Timothy J. Hollowood

Massive Vacua of Script 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.

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.

Universal aspects of gravity localized on thick branes

Abstract We study gravity in backgrounds that are smooth generalizations of the Randall–Sundrum model, with and without scalar fields. These generalizations include three-branes in higher dimensional spaces which are not necessarily anti-de Sitter far from the branes, intersecting brane configurations and configurations involving negative tension branes. We show that under certain mild assumptions there is a universal equation for the gravitational fluctuations. We study both the graviton ground state and the continuum of Kaluza–Klein modes and we find that the four-dimensional gravitational mode is localized precisely when the effects of the continuum modes decouple at distances larger than the fundamental Planck scale. The decoupling is contingent only on the long-range behaviour of the metric from the brane and we find a universal form for the corrections to Newtons law. We also comment on the possible contribution of resonant modes. Given this, we find general classes of metrics which maintain localized four-dimensional gravity. We find that three-brane metrics in five dimensions can arise from a single scalar field source, and we rederive the BPS type conditions without any a priori assumptions regarding the form of the scalar potential. We also show that a single scalar field cannot produce conformally-flat locally intersecting brane configurations or a p -brane in greater than ( p +2)-dimensions.

Matrix models, geometric engineering and elliptic genera

We compute the prepotential of = 2 supersymmetric gauge theories in four dimensions obtained by toroidal compactifications of gauge theories from 6 dimensions, as a function of Kahler and complex moduli of T2. We use three different methods to obtain this: matrix models, geometric engineering and instanton calculus. Matrix model approach involves summing up planar diagrams of an associated gauge theory on T2. Geometric engineering involves considering F-theory on elliptic threefolds, and using topological vertex to sum up worldsheet instantons. Instanton calculus involves computation of elliptic genera of instanton moduli spaces on R4. We study the compactifications of = 2* theory in detail and establish equivalence of all these three approaches in this case. As a byproduct we geometrically engineer theories with massive adjoint fields. As one application, we show that the moduli space of mass deformed M5-branes wrapped on T2 combines the Kahler and complex moduli of T2 and the mass parameter into the period matrix of a genus 2 curve.

Solitons in affine Toda field theories

Abstract The solutions of classical A r affine Toda field theories, with imaginary coupling constant, are investigated. Soliton solutions are found, which, despite the complex form of the hamiltonian, have real classical masses. The solitons have the same gross spectrum as the fundamental Toda particles, although in addition to their mass they carry a non-trivial topological charge, which is a weight of one of the fundamental representations of A r . Multiple-soliton solutions are constructed which describe the classical scattering theory of the solitons. The generalization to other simply laced theories is briefly considered, with some explicit results forD 4 .

Gluino condensate and magnetic monopoles in supersymmetric gluodynamics

Abstract We examine supersymmetric SU(N) gauge theories on R 3 × S 1 with a circle of circumference β. These theories interpolate between four-dimensional N = 1 pure gauge theory for β = ∞ and three-dimensional N = 2 gauge theory for β = 0. The dominant field configurations of the R 3 × S 1 SU(N) theories in the semi-classical regime arise from N varieties of monopole. Periodic instanton configurations correspond to mixed configurations of N single monopoles of the N different types. We semi-classically evaluate the non-perturbatively generated superpotential of the R 3 × S 1 theory and hence determine its vacuum structure. to the gluino condensate in these theories and take the decompactification limit β = ∞. In this way we obtain a value for the gluino condensate in the four-dimensional N = 1 supersymmetric SU(N) Yang-Mills theory, which agrees with the previously known ‘weak coupling’ expression but not with the ‘strong coupling’ expression derived in the early literature solely from instanton considerations. Moreover, we discover that the superpotential gives a mass to the dual (magnetic) photon, which implies confinement of the original electric photon and disappearance of all the massless modes.

Multi-instanton calculus and the AdS / CFT correspondence in N=4 superconformal field theory

Abstract We present a self-contained study of ADHM multi-instantons in SU(N) gauge theory, especially the novel interplay with supersymmetry and the large-N limit. We give both field- and string-theoretic derivations of the N = 4 supersymmetric multi-instanton action and collective coordinate integration measure. As a central application, we focus on certain n-point functions Gn, n = 16, 8 or 4, in N = 4 SU(N) gauge theory at the conformal point (as well as on related higher-partial-wave correlators); these are correlators in which the 16 exact supersymmetric and superconformal fermion zero-modes are saturated. In the large-N limit, for the first time in any four-dimensional theory, we are able to evaluate all leading-order multi-instanton contributions exactly. We find compelling evidence for Maldacenas conjecture: (1) The large = N k-instanton collective coordinate space has the geometry of a single copy of AdS5 × S5. (2) The integration measure on this space includes the partition function of ten-dimensional N = 1 SU(k) gauge theory dimensionally reduced to zero dimensions, matching the description of D-instantons in Type IIB string theory. (3) In exact agreement with Type IIB string calculations, at the k-instantonlevel, G n = √N g 8 k n− 7 2 e 2πikτ Σ d/k d −2 · F n (x 1 , …, x n ) , where Fn is identical to a convolution of n bulk-to-boundary supergravity propagators.

rational conformal field theories at, and away from, criticality as Toda field theories

Abstract Recently, Zamolodchikov has shown that certain minimal conformal field theories preserve their integrability away from the critical point. In an attempt to understand these results we consider a class of integrable field theories, namely two-dimensional Toda field theories. It is found that for particular values of the coupling constant these theories describe minimal models, in particular the Ising model can be described both by an E 8 and A 1 Toda field theory. The affine versions of these theories then represent the model away from criticality, for instance the Ising model in a magnetic field is described by the affine E 8 Toda field theory, whereas the affine A 1 theory describes a thermal perturbation. This generalizes to a deformation of all minimal models.

An integrable deformation of the AdS5×S5superstring

The S-matrix on the world-sheet theory of the string in AdS has previously been shown to admit a deformation where the symmetry algebra is replaced by the associated quantum group. The case where q is real has been identified as a particular deformation of the Green–Schwarz sigma model. An interpretation of the case with q a root of unity has, until now, been lacking. We show that the Green–Schwarz sigma model admits a discrete deformation which can be viewed as a rather simple deformation of the gauged WZW model, where . The deformation parameter q is then a kth root of unity where k is the level. The deformed theory has the same equations-of-motion as the Green–Schwarz sigma model but has a different symplectic structure. We show that the resulting theory is integrable and has just the right amount of kappa-symmetries that appear as a remnant of the fermionic part of the original gauge symmetry. This points to the existence of a fully consistent deformed string background.

The BPS spectra of gauge theories in two and four dimensions

We study = (2,2) supersymmetric abelian gauge theories in two dimensions. The exact BPS spectrum of these models is shown to coincide with the spectrum of massive hypermultiplets of certain = 2 supersymmetric gauge theories in four dimensions. A special case of these results involves a surprising connection between four-dimensional = 2 SQCD with N colours and N