variables of the integrable system are identified with certain filling fractions which play an important role in the context of the AdS/CFT correspondence.

S-duality of the Leigh-Strassler Deformation via Matrix Models

We investigate an exactly marginal = 1 supersymmetric deformation of SU(N) = 4 supersymmetric Yang-Mills theory discovered by Leigh and Strassler. We use a matrix model to compute the exact superpotential for a further massive deformation of the U(N) Leigh-Strassler theory. We then show how the exact superpotential and eigenvalue spectrum for the SU(N) theory follows by a process of integrating-in. We find that different vacua are related by an action of the SL(2,) modular group on the bare couplings of the theory extending the action of electric-magnetic duality away from the = 4 theory. We perform non-trivial tests of the matrix model results against semiclassical field theory analysis. We also show that there are interesting points in parameter space where condensates can diverge and vacua disappear. Based on the matrix model results, we propose an exact elliptic superpotential to describe the theory compactified on a circle of finite radius.

Exact Superpotentials from Matrix Models

Dijkgraaf and Vafa (DV) have conjectured that the exact superpotential for a large class of = 1 SUSY gauge theories can be extracted from the planar limit of a certain holomorphic matrix integral. We test their proposal against existing knowledge for a family of deformations of = 4 SUSY Yang-Mills theory involving an arbitrary polynomial superpotential for one of the three adjoint chiral superfields. Specifically, we compare the DV prediction for these models with earlier results based on the connection between SUSY gauge theories and integrable systems. We find complete agreement between the two approaches. In particular we show how the DV proposal allows the extraction of the exact eigenvalues of the adjoint scalar in the confining vacuum and hence computes all related condensates of the finite-N gauge theory. We extend these results to include Leigh-Strassler deformations of the = 4 theory.

A symplectic structure for string theory on integrable backgrounds

We define regularised Poisson brackets for the monodromy matrix of classical string theory on × S3. The ambiguities associated with Non-Ultra Locality are resolved using the symmetrisation prescription of Maillet. The resulting brackets lead to an infinite tower of Poisson-commuting conserved charges as expected in an integrable system. The brackets are also used to obtain the correct symplectic structure on the moduli space of finite-gap solutions and to define the corresponding action-angle variables. The canonically-normalised action variables are the filling fractions associated with each cut in the finite-gap construction. Our results are relevant for the leading-order semiclassical quantisation of string theory on AdS5 × S5 and lead to integer-valued filling fractions in this context.

The asymptotic spectrum of the Script N = 4 super-Yang-Mills spin chain

In this paper we discuss the asymptotic spectrum of the spin chain description of planar = 4 SUSY Yang-Mills. The states appearing in the spectrum belong to irreducible representations of the unbroken supersymmetry SU(2|2) × SU(2|2) with non-trivial extra central extensions. The elementary magnon corresponds to the bifundamental representation while boundstates of Q magnons form a certain short representation of dimension 16Q2. Generalising the Beiserts analysis of the Q = 1 case, we derive the exact dispersion relation for these states by purely group theoretic means.

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.

Quantum scattering of giant magnons

We perform a first-principles semi-classical computation of the one-loop corrections to the dispersion relation and S-matrix of Giant Magnons in AdS5 × S5 string theory. The results agree exactly with expectations based on the strong coupling expansion of the exact Asymptotic Bethe Ansatz equations. In particular we reproduce the Hernandez-Lopez term in the dressing phase.

Massive Vacua of N=1* Theory and S Duality from Matrix Models

In this note we show how Dijkgraaf and Vafas hypothesis relating the exact superpotential of an = 1 theory to a matrix model can be used to describe all the massive vacua of the = 1*, or mass deformed = 4, theory including the Higgs vacuum. The matrix model computation of the superpotential for each massive vacuum independently yields a modular function of the associated effective coupling in that vacuum which agrees with previously derived results up to a vacuum-independent additive constant. The results in the different massive vacua can be related by the action of SL(2,) on the = 4 coupling, thus providing evidence for modular invariance of the underlying = 4 theory.