Showing papers in "Advances in Theoretical and Mathematical Physics in 2004"

The Hagedorn - deconfinement phase transition in weakly coupled large N gauge theories

Abstract: We demonstrate that weakly coupled, large $N, d$-dimensional $SU(N)$ gauge theories on a class of compact spatial manifolds (including $S^{d-1}\times {\rm time}$) undergo deconfinement phase transitions at temperatures proportional to the inverse length scale of the manifold in question The low temperature phase has a free energy of order one, and is characterized by a stringy (Hagedorn) growth in its density of states The high temperature phase has a free energy of order $N^2$ These phases are separated either by a single first order transition that generically occurs below the Hagedorn temperature or by two continuous phase transitions, the first of which occurs at the Hagedorn temperature These phase transitions could perhaps be continuously connected to the usual flat space deconfinement transition in the case of confining gauge theories, and to the Hawking-Page nucleation of $AdS_5$ black holes in the case of the $\mathcal{N}=4$ supersymmetric Yang-Mills theory We suggest that deconfinement transitions may generally be interpreted in terms of black hole formation in a dual string theory Our analysis proceeds by first reducing the Yang-Mills partition function to a $(0+0)$-dimensional integral over a unitary matrix $U$, which is the holonomy (Wilson loop) of the gauge field around the thermal time circle in Euclidean space; deconfinement transitions are large $N$ transitions in this matrix integral

Sasaki-Einstein Metrics on S^2 x S^3

Abstract: We present a countably infinite number of new explicit co-homogeneity one Sasaki-Einstein metrics on $S^2\times S^3$ of both quasi-regular and irregular type. These give rise to new solutions of type IIB supergravity which are expected to be dual to $\mathcal{N}=1$ superconformal field theories in four dimensions with compact or non-compact $R$-symmetry and rational or irrational central charges, respectively.

A New infinite class of Sasaki-Einstein manifolds

Abstract: We show that for every positive curvature Kahler-Einstein manifold in dimension 2n there is a countably infinite class of associated Sasaki-Einstein manifolds X_{2n+3} in dimension 2n+3. When n=1 we recover a recently discovered family of supersymmetric AdS_5 x X_5 solutions of type IIB string theory, while when n=2 we obtain new supersymmetric AdS_4 x X_7 solutions of D=11 supergravity. Both are expected to provide new supergravity duals of superconformal field theories.

On the classification of asymptotic quasinormal frequencies for $d$-dimensional black holes and quantum gravity

Abstract: We provide a complete classification of asymptotic quasinormal frequencies for static, spherically symmetric black hole spacetimes in d dimensions This includes all possible types of gravitational perturbations (tensor, vector and scalar type) as described by the Ishibashi–Kodama master equations The frequencies for Schwarzschild are dimension independent, while for Reissner–Nordstrom are dimension dependent (the extremal Reissner–Nordstrom case must be considered separately from the non–extremal case) For Schwarzschild de Sitter, there is a dimension independent formula for the frequencies, except in dimension d = 5 where the formula is different For Reissner–Nordstrom de Sitter there is a dimension dependent formula for the frequencies, except in dimension d = 5 where the formula is different Schwarzschild and Reissner–Nordstrom Anti–de Sitter black hole spacetimes are simpler: the formulae for the frequencies will depend upon a parameter related to the tortoise coordinate at spatial infinity, and scalar type perturbations in dimension d = 5 lead to a continuous spectrum for the quasinormal frequencies We also address non–black hole spacetimes, such as pure de Sitter spacetime—where there are quasinormal modes only in odd dimensions—and pure Anti–de Sitter spacetime—where again scalar type perturbations in dimension d = 5 lead to a continuous spectrum for the normal frequencies Our results match previous numerical calculations with great accuracy Asymptotic quasinormal frequencies have also been applied in the framework of quantum gravity for black holes Our results show that it is only in the simple Schwarzschild case which is possible to obtain sensible results concerning area quantization or loop quantum gravity In an effort to keep this paper self–contained we also review earlier results in the literature

Search for a holographic dual to AdS_3 x S^3 x S^3 x S^1

Abstract: The problem of finding a holographic dual to string theory on AdS_3ק^3ק^3ק^1 is examined in depth. This background supports a large \CN=4 superconformal symmetry. While in some respects similar to the familiar small \CN=4 systems on AdS_3ק^3×K^3 and AdS_3ק^3×T^4, there are important qualitative differences. Using an analogue of the elliptic genus for large CN=4 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.

Parity Invariance For String In Twistor Space

Abstract: Topological string theory with twistor space as the target makes visible some otherwise difficult to see properties of perturbative Yang-Mills theory. But left-right symmetry, which is obvious in the standard formalism, is highly unclear from this point of view. Here we prove that tree diagrams computed from connected $D$-instanton configurations are parity-symmetric. The main point in the proof also works for loop diagrams.

Fractional Branes in Landau-Ginzburg Orbifolds

Abstract: We construct fractional branes in Landau-Ginzburg orbifold categories and study their behavior under marginal closed string perturbations. This approach is shown to be more general than the rational boundary state construction. In particular we find new D-branes on the quintic -- such as a single D0-brane -- which are not restrictions of bundles on the ambient projective space. We also exhibit a family of deformations of the D0-brane in the Landau-Ginzburg category parameterized by points on the Fermat quintic.

Spectra of D-branes with Higgs vevs

Abstract: In this paper we continue previous work on counting open string states between D-branes by considereing open strings between D-branes with nonzero Higgs vevs, and in particular, nilpotent Higgs vevs, as arise, for example, when studying D-branes in orbifolds. Ordinarily Higgs vevs can be interpreted as moving the D-brane, but nilpotent Higgs vevs have zero eigenvalues, and so their interpretation is more interesting - for example, they often correspond to nonreduced schemes, which furnishes an important link in understanding old results relating classical D-brane moduli spaces in orbifolds to Hilbert schemes, resolutions of quotient spaces, and the McKay correspondence. We give a sheaf-theoretic description of D-branes with Higgs vevs, including nilpotent Higgs vevs, and check that description by noting that Ext groups between the sheaves modelling the D-branes, do in fact correctly count open string states. In particular, our analysis expands the types of sheaves which admit on-shell physical interpretations, which is an important step for making derived categories useful for physics.

M-theory, type IIA superstrings, and elliptic cohomology

Abstract: The topological part of the M-theory partition function was shown by Witten to be encoded in the index of an E8 bundle in eleven dimensions. This partition function is, however, not automatically anomaly-free. We observe here that the vanishing W_7=0 of the Diaconescu-Moore-Witten anomaly in IIA and compactified M-theory partition function is equivalent to orientability of spacetime with respect to (complex-oriented) elliptic cohomology. Motivated by this, we define an elliptic cohomology correction to the IIA partition function, and propose its relationship to interaction between 2-branes and 5-branes in the M-theory limit.

The notion of observable in the covariant Hamiltonian formalism for the calculus of variations with several variables

Abstract: This papers is concerned with multisymplectic formalisms which are the frameworks for Hamiltonian theories for fields theory. Our main purpose is to study the observable (n − 1)-forms which allows one to construct observable functionals on the set of solutions of the Hamilton equations by integration. We develop here two different points of view: generalizing the law {p, q} = 1 or the law dF/dt = {H, F }. This leads to two possible definitions; we explore the relationships and the differences between these two concepts. We show that — in contrast with the de Donder–Weyl theory — the two definitions coincides in the Lepage–Dedecker theory.

The Trautman-Bondi mass of hyperboloidal initial data sets

Abstract: We give a definition of mass for conformally compactifiable initial data sets. The asymptotic conditions are compatible with existence of gravitational radiation, and the compactifications are allowed to be polyhomogeneous. We show that the resulting mass is a geometric invariant, and we prove positivity thereof in the case of a spherical conformal infinity. When R(g) -- or, equivalently, trgK -- tends to a negative constant to order one at infinity, the definition is expressed purely in terms of three-dimensional or two-dimensional objects.

Covariant Hamiltonian formalism for the calculus of variations with several variables: Lepage-Dedecker versus De Donder-Weyl

Abstract: The main purpose in the present paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity. For this we consider detailed geometric pictures of Lepage theories in the spirit of Dedecker and try to stress out the interplay between the Lepage-Dedecker (LP) description and the (more usual) De DonderWeyl (DDW) one. One of the main points is the fact that the Legendre transform in the DDW approach is replaced by a Legendre correspondence in the LP theory (this correspondence behaves differently: ignoring the singularities whenever the Lagrangian is degenerate).

Supersymmetric Kaluza-Klein reductions of AdS backgrounds

Abstract: This paper contains a classification of smooth Kaluza--Klein reductions (by one-parameter subgroups) of the maximally supersymmetric anti de Sitter backgrounds of supergravity theories. We present a classification of one-parameter subgroups of isometries of anti de Sitter spaces, discuss the causal properties of their orbits on these manifolds, and discuss their action on the space of Killing spinors. We analyse the problem of which quotients admit a spin structure. We then apply these results to write down the list of smooth everywhere spacelike supersymmetric quotients of AdS_3 x S^3 (x R^4), AdS_4 x S^7, AdS_5 x S^5 and AdS_7 x S^4, and the fraction of supersymmetry preserved by each quotient. The results are summarised in tables which should be useful on their own. The paper also includes a discussion of supersymmetry of singular quotients.

Matroids and p -Branes

Abstract: A link between matroid theory and $p$-branes is discussed. The Schild type action for $p$-branes and matroid bundle notion provide the two central structures for such a link. We use such a connection to bring the duality concept in matroid theory to $p$-branes physics. Our analysis may be of particular interest in M-theory and in matroid bundle theory.

On quantum symmetries of ADE graphs

Abstract: The double triangle algebra(DTA) associated to an ADE graph is considered. A description of its bialgebra structure based on a reconstruction approach is given. This approach takes as initial data the representation theory of the DTA as given by Ocneanu's cell calculus. It is also proved that the resulting DTA has the structure of a weak *-Hopf algebra. As an illustrative example, the case of the graph A3 is described in detail.

On the Structure os Asymptotically de Sitter and Anti-de Sitter Spaces

Abstract: We discuss several aspects of the relation between asymptotically AdS and asymptotically dS spacetimes including: the continuation between these types of spaces, the global stability of asymptotically dS spaces and the structure of limits within this class, holographic renormalization, and the maximal mass conjecture of Balasubramanian-deBoer-Minic.

A New Approach to Quantising Space-Time: III. State Vestors of Functions on Arrows

Abstract: In [1], a new approach was suggested for quantising space-time, or space. This involved developing a procedure for quantising a system whose configuration space--or history-theory analogue--is the set of objects, Ob(Q), in a (small) category Q. The quantum states in this approach are cross-sections of a bundle A is in K[A] of Hilbert spaces over Ob(Q). The Hilbert spaces K[A], A are in Ob(Q)], depend strongly on the object A, and have to be chosen so as to get an irreducible, faithful, representation of the basic `category quantisation monoid'. In the present paper, we develop a different approach in which the state vectors are complex-valued functions on the set of arrows in Q. This throws a new light on the Hilbert bundle scheme: in particular, we recover the results of that approach in the, physically important, example when Q is a small category of finite sets.

Matrix Quantum Mechanics and Soliton Regularization of Noncommutative Field Theory

Abstract: We construct an approximation to field theories on the noncommutative torus based on soliton projections and partial isometries which together form a matrix algebra of functions on the sum of two circles. The matrix quantum mechanics is applied to the perturbative dynamics of scalar field theory, to tachyon dynamics in string field theory, and to the Hamiltonian dynamics of noncommutative gauge theory in two dimensions. We also describe the adiabatic dynamics of solitons on the noncommutative torus and compare various classes of noncommutative solitons on the torus and the plane.

Limiting behavior of local Calabi-Yau metrics

Abstract: We use a generalization of the Gibbons-Hawking ansatz to study the behavior of certain non-compact Calabi-Yau manifolds in the large complex structure limit. This analysis provides an intermediate step toward proving the metric collapse conjecture for toric hypersurfaces and complete intersections.

Mirror symmetry and supermanifold

Abstract: We develop techniques for obtaining the mirror of Calabi-Yau supermanifolds as super Landau-Ginzburg theories. In some cases the dual can be equivalent to a geometry. We apply this to some examples. In particular we show that the mirror of the twistorial Calabi-Yau CP becomes equivalent to a quadric in CP × CP as had been recently conjectured (in the limit where the Kähler parameter of CP, t → ±∞). Moreover we show using these techniques that there is a non-trivial Z2 symmetry for the Kähler parameter, t → −t, which exchanges the opposite helicity states. As another class of examples, we show that the mirror of certain weighted projective (n + 1|1) superspaces is equivalent to compact Calabi-Yau hypersurfaces in weighted projective n space. e-print archive: http://lanl.arXiv.org/abs/hep-th/0403192 940 MIRROR SYMMETRY AND SUPERMANIFOLD

Effective Stringy Description of Schwarzschild Black Holes

Abstract: We start by pointing out that certain Riemann surfaces appear rather naturally in the context of wave equations in the black hole background. For a given black hole there are two closely related surfaces. One is the Riemann surface of complexified ``tortoise'' coordinate. The other Riemann surface appears when the radial wave equation is interpreted as the Fuchsian differential equation. We study these surfaces in detail for the BTZ and Schwarzschild black holes in four and higher dimensions. Topologically, in all cases both surfaces are a sphere with a set of marked points; for BTZ and 4D Schwarzschild black holes there is 3 marked points. In certain limits the surfaces can be characterized very explicitly. We then show how properties of the wave equation (quasi-normal modes) in such limits are encoded in the geometry of the corresponding surfaces. In particular, for the Schwarzschild black hole in the high damping limit we describe the Riemann surface in question and use this to derive the quasi-normal mode frequencies with the log(3) as the real part. We then argue that the surfaces one finds this way signal an appearance of an effective string. We propose that a description of this effective string propagating in the black hole background can be given in terms of the Liouville theory living on the corresponding Riemann surface. We give such a stringy description for the Schwarzschild black hole in the limit of high damping and show that the quasi-normal modes emerge naturally as the poles in 3-point correlation function in the effective conformal theory.

Transition from big crunch to big bang in brane cosmology

Abstract: We consider branes $N=I \times \mathcal{S}_0$, where $\mathcal{S}_0$ is an $n$-dimensional space form, not necessarily compact, in a Schwarzschild-AdS$_{(n+2)}$ bulk $\mathcal{N}$ The branes have a big crunch singularity If a brane is an ARW space, then, under certain conditions, there exists a smooth natural transition flow through the singularity to a reflected brane $\hat{N}$, which has a big bang singularity and which can be viewed as a brane in a reflected Schwarzschild-AdS$_{(n+2)}$ bulk $\hat{\mathcal{N}}$ The joint branes $N \cup \hat{N}$ can thus be naturally embedded in $\mathbb{R}^2 \times \mathcal{S}_0$, hence there exists a second possibility of defining a smooth transition from big crunch to big bang by requiring that $N \cup \hat{N}$ forms a $C^\infty$-hypersurface in $\mathbb{R}^2 \times \mathcal{S}_0$ This last notion of a smooth transition also applies to branes that are not ARW spaces, allowing a wide range of possible equations of state

Effective Superpotentials, Geometry and Integrable Systems

Abstract: We consider the effective superpotentials of N; = 1 SU(Nc) and U(Nc) supersymmetric gauge theories that are obtained from the N = 2 theory by adding a tree-level superpotential. We show that several of the techniques for computing the effective superpotential are implicitly regularized by 2Nc massive chiral multiplets in the fundamental representation, i.e. the gauge theory is embedded in the finite theory with nontrivial UV fixed point. In the maximally confining phase we obtain explicit general formulae for the effective superpotential, which reduce to previously known results in particular cases. In order to study N = 1 and N = 2 theories with fundamentals, we explicitly factorize the Seiberg-Witten curve for 0 ≤ Nf < 2N c and use the results to rederive the N = 1 superpotential. N = 2 gauge theories have an underlying integrable structure, and we obtain results on a new Lax matrix for Nf = Nc .

On quantum symmetries of the non--ADE graph F4

Abstract: We describe quantum symmetries associated with the F4 Dynkin diagram. Our study stems from an analysis of the (Ocneanu) modular splitting equation applied to a partition function which is invariant under a particular congruence subgroup of the modular group.