Showing papers in "Advances in Theoretical and Mathematical Physics in 1999"

Renormalization group flows from holography supersymmetry and a c theorem

Abstract: We obtain first order equations that determine a supersymmetric kink solution in fivedimensional N = 8 gauged supergravity. The kink interpolates between an exterior anti-de Sitter region with maximal supersymmetry and an interior anti-de Sitter region with one quarter of the maximal supersymmetry. One eighth of supersymmetry is preserved by the kink as a whole. We interpret it as describing the renormalization group flow in N = 4 super-Yang-Mills theory broken to an N = 1 theory by the addition of a mass term for one of the three adjoint chiral superfields. A detailed correspondence is obtained between fields of bulk supergravity in the interior anti-de Sitter region and composite operators of the infrared field theory. We also point out that the truncation used to find the reduced symmetry critical point can be extended to obtain a new N = 4 gauged supergravity theory holographically dual to a sector of N = 2 gauge theories based on quiver diagrams. We consider more general kink geometries and construct a c-function that is positive and monotonic if a weak energy condition holds in the bulk gravity theory. For evendimensional boundaries, the c-function coincides with the trace anomaly coefficients of the holographically related field theory in limits where conformal invariance is recovered.

On the Gauge Theory/Geometry Correspondence

Abstract: The 't Hooft expansion of SU(N) Chern-Simons theory on $S^3$ is proposed to be exactly dual to the topological closed string theory on the $S^2$ blow up of the conifold geometry. The $B$-field on the $S^2$ has magnitude $Ng_s=\lambda$, the 't Hooft coupling. We are able to make a number of checks, such as finding exact agreement at the level of the partition function computed on {\it both} sides for arbitrary $\lambda$ and to all orders in 1/N. Moreover, it seems possible to derive this correspondence from a linear sigma model description of the conifold. We propose a picture whereby a perturbative D-brane description, in terms of holes in the closed string worldsheet, arises automatically from the coexistence of two phases in the underlying U(1) gauge theory. This approach holds promise for a derivation of the AdS/CFT correspondence.

Non-spherical horizons, I

Abstract: We formulate an extension of Maldacena’s AdS/CFT conjectures to the case of branes located at singular points in the ambient transverse space. For singularities which occur at finite distance in the moduli space of M or F theory models with spacetime-filling branes, the conjectures identify the worldvolume theory on the p-branes with a compactification of M or IIB theory on AdSp+2 × HD−p−2. We show how the singularity determines the horizon H, and demonstrate the relationship between global symmetries on the worldvolume and gauge symmetries in the AdS model. As a first application, we study some singularities relevant to the D3-branes required in four-dimensional F -theory. For these we are able to explicitly derive the low-energy field theory on the worldvolume and compare its properties to predictions from the dual AdS model. In particular, we examine the baryon spectra of the models and the fate of the Abelian factors in the gauge group. October 1998 ∗ On leave from Department of Particle Physics, Weizmann Institute of Science, Rehovot, Israel Spacetime-filling branes have emerged as an essential feature of string and M-theory compactifications in at least three contexts: (1) new branches of the heterotic string in six dimensions with “extra” tensor multiplets, which can be represented by a Hořava– Witten-type compactification of M-theory on (S/Z2) × K3 but with extra spacetimefilling M5-branes representing the extra tensor multiplets [1,2,3]; (2) F -theory models in four dimensions (which can be regarded as compactifications of the IIB string with D7branes included) which in general require spacetime-filling D3-branes to cancel a tadpole anomaly [4]; and (3) M-theory models in three dimensions, which require spacetime-filling M2-branes to cancel a similar tadpole anomaly [4]. In each of these cases, the spacetimefilling brane meets the compactifying space at a single point, and the string or M-theory remains finite near the brane. Remarkably, this short list of branes (M5, D3, and M2) is precisely the list of branes for which a certain scaling limit is expected to lead to a “boundary” conformal field theory in the recent AdS/CFT conjectures [5,6,7]. In fact, the scaling limit can be taken even when the space transverse to the branes is curved, as in the compactification scenarios above. The details of the metric far from the location y0 of the brane in the transverse space become irrelevant; for the purposes of studying the scaling limit, the metric on the compactifying space can be approximated by some metric on its tangent space Ty0 at y0. In the scaling limit, the rescaled supergravity metric approaches a metric of the form AdSp+2 × S in which the anti-de Sitter space has been formed out of the worldvolume of the brane and the radial direction within Ty0 , and S k is the unit sphere within Ty0 . Maldacena’s conjecture proposes that the M or string theory on this space AdSp+2 × S, with N units of flux of the supergravity k-form field strength through S, is dual to a specific conformal field theory on the boundary of AdSp+2. The conjecture applies to the large N limit when a large number of these branes have been brought together; in the compactification context, fairly large values of N can be obtained by bringing together all available branes in a given model. Virtually all points in the compactifying space have identical behavior in this scaling limit. That situation changes, however, if we consider a compactifying space which itself 1 This last requirement excludes consideration of F -theory models in eight dimensions with the D7-brane being spacetime-filling. 2 The exception is the four-dimensional F -theory models, where points located along the D7branes behave differently; in particular, the string coupling becomes infinite at such points. The behavior of D3-branes at such points has recently been determined [8,9,10], and we will not

M theory, topological strings and spinning black holes

Abstract: We consider M-theory compactification on Calabi-Yau threefolds. The recently discovered connection between the BPS states of wrapped M2 branes and the topological string amplitudes on the threefold is used both as a tool to compute topological string amplitudes at higher genera as well as to unravel the degeneracies and quantum numbers of BPS states. Moduli spaces of $k$-fold symmetric products of the wrapped M2 brane play a crucial role. We also show that the topological string partition function is the Calabi-Yau version of the elliptic genus of the symmetric product of $K3$'s and use the macroscopic entropy of spinning black holes in 5 dimensions to obtain new predictions for the asymptotic growth of the topological string amplitudes at high genera.

Local mirror symmetry: Calculations and interpretations

Abstract: We describe local mirror symmetry from a mathematical point of view and make several A-model calculations using the mirror principle (localization). Our results agree with B-model computations from solutions of Picard-Fuchs differential equations constructed form the local geometry near a Fano surface within a Calabi-Yau manifold. We interpret the Gromov-Witten-type numbers from an enumerative point of view. We also describe the geometry of singular surfaces and show how the local invariants of singular surfaces agree with the smooth cases when they occur as complete intersections. † email: chi@math.harvard.edu, yau@math.harvard.edu ∗ email: klemm@ias.edu ∗∗ email: zaslow@math.nwu.edu

Connectedness of the boundary in the AdS / CFT correspondence

Abstract: Let M be a complete Einstein manifold of negative curvature, and assume that (as in the AdS/CFT correspondence) it has a Penrose compactification with a conformal boundary N of positive scalar curvature. We show that under these conditions, H_n(M;Z) = 0 and in particular N must be connected. These results resolve some puzzles concerning the AdS/CFT correspondence.

Isolated horizons: The Classical phase space

Abstract: A Hamiltonian framework is introduced to encompass non-rotating (but possibly charged) black holes that are “isolated” near future time-like infinity or for a finite time interval The underlying space-times need not admit a stationary Killing field even in a neighborhood of the horizon; rather, the physical assumption is that neither matter fields nor gravitational radiation fall across the portion of the horizon under consideration A precise notion of non-rotating isolated horizons is formulated to capture these ideas With these boundary conditions, the gravitational action fails to be differentiable unless a boundary term is added at the horizon The required term turns out to be precisely the Chern-Simons action for the self-dual connection The resulting symplectic structure also acquires, in addition to the usual volume piece, a surface term which is the Chern-Simons symplectic structure We show that these modifications affect in subtle but important ways the standard discussion of constraints, gauge and dynamics In companion papers, this framework serves as the point of departure for quantization, a statistical mechanical calculation of black hole entropy and a derivation of laws of black hole mechanics, generalized to isolated horizons It may also have applications in classical general relativity, particularly in the investigation of of analytic issues that arise in the numerical studies of black hole collisions Typeset using REVTEX

Wilson loops from multicentre and rotating branes, mass gaps and phase structure in gauge theories

Abstract: Within the AdS/CFT correspondence we use multicentre D3-brane metrics to investigate Wilson loops and compute the associated heavy quark–antiquark potentials for the strongly coupled SU(N) super-Yang–Mills gauge theory, when the gauge symmetry is broken by the expectation values of the scalar fields. For the case of a uniform distribution of D3-branes over a disc, we find that there exists a maximum separation beyond which there is no force between the quark and the antiquark, i.e. the screening is complete. We associate this phenomenon with the possible existence of a mass gap in the strongly coupled gauge theory. In the finite-temperature case, when the corresponding supergravity solution is a rotating D3-brane solution, there is a class of potentials interpolating between a Coulombic and a confining behaviour. However, above a certain critical value of the mass parameter, the potentials exhibit a behaviour characteristic of statistical systems undergoing phase transitions. The physical path preserves the concavity property of the potential and minimizes the energy. Using the same rotating-brane solutions, we also compute spatial Wilson loops, associated with the quark–antiquark potential in models of three-dimensional gauge theories at zero temperature, with similar results.

The Quantum Tetrahedron in 3 and 4 Dimensions

Abstract: Recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the `quantum tetrahedron'. Starting with a classical phase space whose points correspond to geometries of the tetrahedron in R^3, we use geometric quantization to obtain a Hilbert space of states. This Hilbert space has a basis of states labeled by the areas of the faces of the tetrahedron together with one more quantum number, e.g. the area of one of the parallelograms formed by midpoints of the tetrahedron's edges. Repeating the procedure for the tetrahedron in R^4, we obtain a Hilbert space with a basis labelled solely by the areas of the tetrahedron's faces. An analysis of this result yields a geometrical explanation of the otherwise puzzling fact that the quantum tetrahedron has more degrees of freedom in 3 dimensions than in 4 dimensions.

Symmetry breaking boundary conditions and WZW orbifolds

Abstract: Symmetry breaking boundary conditions for WZW theories are discussed. We derive explicit formulae for the reflection coefficients in the presence of boundary conditions that preserve only an orbifold subalgebra with respect to an involutive automorphism of the chiral algebra. The characters and modular transformations of the corresponding orbifold theories are computed. Both inner and outer automorphisms are treated.

BF description of higher dimensional gravity theories

Abstract: It is well known that, in the first-order formalism, pure three-dimensional gravity is just the BF theory. Similarly, four-dimensional general relativity can be formulated as BF theory with an additional constraint term added to the Lagrangian. In this paper we show that the same is true also for higher-dimensional Einstein gravity: in any dimension gravity can be described as a constrained BF theory. Moreover, in any dimension these constraints are quadratic in the B field. After describing in details the structure of these constraints, we scketch the ``spin foam'' quantization of these theories, which proves to be quite similar to the spin foam quantization of general relativity in three and four dimensions. In particular, in any dimension, we solve the quantum constraints and find the so-called simple representations and intertwiners. These exhibit a simple and beautiful structure that is common to all dimensions.

Chen’s iterated integral represents the operator product expansion

Abstract: The recently discovered formalism underlying renormalization theory, the Hopf algebra of rooted trees, allows to generalize Chen’s lemma In its generalized form it describes the change of a scale in Green functions, and hence relates to the operator product expansion Hand in hand with this generalization goes the generalization of the ordinary factorial n! to the tree factorial t Various identities on tree-factorials are derived which clarify the relation between Connes-Moscovici weights and Quantum Field Theory

M theory, Joyce orbifolds and super Yang–Mills

Abstract: We geometrically engineer d=4 N=1 supersymmetric Yang-Mills theories by considering M theory on various Joyce orbifolds. We argue that the superpotential of these models is generated by fractional membrane instantons. The relation of this superpotential to membrane anomalies is also discussed.

Shortening of primary operators in N extended SCFT(4) and harmonic superspace analyticity

Abstract: We present the analysis of all possible shortenings which occur for composite gauge invariant conformal primary superfields in SU(2,2/N) invariant gauge theories. These primaries have top-spin range N/2 \\leq J_{max} < N with J_{max} = J_1 + J_2, (J_1,J_2) being the SL(2,C) quantum numbers of the highest spin component of the superfield. In Harmonic superspace, analytic and chiral superfields give J_{max}= N/2 series while intermediate shortenings correspond to fusion of chiral with analytic in N=2, or analytic with different analytic structures in N=3,4. In the AdS/CFT language shortenings of UIR's correspond to all possible BPS conditions on bulk states. An application of this analysis to multitrace operators, corresponding to multiparticle supergravity states, is spelled out.

Holomorphic anomaly equation and BPS state counting of rational elliptic surface

Abstract: We consider the generating function (prepotential) for Gromov-Witten invariants of rational elliptic surface. We apply the local mirror principle to calculate the prepotential and prove a certain recursion relation, holomorphic anomaly equation, for genus 0 and 1. We propose the holomorphic anomaly equation for all genera and apply it to determine higher genus Gromov-Witten invariants and also the BPS states on the surface. Generalizing G\"ottsche's formula for the Hilbert scheme of $g$ points on a surface, we find precise agreement of our results with the proposal recently made by Gopakumar and Vafa(hep-th/9812127).

Rotating D3-branes and QCD in three dimensions

Abstract: We investigate the rotating D3-brane solution with maximum number of angular momentum parameters. After determining the angular velocities, Hawking temperature, ADM mass and entropy, we use this geometry to construct general three-parameter models of non-supersymmetric pure SU(N) Yang-Mills theories in 2+1 dimensions. We calculate glueball masses in the WKB approximation and obtain closed analytic expressions for generic values of the parameters. We also determine the masses of Kaluza--Klein states associated with internal parts of the ten-dimensional metric and investigate the parameter region where some of these states are decoupled. To leading order in 1/\lambda and 1/N (where \lambda is the 't Hooft coupling) we find a global U(1)^3 symmetry and states with masses comparable to glueball masses, which have no counterpart in the more familiar (finite \lambda, N) Yang-Mills theories.

Constructing D-branes from K theory

Abstract: A detailed review of recent developments in the topological classification of D-branes in superstring theory is presented. Beginning with a thorough, self-contained introduction to the techniques and applications of topological K-theory, the relationships between the classic constructions of K-theory and the recent realizations of D-branes as tachyonic solitons, coming from bound states of higher dimensional systems of unstable branes, are described. It is shown how the K-theory formalism naturally reproduces the known spectra of BPS and non-BPS D-branes, and how it can be systematically used to predict the existence of new states. The emphasis is placed on the new interpretations of D-branes as conventional topological solitons in other brane worldvolumes, how the mathematical formalism can be used to deduce the gauge field content on both supersymmetric and non-BPS branes, and also how K-theory predicts new relationships between the various superstring theories and their D-brane spectra. The implementations of duality symmetries as natural isomorphisms of K-groups are discussed. The relationship with the standard cohomological classification is presented and used to derive an explicit formula for D-brane charges. Some string theoretical constructions of the K-theory predictions are also briefly described.

The asymptotics of an amplitude for the 4-simplex

Abstract: An expression for the oscillatory part of an asymptotic formula for the relativistic spin network amplitude for a 4-simplex is given The amplitude depends on specified areas for each two-dimensional face in the 4-simplex The asymptotic formula has a contribution from each flat Euclidean metric on the 4-simplex which agrees with the given areas The oscillatory part of each contribution is determined by the Regge calculus Einstein action for that geometry

D-branes, T-duality, and Index Theory

Abstract: We show that the transformation of D-branes under T-duality on four-torus is represented by Nahm transform of instantons. The argument for this allows us to generalize Nahm transform to the case of orthogonal and symplectic gauge groups as well as to instantons on Z_2 orbifold of four-torus. In addition, we identify the isomorphism of K-theory groups that realizes the transformation of D-brane charges under T-duality on torus of arbitrary dimensions. By the isomorphism we are lead to identify the correct K-theory group that classifies D-brane charges in Type II orientifold.

Uncovering the symmetries on $[p,q]$ 7-branes: Beyond the Kodaira classification

Abstract: We begin a classification of the symmetry algebras arising on configurations of type IIB [p,q] 7-branes. These include not just the Kodaira symmetries that occur when branes coalesce into a singularity, but also algebras associated to other physically interesting brane configurations that cannot be collapsed. We demonstrate how the monodromy around the 7-branes essentially determines the algebra, and thus 7-brane gauge symmetries are classified by conjugacy classes of the modular group SL(2,Z). Through a classic map between the modular group and binary quadratic forms, the monodromy fixes the asymptotic charge form which determines the representations of the various (p,q) dyons in probe D3-brane theories. This quadratic form also controls the change in the algebra during transitions between different brane configurations. We give a unified description of the brane configurations extending the D_N, E_N and Argyres-Douglas H_N series beyond the Kodaira cases. We anticipate the appearance of affine and indefinite infinite-dimensional algebras, which we explore in a sequel paper.

Boundaries of zero scalar curvature in the AdS / CFT correspondence

Abstract: In hep-th/9910245, Witten and Yau consider the AdS/CFT correspondence in the context of a Riemannian Einstein manifold $M^{n+1}$ of negative Ricci curvature which admits a conformal compactification with conformal boundary $N^n$. They prove that if the conformal class of the boundary contains a metric of positive scalar curvature, then $M$ and $N$ have several desirable properties: (1) $N$ is connected, (2) the $n$th homology of the compactified $M$ vanishes, and (3) the fundamental group of $M$ is "bounded by" that of $N$. Here it is shown that all of these results extend to the case where the conformal class of the boundary contains a metric of nonnegative scalar curvature. (The case of zero scalar curvature is of interest as it is borderline for the stability of the theory.) The proof method used here is different from, and in some sense dual to, that used by Witten and Yau. While their method involves minimizing the co-dimension one brane action on $M$, and requires the machinery of geometric measure theory, the main arguments presented here use only geodesic geometry.

Uncovering Infinite Symmetries on [p,q] 7-branes: Kac-Moody Algebras and Beyond

Abstract: In a previous paper we explored how conjugacy classes of the modular group classify the symmetry algebras that arise on type IIB [p,q] 7-branes. The Kodaira list of finite Lie algebras completely fills the elliptic classes as well as some parabolic classes. Loop algebras of E_N fill additional parabolic classes, and exotic finite algebras, hyperbolic extensions of E_N and more general indefinite Lie algebras fill the hyperbolic classes. Since they correspond to brane configurations that cannot be made into strict singularities, these non-Kodaira algebras are spectrum generating and organize towers of massive BPS states into representations. The smallest brane configuration with unit monodromy gives rise to the loop algebra \hat{E}_9 which plays a central role in the theory. We elucidate the patterns of enhancement relating E_8, E_9, \hat{E}_9 and E_10. We examine configurations of 24 7-branes relevant to type IIB compactifications on a two-sphere, or F-theory on K3. A particularly symmetric configuration separates the 7-branes into two groups of twelve branes and the massive BPS spectrum is organized by E_10 + E_10.

On discrete twist and four flux in N=1 heterotic / F theory compactifications

Abstract: We give an indirect argument for the matching G2 = −π∗γ of four-flux and discrete twist in the duality between N = 1 heterotic string and F -theory. This treats in detail the Euler number computation for the physically relevant case of a Calabi-Yau fourfold with singularities. bandreas@physics.unc.edu, supported by U.S. DOE grant DE-FG05-85ER40219/Task A. curio@ias.edu, supported by NSF grant DMS9627351. E-mail address after 1.July: curio@physik.hu-berlin.de

F-Theory with Quantized Fluxes

Abstract: We present evidence that the CHL string in eight dimensions is dual to F-theory compactified on elliptic K3 with a 0(2) monodromy group. The monodromy group 0(2) allows one to turn on the flux of an antisymmetric two form along the base. The Bµ� flux is quantized and therefore the moduli space of the CHL string is disconnected from the moduli space of F-theory/Heterotic strings (as expected). The non-zero Bµ� flux obstructs certain deformations restricting the moduli of elliptic K3 to a 10 dimensional moduli space. We also discuss how one can reconstruct the gauge groups from the elliptic fibration structure.

Domain walls of gauged supergravity, M-branes, and algebraic curves

Abstract: We provide an algebraic classification of all supersymmetric domain wall solutions of maximal gauged supergravity in four and seven dimensions, in the presence of non-trivial scalar fields in the coset SL(8,R)/SO(8) and SL(5,R)/SO(5) respectively. These solutions satisfy first-order equations, which can be obtained using the method of Bogomol'nyi. From an eleven-dimensional point of view they correspond to various continuous distributions of M2- and M5-branes. The Christoffel-Schwarz transformation and the uniformization of the associated algebraic curves are used in order to determine the Schrodinger potential for the scalar and graviton fluctuations on the corresponding backgrounds. In many cases we explicitly solve the Schrodinger problem by employing techniques of supersymmetric quantum mechanics. The analysis is parallel to the construction of domain walls of five-dimensional gauged supergravity, with scalar fields in the coset SL(6,R)/SO(6), using algebraic curves or continuous distributions of D3-branes in ten dimensions. In seven dimensions, in particular, our classification of domain walls is complete for the full scalar sector of gauged supergravity. We also discuss some general aspects of D-dimensional gravity coupled to scalar fields in the coset SL(N,R)/SO(N).

Supersymmetric fivebrane solitons

Abstract: We study the conditions for the fivebrane worldvolume theory in D=11 to admit supersymmetric solitons with non-vanishing self-dual three-form. We construct some new soliton solutions consisting of “superpositions” of calibrated surfaces, self-dual strings and instantons.

On the construction of gauge theories from non critical type 0 strings

Abstract: We investigate Polyakov's proposal of constructing Yang-Mills theories by using non critical type 0 strings We break conformal invariance by putting the system at finite temperature and find that the entropy of the cosmological solutions for these theories matches that of a gas of weakly interacting Yang-Mills bosons, up to a numerical constant The computation of the entropy using the effective action approach presents some novelties in that the whole contribution comes from the RR fields We also find an area law and a mass gap in the theory and show that such behavior persists for $p>4$ We comment on the possible physical meaning of this result

Quartic gauge couplings from K3 geometry

Abstract: We show how certain F^4 couplings in eight dimensions can be computed using the mirror map and K3 data. They perfectly match with the corresponding heterotic one-loop couplings, and therefore this amounts to a successful test of the conjectured duality between the heterotic string on T^2 and F-theory on K3. The underlying quantum geometry appears to be a 5-fold, consisting of a hyperk"ahler 4-fold fibered over a IP^1 base. The natural candidate for this fiber is the symmetric product Sym^2(K3). We are lead to this structure by analyzing the implications of higher powers of E_2 in the relevant Borcherds counting functions, and in particular the appropriate generalizations of the Picard-Fuchs equations for the K3.

On open/closed string duality

Abstract: It was recently shown, using the AdS/CFT correspondence, that the low energy effective action of a large $N$ open string theory satisfies a holographic RG flow equation closely related to the Hamilton-Jacobi equation of 5-d supergravity. In this paper we re-obtain the same flow equation in the dual regime of small 't Hooft coupling $\lambda\ll 1$. Our derivation makes use of the conformal equivalence between planar open string diagrams and closed string tree diagrams. This equivalence can be viewed as a microscopic explanation of the open/closed string duality that underlies the AdS/CFT correspondence.