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We study a limit in which a relativistic CFT reduces to conformal quantum mechanics, and relate the partition functions of the two theories. When the initial CFT is holographic, our limit coincides with an ultra-spinning limit in the gravity dual. We therefore propose that ultra-spinning black holes are dual to an appropriate ensemble in finite-dimensional conformal quantum mechanics. The limit is studied in detail for SCFTs in four and six dimensions. These theories have a superconformal index which can be computed by gluing together two or more blocks. Applying our limit to the index effectively isolates a single such block. Our results therefore suggest that ultra-spinning black holes play the role of blocks in the gravitational dual of holomorphic factorisation., Comment: 57 pages. Minor clarifications, citations added. Matches published version

We give a geometric interpretation for superconformal quantum mechanics defined on a hyper-Kahler cone which has an equivariant symplectic resolution. BPS states are identified with certain twisted Dolbeault cohomology classes on the resolved space and their index degeneracies can also be related to the Euler characteristic computed in equivariant sheaf cohomology. In the special case of the Hilbert scheme of K points on C2, we obtain a rigorous estimate for the exponential growth of the index degeneracies of BPS states as K goes to infinity. This growth serves as a toy model for our recently proposed duality between a seven dimensional black hole and superconformal quantum mechanics.

We provide evidence for a holographic duality between superconformal quantum mechanics on the moduli space of Yang-Mills instantons and M-theory in certain asymptotically $AdS_{7}\times S^{4}$ backgrounds with a plane-wave boundary metric. We show that the gravitational background admits a supersymmetric black hole solution whose entropy is precisely reproduced by the superconformal index of the dual quantum mechanics.

We study the hemisphere partition function of a three-dimensional $\mathcal{N}=4$ supersymmetric $U(N)$ gauge theory with one adjoint and one fundamental hypermultiplet -- the ADHM quiver theory. In particular, we propose a distinguished set of UV boundary conditions which yield Verma modules of the quantised chiral rings of the Higgs and Coulomb branches. In line with a recent proposal by two of the authors in collaboration with M. Bullimore, we show explicitly that the hemisphere partition functions recover the characters of these modules in two limits, and realise blocks gluing exactly to the partition functions of the theory on closed three-manifolds. We study the geometry of the vortex moduli space and investigate the interpretation of the vortex partition functions as equivariant indices of quasimaps to the Hilbert scheme of points in $\mathbb{C}^2$. We also investigate half indices of the ADHM quiver gauge theory in the presence of a line operator and discuss their geometric interpretation. Along the way we find interesting relations between our hemisphere blocks and related quantities in topological string theory and equivariant quantum K-theory., Comment: 32+26 pages, 6 figures. v2: Discussion of Neumann boundary conditions clarified, JHEP version

We study the twisted indices of $\mathcal{N}=4$ supersymmetric gauge theories in three dimensions on spatial $S^{2}$ with an angular momentum refinement. We demonstrate factorisation of the index into holomorphic blocks for the $T[SU(N)]$ theory in the presence of generic fluxes and fugacities. We also investigate the relation between the twisted index, Hilbert series and the moduli space of vortices. In particular, we show that each holomorphic block coincides with a generating function for the $\chi_{t}$ genera of the moduli spaces of "local" vortices. The twisted index itself coincides with a corresponding generating function for the $\chi_{t}$ genera of moduli spaces of "global" vortices in agreement with a proposal of Bullimore et. al. We generalise this geometric interpretation of the twisted index to include fluxes and Chern-Simons levels. For the $T[SU(N)]$ theory, the relevant moduli spaces are the local and global versions of Laumon space respectively and we demonstrate the proposed agreements explicitly using results from the mathematical literature. Finally, we exhibit a precise relation between the Coulomb branch Hilbert series and the Poincar\'e polynomials of the corresponding vortex moduli spaces., Comment: 56 pages, 4 figures. v2: published version, references added

We consider supersymmetric quantum mechanics on a K\"{a}hler cone, regulated via a suitable resolution of the conical singularity. The unresolved space has a $\mathfrak{u}(1,1|2)$ superconformal symmetry and we propose the existence of an associated quantum mechanical theory with a discrete spectrum consisting of unitary, lowest weight representations of this algebra. We define a corresponding superconformal index and compute it for a wide range of examples., Comment: 49 pages, 6 figures; references added

We study quantum mechanical systems with $\mathfrak{osp}(4^{*}|4)$ superconformal symmetry. We classify unitary lowest-weight representations of this superconformal algebra and define an index which receives contributions from short and semi-short multiplets only. We consider the example of a quantum mechanical $\sigma$-model with hyper-K\"{a}hler target $\mathcal{M}$ equipped with a triholomorphic homothety. The superconformal index coincides with the Witten index of a novel form of supersymmetric quantum mechanics for a particle moving on $\mathcal{M}$ in a background magnetic field in which an unbroken $\mathfrak{su}(1|2)$ subalgebra of the superconformal algebra is linearly realised as a global symmetry., Comment: 21 pages

We define an index for $\mathfrak{osp}(4^{*}|4)$ superconformal quantum mechanics on a hyperK\"{a}hler cone. The index is defined on an equivariant symplectic resolution of the cone, which acts as a regulator. We present evidence that the index does not depend on the choice of resolution parameters and encodes information about the spectrum of (semi-) short representations of the superconformal algebra of the unresolved space. In particular, there are two types of multiplet which can be counted exactly using the index. These correspond to holomorphic functions on the cone and to the generators of the Borel-Moore homology on the resolved space respectively. We calculate the resulting index by localisation for a large class of examples., Comment: 39 pages

Yang-Mills instantons are solitonic particles in d=4+1 dimensional gauge theories. We construct and analyse the quantum Hall states that arise when these particles are restricted to the lowest Landau level. We describe the ground state wavefunctions for both Abelian and non-Abelian quantum Hall states. Although our model is purely bosonic, we show that the excitations of this 4d quantum Hall state are governed by the Nekrasov partition function of a certain five dimensional supersymmetric gauge theory with Chern-Simons term. The partition function can also be interpreted as a variant of the Hilbert series of the instanton moduli space, counting holomorphic sections rather than holomorphic functions. It is known that the Hilbert series of the instanton moduli space can be rewritten using mirror symmetry of 3d gauge theories in terms of Coulomb branch variables. We generalise this approach to include the effect of a five dimensional Chern-Simons term. We demonstrate that the resulting Coulomb branch formula coincides with the corresponding Higgs branch Molien integral which, in turn, reproduces the standard formula for the Nekrasov partition function., Comment: 50 pages, 2 figures

We study a U(N) gauged matrix quantum mechanics which, in the large N limit, is closely related to the chiral WZW conformal field theory. This manifests itself in two ways. First, we construct the left-moving Kac-Moody algebra from matrix degrees of freedom. Secondly, we compute the partition function of the matrix model in terms of Schur and Kostka polynomials and show that, in the large $N$ limit, it coincides with the partition function of the WZW model. This same matrix model was recently shown to describe non-Abelian quantum Hall states and the relationship to the WZW model can be understood in this framework., Comment: 28 pages + appendices. v2: references added. v3: another reference, version to be published in JHEP

We propose a matrix quantum mechanics for a class of non-Abelian quantum Hall states. The model describes electrons which carry an internal SU(p) spin. The ground states of the matrix model include spin-singlet generalisations of the Moore-Read and Read-Rezayi states and, in general, lie in a class previously introduced by Blok and Wen. The effective action for these states is a U(p) Chern-Simons theory. We show how the matrix model can be derived from quantisation of the vortices in this Chern-Simons theory and how the matrix model ground states can be reconstructed as correlation functions in the boundary WZW model., Comment: 35 pages, 2 figures. v2: references added. v3: minor changes, version to be published in Phys. Rev. B

We construct new integrable systems describing particles with internal spin from four-dimensional $\mathcal{N}=2$ quiver gauge theories. The models can be quantized and solved exactly using the quantum inverse scattering method and also using the Bethe/Gauge correspondence., Comment: 35 pages + appendices, 6 figures. v2: minor corrections, version submitted to journal

We study supersymmetric quantum mechanics on the moduli space of Yang-Mills instantons on R^2 x T^2 and its application to the discrete light-cone quantisation (DLCQ) of N=4 SUSY Yang-Mills. In the presence of a target space magnetic field, the model has a discrete spectrum with the wavefunctions of generic energy eigenstates supported away from the singular points of the moduli space. The corresponding Hamiltonian is part of an osp(1,1|4) superalgebra which enlarges to su(1,1|4) superconformal invariance in the sector corresponding to the N=4 theory. The Hamiltonian is isospectral to the light-cone dilatation operator of the N=4 theory in this sector. The model also has an interesting scaling limit where it becomes integrable. We determine the semiclassical spectrum in this limit. We discuss a possible approach to constructing the dilatation operator of N=4 supersymmetric Yang-Mills theory in DLCQ., Comment: 33 pages

We study the quantum mechanical sigma model arising in the discrete light-cone quantisation of N=4 supersymmetric Yang-Mills theory. The target space is a certain torus fibration over a scale-invariant special Kahler manifold. We show that the expected SU(1,1|4) light-cone superconformal invariance of the N=4 theory emerges in a limit where the volume of the fibre goes to zero and give an explicit construction of the generators. The construction given here yields a large new family of superconformal quantum mechanical models with SU(1,1|4) invariance., Comment: 32 pages

We study the BPS spectrum and walls of marginal stability of the $\mathcal{N}=2$ supersymmetric theory in four dimensions with gauge group SU(n) and $n\le N_{f}<2n$ fundamental flavours at the root of the Higgs branch. The strong-coupling spectrum of this theory was conjectured in hep-th/9902134 to coincide with that of the two-dimensional supersymmetric $\mathbb{CP}^{2n-N_{f}-1}$ sigma model. Using the Kontsevich--Soibelman wall-crossing formula, we start with the conjectured strong-coupling spectrum and extrapolate it to all other regions of the moduli space. In the weak-coupling regime, our results precisely agree with the semiclassical analysis of hep-th/9902134: in addition to the usual dyons, quarks, and $W$ bosons, if the complex masses obey a particular inequality, the resulting weak-coupling spectrum includes a tower of bound states consisting of a dyon and one or more quarks. In the special case of $\mathbb{Z}_{n}$-symmetric masses, there are bound states with one quark for odd $n$ and no bound states for even $n$., Comment: 11 pages, 4 figures

We study the interplay between wall-crossing in four-dimensional gauge theory and instanton contributions to the moduli space metric of the same theory on $\mathbb{R}^{3}\times S^{1}$. We consider $\mathcal{N}=2$ SUSY Yang--Mills with gauge group SU(n) and focus on walls of marginal stability which extend to weak coupling. By comparison with explicit field theory results we verify the Kontsevich--Soibelman formula for the change in the BPS spectrum at these walls and check the smoothness of the metric in the corresponding compactified theory. We also verify in detail the predictions for the one instanton contribution to the metric coming from the non-linear integral equations of Gaiotto, Moore and Nietzke., Comment: 26 pages, no figures

We study scalar excitations of high spin operators in N=4 super Yang-Mills theory, which are dual to solitons propagating on a long folded string in AdS_3 x S^1. In the spin chain description of the gauge theory, these are associated to holes in the magnon distribution in the sl(2,R) sector. We compute the all-loop hole S-matrix from the asymptotic Bethe ansatz, and expand in leading orders at weak and strong coupling. The worldsheet S-matrix of solitonic excitations on the GKP string is calculated using semiclassical quantization. We find an exact agreement between the gauge theory and string theory results., Comment: 13 pages. v2: minor corrections, references added

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 added

We present a new duality between the F-terms of supersymmetric field theories defined in two- and four-dimensions respectively. The duality relates N=2 supersymmetric gauge theories in four dimensions, deformed by an Omega-background in one plane, to N=(2,2) gauged linear sigma-models in two dimensions. On the four dimensional side, our main example is N=2 SQCD with gauge group SU(L) and 2L fundamental flavours. Using ideas of Nekrasov and Shatashvili, we argue that the Coulomb branch of this theory provides a quantization of the classical Heisenberg SL(2) spin chain. Agreement with the standard quantization via the Algebraic Bethe Ansatz implies the existence of an isomorphism between the chiral ring of the 4d theory and that of a certain two-dimensional theory. The latter can be understood as the worldvolume theory on a surface operator/vortex string probing the Higgs branch of the same 4d theory. We check the proposed duality by explicit calculation at low orders in the instanton expansion. One striking consequence is that the Seiberg-Witten solution of the 4d theory is captured by a one-loop computation in two dimensions. The duality also has interesting connections with the AGT conjecture, matrix models and topological string theory where it corresponds to a refined version of the geometric transition., Comment: 51 pages, 7 figures. Additional comments, minor improvements and references added

We analyse semiclassical strings in AdS in the limit of one large spin. In this limit, classical string dynamics is described by a finite number of collective coordinates corresponding to spikes or cusps of the string. The semiclassical spectrum consists of two branches of excitations corresponding to "large" and "small" spikes respectively. We propose that these states are dual to the excitations known as large and small holes in the spin chain description of N=4 SUSY Yang-Mills. The dynamics of large spikes in classical string theory can be mapped to that of a classical spin chain of fixed length. In turn, small spikes correspond to classical solitons propagating on the background formed by the large spikes. We derive the dispersion relation for these excitations directly in the finite gap formalism., Comment: 36 pages, 9 figures

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 figures

We compute the force and torque between well-separated, slowly-moving Giant Magnons with arbitrary orientations on S^5. We propose an effective Hamiltonian for Giant Magnons in this regime.

We study the semiclassical spectrum of excitations around a long spinning string in AdS_3. In addition to the usual small fluctuations, we find the spectrum contains a branch of solitonic excitations of finite energy. We determine the dispersion relation for these excitations. This has a relativistic form at low energies but also matches the dispersion relation for the "hole" of the dual gauge theory spin chain at high energies. The low-energy behaviour is consistent with the hypothesis that the solitonic excitations studied here are continuously related to the elementary excitations of the string., Comment: 17 pages, 1 figure; additional references included

We determine spectral curves for the known spiky string solutions in AdS space in the limit of large angular momentum. We also construct generic multi-spike solutions in this limit and compute the corresponding spectral data. The resulting spectral curves precisely match those of the classical spin chain describing the dual operators in one-loop gauge theory. Our results confirm the map between string theory and gauge theory degrees of freedom proposed in arXiv:0805.4387 [hep-th]., Comment: 51 pages, 5 figures; added 1 reference, corrected typos

We study the semiclassical spectrum of bosonic string theory on AdS_3 x S^1 in the limit of large AdS angular momentum. At leading semiclassical order, this is a subsector of the IIB superstring on AdS_5 x S^5. The theory includes strings with K spikes which approach the boundary in this limit. We show that, for all K, the spectrum of these strings exactly matches that of the corresponding operators in the dual gauge theory up to a single universal prefactor which can be identified with the cusp anomalous dimension. We propose a precise map between the dynamics of the spikes and the classical SL(2,R) spin chain which arises in the large-spin limit of N=4 Super Yang-Mills theory., Comment: 37 Pages, 8 figures

We study the conjectured exact S-matrix for the scattering of BPS magnon boundstates in the spin-chain description of planar N=4 SUSY Yang-Mills. The conjectured S-matrix exhibits both simple and double poles at complex momenta. Some of these poles lie parametrically close to the real axis in momentum space on the branch where particle energies are positive. We show that all such poles are precisely accounted for by physical processes involving one or more on-shell intermediate particles belonging to the known BPS spectrum., Comment: 32 pages, 9 figures

We perform a first-principles semi-classical computation of the one-loop corrections to the dispersion relation and S-matrix of Giant Magnons in AdS_5 x S^5 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., Comment: 47 pages, 2 figures. References added, to be submitted to JHEP

We investigate the analytic structure of the magnon S-matrix in the spin-chain description of planar ${\cal N}=4$ SUSY Yang-Mills/$AdS_{5}\times S^{5}$ strings. Semiclassical analysis suggests that the exact S-matrix must have a large family of poles near the real axis in momentum space. In this article we show that these are double poles corresponding to the exchange of pairs of BPS magnons. Their locations in the complex plane are uniquely fixed by the known dispersion relation for the BPS particles. The locations precisely agree with the recent conjecture for the $S$ matrix by Beisert, Hernandez, Lopez, Eden and Staudacher (hep-th/0609044 and hep-th/0610251). These poles do not signal the presence of new bound states. In fact, a certain non-BPS localized classical solution, which was thought to give rise to new bound states, can actually decay into a pair of BPS magnons., Comment: 40 pages, 14 figures; typos corrected, references added

In this paper we discuss the asymptotic spectrum of the spin chain description of planar N=4 SUSY Yang-Mills. The states appearing in the spectrum belong to irreducible representations of the unbroken supersymmetry SU(2|2) x 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 16Q^{2}. Generalising the Beisert's analysis of the Q=1 case, we derive the exact dispersion relation for these states by purely group theoretic means., Comment: 17 pages, No figure

We study the scattering of magnon boundstates in the spin-chain description of planar N=4 SUSY Yang-Mills. Starting from the conjectured exact S-matrix for magnons in the SU(2) sector, we calculate the corresponding S-matrix for boundstates with an arbitrary number of constituent magnons. The resulting expression has an interesting analytic structure with both simple and double poles. We also calculate the semiclassical S-matrix for the scattering of the corresponding excitations on the string worldsheet known as Dyonic Giant Magnons. We find precise agreement with the magnon boundstate S-matrix in the limit of large 't Hooft coupling., Comment: 14 pages

We define regularised Poisson brackets for the monodromy matrix of classical string theory on R x S^3. 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 AdS_5 x S^5 and lead to integer-valued filling fractions in this context., Comment: 41 pages, 2 figures; added references, corrected typos, improved discussion of Hamiltonian constraints

We study the classical spectrum of string theory on AdS_5 X S^5 in the Hofman-Maldacena limit. We find a family of classical solutions corresponding to Giant Magnons with two independent angular momenta on S^5. These solutions are related via Pohlmeyer's 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 N=4 SUSY Yang-Mills. The exact dispersion relation for these states is obtained from a purely classical calculation in string theory., Comment: LaTeX file, 16 pages. One figure. Corrected references

We study the spectrum of asymptotic states in the spin-chain description of planar N=4 SUSY Yang-Mills. In addition to elementary magnons, the asymptotic spectrum includes an infinite tower of multi-magnon bound states with exact dispersion relation, Delta-J_{1} = sqrt{Q^{2}+(lambda/pi^2)sin^2(p/2)}, where the positive integer Q is the number of constituent magnons. These states account precisely for the known poles in the exact S-matrix. Like the elementary magnon, they transform in small representations of supersymmetry and are present for all values of the 't Hooft coupling. At strong coupling we identify the dual states in semiclassical string theory., Comment: LaTeX File, 14 Pages. Minor improvements and corrected references

We study the dynamics of finite-gap solutions in classical string theory on R x S^3. Each solution is characterised by a spectral curve, \Sigma, of genus g and a divisor, \gamma, 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(\Sigma). 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., Comment: 64 Pages, 5 Figures; typos corrected and references added

We study large-N double-scaling limits of U(N) gauge theories in four dimensions. We focus on theories in a partially confining phase where an abelian subgroup $\hat{G}$ of the gauge group remains unconfined. Double-scaling is defined near critical points in the parameter/moduli space where states charged under $\hat{G}$ become massless. In specific cases, we present evidence that the double-scaled theory is dual to a non-critical superstring background. Models studied include the $\beta$-deformation of ${\cal N}=4$ SUSY Yang-Mills which leads to a non-critical string theory with sixteen supercharges. We also study ${\cal N}=1$ SUSY Yang-Mills theory coupled to a single chiral superfield with a polynomial superpotential which leads to a related string theory with eight supercharges. In both cases the string coupling is small and the background is free from Ramond-Ramond flux., Comment: 75 pages, 5 figures

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 pages

We present evidence for a new deconstruction of Little String Theory (LST). The starting point is a four-dimensional conformal field theory on its Higgs branch which provides a lattice regularization of six-dimensional gauge theory. We argue that the corresponding continuum limit is a 't Hooft large-N limit of the same four-dimensional theory on an S-dual confining branch. The AdS/CFT correspondence is then used to study this limit in a controlled way. We find that the limit yields LST compactified to four dimensions on a torus of fixed size. The limiting theory also contains other massive and massless states which are completely decoupled. The proposal can be adapted to deconstruct Double-Scaled Little String Theory and provides the first example of a large-N confining gauge theory in four dimensions with a fully tractable string theory dual., Comment: LaTeX file, 50 pages. Minor improvements, references added

We study an exactly marginal deformation of N=4 SUSY Yang-Mills with gauge group U(N) using field theory and string theory methods. The classical theory has a Higgs branch for rational values of the deformation parameter. We argue that the quantum theory also has an S-dual confining branch which cannot be seen classically. The low-energy effective theory on these branches is a six-dimensional non-commutative gauge theory with sixteen supercharges. Confinement of magnetic and electric charges, on the Higgs and confining branches respectively, occurs due to the formation of BPS-saturated strings in the low energy theory. The results also suggest a new way of deconstructing Little String Theory as a large-N limit of a confining gauge theory in four dimensions., Comment: LaTeX file, 59 pages, no figures. Minor improvements, references added

We investigate an exactly marginal N=1 supersymmetric deformation of SU(N) 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,Z) modular group on the bare couplings of the theory extending the action of electric-magnetic duality away from the N=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., Comment: 33 pages, 2 figures, uses JHEP.cls; minus sign typo corrected and 1 reference added, submitted to JHEP

In this note we show how Dijkgraaf and Vafa's hypothesis relating the exact superpotential of an N=1 theory to a matrix model can be used to describe all the massive vacua of the N=1*, or mass deformed N=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,Z) on the N=4 coupling, thus providing evidence for modular invariance of the underlying N=4 theory., Comment: 9 pages, JHEP.cls

Dijkgraaf and Vafa (DV) have conjectured that the exact superpotential for a large class of N=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 N=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 N=4 theory., Comment: 28 pages, 1 figure, latex with JHEP.cls, replaced with typos corrected and one clarifying comment

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 briefly 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., Comment: 230 pages, amstex, draft review (comments welcome)

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 pages

We study relevant deformations of the N=2 superconformal theory on the world-volume of N D3 branes at an A_{k-1} singularity. In particular, we determine the vacuum structure of the mass-deformed theory with N=1 supersymmetry and show how the different vacua are permuted by an extended duality symmetry. We then obtain exact, modular covariant formulae (for all k, N and arbitrary gauge couplings) for the holomorphic observables in the massive vacua in two different ways: by lifting to M-theory, and by compactification to three dimensions and subsequent use of mirror symmetry. In the latter case, we find an exact superpotential for the model which coincides with a certain combination of the quadratic Hamiltonians of the spin generalization of the elliptic Calogero-Moser integrable system., Comment: 55 pages, 5 figures, latex with JHEP.cls

We review four basic examples where string theory and/or field theory dualities predict the existence of soliton bound-states. These include the existence of threshold bound-states of D0 branes required by IIA/M duality and the closely-related bound-states of instantons in the maximally supersymmetric five dimensional gauge theory. In the IIB theory we discuss (p,q)-strings as bound-states of D and F strings, as well as the corresponding bound-states of monopoles and dyons in N=4 supersymmetric Yang-Mills theory whose existence was predicted by Sen. In particular we consider the L^2-index theory relevant for counting these states. In each case we show that the bulk contribution to the index can be evaluated by relating it to an instanton effect in the corresponding theory with a compact Euclidean time dimension. The boundary contribution to the index can be determined by considering the asymptotic regions of the relevant moduli space., Comment: 36 pages, latex with JHEP.cls

The D-instanton partition function is a fascinating quantity because in the presence of N D3-branes, and in a certain decoupling limit, it reduces to the functional integral of N=4 U(N) supersymmetric gauge theory for multi-instanton solutions. We study this quantity as a function of non-commutativity in the D3-brane theory, VEVs corresponding to separating the D3-branes and alpha'. Explicit calculations are presented in the one-instanton sector with arbitrary N, and in the large-N limit for all instanton charge. We find that for general instanton charge, the matrix theory admits a nilpotent fermionic symmetry and that the action is Q-exact. Consequently the partition function localizes on the minima of the matrix theory action. This allows us to prove some general properties of these integrals. In the non-commutative theory, the contributions come from the ``Higgs Branch'' and are equal to the Gauss-Bonnet-Chern integral of the resolved instanton moduli space. Separating the D3-branes leads to additional localizations on products of abelian instanton moduli spaces. In the commutative theory, there are additional contributions from the ``Coulomb Branch'' associated to the small instanton singularities of the instanton moduli space. We also argue that both non-commutativity and alpha'-corrections play a similar role in suppressing the contributions from these singularities. Finally we elucidate the relation between the partition function and the Euler characteristic of the instanton moduli space., Comment: 32 pages, JHEP.cls, some extra comments and typos fixed