Publications

We discuss in detail the 1+1-dimensional superconformal field theory dual to type II string theory on AdS3×S3×T4, emphasizing the string theoretic aspects of this duality. For one unit of Neveu-Schwarz (NS-NS) 5-brane flux (Q5=1), this string theory has been suggested to be dual to a grand-canonical ensemble of T4N/SN free symmetric orbifold conformal field theories (CFTs). We show how the string genus expansion emerges to all orders for the free orbifold grand-canonical correlation functions. We also discuss how the strong coupling limit of the NS-NS string theory arises (even at large N) in the free orbifold description, and argue why this limit does not have a weakly coupled RR description. The dual conformal field theory (CFT) includes (for all values of Q5) an extra T4 factor that is decoupled from perturbative string theory. We discuss the exactly marginal deformations that relate the different values of Q5, including the precise JJ¯ deformations mixing this extra T4 with the symmetric orbifold.

We provide a precise relation between an ensemble of Narain conformal field theories (CFTs) with central charge c = n, and a sum of (U(1) × U(1))ⁿ Chern-Simons theories on different handlebody topologies. We begin by reviewing the general relation of additive codes to Narain CFTs. Then we describe a holographic duality between any given Narain theory and a pure Chern-Simons theory on a handlebody manifold. We proceed to consider an ensemble of Narain theories, defined in terms of an ensemble of codes of length n over ℤ_k × ℤ_k for prime k. We show that averaging over this ensemble is holographically dual to a level-k (U(1) × U(1))ⁿ Chern-Simons theory, summed over a finite number of inequivalent classes of handlebody topologies. In the limit of large k the ensemble approaches the ensemble of all Narain theories, and its bulk dual becomes equivalent to \u201cU(1)-gravity\u201d the sum of the pertubative part of the Chern-Simons wavefunction over all possible handlebodies providing a bulk microscopic definition for this theory. Finally, we reformulate the sum over handlebodies in terms of Hecke operators, paving the way for generalizations.

The Charge Convexity Conjecture (CCC) states that in a unitary conformal field theory in d ≥ 3 dimensions with a global symmetry, the minimal dimension of operators in certain representations of the symmetry, as a function of the charge q of the representation (or a generalized notion of it), should be convex. More precisely, this was conjectured to be true when q is restricted to positive integer multiples of some integer q₀. The CCC was tested on a number of examples, most of which are in d 0 is taken to be the charge of the lowest-dimension positively-charged operator was shown to hold in all of them. In this paper we test the conjecture in a non-trivial example of a d = 4 theory, which is the family of Caswell-Banks-Zaks IR fixed points of SU(N_c) gauge theory coupled to N_f massless fermions and N_s massless scalars. In these theories, the lowest-dimension gauge-invariant operators that transform non-trivially under the global symmetry are mesons. These may consist of two scalars, two fermions or one of each. We find that the CCC holds in all applicable cases, providing significant new evidence for its validity, and suggesting a stronger version for non-simple global symmetry groups.

We study the low-energy limit of Wilson lines (charged impurities) in conformal gauge theories in 2+1 and 3+1 dimensions. As a function of the representation of the Wilson line, certain defect operators can become marginal, leading to interesting renormalization group flows and for sufficiently large representations to complete or partial screening by charged fields. This result is universal: in large enough representations, Wilson lines are screened by the charged matter fields. We observe that the onset of the screening instability is associated with fixed-point mergers. We study this phenomenon in a variety of applications. In some cases, the screening of the Wilson lines takes place by dimensional transmutation and the generation of an exponentially large scale. We identify the space of infrared conformal Wilson lines in weakly coupled gauge theories in 3+1 dimensions and determine the screening cloud due to bosons or fermions. We also study QED in 2+1 dimensions in the large Nf limit and identify the nontrivial conformal Wilson lines. We briefly discuss 't Hooft lines in 3+1-dimensional gauge theories and find that they are screened in weakly coupled gauge theories with simply connected gauge groups. In non-Abelian gauge theories with S duality, this together with our analysis of the Wilson lines gives a compelling picture for the screening of the line operators as a function of the coupling.

The weak gravity conjecture is typically stated as a bound on the mass-to-charge ratio of a particle in the theory. Alternatively, it has been proposed that its natural formulation is in terms of the existence of a particle which is self-repulsive under all long-range forces. We propose a closely related, but distinct, formulation, which is that it should correspond to a particle with non-negative self-binding energy. This formulation is particularly interesting in antide Sitter space, because it has a simple conformal field theory (CFT) dual formulation: let Δ(q) be the dimension of the lowest-dimension operator with charge q under some global U(1) symmetry, then Δ(q) must be a convex function of q. This formulation avoids any reference to holographic dual forces or even to locality in spacetime, and so we make a wild leap, and conjecture that such convexity of the spectrum of charges holds for any (unitary) conformal field theory, not just those that have weakly coupled and weakly curved duals. This charge convexity conjecture, and its natural generalization to larger global symmetry groups, can be tested in various examples where anomalous dimensions can be computed, by perturbation theory, 1/N expansions and semiclassical methods. In all examples that we tested we find that the conjecture holds. We do not yet understand from the CFT point of view why this is true.

We construct holographic backgrounds that are dual by the AdS/CFT correspondence to Euclidean conformal field theories on products of spheres Sd1xSd2, for conformal field theories whose dual may be approximated by classical Einstein gravity (typically these are large N strongly coupled theories). For d(2) = 1 these backgrounds correspond to thermal field theories on Sd1, and Hawking and Page found that there are several possible bulk solutions, with two different topologies, that compete with each other, leading to a phase transition as the relative size of the spheres is modified. By numerically solving the Einstein equations we find similar results also for d(2)> 1, with bulk solutions in which either one or the other sphere shrinks to zero smoothly at a minimal value of the radial coordinate, and with a first order phase transition (for d(1) + d(2)

We consider a string dual of a partially topological U(N) Chern-Simons-matter (PTCSM) theory recently introduced by Aganagic, Costello, McNamara and Vafa. In this theory, fundamental matter fields are coupled to the Chern-Simons theory in a way that depends only on a transverse holomorphic structure on a manifold; they are not fully dynamical, but the theory is also not fully topological. One description of this theory arises from topological strings on the deformed conifold TS3 with N Lagrangian 3-branes and additional coisotropic flavor' 5-branes. Applying the idea of the Gopakumar-Vafa duality to this setup, we suggest that this has a dual description as a topological string on the resolved conifold CP1, in the presence of coisotropic 5-branes. We test this duality by computing the annulus amplitude on the deformed conifold and the disc amplitude on the resolved conifold via equivariant localization, and we find an agreement between the two. We find a small discrepancy between the topological string results and the large N limit of the partition function of the PTCSM theory arising from the deformed conifold, computed via field theory localization by a method proposed by Aganagic et al. We discuss possible origins of the mismatch.

Three-dimensional Chern-Simons vector models display an approximate higher spin symmetry in the large N limit. Their single-trace operators consist of a tower of weakly broken currents, as well as a scalar a of approximate twist 1 or 2. We study the consequences of crossing symmetry for the four-point correlator of a in a 1/N expansion, using analytic bootstrap techniques. To order 1/N we show that crossing symmetry fixes the contribution from the tower of currents, providing an alternative derivation of well-known results by Maldacena and Zhiboedov. When sigma has twist 1 its OPE receives a contribution from the exchange of a itself with an arbitrary coefficient, due to the existence of a marginal sextic coupling. We develop the machinery to determine the corrections to the OPE data of double-trace operators due to this, and to similar exchanges. This in turns allows us to fix completely the correlator up to three known truncated solutions to crossing. We then proceed to study the problem to order 1/N-2. We find that crossing implies the appearance of odd-twist double-trace operators, and calculate their OPE coefficients in a large spin expansion. Also, surprisingly, crossing at order 1/N-2, implies non-trivial O(1/N) anomalous dimensions for even-twist double-trace operators, even though such contributions do not appear in the four-point function at order 1/N (in the case where there is no scalar exchange). We argue that this phenomenon arises due to operator mixing. Finally, we analyse the bosonic vector model with a sextic coupling without gauge interactions, and determine the order 1/N-2 corrections to the dimensions of twist-2 double-trace operators.

In this paper, we analyze the renormalization group (RG) flow of field theories with quenched disorder, in which the couplings vary randomly in space. We analyze both classical (Euclidean) disorder and quantum disorder, emphasizing general properties rather than specific cases. The RG flow of the disorder-averaged theories takes place in the space of their coupling constants and also in the space of distributions for the disordered couplings, and the two mix together. We write down a generalization of the CallanSymanzik equation for the flow of disorder-averaged correlation functions. We find that local operators can mix with the response of the theory to local changes in the disorder distribution and that the generalized Callan-Symanzik equation mixes the disorder averages of several different correlation functions. For classical disorder, we show that this can lead to new types of anomalous dimensions and to logarithmic behavior at fixed points. For quantum disorder, we find that the RG flow always generates a rescaling of time relative to space, which at a fixed point generically leads to Lifshitz scaling. The dynamical scaling exponent z behaves as an anomalous dimension (as in other nonrelativistic RG flows), and we compute it at leading order in perturbation theory in the disorder for a general theory. Our results agree with a previous perturbative computation by Boyanovsky and Cardy, and with a holographic disorder computation of Hartnoll and Santos. We also find in quantum disorder that local operators mix with nonlocal (in time) operators under the RG, and that there are critical exponents associated with the disorder distribution that have not previously been discussed. In large-N theories, the disorder averages may be computed exactly, and we verify that they are consistent with the generalized Callan-Symanzik equations.

We propose and demonstrate a new use for conformal field theory (CFT) crossing equations in the context of AdS/CFT: the computation of loop amplitudes in AdS, dual to non-planar correlators in holographic CFTs. Loops in AdS are largely unexplored, mostly due to technical difficulties in direct calculations. We revisit this problem, and the dual 1/N expansion of CFTs, in two independent ways. The first is to show how to explicitly solve the crossing equations to the first subleading order in 1/N², given a leading order solution. This is done as a systematic expansion in inverse powers of the spin, to all orders. These expansions can be resummed, leading to the CFT data for finite values of the spin. Our second approach involves Mellin space. We show how the polar part of the four-point, loop-level Mellin amplitudes can be fully reconstructed from the leading-order data. The anomalous dimensions computed with both methods agree. In the case of ϕ⁴ theory in AdS, our crossing solution reproduces a previous computation of the one-loop bubble diagram. We can go further, deriving the four-point scalar triangle diagram in AdS, which had never been computed. In the process, we show how to analytically derive anomalous dimensions from Mellin amplitudes with an infinite series of poles, and discuss applications to more complicated cases such as the N = 4 super-Yang-Mills theory.

In this paper we study supersymmetric field theories on an AdS(p) x S-q space-time that preserves their full supersymmetry. This is an interesting example of supersymmetry on a non-compact curved space. The supersymmetry algebra on such a space is a (p - 1)-dimensional superconformal algebra, and we classify all possible algebras that can arise for p >= 3. In some AdS(3) cases more than one superconformal algebra can arise from the same field theory. We discuss in detail the special case of four dimensional field theories with N = 1 and N = 2 supersymmetry on AdS(3) x S-1.

In this note we study four dimensional theories with N = 3 superconformal symmetry, that do not also have N = 4 supersymmetry. No examples of such theories are known, but their existence is also not ruled out. We analyze several properties that such theories must have. We show that their conformal anomalies obey a = c. Using the N = 3 superconformal algebra, we show that they do not have any exactly marginal deformations preserving N = 3 supersymmetry, or global symmetries (except for their R-symmetries). Finally, we analyze the possible dimensions of chiral operators labeling their moduli space.

Field theories with weakly coupled holographic duals, such as large N gauge theories, have a natural separation of their operators into 'single-trace operators' (dual to single-particle states) and 'multi-trace operators (dual to multi-particle states). There are examples of large N gauge theories where the beta functions of single-trace coupling constants all vanish, but marginal multi-trace coupling constants have non-vanishing beta functions that spoil conformal invariance (even when all multi-trace coupling constants vanish). The holographic dual of such theories should be a classical solution in anti-de Sitter space, in which the boundary conditions that correspond to the multi-trace coupling constants depend on the cutoff scale, in a way that spoils conformal invariance. We argue that this is realized through specific bulk coupling constants that lead to a running of the multi-trace coupling constants. This fills a missing entry in the holographic dictionary.

Many examples of low-energy dualities have been found in supersymmetric gauge theories with four supercharges, both in four and in three space-time dimensions. In these dualities, two theories that are different at high energies have the same low-energy limit. In this paper we clarify the relation between the dualities in four and in three dimensions. We show that every four dimensional duality gives rise to a three dimensional duality between theories that are similar, but not identical, to the dimensional reductions of the four dimensional dual gauge theories to three dimensions. From these specific three dimensional dualities one can flow to many other low-energy dualities, including known three dimensional dualities and many new ones. We discuss in detail the case of three dimensional SU(N _c) supersymmetric QCD theories, showing how to derive new duals for these theories from the four dimensional duality.

Starting with a choice of a gauge group in four dimensions, there is often freedom in the choice of magnetic and dyonic line operators. Different consistent choices of these operators correspond to distinct physical theories, with the same correlation functions of local operators in R⁴. In some cases these choices are permuted by shifting the θ-angle by 2π. In other cases they are labeled by new discrete θ-like parameters. Using this understanding we gain new insight into the dynamics of four-dimensional gauge theories and their phases. The existence of these distinct theories clarifies a number of issues in electric/magnetic dualities of supersymmetric gauge theories, both for the conformal N = 4 theories and for the low-energy dualities of N = 1 theories.

We study the low-energy effective action governing the transverse fluctuations of a long string, such as a confining flux tube in QCD. We work in the static gauge where this action contains only the transverse excitations of the string. The static gauge action is strongly constrained by the requirement that the Lorentz symmetry, that is spontaneously broken by the long string vacuum, is nonlinearly realized on the Nambu-Goldstone bosons. One solution to the constraints (at the classical level) is the Nambu-Goto action, and the general solution contains higher derivative corrections to this. We show that in 2 + 1 dimensions, the first allowed correction to the Nambu-Goto action is proportional to the squared curvature of the induced metric on the worldsheet. In higher dimensions, there is a more complicated allowed correction that appears at lower order than the curvature squared. We argue that this leading correction is similar to, but not identical to, the one-loop determinant √-hR□- ¹R computed by Polyakov for the bosonic fundamental string.

We consider the conformal field theory of N complex massless scalars in 2 + 1 dimensions, coupled to a U(N) Chern-Simons theory at level k. This theory has a 't Hooft large N limit, keeping fixed λ ≡ N/k. We compute some correlation functions in this theory exactly as a function of λ, in the large N (planar) limit. We show that the results match with the general predictions of Maldacena and Zhiboedov for the correlators of theories that have high-spin symmetries in the large N limit. It has been suggested in the past that this theory is dual (in the large N limit) to the Legendre transform of the theory of fermions coupled to a Chern-Simons gauge field, and our results allow us to find the precise mapping between the two theories. We find that in the large N limit the theory of N scalars coupled to a U(N) _k Chern-Simons theory is equivalent to the Legendre transform of the theory of k fermions coupled to a U(k) _N Chern-Simons theory, thus providing a bosonization of the latter theory. We conjecture that perhaps this duality is valid also for finite values of N and k, where on the fermionic side we should now have (for N _f flavors) a U(k)N-{N_f/2 theory. Similar results hold for real scalars (fermions) coupled to the O(N) _k Chern-Simons theory.

In this paper we discuss the dynamics of conformal field theories on anti-de Sitter space, focussing on the special case of the N = 4 supersymmetric Yang-Mills theory on AdS₄. We argue that the choice of boundary conditions, in particular for the gauge field, has a large effect on the dynamics. For example, for weak coupling, one of two natural choices of boundary conditions for the gauge field leads to a large N deconfinement phase transition as a function of the temperature, while the other does not. For boundary conditions that preserve supersymmetry, the strong coupling dynamics can be analyzed using S-duality (relevant for g_{y m}1), utilizing results of Gaiotto and Witten, as well as by using the AdS/CFT correspondence (relevant for large N and large 't Hooft coupling). We argue that some very specific choices of boundary conditions lead to a simple dual gravitational description for this theory, while for most choices the gravitational dual is not known. In the cases where the gravitational dual is known, we discuss the phase structure at large 't Hooft coupling.

We study three dimensional N = 2 supersymmetric QCD theories with O(N _c) gauge groupsand with N_f chiral multiplets in the vector representation. We argue that for N_fc - 2 there is a runaway potential on the moduli space and no vacuum. For N_f ≥ N_c - 2 there is a moduli space also in the quantum theory, and for N_f ≥ N_c - 1 there is a superconformal fixed point at the origin of this moduli space that has a dual description as the low-energy fixed point of an O(N_f - N_c + 2) gauge theory. We test this duality in various ways;in some cases the duality for an O(2) gauge theory may be related to the known duality for U(1) gauge theories. We also discuss real mass deformations, which allow to connect theories with a different Chern-Simons level. This allows us to connect our duality with the known duality in O(N_c) theories with a Chern-Simons term of level k, where the dual gauge group is given by O(N_f + |k| - N_c + 2).

We study a brane configuration of D4-branes and NS5-branes in weakly coupled type IIA string theory, which describes in a particular limit d=4 N=1 SU(N+p) supersymmetric QCD with 2N flavors and a quartic superpotential. We describe the geometric realization of the supersymmetric vacuum structure of this gauge theory. We focus on the confining vacua of the gauge theory, whose holographic description is given by the MQCD brane configuration in the near-horizon geometry of N D4-branes. This description, which gives an embedding of MQCD into a field theory decoupled from gravity, is valid for 1pN, in the limit of large five-dimensional 't Hooft couplings for the color and flavor groups. We analyze various properties of the theory in this limit, such as the spectrum of mesons, the finite temperature behavior, and the quark-antiquark potential. We also discuss the same brane configuration on a circle, where it gives a geometric description of the moduli space of the Klebanov-Strassler cascading theory, and some nonsupersymmetric generalizations.

We perform a systematic analysis of the D-brane charges associated with string theory realizations of d = 3 gauge theories, focusing on the examples of the N = 4 supersymmetric U(N) × U(N + M) Yang-Mills theory and the N = 3 supersymmetric U(N)×U(N +M) Yang-Mills-Chern-Simons theory. We use both the brane construction of these theories and their dual string theory backgrounds in the supergravity approximation. In the N = 4 case we generalize the previously known gravitational duals to arbitrary values of the gauge couplings, and present a precise mapping between the gravity and field theory parameters. In the N = 3 case, which (for some values of N and M) flows to an N = 6 supersymmetric Chern-Simons-matter theory in the IR, we argue that the careful analysis of the charges leads to a shift in the value of the B ₂ field in the IR solution by 1/2, in units where its periodicity is one, compared to previous claims. We also suggest that the N = 3 theories may exhibit, for some values of N and M, duality cascades similar to those of the Klebanov-Strassler theory.

We construct three dimensional Chern-Simons-matter theories with gauge groups U(N) × U(N) and SU(N) × SU(N) which have explicit = 6 superconformal symmetry. Using brane constructions we argue that the U(N) × U(N) theory at level k describes the low energy limit of N M2-branes probing a C ⁴/Z _k singularity. At large N the theory is then dual to M-theory on AdS ₄ × S ⁷/Z _k. The theory also has a 't Hooft limit (of large N with a fixed ratio N/k) which is dual to type IIA string theory on AdS ₄ × CP ³. For k = 1 the theory is conjectured to describe N M2-branes in flat space, although our construction realizes explicitly only six of the eight supersymmetries. We give some evidence for this conjecture, which is similar to the evidence for mirror symmetry in d = 3 gauge theories. When the gauge group is SU(2) × SU(2) our theory has extra symmetries and becomes identical to the Bagger-Lambert theory.

We study the holographic map between long open strings, which stretch between D-branes separated in the bulk space-time, and operators in the dual boundary theory. We focus on a generalization of the Sakai-Sugimoto holographic model of QCD, where the simplest chiral condensate involves an operator of this type. Its expectation value is dominated by a semiclassical string world sheet, as for Wilson loops. We also discuss the deformation of the model by this operator, and, in particular, its effect on the meson spectrum. This deformation can be thought of as a generalization of a quark mass term to strong coupling. It leads to the first top-down holographic model of QCD with a non-Abelian chiral symmetry which is both spontaneously and explicitly broken, as in QCD. Other examples we study include half-supersymmetric open Wilson lines, and systems of D-branes ending on NS5-branes, which can be analyzed using world sheet methods.

We study the conformal field theory dual of the type IIA flux compactification model of DeWolfe, Giryavets, Kachru and Taylor, with all moduli stabilized. We find its central charge and properties of its operator spectrum. We concentrate on the moduli space of the conformal field theory, which we investigate through domain walls in the type IIA string theory. The moduli space turns out to consist of many different branches. We use Bezout's theorem and Bernstein's theorem to enumerate the different branches of the moduli space and estimate their dimension.

We numerically construct black hole solutions corresponding to the deconfined, chirally symmetric phase of strongly coupled cascading gauge theories at various temperatures. We compute the free energy as a function of the temperature, and we show that it becomes positive below some critical temperature, indicating the possibility of a first order phase transition at which the theory deconfines and restores the chiral symmetry.

We analyze the finite temperature behavior of the Sakai-Sugimoto model, which is a holographic dual of a theory which spontaneously breaks a U(N_f)_L × U(N_f)_R chiral flavor symmetry at zero temperature. The theory involved is a 4 + 1 dimensional supersymmetric SU(N_c) gauge theory compactified on a circle of radius R with anti-periodic boundary conditions for fermions, coupled to N_f left-handed quarks and N_f right-handed quarks which are localized at different points on the compact circle (separated by a distance L). In the supergravity limit which we analyze (corresponding in particular to the large N_c limit of the gauge theory), the theory undergoes a deconfinement phase transition at a temperature T_d = 1/2πR. For quark separations obeying L > L_c ≃ 0.97 * R the chiral symmetry is restored at this temperature, but for L c ≃ 0.97 * R there is an intermediate phase which is deconfined with broken chiral symmetry, and the chiral symmetry is restored at T_χSB ≃ 0.154/L. All of these phase transitions are of first order.

We discuss the moduli space of nine dimensional ≤ 1 supersymmetric compactifications of M theory / string theory with reduced rank (rank 10 or rank 2), exhibiting how all the different theories (including M theory compactified on a Klein bottle and on a Möbius strip, the Dabholkar-Park background, CHL strings and asymmetric orbifolds of type II strings on a circle) fit together, and what are the weakly coupled descriptions in different regions of the moduli space. We argue that there are two disconnected components in the moduli space of theories with rank 2. We analyze in detail the limits of the M theory compactifications on a Klein bottle and on a Möbius strip which naively give type IIA string theory with an uncharged orientifold 8-plane carrying discrete RR flux. In order to consistently describe these limits we conjecture that this orientifold non-perturbatively splits into a D8-brane and an orientifold plane of charge (-1) which sits at infinite coupling. We construct the M(atrix) theory for M theory on a Klein bottle (and the theories related to it), which is given by a 2+1 dimensional gauge theory with a varying gauge coupling compactified on a cylinder with specific boundary conditions. We also clarify the construction of the M(atrix) theory for backgrounds of rank 18, including the heterotic string on a circle.

Following recent developments in model building we construct a simple, natural and controllable model of gauge-mediated supersymmetry breaking.

We argue for the existence of plasma balls - metastable, nearly homogeneous lumps of gluon plasma at just above the deconfinement energy density - in a class of large-N confining gauge theories that undergo first-order deconfinement transitions. Plasma balls decay over a time scale of order N² by thermally radiating hadrons at the deconfinement temperature. In gauge theories that have a dual description that is well approximated by a theory of gravity in a warped geometry, we propose that plasma balls map to a family of classically stable finite-energy black holes localized in the IR. We present a conjecture for the qualitative nature of large-mass black holes in such backgrounds and numerically construct these black holes in a particular class of warped geometries. These black holes have novel properties; in particular, their temperature approaches a nonzero constant value at large mass. Black holes dual to plasma balls shrink as they decay by Hawking radiation; towards the end of this process, they resemble ten-dimensional Schwarzschild black holes, which we propose are dual to small plasma balls. Our work may find practical applications in the study of the physics of localized black holes from a dual viewpoint.

We give a direct path-integral calculation of the partition function for pure 3+1 dimensional U(N) Yang-Mills theory at large N on a small S3, up to two loop order in perturbation theory. From this, we calculate the one loop shift in the Hagedorn/deconfinement temperature for the theory at small volume, finding that it increases (in units of the inverse sphere radius) as we go to larger coupling (larger volume). Our results also allow us to read off the sum of one loop anomalous dimensions for all operators with a given engineering dimension in planar Yang-Mills theory on R4. As checks on our calculation, we reproduce both the Hagedorn shift and some of the anomalous dimension sums by independent methods using the results of Beisert et al. and Spradlin and Volovich to establish a first-order deconfinement transition for pure Yang-Mills theory on a small S3.

The AdS/CFT correspondence relates deformations of the CFT by "multi-trace operators" to "non-local string theories". The deformed theories seem to have non-local interactions in the compact directions of space-time; in the gravity approximation the deformed theories involve modified boundary conditions on the fields which are explicitly non-local in the compact directions. In this note we exhibit a particular non-local property of the resulting space-time theory. We show that in the usual backgrounds appearing in the AdS/CFT correspondence, the commutator of two bulk scalar fields at points with a large enough distance between them in the compact directions and a small enough time-like distance between them in AdS vanishes, but this is not always true in the deformed theories. We discuss how this is consistent with causality.

We give an analytic demonstration that the 3+1-dimensional large N SU(N) pure Yang-Mills theory, compactified on a small S3 so that the coupling constant at the compactification scale is very small, has a first order deconfinement transition as a function of temperature. We do this by explicitly computing the relevant terms in the canonical partition function up to three-loop order; this is necessary because the leading (one-loop) result for the phase transition is precisely on the border line between a first order and a second order transition. Since numerical work strongly suggests that the infinite-volume large N theory also has a first order deconfinement transition, we conjecture that the phase structure is independent of the size of the S3. To deal with divergences in our calculations, we are led to introduce a novel method of regularization useful for non-Abelian gauge theory on S3.

We demonstrate that weakly coupled, large N, d-dimensional SU(N) gauge theories on a class of compact spatial manifolds (including S^d-1× time) un-dergo deconfinement phase transitions at temperatures proportional to the inverse length scale of the manifold in question. The low temperature phase has a free energy of order one, and is characterized by a stringy (Hage-dorn) growth in its density of states. The high temperature phase has a free energy of order N². These phases are separated either by a single first order transition that generically occurs below the Hagedorn temperature or by two continuous phase transitions, the first of which occurs at the Hage-dorn temperature. These phase transitions could perhaps be continuously connected to the usual flat space deconfinement transition in the case of confining gauge theories, and to the Hawking-Page nucleation of AdS₅ black holes in the case of the N = 4 supersymmetric Yang-Mills theory. We sug-gest that deconfinement transitions may generally be interpreted in terms of black hole formation in a dual string theory. Our analysis proceeds by first reducing the Yang-Mills partition function to a (0 + 0)-dimensional in-tegral over a unitary matrix U, which is the holonomy (Wilson loop) of the gauge field around the thermal time circle in Euclidean space; deconfinement transitions are large N transitions in this matrix integral.

Little string theory (LST) is a still somewhat mysterious theory that describes the dynamics near a certain class of time-like singularities in string theory. In this paper we discuss the topological version of LST, which describes topological strings near these singularities. For (5+1)-dimensional LSTs with sixteen supercharges, the topological version may be described holographically in terms of the N=4 topological string (or the N=2 string) on the transverse part of the near-horizon geometry of NS5-branes. We show that this topological string can be used to efficiently compute the half-BPS F⁴ terms in the low-energy effective action of the LST. Using the strong-weak coupling string duality relating type IIA strings on K3 and heterotic strings on T⁴, the same terms may also be computed in the heterotic string near a point of enhanced gauge symmetry. We study the F⁴ terms in the heterotic string and in the LST, and show that they have the same structure, and that they agree in the cases for which we compute both of them. We also clarify some additional issues, such as the definition and role of normalizable modes in holographic linear dilaton backgrounds, the precise identifications of vertex operators in these backgrounds with states and operators in the supersymmetric Yang-Mills theory that arises in the low energy limit of LST, and the normalization of two-point functions.

In these lecture notes from the 2002 Cargese summer school we review the progress that has been made towards finding a string theory for QCD (or for pure (super)Yang-Mills theory) following the discovery of the AdS/CFT correspondence. We start with a brief review of the AdS/CFT correspondence and a general discussion of its application to the construction of a string theory for QCD. We then discuss in detail two possible paths towards a QCD string theory, one which uses a mass deformation of the N = 4 super Yang-Mills theory (the Polchinski-Strassler background) and the other using a compactification of "little string theory" on S-2 (the Maldacena-Nufiez solution). A third approach (the Klebanov-Strassler solution) is described in other lectures of this school. We briefly assess the advantages and disadvantages of all three approaches.

N = 4 supersymmetric Yang-Mills theory with gauge group SU(N) (N greater than or equal to 3) is believed to have two exactly marginal deformations which break the supersymmetry to N = 1. We discuss the construction of the string theory dual to these deformations, in the supergravity approximation, in a perturbation series around the AdS(5) x S-5 solution. We construct explicitly the deformed solution at second order in the deformation. We show that deformations which are marginal but not exactly marginal lead to a non-conformal solution with a logarithmically running coupling constant. Surprisingly, at third order in the deformation we find the same beta functions for the couplings in field theory and in supergravity, suggesting that the leading order beta functions (or anomalous dimensions) do not depend on the gauge coupling (the coefficient is not renormalized)

In this paper we study exactly marginal deformations of field theories living on D3-branes at low energies. These theories include N = 4 supersymmetric Yang-Mills theory and theories obtained from it via the orbifolding procedure. We restrict ourselves only to orbifolds and deformations which leave some supersymmetry unbroken. A number of new families of N = 1 superconformal field theories are found. We analyze the deformations perturbatively, and also by using general arguments for the dimension of the space of exactly marginal deformations. We find some cases where the space of perturbative exactly marginal deformations is smaller than the prediction of the general analysis (at least up to three-loop order), and other cases where the perturbative result (at low orders) has a non-generic form.

We exhibit a simple class of exactly marginal "double-trace" deformations of two-dimensional conformal field theories (CFTs) which have AdS₃ duals, in which the deformation is given by a product of left- and right-moving U(1) currents. In this special case the deformation on AdS₃ is generated by a local boundary term in three dimensions, which changes the physics also in the bulk via bulk-boundary propagators. However, the deformation is nonlocal in six dimensions and on the string world sheet, as in generic nonlocal string theories. Because of the simplicity of the deformation we can explicitly make computations in the nonlocal string theory and compare them to CFT computations, and we obtain precise agreement. We discuss the effect of the deformation on closed strings and on D branes. The examples we analyze include a supersymmetry-breaking but exactly marginal "double-trace" deformation, which is dual to a string theory in which no destabilizing tadpoles are generated for moduli nonperturbatively in all couplings, despite the absence of supersymmetry. We explain how this cancellation works on the gravity side in string perturbation theory, and also nonperturbatively at leading order in the deformation parameter. We also discuss possible flat space limits of our construction.

We study the (2, 0) superconformal theories in six dimensions, which arise from the lowenergy limit of k coincident 5-branes, using their discrete light-cone formulation as a superconformal quantum mechanical sigma model. We analyze the realization of the superconformal symmetry in the quantum mechanics, and the realization of primary operators. As an example we compute the spectrum of chiral primary states in symmetric Spin(5)R representations. To facilitate the analysis we introduce and briefly discuss a new class of Lorentz non-invariant theories, which flow in the IR to the (2, 0) superconformal field theories but differ from them in the UV.