From Fields to Strings: Circumnavigating Theoretical Physics

The following sections are included:

• FROM THE EDITORS

• LIST OF CONTRIBUTORS

• CONTENTS

Everyone has their own private universe. Ian's was immense and diverse. But there are two main parts that determined his world.

One was of course PHYSICS. He was one of the rare breed for whom there was only one possible way of life. 10 years ago when his job prospects were bleak he was thinking of quitting physics and becoming a “taxi driver” which meant a financial analyst, a programmer, anything. For him all professions divided into two categories — physics and non-physics, a “taxi driver”…

The following sections are included:

• Part I – End of Childhood
  • Chapter 1 – A Thousand Years Before Our Era. June 15, 1941
  • Chapter 2 – Farewell Childhood! August 18, 1941
  • Chapter 3 – And They Came, Scourged By the Sun… Germans August 1941-June 1942
  • Chapter 4 – Hell. Third Month in Hell. June 17, 1942 – October 31, 1942
  • Chapter 5 – Third Day of the New Era. Policeman. November 3, 1942
  • Chapter 6 – Happy Holiday, My Son! November 7, 1942
  • Chapter 7 – My Dear Grachiki! November 13, 1942
  • Chapter 8 – Mikhailovna and Pronya. Kettle. March 1943
  • Chapter 9 – The Last…

• Part II – Stalingrad

• Part III – Glazov

• Part IV – Kaddish

When some people leave us, the world becomes emptier and colder…

Ian Kogan was born into a Jewish family on September 14, 1958 in Glazov, a small town in the Northern Urals, far away from all the cultural centers of what was then the Soviet Union. All life in this town revolved around a uranium plant where Ian's parents worked for 41 years…

Ian was born and spent his school years in Glazov, a provincial town in the Eastern part of European Russia, between the Volga and the Urals. His parents worked there in a big factory, which belonged to the Russian Ministry of Middle Machine Building (later – the Ministry of Atomic Energy). So they had some relation to physics and, because this factory dominated the town, school education was also oriented in this direction. Ian became interested in physics at school, educated himself a lot and succeeded in becoming a student of Moscow Institute of Physics and Technology – MIPT. (This was not easy for a Jew! I know this well from the experience of my own son). On becoming a student he realized that his knowledge of physics as well as his general culture level were lower than that of some of his classmates who came from Moscow. So he worked hard – and succeeded again…

Following his Doctorate in Moscow and Fellowships at the Universities of British Columbia and Princeton, Ian came to Oxford in 1994 as University Lecturer in Theoretical Physics and as Fellow of Balliol.

Always amused by our peculiar ways, Ian was instantly at home in Balliol with its large school of Physics and of Physics and Philosophy. Students over the last nine years have left with vivid memories of rapid-fire exchanges on a limitless range of subjects, of detailed and painstaking one-to-one teaching, of widely ranging tutorials and classes, and of Ian's innate feeling for the mathematical beauty of his subject. The wiser among them will have begun to appreciate the privilege of contact with a man of profound and towering intellect…

When Ian came to Oxford almost a decade ago he brought with him experience of very different institutions but also respect and sensitivity towards this one. It did not take him long to appreciate the fundamental good sense of the English custom of the coffee break at 11 and the tea break at 4. But Ian was a dynamo and we learned that the true purpose of the coffee break is to initiate a physics discussion which, had it not been for the intervention of lunch, would still have been going on at tea time. Ian was a man at ease with himself and forever generous to others. He was an ever-present inhabitant of the Discussion Room in Keble Road and loved to include anyone who was interested in the subject under consideration. Students, post-docs, staff; we have spent many happy hours exploring the magical world of theoretical physics in Ian's company…

Attraction between quarks is a fundamental aspect of QCD. It is plausible that several of the most profound aspects of low-energy QCD dynamics are connected to diquark correlations, including: paucity of exotics (which is the foundation of the quark model and of traditional nuclear physics), similarity of mesons and baryons, color superconductivity at high density, hyperfine splittings, ΔI = 1/2 rule, and some striking features of structure and fragmentation functions. After a brief overview of these issues, I discuss how diquarks can be studied in isolation, both phenomenologically and numerically, and present approximate mass differences for diquarks with different quantum numbers. The mass-loaded generalization of the Chew–Frautschi formula provides an essential tool.

A qualitative analysis of the chiral phase transition in QCD with two massless quarks and non-zero baryon density is performed. It is assumed that at zero baryonic density, ρ = 0, the temperature phase transition is of second order. Due to a specific power dependence of the baryon masses on the chiral condensate the phase transition becomes of first order at temperature T = T_ph(ρ) for ρ > 0. At temperatures T_cont(ρ) > T > T_ph(ρ) there is a mixed phase consisting of the quark phase (stable) and the hadron phase (unstable). At the temperature T = T_cont(ρ) the system experiences a continuous transition to the pure chirally symmetric phase.

This paper reviews some recent progress on QCD functional integrals at nonzero chemical potentials. One issue discussed is the use of QCD inequalities for this regime. In particular, the positivity of the integrand of particular Euclidean space functional integrals for two-flavor QCD with degenerate quark masses is used to demonstrate that the free energy per unit volume for QCD with a baryon chemical potential μ_B (and zero isospin chemical potential) is necessarily greater than the free energy with isospin chemical potential (and zero baryon chemical potential). This result may be of use in model finite density systems. A corollary to this result is a rigorous ab initio bound on the nucleon mass. The second major issue addressed is the so-called “Silver Blaze” problem: the fact that at zero temperature and chemical potentials less than some critical value the free energy remains as that of the vacuum. This is puzzling in the context of a functional integral since a chemical potential affects the functional determinant of the Dirac operator and any nonzero y, changes every eigenvalue of the Dirac operator compared to the μ = 0 value. The isospin Silver Blaze problem is solved through the study of the spectrum of the operator . The status of the baryon Silver Blaze problem is briefly discussed.

We review attempts to apply the variational principle to understand the vacuum of non-abelian gauge theories. In particular, we focus on the method explored by Ian Kogan and collaborators, which imposes exact gauge invariance on the trial Gaussian wave functional prior to the minimization of energy. We describe the application of the method to a toy model, confining compact QED in 2+1 dimensions, where it works wonderfully and reproduces all known non-trivial results. We then follow its applications to pure Yang-Mills theory in 3+1 dimensions at zero and finite temperature. Among the results of the variational calculation are dynamical mass generation and the analytic description of the deconfinement phase transition.

In this review, we discuss the confining and finite-temperature properties of the 3D SU(N) Georgi–Glashow model, and of 4D compact QED. At zero temperature, we derive string representations of both theories, thus constructing the SU(N) version of Polyakov's theory of confining strings. We discuss the geometric properties of confining strings, as well as the appearance of the string θ term from the field-theoretical one in 4D, and k-string tensions at N > 2. In particular, we point out the relevance of negative stiffness for stabilizing confining strings, an effect recently re-discovered in material science. At finite temperature, we present a derivation of the confining-string free energy and show that, at the one-loop level and for a certain class of string models in the large-D limit, it matches that of QCD at large N. This crucial matching is again a consequence of the negative stiffness. In the discussion of the finite–temperature properties of the 3D Georgi–Glashow model, in order to be closer to QCD, we mostly concentrate at the effects produced by some extensions of the model by external matter fields, such as dynamical fundamental quarks or photinos, in the supersymmetric generalization of the model.

Yang–Mills theories in four space-time dimensions possess a hidden symmetry which does not exhibit itself as a symmetry of classical Lagrangians, but is only revealed on the quantum level. It turns out that the effective Yang–Mills dynamics in several important limits is described by completely integrable systems that prove to be related to the celebrated Heisenberg spin chain and its generalizations. In this review, we explain the general phenomenon of complete integrability and its realization in several different situations. As a prime example, we consider in some detail the scale dependence of composite (Wilson) operators in QCD and super-Yang–Mills (SYM) theories. Highenergy (Regge) behavior of scattering amplitudes in QCD is also discussed and provides one with another realization of the same phenomenon that differs, however, from the first example in essential details. As a third example, we address the low-energy effective action in a theory which, contrary to the previous two cases, corresponds to a classical integrable model. Finally, we include a short overview of recent attempts to use gauge/string duality in order to relate integrability of Yang–Mills dynamics with the hidden symmetry of a string theory on a curved background.

We review and extend our recent work on the planar (large N) equivalence between gauge theories with varying degrees of supersymmetry. The main emphasis is on the planar equivalence between gluodynamics (super-Yang–Mills theory) and a non-supersymmetric “orientifold field theory”. We outline an “orientifold” large N expansion, analyze its possible phenomenological consequences in one-flavor massless QCD, and make a first attempt at extending the correspondence to three massless flavors. An analytic calculation of the quark condensate in one-flavor QCD starting from the gluino condensate in gluodynamics is thoroughly discussed. We also comment on a planar equivalence involving supersymmetry, on “chiral rings” in non-supersymmetric theories, and on the origin of planar equivalence from an underlying, non-tachyonic type-0 string theory. Finally, possible further directions of investigation, such as the gauge/gravity correspondence in large-N orientifold field theory, are briefly discussed.

I present a pedagogical review of Heisenberg–Euler effective Lagrangians, beginning with the original work of Heisenberg and Euler, and Weisskopf, for the one loop effective action of quantum electrodynamics in a constant electromagnetic background field, and then summarizing some of the important applications and generalizations to inhomogeneous background fields, nonabelian backgrounds, and higher loop effective Lagrangians.

We consider the theories obtained by dimensional reduction to D = 1,2,3 of 4D super-symmetric Yang–Mills theories and calculate there the effective low-energy lagrangians describing moduli space dynamics — the low-dimensional analogs of the Seiberg–Witten effective lagrangian. The effective theories thus obtained are rather beautiful and interesting from a mathematical viewpoint. In addition, their study allows one to understand better some essential features of 4D supersymmetric theories, in particular the non-renormalization theorems.

Quantum fluctuations generate in three-dimensional gauge theories not only radiative corrections to the Chern–Simons coupling but also non-analytic terms in the effective action. We review the role of those terms in gauge theories with massless fermions and Chern–Simons theories. The explicit form of non-analytic terms turns out to be dependent on the regularization scheme and in consequence the very existence of phenomena like parity and framing anomalies becomes regularization dependent. In particular we find regularization regimes where both anomalies are absent. Due to the presence of non-analytic terms the effective action becomes not only discontinuous but also singular for some background gauge fields which include sphalerons. The appearance of these types of singularity is linked to the existence of nodal configurations in physical states and tunneling suppression at some classical field configurations. In the topological field theory the number of physical states may also become regularization dependent. Another consequence of the peculiar behavior of three-dimensional theories under parity odd regularizations is the existence of a simple mechanism of generation of a mass gap in pure Yang–Mills theory by a suitable choice of regularization scheme. The generic value of this mass does agree with the values obtained in Hamiltonian and numerical analysis. Finally, the existence of different regularization regimes unveils the difficulties of establishing a Zamolodchikov c-theorem for three-dimensional field theories in terms of the induced gravitational Chern–Simons couplings.

We present new evidence for the conjecture that BPS correlation functions in supersymmetric gauge theories are described by an auxiliary two dimensional conformal field theory. We study deformations of the supersymmetric gauge theory by all gauge-invariant chiral operators. We calculate the partition function of the theory on ℝ⁴ with appropriately twisted boundary conditions. For the U(1) theory with instantons (either noncommutative, or D-instantons, depending on the construction) the partition function has a representation in terms of the theory of free fermions on a sphere, and coincides with the tau-function of the Toda lattice hierarchy. Using this result we prove to all orders in the string loop expansion that the effective prepotential (for U(1) with all chiral couplings included) is given by the free energy of the topological string on ℂℙ¹. Gravitational descendants play an important rôle in the gauge fields/string correspondence. The dual string is identified with the little string bound to the fivebrane wrapped on the two-sphere. We also discuss the theory with fundamental matter hypermultiplets.

We discuss a remarkable new approach initiated by Cachazo, Svrcek and Witten for calculating gauge theory amplitudes. The formalism amounts to an effective scalar perturbation theory which in many cases offers a much simpler alternative to the usual Feynman diagrams for deriving n-point amplitudes in gauge theory. At tree level the formalism works in a generic gauge theory, with or without supersymmetry, and for a finite number of colors. There is also growing evidence that the formalism works for loop amplitudes.

We briefly review some examples of confinement which arise in condensed matter physics. We focus on two instructive cases: the off-critical Ising model in a magnetic field, and an array of weakly coupled (extended) Hubbard chains in the Wigner crystal phase. In the appropriate regime, the elementary excitations in these 1 + 1 and quasi-one-dimensional systems are confined into ‘mesons’. Although the models are generically non-integrable, quantum mechanics and form-factor techniques yield valuable information.

We review applications of the sine-Gordon model, the O(3) non-linear sigma model, the U(1) Thirring model, and the O(N) Gross–Neveu model to quasi one-dimensional quantum magnets, Mott insulators, and carbon nanotubes. We focus upon the determination of dynamical response functions for these problems. These quantities are computed by means of form factor expansions of quantum correlation functions in integrable quantum field theories. This approach is reviewed here in some detail.

This is a short review of recent work on fuzzy spaces in their relation to the M(atrix)theory and the Quantum Hall Effect. We give an introduction to fuzzy spaces and how the limit of large matrices is obtained. The complex projective spaces CP^k, and to a lesser extent spheres, are considered. The Quantum Hall Effect and the behavior of edge excitations of a droplet of fermions on these spaces and their relation to fuzzy spaces are also discussed.

The Standard Model partially unifies the strong, electromagnetic and weak interactions, suggesting a common origin for them. A more fundamental theory, a Grand Unified theory or a string theory, can complete this unification and explain many of the features which in the Standard Model are put in by hand. However many possible implementations of such a theory have been suggested. We emphasize the importance of the prediction for gauge coupling unification in distinguishing between these implementations and how it can select a particular string profile. We discuss how the superstring can extend the unification to include gravity leading to a testable relation between the gravitational and Standard Model interactions. A calculation of the threshold effects to be expected from the superheavy modes shows that the relation given by the weakly coupled heterotic string is in good agreement with experiment. We discuss the resulting profile of the supersymmetric extension of the Standard Model which is consistent with these unification predictions. Its phenomenological implications should be testable by future precision experiments looking for rare flavor changing processes and also more directly by direct supersymmetric particle searches at the Large Hadron collider.

We review the finding that the general solution of many physically relevant (bosonic) Einstein-matter systems in the vicinity of a cosmological, i.e. space-like, singularity exhibit a remarkable mixture of chaos and symmetry. On the one hand, the asymptotic behavior, as one approaches the singularity, of the general solution is found to be describable, at each (generic) spatial point, as a billiard motion in an auxiliary Lorentzian space. After a suitable “radial” projection, this evolution is described by a billiard motion on hyperbolic space (which is “chaotic” in many physically interesting cases). On the other hand, for certain Einstein-matter systems, notably for pure Einstein background (Ricci tensor = 0) in any space-time dimension D and for particular Einstein-matter systems arising in string theory, the billiard tables describing asymptotic cosmological behavior are found to be identical to the Weyl chambers of some Lorentzian Kač-Moody algebras. In the case of the bosonic sector of supergravity in 11-dimensional space-time the underlying Lorentzian algebra is that of the hyperbolic Kač-Moody group E₁₀, and there exists some evidence of a correspondence between the general solution of the Einstein-three-form system and a null geodesic in the infinite-dimensional coset space E₁₀/K(E₁₀), where K(E₁₀) is the maximal compact subgroup of E₁₀.

This is a pedagogical introduction to theories with branes and extra dimensions. We first This is a pedagogical introduction to theories with branes and extra dimensions. We first discuss the construction of such models from an effective field theory point of view, and then discuss large extra dimensions and some of their phenomenological consequences. Various possible phenomena (split fermions, mediation of supersymmetry breaking and orbifold breaking of symmetries) are discussed next. The second half of this review is entirely devoted to warped extra dimensions, including the construction of the Randall–Sundrum solution, intersecting branes, radius stabilization, Kaluza–Klein phenomenology and bulk gauge bosons.

I briefly review the arguments why the braneworld models with infinite-volume extra dimensions could solve the cosmological constant problem, evading Weinberg's no-go theorem. Then I discuss in detail the established properties of these models, as well as the features which should be studied further in order to conclude whether these models can truly solve the problem.

We propose a new class of inflationary solutions to the standard cosmological problems (horizon, flatness, monopole,…), based on a modification of old inflation. These models do not require a potential which satisfies the normal inflationary slow-roll conditions. Our universe arises from a single tunneling event as the inflation leaves the false vacuum. Subsequent dynamics (arising from either the oscillations of the inflation field or thermal effects) keep a second field trapped in a false minimum, resulting in an evanescent period of inflation (with roughly 50 e-foldings) inside the bubble. This easily allows the bubble to grow sufficiently large to contain our present horizon volume. Reheating is accomplished when the inflation driving the last stage of inflation rolls down to the true vacuum, and adiabatic density perturbations arise from moduli-dependent Yukawa couplings of the inflation to matter fields. Our scenario has several robust predictions, including virtual absence of gravity waves, a possible absence of tilt in scalar perturbations, and a higher degree of non-Gaussianity than other models. It also naturally incorporates a solution to the cosmological moduli problem.

In this article we discuss gauge/string correspondence based on the non-critical strings With this goal we present several remarkable sigma models with the AdS target spaces. The models have kappa symmetry and are completely integrable. The radius of the AdS space is fixed and thus they describe isolated conformal fixed points of gauge theories in various dimensions. This work is dedicated to the memory of Ian Kogan.

On the space of three-dimensional conformal field theories with U(1) symmetry and a chosen coupling to a background gauge field, there is a natural action of the group SL(2, Z). The generator S of SL(2, Z) acts by letting the background gauge field become dynamical, an operation considered recently by Kapustin and Strassler in explaining three-dimensional mirror symmetry. The other generator T acts by shifting the Chern–Simons coupling of the background field. This SL(2, Z) action in three dimensions is related by the AdS/CFT correspondence to SL(2, Z) duality of low energy U(1) gauge fields in four dimensions.

We review some aspects of logarithmic conformal field theories which might shed some light on the geometrical meaning of logarithmic operators. We consider an approach, put forward by V. Knizhnik, where computation of correlation functions on higher genus Riemann surfaces can be replaced by computations on the sphere under certain circumstances. We show that this proposal naturally leads to logarithmic conformal field theories, when the additional vertex operator insertions, which simulate the branch points of a ramified covering of the sphere, are viewed as dynamical objects in the theory.

We study the Seiberg–Witten solution of supersymmetric low energy effective field theory as an example where physically interesting quantities, the periods of a meromorphic one-form, can be effectively computed within this conformal field theory setting. We comment on the relation between correlation functions computed on the plane, but with insertions of twist fields, and torus vacuum amplitudes.

I review a particular class of physical applications of Logarithmic Conformal Field Theory in strings propagating in changing (not necessarily conformal) backgrounds, namely D-brane recoil in flat or time dependent cosmological backgrounds. The role of recoil logarithmic vertex operators as non-conformal deformations requiring in some cases Liouville dressing is pointed out. It is also argued that, although in the case of nonsupersymmetric recoil deformations the representation of target time as a Liouville zero mode may lead to non-linear quantum mechanics for stringy defects, such non-linearities disappear (or, at least, are strongly suppressed) after world-sheet supersymmetrization. A possible link is therefore suggested between (world-sheet) supersymmetry and linearity of quantum mechanics in this framework.

We study states of large charge density in integrable conformal coset models. For the O(2) coset, we consider two different S-matrices, one corresponding to a Thirring mass perturbation and the other to the continuation to O(2+ε). The former leads to simplification in the conformal limit; the latter gives a more complicated description of the O(2) system, with a large zero mode sector in addition to the right- and left-movers. We argue that for the conformal O(2+2M|2M) supergroup coset, the S-matrix is given by the analog of the O(2+ε) construction.

We examine two-dimensional conformal field theories (CFTs) at central charge c = 0. These arise typically in the description of critical systems with quenched disorder, but also in other contexts including dilute self-avoiding polymers and percolation. We show that such CFTs must in general possess, in addition to their stress energy tensor T(z), an extra field whose holomorphic part, t(z), has conformal weight two. The singular part of the Operator Product Expansion (OPE) between T(z) and t(z) is uniquely fixed up to a single number b, defining a new ‘anomaly’ which is a characteristic of any c = 0 CFT, and which may be used to distinguish between different such CFTs. The extra field t(z) is not primary (unless b = 0), and is a so-called ‘logarithmic operator’ except in special cases which include affine (Kač-Moody) Lie-super current algebras. The number b controls the question of whether Virasoro null-vectors arising at certain conformal weights contained in the c = 0 Kač table may be set to zero or not, in these nonunitary theories. This has, in the familiar manner, implications on the existence of differential equations satisfied by conformal blocks involving primary operators with Kač-table dimensions. It is shown that c = 0 theories where t(z) is logarithmic, contain, besides T and t, additional fields with conformal weight two. If the latter are a fermionic pair, the OPEs between the holomorphic parts of all these conformal weight-two operators are automatically covariant under a global U(1|1) supersymmetry. A full extension of the Virasoro algebra by the Laurent modes of these extra conformal weight-two fields, including t(z), remains an interesting question for future work.

We review the free-field formalism for boundary states. The multi-component free-field formalism is then used to study the boundary states of (p′,p) rational conformal field theories having a W symmetry of the type A_r. We show how the classification of primary fields for these models is obtained by demanding modular covariance of cylinder amplitudes and that the resulting modular S matrix satisfies all the necessary conditions. Basis states satisfying the boundary conditions are found in the form of coherent states and as expected we find that W violating states can be found for all these models. We construct consistent physical boundary states for all the rank 2 (p + 1,p) models (of which the already known case of the 3-state Potts model is the simplest example) and find that the W violating sector possesses a direct analogue of the Verlinde formula.

We give a short introduction to the AdS/CFT correspondence and its plane wave limit.

The problem of finding a holographic CFT dual to string theory on AdS₃ × S³ × S³ × S¹ is examined in depth. This background supports a large superconformal symmetry. While in some respects similar to the familiar small systems on AdS₃ × S³ × K³ and AdS₃ × S³ × T⁴, there are important qualitative differences. Using an analog of the elliptic genus for large theories we rule out all extant proposals – in their simplest form – for a holographic duality to supergravity at generic values of the background fluxes. Modifications of these extant proposals and other possible duals are discussed.

The bulk partition function of pure Chern–Simons theory on a three-manifold is a state in the space of conformal blocks of the dual boundary RCFT, and therefore transforms non-trivially under the boundary modular group. In contrast the bulk partition function of AdS₃ string theory is the modular-invariant partition function of the dual CFT on the boundary. This is a puzzle because AdS₃ string theory formally reduces to pure Chern–Simons theory at long distances. We study this puzzle in the context of massive Chern–Simons theory. We show that the puzzle is resolved in this context by the appearance of a chiral “spectator boson” in the boundary CFT which restores modular invariance. It couples to the conformal metric but not to the gauge field on the boundary. Consequently, we find a generalization of the standard Chern–Simons/RCFT correspondence involving “nonholomorphic conformal blocks” and nonrational boundary CFTs. These generalizations appear in the long-distance limit of AdS₃ string theory, where the role of the spectator boson is played by other degrees of freedom in the theory.

We review a special class of seraiclassical string states in AdS₅ × S⁵ which have a regular expansion of their energy in integer powers of the ratio of the square of the string tension ('t Hooft coupling) and the square of the large angular momentum in S⁵. They allow one to check AdS/CFT duality quantitatively for states in the non-supersymmetric sector and also help to uncover the role of integrable structures on both sides of the string theory – gauge theory duality.

We study the flow of central charges in supersymmetric gauge theories which admit large N supergravity duals. For theories lying in the conformal window, supersymmetry relates this evolution to a renormalization of the R-charges, and this information can, in many cases, be extracted from the dual geometry making use of the conserved U(1)_S current along the flow. In particular, we argue that these charges are determined purely by the complex structure of the 6-dimensional manifold transverse to the D-branes which source the geometry. We demonstrate this relationship in a class of flows between A_k orbifolds and generalized conifolds.

The three string vertex for Type IIB superstrings in a maximally supersymmetric planewave background can be constructed in a light-cone gauge string field theory formalism. The detailed formula contains certain Neumann coefficients, which are functions of a momentum fraction y and a mass parameter μ. This paper reviews the derivation of useful explicit expressions for these Neumann coefficients generalizing flat-space (μ = 0) results obtained long ago. These expressions are then used to explore the large μ asymptotic behavior, which is required for comparison with dual perturbative gauge theory results. The asymptotic formulas, exact up to exponentially small corrections, turn out to be surprisingly simple.

In this paper I attempt to address a serious criticism of the “Anthropic Landscape” and “Discretuum” approach to cosmology, leveled by Banks, Dine and Gorbatov. I argue that in this new and unfamiliar setting, the gauge Hierarchy may not favor low energy supersymmetry.

While most theorists are tied to the mast of four dimensions, some have found it irresistible to speculate about eleven dimensions, the domain of M-theory. We outline a program which starts from the light-cone description of supergravity, and tracks its divergences to suggest the existence of an infinite component theory which in the lightcone relies on the coset F₄/SO(9), long known to be linked to the Exceptional Jordan Algebra

We consider two-dimensional supergravity coupled to ĉ = 1 matter. This system can also be interpreted as noncritical type 0 string theory in a two-dimensional target space. After reviewing and extending the traditional descriptions of this class of theories, we provide a matrix model description. The 0B theory is similar to the realization of two-dimensional bosonic string theory via matrix quantum mechanics in an inverted harmonic oscillator potential; the difference is that we expand around a non-perturbatively stable vacuum, where the matrix eigenvalues are equally distributed on both sides of the potential. The 0A theory is described by a quiver matrix model.

We study closed string tachyon condensation using the RG flow of the worldsheet theory. In many cases the worldsheet theory enjoys supersymmetry, which provides analytic control over the flow, due to non-renormalization theorems. Moreover, Mirror symmetry sheds light on the RG flow in such cases. We discuss the relevant tachyon condensation in the context of both compact and non-compact situations which lead to very different conclusions. Furthermore, the tachyon condensation leads to non-trivial dualities for non-supersymmetric probe theories.

Flux compactifications of string theory exhibiting the possibility of discretely tuning the cosmological constant to small values have been constructed. The highly tuned vacua in this discretuum have curvature radii which scale as large powers of the flux quantum numbers, exponential in the number of cycles in the compactiflcation. By the arguments of Susskind/Witten (in the AdS case) and Gibbons/Hawking (in the dS case), we expect correspondingly large entropies associated with these vacua. If they are to provide a dual description of these vacua on their Coulomb branch, branes traded for the flux need to account for this entropy at the appropriate energy scale. In this note, we argue that simple string junctions and webs ending on the branes can account for this large entropy, obtaining a rough estimate for junction entropy that agrees with the existing rough estimates for the spacing of the discretuum. In particular, the brane entropy can account for the (A)dS entropy far away from string scale correspondence limits.

We derive the non-perturbative corrections to the free energy of the two-matrix model in terms of its algebraic curve. The eigenvalue instantons are associated with the vanishing cycles of the curve. For the (p, q) critical points our results agree with the geometrical interpretation of the instanton effects recently discovered in the CFT approach. The form of the instanton corrections implies that the linear relation between the FZZT and ZZ disc amplitudes is a general property of the 2D string theory and holds for any classical background. We find that the agreement with the CFT results holds in the presence of infinitesimal perturbations by order operators and observe that the ambiguity in the interpretation of the eigenvalue instantons as ZZ-branes (four different choices for the matter and Liouville boundary conditions lead to the same result) is not lifted by the perturbations. We find similar results to the c = 1 string theory in the presence of tachyon perturbations.

We show that the renormahzation group flows of massless superstring modes in the presence of fluctuating D-branes satisfy the equations of fluid dynamics. In particular, we show that the D-brane's U(1) field is related to the velocity function in the Navier-Stokes equation while the dilaton plays the role of the passive scalar advected by the turbulent liquid. This leads us to suggest a possible isomorphism between the off-shell superstring theory in the presence of fluctuating D-branes and fluid mechanical degrees of freedom.

We present an overview of the intimate relationship between string and D-brane dynamics, and the dynamics of gauge and gravitational fields in three spacetime dimensions. The successes, prospects and open problems in describing both perturbative and non-perturbative aspects of string theory in terms of three-dimensional quantum field theory are highlighted.

We review aspects of the Hagedorn regime in critical string theories, from basic facts about the ideal gas approximation to the proposal of a global picture inspired by general ideas of holography. It was suggested that the condensation of thermal winding modes triggers a first order phase transition. We propose, by an Euclidean analogue of the string/black hole correspondence principle, that the transition is actually related to a topology change in spacetime. Similar phase transitions induced by unstable winding modes can be studied in toy models. There, using T-duality of supersymmetric cycles, one can identify a topology change of the Gregory–Laflamme type, which we associate with large-N phase transitions of Yang–Mills theories on tori. This essay is dedicated to the memory of Ian Kogan.

Some recent work on the thermodynamic behavior of the matrix model of M-theory on a pp-wave background is reviewed. We examine a weak coupling limit where computations can be done explicitly. In the large N limit, we find a phase transition between two distinct phases which resembles a “confinement-deconfinement” transition in gauge theory and which we speculate must be related to a geometric transition in M-theory. We review arguments that the phase transition is also related to the Hagedorn transition of little string theory in a certain limit of the 5-brane geometry.

We give a non-technical review of some of the recent developments in our understanding of the tachyon in string theory. We also illustrate the conjecture that open string theory provides a complete description of the dynamics of unstable D-branes.

Lower-dimensional (hyper)surfaces that can carry gauge or gauge/gravitational anomalies occur in many areas of physics: one-plus-one-dimensional boundaries or two-dimensional defect surfaces in condensed matter systems, four-dimensional brane-worlds in higher-dimensional cosmologies or various branes and orbifold planes in string or Mtheory. In all cases we may have (quantum) anomalies localized on these hypersurfaces that are only cancelled by “anomaly inflow” from certain topological interactions in the bulk. Proper cancellation between these anomaly contributions of different origin requires a careful treatment of factors and signs. We review in some detail how these contributions occur and discuss applications in condensed matter (Quantum Hall Effect) and M-theory (five-branes and orbifold planes).

We find the dependence of the maximum cut-off on the fermion mass in a nonrenormalizable field theory of a single massive vector boson with an axial vector coupling to a fermion with a small but non-zero mass.

Two quantum quartic anharmonic many-body oscillators are introduced. One of them is the celebrated Calogero model (rational A_n model) modified by quartic anharmonic twobody interactions which support the same symmetry as the Calogero model. Another model is the three-body Wolfes model (rational G₂ model) with quartic anharmonic interaction added which has the same symmetry as the Wolfes model. Both models are studied in the framework of algebraic perturbation theory and by the variational method.

We consider the quantum mechanics of a particle on the coset superspace SU(2|1)/(U(1) × 17(1)], which is a super-flag manifold with SU(2)/U(1) ≅ S² ‘body'. By incorporating the Wess-Zumino terms associated with the U(1) × U(1) stability group, we obtain an exactly solvable super-generalization of the Landau model for a charged particle on the sphere. We solve this model using the factorization method. Remarkably, the physical Hilbert space is finite-dimensional because the number of admissible Landau levels is bounded by a combination of the U(1) charges. The level saturating the bound has a wavefunction in a shortened, degenerate, irrep of SU(2∣1).

We briefly review the concept of a parallel ‘mirror’ world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should have a smaller temperature than the ordinary one. For this reason its evolution should substantially deviate from the standard cosmology as far as the crucial epochs like baryogenesis, nucleosynthesis etc. are concerned. In particular, we show that in the context of certain baryogenesis scenarios, the baryon asymmetry in the mirror world should be larger than in the observable one. Moreover, we show that mirror baryons could naturally constitute the dominant dark matter component of the Universe, and discuss its cosmological implications.

We consider weak radiative decays of hyperons. It is shown that there exists an exact unitary lower bound for the decay probability, for example, BR(Ξ⁻ → Σ⁻γ)≥1.0 · 10⁻⁴. We show that the real part of the amplitude is singular in the chiral limit, i.e. it containsterms ∼ ln m_q, where m_q is the current quark mass. The coefficient of the logarithmic terms is fixed uniquely. In the case of the decays Ξ⁻ → Σ⁻γ and Ω⁻ → Ξ⁻γ the nonsingular (model-dependent) terms are relatively small, and it is possible to obtain a reasonably accurate estimate for the real part of the amplitudes Ξ⁻ → Σ⁻γ and Ω ⁻→ Ξ ⁻γ. Taking the real part into account, BR(Ξ⁻ → Σ⁻γ) ≈ 1.7 • 10⁻⁴.

Vortices on the world sheet of a string which is moving in a multiply connected space lead to a continuous dependence of the critical indices, in particular, the critical dimensionality, on the parameters of cycles which cannot be contracted. There exist limiting values of the latter quantities because of a phase transition associated with a loss of conformal symmetry. Numerous applications are duscussed.

We study the model of (2 + 1)-dimensional relativistic fermions in a random non-Abelian gauge potential at criticality. The exact solution shows that the operator expansion contains a conserved current - a generator of a continuous symmetry. The presence of this operator changes the operator product expansion and gives rise to logarithmic contributions to the correlation functions at the critical point. We calculate the distribution function of the local density of states in this model and find that it follows the famous log-normal law.

We consider 5-dimensional cosmological solutions of a single brane. The correct cosmology on the brane, i.e., governed by the standard 4-dimensional Priedmann equation, and stable compactification of the extra dimension is guaranteed by the existence of a nonvanishing which is proportional to the 4-dimensional trace of the energy-momentum tensor. We show that this component of the energy-momentum tensor arises from the backreaction of the dilaton coupling to the brane. The same positive features are exhibited in solutions found in the presence of non-vanishing cosmological constants both on the brane (Λ_br) and in the bulk (Λ_B). Moreover, the restoration of the Friedmann equation, with the correct sign, takes place for both signs of Λ_B so long as the sign of Λ_br is opposite Λ_B in order to cancel the energy densities of the two cosmological constants. We further extend our single-brane thin-wall solution to allow a brane with finite thickness.

We consider a modification of gravity at large distances in a Brane Universe which was discussed recently [1–4]. In these models the modification of gravity at large distances is ultimately connected to existence of negative tension brane(s) and exponentially small tunneling factor. We discuss a general model which interpolates between Bi-gravity + − + model [1] and GRS model [2]. We also discuss the possible mechanism of stabilization of negative tension branes in AdS background. Finally we show that extra degrees of freedom of massive gravitons do not lead to disastrous contradiction with General Relativity if the stabilization condition is implemented.

We discuss the stringy properties of high-energy QCD using its hidden integrability in the Regge limit and on the light-cone. It is shown that multi-color QCD in the Regge limit belongs to the same universality class as superconformal with N_f = 2N_c at the strong coupling orbifold point. The analogy with the integrable structure governing the low energy sector of gauge theories is used to develop the brane picture for the Regge limit. In this picture the scattering process is described by a single M2 brane wrapped around the spectral curve of the integrable spin chain and unifying hadrons and reggeized gluons involved in the process. New quasiclassical quantization conditions for the complex higher integrals of motion are suggested which are consistent with the S-duality of the multi-reggeon spectrum. The derivation of the anomalous dimensions of the lowest twist operators is formulated in terms of the Riemann surfaces.

The following sections are included:

• SCIENTIFIC PUBLICATIONS OF IAN KOGAN