Showing papers in "Advances in Theoretical and Mathematical Physics in 2009"

S-duality of boundary conditions in N=4 super Yang-Mills theory

Abstract: By analyzing brane configurations in detail, and extracting general lessons, we develop methods for analyzing S-duality of supersymmetric boundary conditions in N=4 super Yang-Mills theory. In the process, we find that S-duality of boundary conditions is closely related to mirror symmetry of three-dimensional gauge theories, and we analyze the IR behavior of large classes of quiver gauge theories.

Perturbative Algebraic Quantum Field Theory and the Renormalization Groups

Abstract: A new formalism for the perturbative construction of algebraic quantum field theory is developed. The formalism allows the treatment of low-dimensional theories and of non-polynomial interactions. We discuss the connection between the Stuckelberg–Petermann renormalization group which describes the freedom in the perturbative construction with the Wilsonian idea of theories at different scales. In particular, we relate the approach to renormalization in terms of Polchinski’s Flow Equation to the Epstein–Glaser method. We also show that the renormalization group in the sense of Gell–Mann–Low (which characterizes the behaviour of the theory under the change of all scales) is a one-parametric subfamily of the Stuckelberg–Petermann group and that this subfamily is in general only a cocycle. Since the algebraic structure of the Stuckelberg–Petermann group does not depend on global quantities, this group can be formulated in the (algebraic) adiabatic limit without meeting any infrared divergencies. In particular we derive an algebraic version of the Callan–Symanzik equation and define the β-function in a state independent way.

Branes and Quantization

Abstract: The problem of quantizing a symplectic manifold (M,ω) can be formulated in terms of the A-model of a complexification of M. This leads to an interesting new perspective on quantization. From this point of view, the Hilbert space obtained by quantization of (M,ω) is the space of (B_(cc), B) strings, where B_(cc) and B are two A-branes; B is an ordinary Lagrangian A-brane, and Bcc is a space-filling coisotropic A-brane. B is supported on M, and the choice of ω is encoded in the choice of B_(cc). As an example, we describe from this point of view the representations of the group SL(2,R). Another application is to Chern–Simons gauge theory.

Exact chiral ring of AdS(3) / CFT(2)

Abstract: We carry out an exact worldsheet computation of tree level three-point correlators of chiral operators in string theory on AdS(3) x S^3 x T^4 with NS-NS flux. We present a simple representation for the string chiral operators in the coordinate basis of the dual boundary CFT. Striking cancelations occur between the three-point functions of the H3+ and the SU(2) WZW models which result in a simple factorized form for the final correlators. We show, by fixing a single free parameter in the H3+ WZW model, that the fusion rules and the structure constants of the N=2 chiral ring in the bulk are in precise agreement with earlier computations in the boundary CFT of the symmetric product of T^4 at the orbifold point in the large N limit.

D-branes and normal functions

Abstract: We explain the B-model origin of extended Picard-Fuchs equations satisfied by the D-brane superpotential on compact Calabi-Yau three- folds. The domainwall tension is identified with a Poincare normal func- tion — a transversal holomorphic section of the Griffiths intermediate Jacobian — via the Abel-Jacobi map. Within this formalism, we derive the extended Picard-Fuchs equation associated with the mirror of the real quintic.

The cosmological billiard attractor

Abstract: This article is devoted to a study of the asymptotic dynamics of generic solutions of the Einstein vacuum equations toward a generic spacelike singularity. Starting from fundamental assumptions abo ...

Notes on certain other (0,2) correlation functions

Abstract: In this paper, we shall describe some correlation function computations in perturbative heterotic strings that generalize B model computations. On the (2,2) locus, correlation functions in the B model receive no quantum corrections, but off the (2,2) locus, that can change. Classically, the (0,2) analogue of the B model is equivalent to the previously discussed (0,2) analogue of the A model, but with the gauge bundle dualized — our generalization of the A model also simultaneously generalizes the B model. The A and B analogues sometimes have different regularizations, however, which distinguish them quantum-mechanically. We discuss how properties of the (2,2) B model, such as the lack of quantum corrections, are realized in (0,2) A model language. In an appendix, we also extensively discuss how the Calabi–Yau condition for the closed string B model (uncoupled to topological gravity) can be weakened slightly, a detail which does not seem to have been covered in the literature previously. That weakening also manifests in the description of the (2,2) B model as a (0,2) A model.

Quantum 't Hooft operators and S-duality in N=4 super Yang-Mills

Abstract: We provide a quantum path integral definition of an 't Hooft loop operator, which inserts a pointlike monopole in a four dimensional gauge theory We explicitly compute the expectation value of the circular 't Hooft operators in N=4 super Yang-Mills with arbitrary gauge group G up to next to leading order in perturbation theory We also compute in the strong coupling expansion the expectation value of the circular Wilson loop operators The result of the computation of an 't Hooft loop operator in the weak coupling expansion exactly reproduces the strong coupling result of the conjectured dual Wilson loop operator under the action of S-duality This paper demonstrates - for the first time - that correlation functions in N=4 super Yang-Mills admit the action of S-duality

Geometric structures on G 2 and Spin (7)-manifolds

Abstract: This article studies the geometry of moduli spaces of G2-manifolds, associative cycles, coassociative cycles and deformed Donaldson-Thomas bundles. We introduce natural symmetric cubic tensors and differential forms on these moduli spaces. They correspond to Yukawa couplings and correlation functions in M-theory. We expect that the Yukawa coupling characterizes (co-)associative fibrations on these manifolds. We discuss the Fourier transformation along such fibrations and the analog of the Strominger-Yau-Zaslow mirror conjecture for G2-manifolds. We also discuss similar structures and transformations for Spin(7)manifolds.

Towards open string mirror symmetry for one-parameter Calabi–Yau hypersurfaces

Abstract: This work is concerned with branes and differential equations for oneparameter Calabi–Yau hypersurfaces in weighted projective spaces. For a certain class of B-branes, we derive the inhomogeneous Picard–Fuchs equations satisfied by the brane superpotential. In this way, we arrive at a prediction for the real BPS invariants for holomorphic maps of worldsheets with low Euler characteristics, ending on the mirror A-branes.

Gerbe-holonomy for surfaces with defect networks

Abstract: We define the sigma-model action for world-sheets with embedded defect networks in the presence of a three-form field strength. We derive the defect gluing condition for the sigma-model fields and their derivatives, and use it to distinguish between conformal and topological defects. As an example, we treat the WZW model with defects labelled by elements of the centre Z(G) of the target Lie group G; comparing the holonomy for different defect networks gives rise to a 3-cocycle on Z(G). Next, we describe the factorisation properties of two-dimensional quantum field theories in the presence of defects and compare the correlators for different defect networks in the quantum WZW model. This, again, results in a 3-cocycle on Z(G). We observe that the cocycles obtained in the classical and in the quantum computation are cohomologous.

Geometry of particle physics

Abstract: We explain how to construct a large class of new quiver gauge theories from branes at singularities by orientifolding and Higgsing old examples. The new models include the MSSM, decoupled from gravity, as well as some classic models of dynamical SUSY breaking. We also discuss topological criteria for unification.

Defects and D-Brane Monodromies

Abstract: In this paper D-brane monodromies are studied from a world-sheet point of view. More precisely, defect lines are used to describe the parallel transport of D-branes along deformations of the underlying bulk conformal field theories. This method is used to derive B-brane monodromies in Kahler moduli spaces of non-linear sigma models on projective hypersurfaces. The corresponding defects are constructed at Landau-Ginzburg points in these moduli spaces where matrix factorisation tech- niques can be used. Transporting them to the large volume phase by means of the gauged linear sigma model we find that their action on B-branes at large volume can be described by certain Fourier-Mukai transformations which are known from target space geometric considerations to represent the corresponding monodromies.

Siegel modular forms and finite symplectic groups

Abstract: The finite symplectic group $Sp(2g)$ over the field of two elements has a natural representation on the vector space of Siegel modular forms of given weight for the principal congruence subgroup of level two. In this paper we decompose this representation, for various (small) values of the genus and the level, into irreducible representations. As a consequence we obtain uniqueness results for certain modular forms related to the superstring measure, a better understanding of certain modular forms in genus three studied by D’Hoker and Phong as well as a new construction of Miyawaki’s cusp form of weight twelve in genus three.

Non-commutative correspondences, duality and D-branes in bivariant K-theory

Abstract: We describe a categorical framework for the classification of D-branes on non-commutative spaces using techniques from bivariant K-theory of C!-algebras. We present a new description of bivariant K-theory in terms of non-commutative correspondences which is nicely adapted to the study of T-duality in open string theory. We systematically use the diagram calculus for bivariant K-theory as detailed in our previous paper [12]. We explicitly work out our theory for a number of examples of noncommutative manifolds.

On the geometry of quiver gauge theories (Stacking exceptional collections)

Abstract: In this paper we advance the program of using exceptional collections to understand the gauge theory description of a D-brane probing a Calabi-Yau singularity. To this end, we strengthen the connection between strong exceptional collections and fractional branes. To demonstrate our ideas, we derive a strong exceptional collection for every Y^{p,q} singularity, and also prove that this collection is simple.

Dimer models, free fermions and super quantum mechanics

Abstract: This note relates topics in statistical mechanics, graph theory and combinatorics, lattice quantum field theory, super quantum mechanics and string theory. We give a precise relation between the dimer model on a graph embedded on a torus and the massless free Majorana fermion living on the same lattice. A loop expansion of the fermion determinant is performed, where the loops turn out to be compositions of two perfect matchings. These loop states are sorted into co‐chain groups using categorification techniques similar to the ones used for categorifying knot polynomials. The Euler characteristic of the resulting co‐ chain complex recovers the Newton polynomial of the dimer model. We re‐interpret this system as supersymmetric quantum mechanics, where configurations with vanishing net winding number form the ground states. Finally, we make use of the quiver gauge theory ‐ dimer model correspondence to obtain an interpretation of the loops in terms of the physics of D‐branes probing a toric Calabi‐Yau singularity.

Quasiconformal Realizations of $E_{6(6)}, E_{7(7)}, E_{8(8)}$ and $SO(n+3,m+3), N=4 and N>4$ Supergravity and Spherical Vectors

Abstract: After reviewing the underlying algebraic structures we give a unified realization of split exceptional groups F_{4(4)},E_{6(6)}, E_{7(7)}, E_{8(8)} and of SO(n+3,m+3) as quasiconformal groups that is covariant with respect to their (Lorentz) subgroups SL(3,R), SL(3,R)XSL(3,R), SL(6,R), E_{6(6)} and SO(n,m)XSO(1,1), respectively. We determine the spherical vectors of quasiconformal realizations of all these groups twisted by a unitary character $ u$. We also give their quadratic Casimir operators and determine their values in terms of $ u$ and the dimension $n_V$ of the underlying Jordan algebras. For $ u= -(n_V+2)+i\rho$ the quasiconformal action induces unitary representations on the space of square integrable functions in $(2n_V+3)$ variables, that belong to the principle series. For special discrete values of $ u$ the quasiconformal action leads to unitary representations belonging to the discrete series and their continuations. The manifolds that correspond to "quasiconformal compactifications" of the respective $(2n_V+3)$ dimensional spaces are also given. We discuss the relevance of our results to N=8 supergravity and to N=4 Maxwell-Einstein supergravity theories and, in particular, to the proposal that three and four dimensional U-duality groups act as spectrum generating quasiconformal and conformal groups of the corresponding four and five dimensional supergravity theories, respectively.

The so(d+2,2) Minimal Representation and Ambient Tractors: the Conformal Geometry of Momentum Space

Abstract: Tractor Calculus is a powerful tool for analyzing Weyl invariance; although fundamentally linked to the Cartan connection, it may also be arrived at geometrically by viewing a conformal manifold as the space of null rays in a Lorentzian ambient space. For dimension d conformally flat manifolds we show that the (d+2)-dimensional Fefferman--Graham ambient space corresponds to the momentum space of a massless scalar field. Hence on the one hand the null cone parameterizes physical momentum excitations, while on the other hand, null rays correspond to points in the underlying conformal manifold. This allows us to identify a fundamental set of tractor operators with the generators of conformal symmetries of a scalar field theory in a momentum representation. Moreover, these constitute the minimal representation of the non-compact conformal Lie symmetry algebra of the scalar field with Kostant--Kirillov dimension d+1. Relaxing the conformally flat requirement, we find that while the conformal Lie algebra of tractor operators is deformed by curvature corrections, higher relations in the enveloping algebra corresponding to the minimal representation persist. We also discuss potential applications of these results to physics and conformal geometry.

Decay of solutions of the Teukolsky equation for higher spin in the Schwarzschild geometry

Abstract: We prove that the Schwarzschild black hole is linearly stable under electromagnetic and gravitational perturbations. Our method is to show that for spin s = 1 or s = 2, solutions of the Teukolsky equation with smooth, compactly supported initial data outside the event horizon, decay in L 1 .

Non-linear sigma models via the chiral de Rham complex

Abstract: We propose a physical interpretation of the chiral de Rham complex as a formal Hamiltonian quantization of the supersymmetric non-linear sigma model We show that the chiral de Rham complex on a Ca

Topological T-duality and T-folds

Abstract: We explicitly construct the C*-algebras arising in the formalism of Topological T-duality due to Mathai and Rosenberg from string-theoretic data in several key examples. We construct a continuous-trace algebra with an action of ${\mathbb R}^d$ unique up to exterior equivalence from the data of a smooth ${\mathbb T}^d$-equivariant gerbe on a trivial bundle $X = W \times {\mathbb T}^d$. We argue that the `noncommutative T-duals' of Mathai and Rosenberg, should be identified with the nongeometric backgrounds well-known in string theory. We also argue that the crossed-product C*-algebra ${\mathcal A} \rtimes_{\alpha|_{\KZ^d}} {\mathbb Z}^d$ should be identified with the T-folds of Hull which geometrize these backgrounds. We identify the charge group of D-branes on T-fold backgrounds in the C*-algebraic formalism of Topological T-duality. We also study D-branes on T-fold backgrounds. We show that the $K$-theory bundles studied by Echterhoff, Nest and Oyono-Oyono give a natural description of these objects.

New heterotic non-Kähler geometries

Abstract: New heterotic torsional geometries are constructed as orbifolds of T^2 bundles over K3. The discrete symmetries considered can be freely-acting or have fixed points and/or fixed curves. We give explicit constructions when the base K3 is Kummer or algebraic. The orbifold geometries can preserve N=1,2 supersymmetry in four dimensions or be non-supersymmetric.

Efficient quantum processing of three–manifold topological invariants

Abstract: A quantum algorithm for approximating efficiently three–manifold topological invariants in the framework of $SU(2)$ Chern–Simons–Witten (CSW) topological quantum field theory at finite values of the coupling constant $k$ is provided. The model of computation adopted is the $q$-deformed spin network model viewed as a quantum recognizer in the sense of [1], where each basic unitary transition function can be efficiently processed by a standard quantum circuit. This achievement is an extension of the algorithm for approximating polynomial invariants of colored oriented links found in "Spin networks, quantum automata and link invariants" and "An efficient quantum algorithm for colored Jones polynomials." Thus all the significant quantities — partition functions and observables — of quantum CSW theory can be processed efficiently on a quantum computer, reflecting the intrinsic, field-theoretic solvability of such theory at finite $k$. The paper is supplemented by a critical overview of the basic conceptual tools underlying the construction of quantum invariants of links and three–manifolds and connections with algorithmic questions that arise in geometry and quantum gravity models are discussed.

Kähler-Ricci flow, Morse theory, and vacuum structure deformation of N = 1 supersymmetry in four dimensions

Abstract: We address some aspects of four-dimensional chiral N = 1 supersym- metric theories on which the scalar manifold is described by Kahler geometry and can further be viewed as Kahler-Ricci soliton generating a one-parameter family of Kahler geometries. All couplings and solu- tions, namely the BPS domain walls and their supersymmetric Lorentz invariant vacua turn out to be evolved with respect to the flow parameter related to the soliton. Two models are discussed, namely N = 1 theory on Kahler-Einstein manifold and U(n) symmetric Kahler-Ricci soliton with positive definite metric. In the first case, we find that the evolution of the soliton causes topological change and correspondingly, modifies the

On the mathematics and physics of high genus invariants of $[C^3/Z_3]$

Abstract: This paper wishes to foster communication between mathematicians and physicists working in mirror symmetry and orbifold Gromov-Witten theory. We provide a reader friendly review of the physics computation in [arXiv:hep-th/0607100] that predicts Gromov-Witten invariants of [C^3/Z_3] in arbitrary genus, and of the mathematical framework for expressing these invariants as Hodge integrals. Using geometric properties of the Hodge classes, we compute the unpointed invariants for g=2,3, thus providing the first high genus mathematical check of the physics predictions.

Equivariant cohomology of the chiral de Rham complex and the half-twisted gauged sigma model

Abstract: In this paper, we study the perturbative aspects of the half-twisted variant of Witten's topological A-model coupled to a non-dynamical gauge field with Kahler target space X being a G-manifold. Our main objective is to furnish a purely physical interpretation of the equivariant cohomology of the chiral de Rham complex, recently constructed by Lian and Linshaw in [arXiv:math/0501084], called the "chiral equivariant cohomology". In doing so, one finds that key mathematical results such as the vanishing in the chiral equivariant cohomology of positive weight classes, lend themselves to straightforward physical explanations. In addition, one can also construct topological invariants of X from the correlation functions of the relevant physical operators corresponding to the non-vanishing weight-zero classes. Via the topological invariance of these correlation functions, one can verify, from a purely physical perspective, the mathematical isomorphism between the weight-zero subspace of the chiral equivariant cohomology and the classical equivariant cohomology of X. Last but not least, one can also determine fully, the de Rham cohomology ring of X/G, from the topological chiral ring generated by the local ground operators of the physical model under study.

On LCSFT/MST correspondence

Abstract: It was suggested that light-cone superstring field theory (LCSFT) and matrix string theory (MST) are closely related Especially the bosonic twist fields and the fermionic spin fields in MST correspond to the string interaction vertices in LCSFT Since CFT operators are characterized by their OPEs, in our previous work we realized the most important OPE of the twist fields by computing contractions of the interaction vertices using the bosonic cousin of LCSFT Here using the full LCSFT we generalize our previous work into the realization of OPEs for a vast class of operators