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Motivated by the desire to understand chaos in the $S$-matrix of string theory, we study tree level scattering amplitudes involving highly excited strings. While the amplitudes for scattering of light strings have been a hallmark of string theory since its early days, scattering of excited strings has been far less studied. Recent results on black hole chaos, combined with the correspondence principle between black holes and strings, suggest that the amplitudes have a rich structure. We review the procedure by which an excited string is formed by repeatedly scattering photons off of an initial tachyon (the DDF formalism). We compute the scattering amplitude of one arbitrary excited string and any number of tachyons in bosonic string theory. At high energies and high mass excited state these amplitudes are determined by a saddle-point in the integration over the positions of the string vertex operators on the sphere (or the upper half plane), thus yielding a generalization of the "scattering equations". We find a compact expression for the amplitude of an excited string decaying into two tachyons, and study its properties for a generic excited string. We find the amplitude is highly erratic as a function of both the precise excited string state and of the tachyon scattering angle relative to its polarization, a sign of chaos., Comment: 65 pages, 11 figures; published version

Motivated by $T\bar T$, we introduce and study a wide class of solvable deformations of quantum-mechanical theories. These deformations map the Hamiltonian to a function of itself. We solve these theories by computing all finite-temperature correlation functions of the deformed theory in terms of the correlators of the undeformed theory. Applications to AdS/CFT, SYK, and the Schwarzian theory are considered. We write down the deformed Schwarzian action for an arbitrary Hamiltonian deformation and find that the maximal Lyapunov exponent is unchanged., Comment: 30 pages + appendices. 3 figures. v2: Typos corrected, updated and corrected the discussion in section 2.3 and references added

Quantum field theory in zero spatial dimensions has a rich infrared landscape and a universal ultraviolet, inverting the usual Wilsonian paradigm. For holographic theories this implies a rich landscape of asymptotically AdS$_2$ geometries. Isolating the interiors of these spacetimes suggests a study of the analog of the $T\overline{T}$ deformation in quantum mechanics, which we pursue here. An equivalent description of this deformation in terms of coupling to worldline gravity is proposed, and applications to quantum mechanical systems - including the Schwarzian theory - are studied., Comment: 35 pages plus appendices

Large $N$ melonic theories are characterized by two-point function Feynman diagrams built exclusively out of melons. This leads to conformal invariance at strong coupling, four-point function diagrams that are exclusively ladders, and higher-point functions that are built out of four-point functions joined together. We uncover an incredibly useful property of these theories: the six-point function, or equivalently, the three-point function of the primary $O(N)$ invariant bilinears, regarded as an analytic function of the operator dimensions, fully determines all correlation functions, to leading nontrivial order in $1/N$, through simple Feynman-like rules. The result is applicable to any theory, not necessarily melonic, in which higher-point correlators are built out of four-point functions. We explicitly calculate the bilinear three-point function for $q$-body SYK, at any $q$. This leads to the bilinear four-point function, as well as all higher-point functions, expressed in terms of higher-point conformal blocks, which we discuss. We find universality of correlators of operators of large dimension, which we simplify through a saddle point analysis. We comment on the implications for the AdS dual of SYK., Comment: 67 pages, v2

We point out that there is a simple variant of the SYK model, which we call cSYK, that is $SL(2,R)$ invariant for all values of the coupling. The modification consists of replacing the UV part of the SYK action with a quadratic bilocal term. The corresponding bulk dual is a non-gravitational theory in a rigid AdS$_2$ background. At weak coupling cSYK is a generalized free field theory; at strong coupling, it approaches the infrared of SYK. The existence of this line of fixed points explains the previously found connection between the three-point function of bilinears in these two theories at large $q$., Comment: 26 pages, v2

The SYK model, a quantum mechanical model of $N \gg 1$ Majorana fermions $\chi_i$, with a $q$-body, random interaction, is a novel realization of holography. It is known that the AdS$_2$ dual contains a tower of massive particles, yet there is at present no proposal for the bulk theory. As SYK is solvable in the $1/N$ expansion, one can systematically derive the bulk. We initiate such a program, by analyzing the fermion two, four and six-point functions, from which we extract the tower of singlet, large $N$ dominant, operators, their dimensions, and their three-point correlation functions. These determine the masses of the bulk fields and their cubic couplings. We present these couplings, analyze their structure and discuss the simplifications that arise for large $q$., Comment: 39 pages, v2: Evaluation of integral in Sec. 3.3.2 corrected

The SYK model: fermions with a $q$-body, Gaussian-random, all-to-all interaction, is the first of a fascinating new class of solvable large $N$ models. We generalize SYK to include $f$ flavors of fermions, each occupying $N_a$ sites and appearing with a $q_a$ order in the interaction. Like SYK, this entire class of models generically has an infrared fixed point. We compute the infrared dimensions of the fermions, and the spectrum of singlet bilinear operators. We show that there is always a dimension-two operator in the spectrum, which implies that, like in SYK, there is breaking of conformal invariance and maximal chaos in the infrared four-point function of the generalized model. After a disorder average, the generalized model has a global $O(N_1) \times O(N_2) \times \ldots\times O(N_f)$ symmetry: a subgroup of the $O(N)$ symmetry of SYK; thereby giving a richer spectrum. We also elucidate aspects of the large $q$ limit and the OPE, and solve $q=2$ SYK at finite $N$., Comment: 46 pages, v2 minor changes; published version

Ultrahigh-energy string scattering is investigated to clarify the relative role of string and gravitational effects, and their possible contributions to nonlocal behavior. Different regimes can be characterized by varying the impact parameter at fixed energy. In the regime where momentum transfers reach the string scale, string effects appear subdominant to higher-loop gravitational processes, approximated via the eikonal. At smaller impact parameters, "diffractive" or "tidal" string excitation leads to processes dominated by highly excited strings. However, new evidence is presented that these excitation effects do not play a direct role in black hole formation, which corresponds to breakdown of gravitational perturbation theory and appears to dominate at sufficiently small impact parameters. The estimated amplitudes violate expected bounds on high-energy behavior for local theories., Comment: 19 pages, harvmac. v2: fixed typos, added refs and discussion of longitudinal spread. v3: minor changes to agree with published version

In this paper, we explore the questions of time, locality and causality in the framework of covariant open bosonic string field theory. We show that if an open string field is expressed as a certain local function on spacetime--in particular, a function of the lightcone component of the midpoint and the transverse center of mass degrees of freedom--that cubic string field theory is nonsingular and local in lightcone time. In particular, the theory has a well defined initial value formulation resembling that of an ordinary second order relativistic field theory in lightcone frame. This description can be achieved by a nonsingular unitary transformation on the Fock space, and we demonstrate explicitly that the theory is gauge invariant and the interaction vertex is local in this basis. With an initial value formulation at hand, we are able to construct an explicit second quantized operator formalism for the theory using the Hamiltonian BRST formalism. We also explore issues of causality by considering a singular limit of the theory where all spacetime coordinates are taken to the midpoint. At any stage in this limit, the theory is well-defined and arbitrarily close to being completely local and manifestly causal. We argue that the this limit must account for the macroscopic causality of the string S-matrix., Comment: 50 pages, 8 figures. Minor corrections, references added

We use the \hat c=1 matrix model to compute the potential energy V(C) for (the zero mode of) the RR scalar in two-dimensional type 0B string theory. The potential is induced by turning on a background RR flux, which in the matrix model corresponds to unequal Fermi levels for the two types of fermions. Perturbatively, this leads to a linear runaway potential, but non-perturbative effects stabilize the potential, and we find the exact expression V(C)=\frac{1}{2\pi}\int da\arccos [\cos(C)/\sqrt{1+e^{-2\pi a}}]. We also compute the finite-temperature partition function of the 0B theory in the presence of flux. The perturbative expansion is T-dual to the analogous result in type 0A theory, but non-perturbative effects (which depend on C) do not respect naive R\to 1/R duality. The model can also be used to study scattering amplitudes in background RR fluxes., Comment: 38 pages, 4 figures

Berenstein, Maldacena, and Nastase have proposed, as a limit of the strong form of the AdS/CFT correspondence, that string theory in a particular plane wave background is dual to a certain subset of operators in the N=4 super-Yang-Mills theory. Even though this is a priori a strong/weak coupling duality, the matrix elements of the string theory Hamiltonian, when expressed in gauge theory variables, are analytic in the 't Hooft coupling constant. This allows one to conjecture that, like the masses of excited string states, these can be recovered using perturbation theory in Yang-Mills theory. In this paper we identify the difference between the generator of scale transformations and a particular U(1) R-symmetry generator as the operator dual to the string theory Hamiltonian for nonvanishing string coupling. We compute its matrix elements and find that they agree with the string theory prediction provided that the state-operator map is modified for nonvanishing string coupling. We construct this map explicitly and calculate the anomalous dimensions of the new operators. We identify the component arising from the modification of the state-operator map with the contribution of the string theory contact terms to the masses of string states., Comment: 38 pages, Latex; v2: Comparison with string theory changed in light of corrections to string theory results in hep-th/0206073 v3; state-operator map modified; Physical interpretation and conclusions unchanged

It has been recently proposed that string theory in the background of a plane wave corresponds to a certain subsector of the N=4 supersymmetric Yang-Mills theory. This correspondence follows as a limit of the AdS/CFT duality. As a particular case of the AdS/CFT correspondence, it is a priori a strong/weak coupling duality. However, the predictions for the anomalous dimensions which follow from this particular limit are analytic functions of the 't Hooft coupling constant $\lambda$ and have a well defined expansion in the weak coupling regime. This allows one to conjecture that the correspondence between the strings on the plane wave background and the Yang-Mills theory works at the level of perturbative expansions. In our paper we perform perturbative computations in the Yang-Mills theory that confirm this conjecture. We calculate the anomalous dimension of the operator corresponding to the elementary string excitation. We verify at the two loop level that the anomalous dimension has a finite limit when the R charge $J\to \infty$ keeping $\lambda/J^2$ finite. We conjecture that this is true at higher orders of perturbation theory. We show, by summing an infinite subset of Feynman diagrams, under the above assumption, that the anomalous dimensions arising from the Yang-Mills perturbation theory are in agreement with the anomalous dimensions following from the string worldsheet sigma-model., Comment: 26 pages, LaTeX, added references, small changes in the introduction

Motivated by noncommutative Chern-Simons theory, we construct an infinite class of field theories that satisfy the axioms of Witten's string field theory. These constructions have no propagating open string degrees of freedom. We demonstrate the existence of non-trivial classical solutions. We find Wilson loop-like observables in these examples., Comment: 10 pages, RevTeX

We describe the ghost sector of cubic string field theory in terms of degrees of freedom on the two halves of a split string. In particular, we represent a class of pure ghost BRST operators as operators on the space of half-string functionals. These BRST operators were postulated by Rastelli, Sen, and Zwiebach to give a description of cubic string field theory in the closed string vacuum arising from condensation of a D25-brane in the original tachyonic theory. We find a class of solutions for the ghost equations of motion using the pure ghost BRST operators. We find a vanishing action for these solutions, and discuss possible interpretations of this result. The form of the solutions we find in the pure ghost theory suggests an analogous class of solutions in the original theory on the D25-brane with BRST operator Q_B coupling the matter and ghost sectors., Comment: 44 pages, Latex; v2: reference added, typos fixed, footnote added regarding ghost solution

We propose that the expectation value of a circular BPS-Wilson loop in N=4 SUSYM can be calculated exactly, to all orders in a 1/N expansion and to all orders in g^2 N. Using the AdS/CFT duality, this result yields a prediction of the value of the string amplitude with a circular boundary to all orders in alpha' and to all orders in g_s. We then compare this result with string theory. We find that the gauge theory calculation, for large g^2 N and to all orders in the 1/N^2 expansion does agree with the leading string theory calculation, to all orders in g_s and to lowest order in alpha'. We also find a relation between the expectation value of any closed smooth Wilson loop and the loop related to it by an inversion that takes a point along the loop to infinity, and compare this result, again successfully, with string theory., Comment: LaTeX, 22 pages, 3 figures. Argument corrected and two new sections added

We present a unified treatment of classical solutions of noncommutative gauge theories. We find all solutions of the noncommutative Yang-Mills equations in 2 dimensions; and show that they are labelled by two integers -- the rank of gauge group and the magnetic charge. The magnetic vortex solutions are unstable in 2+1 dimensions, but correspond to the full, stable BPS solutions of N=4 U(1) noncommutative gauge theory in 4 dimensions, that describes N infinite D1 strings that pierce a D3 brane at various points, in the presence of a background B field in the Seiberg-Witten limit. We discuss the behaviour of gauge invariant observables in the background of the solitons. We use these solutions to construct a panoply of BPS and non-BPS solutions of supersymmetric gauge theories that describe various configurations of D-branes. We analyze the instabilities of the non-BPS solutions. We also present an exact analytic solution of noncommutative gauge theory that describes a U(2) monopole., Comment: 37pp. harvmac big mode

We construct gauge invariant operators in non-commutative gauge theories which in the IR reduce to the usual operators of ordinary field theories (e.g. F^2). We show that in the deep UV the two-point functions of these operators admit a universal exponential behavior which fits neatly with the dual supergravity results. We also consider the ratio between n-point functions and two-point functions to find exponential suppression in the UV which we compare to the high energy fixed angle scattering of string theory., Comment: LaTeX2e, 29 pages, 6 figures, minor changes, references added, more references added

We continue our study of solitons in noncommutative gauge theories and present an extremely simple BPS solution of N=4 U(1) noncommutative gauge theory in 4 dimensions, which describes N infinite D1 strings that pierce a D3 brane at various points, in the presence of a background B-field in the Seiberg-Witten limit. We call this solution the N-fluxon. For N=1 we calculate the complete spectrum of small fluctuations about the fluxon and find three kinds of modes: the fluctuations of the superstring in 10 dimensions arising from fundamental strings attached to the D1strings, the ordinary particles of the gauge theory in 4 dimensions and a set of states with discrete spectrum, localized at the intersection point--- corresponding to fundamental strings stretched between the D1string and the D3 brane. We discuss the fluctuations about theN-fluxon as well and derive explicit expressions for the amplitudes of interactions between these various modes. We show that translations in noncommutative gauge theories are equivalent to gauge transformations (plus a constant shift of the gauge field) and discuss the implications for the translational zeromodes of our solitons. We also find the dyonic versions of N-fluxon, as well as of our previous string-monopole solution., Comment: 30pp. ; v2. references added, a few typos corrected

We study some non-perturbative aspects of noncommutative gauge theories. We find analytic solutions of the equations of motion, for noncommutative U(1) gauge theory, that describe magnetic monopoles with a finite tension string attached. These solutions are non-singular, finite and sourceless. We identify the string with the projection of a D-string ending on a D3-brane in the presence of a constant B-field., Comment: harvmac, 37 pp; v2. tension matched, typos corrected, refs added

We construct unstable classical solutions of Yang-Mills theories and their dual unstable states of type IIB on AdS_5. An example is the unstable D0-brane of type IIB located at the center of AdS. This has a field theory dual which is a sphaleron in gauge theories on S^3 x R. We argue that the two are dual because both are sphalerons associated to the topology of the instanton/D-instanton. This agreement provides a non-supersymmetric test of the AdS/CFT duality. As an illustration, many aspects of Sen's hypothesis regarding the unstable branes can be seen easily in the weakly coupled dual field theory description. In Euclidean AdS the D0-branes are dual to gauge theory merons. This implies that the two ends of a D0-brane world-line carry half the charge of a D-instanton. Other examples involve unstable strings in the Coulomb phase., Comment: 23 pages, 9 figures

A systematic approach to the study of semiclassical fluctuations of strings in AdS_5 x S^5 based on the Green-Schwarz formalism is developed. We show that the string partition function is well defined and finite. Issues related to different gauge choices are clarified. We consider explicitly several cases of classical string solutions with the world surface ending on a line, on a circle or on two lines on the boundary of AdS. The first example is a BPS object and the partition function is one. In the third example the determinants we derive should give the first corrections to the Wilson loop expectation value in the strong coupling expansion of the n=4 SYM theory at large N., Comment: 61 pages, harvmac, minor changes

The AdS/CFT correspondence suggests that the Wilson loop of the large N gauge theory with N=4 supersymmetry in 4 dimensions is described by a minimal surface in AdS_5 x S^5. We examine various aspects of this proposal, comparing gauge theory expectations with computations of minimal surfaces. There is a distinguished class of loops, which we call BPS loops, whose expectation values are free from ultra-violet divergence. We formulate the loop equation for such loops. To the extent that we have checked, the minimal surface in AdS_5 x S^5 gives a solution of the equation. We also discuss the zig-zag symmetry of the loop operator. In the N=4 gauge theory, we expect the zig-zag symmetry to hold when the loop does not couple the scalar fields in the supermultiplet. We will show how this is realized for the minimal surface., Comment: 51 pages, 7 figures

On the occasion of the 25th anniversary of Asymptotic Freedom, celebrated at the QCD Euroconference 98 on Quantum Chromodynamics, Montpellier, July 1998, I described the discovery of Asymptotic Freedom and the emergence of QCD., Comment: 35 pages, LaTeX

In this paper we explore some of the features of large N supersymmetric and nonsupersymmetric gauge theories using Maldacena's duality conjectures. We shall show that the resulting strong coupling behavior of the gauge theories is consistent with our qualitative expectations of these theories. Some of these consistency checks are highly nontrivial and give additional evidence for the validity of the proposed dualities., Comment: 31 pages, LaTeX, 11 eps figures, typos corrected

We study the spectrum of 1+1 dimensional large $N$ QCD coupled to an adjoint Majorana fermion of mass $m$. As $m\to 0$ this model makes a transition from confinement to screening. We argue that in this limit the spectrum becomes continuous for mass greater than twice the mass of the lightest bound state. This critical mass is nothing but the threshold for a decay into two lightest states. We present numerical results based on DLCQ that appear to support our claim., Comment: 17 pages, 9 postscript figures, uses psfig.sty, reference added

In this talk, delivered at the Oscar Klein Centenary Symposium in Stockholm, I review the 1938 conference held in Warsaw devoted to \lq \lq New Theories in Physics". I review all of the talks presented at this meeting and discuss in detail Klein's paper where he proposed a unified model of electromagnetism and the nuclear force that foreshadowed the later developments of non-Abelian gauge theories., Comment: A talk delivered at the Oscar Klein Centenary Symposium, 46 pages plus 1 uuencoded eps figures

The basic concepts of non-commutative probability theory are reviewed and applied to the large $N$ limit of matrix models. We argue that this is the appropriate framework for constructing the master field in terms of which large $N$ theories can be written. We explicitly construct the master field in a number of cases including QCD$_2$. There we both give an explicit construction of the master gauge field and construct master loop operators as well. Most important we extend these techniques to deal with the general matrix model, in which the matrices do not have independent distributions and are coupled. We can thus construct the master field for any matrix model, in a well defined Hilbert space, generated by a collection of creation and annihilation operators---one for each matrix variable---satisfying the Cuntz algebra. We also discuss the equations of motion obeyed by the master field., Comment: 46 pages plus 11 uuencoded eps figures

Large $N$ two-dimensional QCD on a cylinder and on a vertex manifold (a sphere with three holes) is investigated. The relation between the saddle-point description and the collective field theory of QCD$_2$ is established. Using this relation, it is shown that the Douglas--Kazakov phase transition on a cylinder is associated with the presence of a gap in the eigenvalue distributions for Wilson loops. An exact formula for the phase transition on disc with an arbitrary boundary holonomy is found. The role of instantons in inducing such transitions is discussed. The zero-area limit of the partition function on a vertex manifold is studied. It is found that this partition function vanishes unless the boundary conditions satisfy a certain selection rule which is an analogue of momentum conservation in field theory., Comment: 48 pages, preprint PUPT-1503

The $1/N$ expansion of the weak coupling phase of two-dimensional QCD on a sphere is constructed. It is demonstrated that the phase transition from the weak to the strong coupling phase is induced by instantons. A double scaling limit of the theory at the point of the phase transition is constructed, the value of string susceptibility is determined to be $\gamma_{str}=-1$. The possibility of instanton induced large $N$ phase transitions in four dimensional QCD is explored., Comment: 24 pages Some changes have been made and references added

Many Texo's have been corrected and a reference added., Comment: 57 pages, CERN-TH. 6827/93, PUPT-1382,LBL-33458, UCB-PTH-93/09