Showing papers in "Advances in Theoretical and Mathematical Physics in 2008"

The su(2|2) dynamic S-matrix

Abstract: We derive and investigate the S-matrix for the su(2|3) dynamic spin chain and for planar N = 4 super Yang–Mills. Due to the large amount of residual symmetry in the excitation picture, the S-matrix turns out to be fully constrained up to an overall phase. We carry on by diagonalizing it and obtain Bethe equations for periodic states. This proves an earlier proposal for the asymptotic Bethe equations for the su(2|3) dynamic spin chain and for N = 4 SYM.

Dimer models from mirror symmetry and quivering amoebae

Abstract: Dimer models are 2-dimensional combinatorial systems that have been shown to encode the gauge groups, matter content and tree-level superpotential of the world-volume quiver gauge theories obtained by placing D3-branes at the tip of a singular toric Calabi-Yau cone. In particular the dimer graph is dual to the quiver graph. However, the string theoretic explanation of this was unclear. In this paper we use mirror symmetry to shed light on this: the dimer models live on a T^2 subspace of the T^3 fiber that is involved in mirror symmetry and is wrapped by D6-branes. These D6-branes are mirror to the D3-branes at the singular point, and geometrically encode the same quiver theory on their world-volume.

Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes

Abstract: The present work extends our short communication L. Andersson, M. Mars and W. Simon, Local existence of dynamical and trapping horizons, Phys. Rev. Lett. 95 (2005), 111102. For smooth marginally outer trapped surfaces (MOTS) in a smooth spacetime, we define stability with respect to variations along arbitrary vectors v normal to the MOTS. After giving some introductory material about linear nonself-adjoint elliptic operators, we introduce the stability operator Lv and we characterize stable MOTS in terms of sign conditions on the principal eigenvalue of Lv. The main result shows that given a strictly stable MOTS S0 ⊂ Σ0 in a spacetime with a reference foliation Σt, there is an open marginally outer trapped tube (MOTT), adapted to the reference foliation, which contains S0. We give conditions under which the MOTT can be completed. Finally, we show that under standard energy conditions on the spacetime, the MOTT must be either locally achronal, spacelike or null.

General boundary quantum field theory: Foundations and probability interpretation

Abstract: We elaborate on the proposed general boundary formulation as an extension of standard quantum mechanics to arbitrary (or no) backgrounds. Temporal transition amplitudes are generalized to amplitudes for arbitrary space-time regions. State spaces are associated to general (not necessarily spacelike) hypersurfaces. We give a detailed foundational exposition of this approach, including its probability interpretation and a list of core axioms. We explain how standard quantum mechanics arises as a special case. We include a discussion of probability conservation and unitarity, showing how these concepts are generalized in the present framework. We formulate vacuum axioms and incorporate space-time symmetries into the framework. We show how the Schrodinger–Feynman approach is a suitable starting point for casting quantum field theories into the general boundary form. We discuss the role of operators.

Resolved Toroidal Orbifolds and their Orientifolds

Abstract: We discuss the resolution of toroidal orbifolds. For the resulting smooth Calabi–Yau manifolds, we calculate the intersection ring and determine the divisor topologies. In a next step, the orientifold quotients are constructed.

Linear Models for Flux Vacua

Abstract: We construct worldsheet descriptions of heterotic flux vacua as the IR limits of N=2 gauge theories. Spacetime torsion is incorporated via a 2d Green-Schwarz mechanism in which a doublet of axions cancels a one-loop gauge anomaly. Manifest (0, 2) supersymmetry and the compactness of the gauge theory instanton moduli space suggest that these models, which include Fu-Yau models, are stable against worldsheet instantons, implying that they, like Calabi-Yaus, may be smoothly extended to solutions of the exact beta functions. Since Fu-Yau compactifications are dual to KST-type flux compactifications, this provides a microscopic description of these IIB RR-flux vacua.

Uniqueness of open/closed rational CFT with given algebra of open states

Abstract: We study the sewing constraints for rational two-dimensional conformal field theory on oriented surfaces with possibly non-empty boundary. The boundary condition is taken to be the same on all segm ...

Triadophilia: A special corner in the landscape

Abstract: It is well known that there are a great many apparently consistent vacua of string theory. We draw attention to the fact that there appear to be very few Calabi--Yau manifolds with the Hodge numbers h^{11} and h^{21} both small. Of these, the case (h^{11}, h^{21})=(3,3) corresponds to a manifold on which a three generation heterotic model has recently been constructed. We point out also that there is a very close relation between this manifold and several familiar manifolds including the `three-generation' manifolds with \chi=-6 that were found by Tian and Yau, and by Schimmrigk, during early investigations. It is an intriguing possibility that we may live in a naturally defined corner of the landscape. The location of these three generation models with respect to a corner of the landscape is so striking that we are led to consider the possibility of transitions between heterotic vacua. The possibility of these transitions, that we here refer to as transgressions, is an old idea that goes back to Witten. Here we apply this idea to connect three generation vacua on different Calabi-Yau manifolds.

The Topological G 2 String

Abstract: We construct new topological theories related to sigma models whose target space is a seven-dimensional manifold of G2 holonomy. We define a new type of topological twist and identify the BRST operator and the physical states. Unlike the more familiar six-dimensional case, our topological model is defined in terms of conformal blocks and not in terms of local operators of the original theory. We also present evidence that one can extend this definition to all genera and construct a seven-dimensional topological string theory. We compute genus zero correlation functions and relate these to Hitchin’s functional for three-forms in seven dimensions. Along the way we develop the analogue of special geometry for G2 manifolds. When the seven-dimensional topological twist is applied to the product of a Calabi–Yau manifold and a circle, the result is an interesting combination of the six-dimensional A and B models.

Exact renormalization of a noncommutative Φ3 model in 6 dimensions

Abstract: The noncommutative self-dual φ 3 model in six dimensions is quantized and essentially solved, by mapping it to the Kontsevich model. The model is shown to be renormalizable and asymptotically free, and solvable genus by genus. It requires both wavefunction and coupling constant renormalization. The exact (“all-order”) renormalization of the bare parameters is determined explicitly, which turns out to depend on the genus 0 sector only. The running coupling constant is also computed exactly, which decreases more rapidly than predicted by the 1-loop beta-function. A phase transition to an unstable phase is found.

Metastable Vacua in Perturbed Seiberg-Witten Theories

Abstract: We show that, for a generic choice of a point on the Coulomb branch of any N = 2 supersymmetric gauge theory, it is possible to find a superpotential perturbation which generates a metastable vacuum at the point. For theories with SU(N) gauge group, such a superpotential can be expressed as a sum of single-trace terms for N = 2 and 3. If the metastable point is chosen at the origin of the moduli space, we can show that the superpotential can be a single-trace operator for any N. In both cases, the superpotential is a polynomial of degree 3N of the vector multiplet scalar field.

Parameter Space of Quiver Gauge Theories

Abstract: Placing a set of branes at a Calabi-Yau singularity leads to an N=1 quiver gauge theory. We analyze F-term deformations of such gauge theories. A generic deformation can be obtained by making the Calabi-Yau non-commutative. We discuss non-commutative generalisations of well-known singularities such as the Del Pezzo singularities and the conifold. We also introduce new techniques for deriving superpotentials, based on quivers with ghosts and a notion of generalised Seiberg duality. The curious gauge structure of quivers with ghosts is most naturally described using the BV formalism. Finally we suggest a new approach to Seiberg duality by adding fields and ghost-fields whose effects cancel each other.

Quantization of flag manifolds and their supersymmetric extensions

Abstract: We first review the description of flag manifolds in terms of Plucker coordinates and coherent states. Using this description, we construct fuzzy versions of the algebra of functions on these spaces in both operatorial and star product language. Our main focus is here on flag manifolds appearing in the double fibration underlying the most common twistor correspondences. After extending the Plucker description to certain supersymmetric cases, we also obtain the appropriate deformed algebra of functions on a number of fuzzy flag supermanifolds. In particular, fuzzy versions of Calabi–Yau supermanifolds are found.

Topological strings, two-dimensional Yang-Mills theory and Chern-Simons theory on torus bundles

Abstract: We study the relations between two-dimensional Yang-Mills theory on the torus, topological string theory on a Calabi-Yau threefold whose local geometry is the sum of two line bundles over the torus, and Chern-Simons theory on torus bundles. The chiral partition function of the Yang-Mills gauge theory in the large N limit is shown to coincide with the topological string amplitude computed by topological vertex techniques. We use Yang-Mills theory as an efficient tool for the computation of Gromov-Witten invariants and derive explicitly their relation with Hurwitz numbers of the torus. We calculate the Gopakumar-Vafa invariants, whose integrality gives a non-trivial confirmation of the conjectured nonperturbative relation between two-dimensional Yang-Mills theory and topological string theory. We also demonstrate how the gauge theory leads to a simple combinatorial solution for the Donaldson-Thomas theory of the Calabi-Yau background. We match the instanton representation of Yang-Mills theory on the torus with the nonabelian localization of Chern-Simons gauge theory on torus bundles over the circle. We also comment on how these results can be applied to the computation of exact degeneracies of BPS black holes in the local Calabi-Yau background.

Causal properties of AdS-isometry groups I: Causal actions and limit sets

Abstract: We study the causality relation in the $3$-dimensional anti-de Sitter space AdS and its conformal boundary $\mbox{Ein}_2$. To any closed achronal subset $\Lambda$ in $\mbox{Ein}_2$ we associate the invisible domain $E(\Lambda)$ from $\Lambda$ in AdS. We show that if $\Gamma$ is a torsion-free discrete group of isometries of AdS preserving $\Lambda$ and is non-elementary (for example, not abelian) then the action of $\Gamma$ on $E(\Lambda)$ is free, properly discontinuous and strongly causal. If $\Lambda$ is a topological circle then the quotient space $M_\Lambda(\Gamma) = \Gamma\backslash{E}(\Lambda)$ is a maximal globally hyperbolic AdS-spacetime admitting a Cauchy surface $S$ such that the induced metric on $S$ is complete. In a forthcoming paper we study the case where $\Gamma$ is elementary and use the results of the present paper to define a large family of AdS-spacetimes including all the previously known examples of BTZ multi-black holes.

Supersymmetric WZW models and twisted K theory of SO(3)

Abstract: We present an encompassing treatment of D-brane charges in supersymmetric SO(3) WZW models. There are two distinct supersymmetric CFTs at each even level: the standard bosonic SO(3) modular invariant tensored with free fermions, as well as a novel twisted model. We calculate the relevant twisted K-theories and find complete agreement with the CFT analysis of D-brane charges. The K-theoretical computation in particular elucidates some important aspects of N=1 supersymmetric WZW models on non-simply connected Lie groups.

Causal properties of AdS-isometry groups II: BTZ multi black-holes

Abstract: This paper is the continuation of [8]. We essentially prove that the familly of strongly causal spacetimes defined in [8] associated to generic achronal subsets in $\mbox{Ein}_2$ contains all the examples of BTZ multi black-holes. It provides new elements for the global description of these multi black-holes. We also prove that any spacetime locally modelled on the anti-de Sitter space admits a well-defined maximal conformal boundary, locally modelled on $\mbox{Ein}_2$, such that any isometry between such spacetimes extends to their conformal boundary.

Instanton calculus and chiral one-point functions in supersymmetric gauge theories

Abstract: We compute topological one-point functions of the chiral operator Tr φk in the maximally confining phase of U(N) supersymmetric gauge theory. These onepoint functions are polynomials in the equivariant parameter ~ and the parameter of instanton expansion q = Λ2N and are of particular interest from gauge/string theory correspondence, since they are related to the Gromov-Witten theory of P . Based on a combinatorial identity that gives summation formula over Young diagrams of relevant functions, we find a relation among chiral one-point functions, which recursively determines the ~ expansion of the generating function of onepoint functions. Using a result from the operator formalism of the Gromov-Witten theory, we also present a vacuum expectation value of the loop operator Tr eitφ.

Mapping the geometry of the F4 group

Abstract: In this paper, we present a construction of the compact form of the exceptional Lie group F4 by exponentiating the corresponding Lie algebra f4. We realize F4 as the automorphisms group of the exceptional Jordan algebra, whose elements are 3 x 3 Hermitian matrices with octonionic entries. We use a parametrization which generalizes the Euler angles for SU(2) and is based on the fibration of F4 via a Spin(9) subgroup as a fiber. This technique allows us to determine an explicit expression for the Haar invariant measure on the F4 group manifold. Apart from shedding light on the structure of F4 and its coset manifold OP2 = F4/Spin(9), the octonionic projective plane, these results are a prerequisite for the study of E6, of which F4 is a (maximal) subgroup.

Heterotic Kähler/non-Kähler Transitions

Abstract: We show how two topologically distinct spaces - the Kahler K3 x T^2 and the non-Kahler T^2 bundle over K3 - can be smoothly connected in heterotic string theory. The transition occurs when the base K3 is deformed to the T^4/Z_2 orbifold limit. The orbifold theory can be mapped via duality to M-theory on K3 x K3 where the transition corresponds to an exchange of the two K3's.

The inverse mean curvature flow in cosmological spacetimes

Abstract: We prove that the leaves of an inverse mean curvature flow provide a foliation of a future end of a cosmological spacetime $N$ under the necessary and sufficient assumptions that $N$ satisfies a future mean curvature barrier condition and a strong volume decay condition. Moreover, the flow parameter t can be used to define a new physically important time function.

Gravitational Collapse to Toroidal and Higher Genus asymptotically AdS Black Holes

Abstract: We match collapsing inhomogeneous as well as spatially homogeneous but anisotropic spacetimes to vacuum static exteriors with a negative cosmological constant and planar or hyperbolic symmetry. The collapsing interiors include the inhomogeneous solutions of Szekeres and of Barnes, which in turn include the Lemâitre-Tolman and the McVittie solutions. The collapse can result in toroidal or higher genus asymptotically AdS black holes.

Topological T-duality and Kaluza–Klein monopoles

Abstract: We study topological T-duality for spaces with a semi-free $S^1-$action with isolated fixed points. Physically, these correspond to spacetimes containing Kaluza-Klein monopoles. We demonstrate that the physical dyonic coordinate of such spaces has an analogue in our formalism. By analogy with the Dirac monopole, we study these spaces as gerbes. We study the effect of Topological T-duality on these gerbes.

Topological Gauge Theories on Local Spaces and Black Hole Entropy Countings

Abstract: We study cohomological gauge theories on total spaces of holomorphic line bundles over complex manifolds and obtain their reduction to the base manifold by $U(1)$-equivariant localization of the path integral. We exemplify this general mechanism by proving via exact path integral localization a reduction for local curves conjectured in hep-th/0411280, relevant to the calculation of black hole entropy/Gromov–Witten invariants. Agreement with the four-dimensional gauge theory is recovered by taking into account in the latter non-trivial contributions coming from one-loop fluctuation determinants at the boundary of the total space. We also study a class of abelian gauge theories on Calabi–Yau local surfaces, describing the quantum foam for the $A$-model, relevant to the calculation of Donaldson–Thomas invariants.

Positive Lyapunov exponents and localization bounds for strongly mixing potentials

Abstract: For a one-dimensional discrete Schrodinger operator with a weakly coupled potential given by a strongly mixing dynamical system with power law decay of correlations, we derive for all energies including the band edges and the band center a perturbative formula for the Lyapunov exponent Under adequate hypothesis, this shows that the Lyapunov exponent is positive on the whole spectrum This in turn implies that the Hausdorff dimension of the spectral measure is zero and that the associated quantum dynamics grows at most logarithmically in time

Matrix model superpotentials and ADE singularities

Abstract: We use F. Ferrari’s methods relating matrix models to Calabi-Yau spaces in order to explain much of Intriligator and Wecht’s ADE classification of N = 1 superconformal theories which arise as RG fixed points of N = 1 SQCD theories with adjoints. We find that ADE superpotentials in the Intriligator–Wecht classification exactly match matrix model superpotentials obtained from Calabi-Yaus with corresponding ADE singularities. Moreover, in the additional b O, b A, b D and b E cases we find new singular geometries. These ‘hat’ geometries are closely related to their ADE counterparts, but feature non-isolated singularities. As a byproduct, we give simple descriptions for small resolutions of Gorenstein threefold singularities in terms of transition functions between just two coordinate charts. To obtain these results we develop an algorithm for blowing down exceptional P 1 s, described in the appendix.

Deformations and D-branes

Abstract: I discuss the relation of Hochschild cohomology to the physical states in the closed topological string. This allows a notion of deformation intrinsic to the derived category. I use this to identify deformations of a quiver gauge theory associated to a D-branes at a singularity with generalized deformations of the geometry of the resolution of the singularity. An explicit map is given from noncommutative deformations (i.e., B-fields) to terms in the superpotential.

The half-twisted orbifold sigma model and the Chiral de Rham complex

Abstract: In this paper, we study the perturbative aspects of the half-twisted variant of Witten’s topological A-model on a complex orbifold X/G, where G is an isometry group of X. The objective is to furnish a purely physical interpretation of the mathematical theory of the Chiral de Rham complex on orbifolds recently constructed by Frenkel and Szczesny in Chiral de Rham complex and orbifolds, Preprint, arXiv: math.AG/ 0307181. In turn, one can obtain a novel understanding of the holomorphic (twisted) N = 2 superconformal structure underlying the untwisted and twisted sectors of the quantum sigma model, purely in terms of an obstruction (or a lack thereof) to a global definition of the relevant physical operators which correspond to G-invariant sections of the sheaf of Chiral de Rham complex on X. Explicit examples are provided to help illustrate this connection, and comparisons with their non-orbifold counterparts are also made in an aim to better understand the action of the G-orbifolding on the original half-twisted sigma model on X.

Invariant bundles on B-fibered Calabi-Yau spaces and the Standard Model

Abstract: We derive the Standard model gauge group together with chiral fermion generations from the heterotic string by turning on a Wilson line on a non-simply connected Calabi-Yau threefold with an SU(5) gauge group. For this we construct stable ${\bf Z_2}$-invariant $SU(4)\times U(1)$ bundles on an elliptically fibered cover Calabi-Yau threefold of special fibration type (the $B$-fibration). The construction makes use of a modified spectral cover approach giving just invariant bundles.

Zero Field Hall Effect in (2+1)-dimensional QED

Abstract: In QED of two space dimensions, a quantum Hall effect occurs in the absence of any magnetic field. We give a simple and transparent explanation. In solid state physics, the Hall conductivity for non-degenerate ground state is expected to be given by an integer, the Chern number. In our field-free situation, however, the conductivity is $\pm 1/2$ in natural units. We fit this half-integral result into the topological setting and give a geometric explanation reconciling the points of view of QFT and solid state physics. For quasi-periodic boundary conditions, we calculate the finite size correction to the Hall conductivity. Applications to graphene and similar materials are discussed.