Showing papers in "Advances in Theoretical and Mathematical Physics in 2003"
TL;DR: In this article, a two-parameter generalization of the Seiberg-Witten prepotential is presented, which is rather natural from the M-theory/five dimensional perspective, and conjecture its relation to the tau-functions of KP/Toda hierarchy.
Abstract: Direct evaluation of the Seiberg-Witten prepotential is accomplished following the localization programme suggested in [1]. Our results agree with all low-instanton calculations available in the literature. We present a two-parameter generalization of the Seiberg-Witten prepotential, which is rather natural from the M-theory/five dimensional perspective, and conjecture its relation to the tau-functions of KP/Toda hierarchy.
2,159 citations
TL;DR: In this article, the precise mathematical structure underlying loop quantum cosmology and the sense in which it implements the full quantization program in a symmetry reduced model has been made explicit, thereby providing a firmer mathematical and conceptual foundation to the subject.
Abstract: Applications of Riemannian quantum geometry to cosmology have had notable successes. In particular, the fundamental discreteness underlying quantum geometry has led to a natural resolution of the big bang singularity. However, the precise mathematical structure underlying loop quantum cosmology and the sense in which it implements the full quantization program in a symmetry reduced model has not been made explicit. The purpose of this paper is to address these issues, thereby providing a firmer mathematical and conceptual foundation to the subject.
794 citations
TL;DR: In this article, a new derivation of the quasinormal frequencies of Schwarzschild black holes in d greater than or equal to 4 and Reissner-Nordstrom in d = 4, in the limit of infinite damping was given.
Abstract: We give a new derivation of the quasinormal frequencies of Schwarzschild black holes in d greater than or equal to 4 and Reissner-Nordstrom black holes in d = 4, in the limit of infinite damping. For Schwarzschild in d greater than or equal to 4 we find that the asymptotic real part is THawkinglog(3) for scalar perturbations and for some gravitational perturbations; this confirms a result previously obtained by other means in the case d = 4. For Reissner-Nordstrom in d = 4 we find a specific generally aperiodic behavior for the quasinormal frequencies, both for scalar perturbations and for electromagnetic-gravitational perturbations. The formulae are obtained by studying the monodromy of the perturbation analytically continued to the complex plane; the analysis depends essentially on the behavior of the potential in the 'unphysical' region near the black hole singularity.
325 citations
TL;DR: In this paper, a deformation of N = 1 supersymmetric gauge theories in four dimensions, called the C-deformation, was considered, where the gluino field satisfies a Clifford-like algebra dictated by a self-dual two-form, instead of the standard Grassmannian algebra.
Abstract: We consider a deformation of N = 1 supersymmetric gauge theories in four dimensions, which we call the C-deformation, where the gluino field satisfies a Clifford-like algebra dictated by a self-dual two-form, instead of the standard Grassmannian algebra. The superpotential of the deformed gauge theory is computed by the full partition function of an associated matrix model (or more generally a bosonic gauge theory), including non-planar diagrams. In this identification, the strength of the two-form controls the genus expansion of the matrix model partition function. For the case of pure N = 1 Yang-Mills this deformation leads to the identification of the all genus partition function of c non-critical bosonic string at self-dual radius as the glueball superpotential. Though the C-deformation violates Lorentz invariance, the deformed F-terms are Lorentz invariant and the Lorentz violation is screened in the IR.
291 citations
TL;DR: In this paper, an open-string generalization of Vafa's formula for closed-string topological correlators has been proposed for the case of Landau-Ginzburg orbifolds.
Abstract: We compute topological correlators in Landau-Ginzburg models on a Riemann surface with arbitrary number of handles and boundaries. The boundaries may correspond to arbitrary topological D-branes of type B. We also allow arbitrary operator insertions on the boundary and in the bulk. The answer is given by an explicit formula which can be regarded as an open-string generalization of C. Vafa's formula for closed-string topological correlators. We discuss how to extend our results to the case of Landau-Ginzburg orbifolds.
218 citations
TL;DR: The instanton partition function of N = 2, D = 4 SU(2) gauge theory is obtained by taking the field theory limit of the topological open string partition function, given by a Chern-Simons theory, of a CY3-fold as discussed by the authors.
Abstract: The instanton partition function of N = 2, D = 4 SU(2) gauge theory is obtained by taking the field theory limit of the topological open string partition function, given by a Chern-Simons theory, of a CY3-fold. The CY3-fold on the open string side is obtained by geometric transition from local IP{sup 1} x IP{sup 1} which is used in the geometric engineering of the SU(2) theory. The partition function obtained from the Chern-Simons theory agrees with the closed topological string partition function of local IP{sup 1} x IP{sup 1} proposed recently by Nekrasov. We also obtain the partition functions for local F{sub 1} and F{sub 2} CY3-folds and show that the topological string amplitudes of all three local Hirzebruch surfaces give rise to the same field theory limit. It is shown that a generalization of the topological closed string partition function whose field theory limit is the generalization of the instanton partition function, proposed by Nekrasov, can be determined easily from the Chern-Simons theory.
182 citations
TL;DR: In this article, the authors consider a somewhat unconventional approach for deriving worldvolume theories for D3 branes probing Calabi-Yau singularities, which consists of extrapolating the calculation of F-terms to the large volume limit.
Abstract: In this paper we consider a somewhat unconventional approach for deriving worldvolume theories for D3 branes probing Calabi-Yau singularities. The strategy consists of extrapolating the calculation of F-terms to the large volume limit. This method circumvents the inherent limitations of more traditional approaches used for orbifold and toric singularities. We illustrate its usefulness by deriving quiver theories for D3 branes probing singularities where a Del Pezzo surface containing four, five or six exceptional curves collapses to zero size. In the latter two cases the superpotential depends explicitly on complex structure parameters. These are examples of probe theories for singularities which can currently not be computed by other means.
158 citations
TL;DR: Liouville theory with a negative norm boson and no screening charge corresponds to an exact classical solution of closed bosonic string theory describing time-dependent bulk tachyon condensation as discussed by the authors.
Abstract: Liouville theory with a negative norm boson and no screening charge corresponds to an exact classical solution of closed bosonic string theory describing time-dependent bulk tachyon condensation. A simple expression for the two point function is proposed based on renormalization/analytic continuation of the known results for the ordinary (positive-norm) Liouville theory. The expression agrees exactly with the minisuperspace result for the closed string pair-production rate (which diverges at finite time). Puzzles concerning the three-point function are presented and discussed.
152 citations
TL;DR: In this article, the chiral ring of the gluino fields is deformed by the supergravity fields, generalizing the C-deformation which was recently introduced. But understanding the gravity corrections to the F-terms requires a non-traditional interpretation of these identities.
Abstract: We study F-terms describing coupling of the supergravity to N = 1 supersymmetric gauge theories which admit large N expansions. We show that these F-terms are given by summing over genus one non-planar diagrams of the large N expansion of the associated matrix model (or more generally bosonic gauge theory). The key ingredient in this derivation is the observation that the chiral ring of the gluino fields is deformed by the supergravity fields, generalizing the C-deformation which was recently introduced. The gravity induced part of the C-deformation can be derived from the Bianchi identities of the supergravity, but understanding gravitational corrections to the F-terms requires a non-traditional interpretation of these identities.
152 citations
TL;DR: In this article, the relation between Newton's constant, the GUT scale, and the value of $\alpha_{GUT}$ was estimated and the implications for proton decay were explored.
Abstract: Models of particle physics based on manifolds of G2 holonomy are in most respects much more complicated than other string-derived models, but as we show here they do have one simplification: threshold corrections to grand unification are particularly simple. We compute these corrections, getting completely explicit results in some simple cases. We estimate the relation between Newton's constant, the GUT scale, and the value of $\alpha_{GUT}$, and explore the implications for proton decay. In the case of proton decay, there is an interesting mechanism which (relative to four-dimensional SUSY GUT's) enhances the gauge boson contribution to $p\to\pi^0e^+_L$ compared to other modes such as $p\to \pi^0e^+_R$ or $p\to \pi^+\bar
u_R$. Because of numerical uncertainties, we do not know whether to intepret this as an enhancement of the $p\to \pi^0e^+_L$ mode or a suppression of the others.
130 citations
TL;DR: In this article, the relation between one-dimensional Schrodinger equation and the vacuum eigenvalues of the Q-operators is extended to their higher-level eigen values.
Abstract: Relation between one-dimensional Schrodinger equation and the vacuum eigenvalues of the Q-operators is extended to their higher-level eigenvalues.
TL;DR: In this paper, the effective Hamiltonians of matrix-valued symbols have been analyzed and compared to the Egorov theorem and the generalized Berry connection, and a limit for effective Hamiltonian limit has been established.
Abstract: 3.1 Almost invariant subspaces
3.2 Mapping to the reference space
3.3 Effective dynamics
3.3.1 Expanding the effective Hamiltonian
3.4 Semiclassical limit for effective Hamiltonians
3.4.1 Semiclassical analysis for matrix-valued symbols
3.4.2 Geometrical interpretation: the generalized Berry connection
3.4.3 Semiclassical observables and an Egorov theorem
TL;DR: In this paper, an integral representation for the Dirac propagator involving the solutions of the ODEs which arise in Chandrasekhar's separation of variables was derived for the Cauchy problem for the massive Dirac equation in the non-extreme Kerr-Newman geometry outside the event horizon.
Abstract: We consider the Cauchy problem for the massive Dirac equation in the non-extreme Kerr-Newman geometry outside the event horizon. We derive an integral representation for the Dirac propagator involving the solutions of the ODEs which arise in Chandrasekhar's separation of variables. It is proved that for initial data in Linfinityloc near the event horizon with L2 decay at infinity, the probability of the Dirac particle to be in any compact region of space tends to zero as t goes to infinity. This means that the Dirac particle must either disappear in the black hole or escape to infinity.
TL;DR: In this article, it was shown that the Hamiltonian describing N nonrelativistic electrons with spin, interacting with the quantized radiation field and several fixed nuclei with total charge Z, has a ground state when N < Z + 1.
Abstract: We show that the Hamiltonian describing N nonrelativistic electrons with spin, interacting with the quantized radiation field and several fixed nuclei with total charge Z, has a ground state when N < Z + 1. The result holds for any value of the fine structure constant α and for any value of the ultraviolet cutoff A on the radiation field. There is no infrared cutoff. The basic mathematical ingredient in our proof is a novel localization of the electromagnetic field in such a way that the errors in the energy are of smaller order than 1/L, where L is the localization radius.
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TL;DR: In this article, dual descriptions of (0, 2) gauged linear sigma models were constructed, and the instanton corrected chiral ring was determined for some illustrative examples.
Abstract: We construct dual descriptions of (0, 2) gauged linear sigma models. In some cases, the dual is a (0, 2) Landau-Ginzburg theory, while in other cases, it is a non-linear sigma model. The duality map defines an analogue of mirror symmetry for (0, 2) theories. Using the dual description, we determine the instanton corrected chiral ring for some illustrative examples. This ring defines a (0, 2) generalization of the quantum cohomology ring of (2, 2) theories.
TL;DR: In this paper, the relation between supersymmetric gauge theories and matrix models beyond the planar limit has been investigated by summing genus one matrix model diagrams, which give the leading $1/N^2$ corrections in the large $N$ limit of the matrix model.
Abstract: We provide evidence of the relation between supersymmetric gauge theories and matrix models beyond the planar limit. We compute gravitational $R^2$ couplings in gauge theories perturbatively, by summing genus one matrix model diagrams. These diagrams give the leading $1/N^2$ corrections in the large $N$ limit of the matrix model and can be related to twist field correlators in a collective conformal field theory. In the case of softly broken $SU(N)\ \cN=2$ super Yang-Mills theories, we find that these exact solutions of the matrix models agree with results obtained by topological field theory methods.
TL;DR: In this article, a Poisson structure on H n+2g, where H is a semidirect product group of the form G ⋉ g ∗, has been shown to arise naturally as a symmetry of the quantum algebra.
Abstract: We quantise a Poisson structure on H n+2g , where H is a semidirect product group of the form G ⋉ g ∗ . This Poisson structure arises in the combinatorial description of the phase space of Chern-Simons theory with gauge group G ⋉ g ∗ on R × Sg,n, where Sg,n is a surface of genus g with n punctures. The quantisation of this Poisson structure is a key step in the quantisation of Chern-Simons theory with gauge group G ⋉ g ∗ . We construct the quantum algebra and its irreducible representations and show that the quantum double D(G) of the group G arises naturally as a symmetry of the quantum algebra.
TL;DR: In this article, the notion of a ''top'' was introduced as half of a three-dimensional reflexive polytope, and a prescription for assigning an affine, possibly twisted Kac-Moody algebra to any such top (and more generally to any elliptic fibration structure) in a precise way that involves the lengths of simple roots and coefficients of null roots.
Abstract: Candelas and Font introduced the notion of a `top' as half of a three dimensional reflexive polytope and noticed that Dynkin diagrams of enhanced gauge groups in string theory can be read off from them. We classify all tops satisfying a generalized definition as a lattice polytope with one facet containing the origin and the other facets at distance one from the origin. These objects torically encode the local geometry of a degeneration of an elliptic fibration. We give a prescription for assigning an affine, possibly twisted Kac-Moody algebra to any such top (and more generally to any elliptic fibration structure) in a precise way that involves the lengths of simple roots and the coefficients of null roots. Tops related to twisted Kac-Moody algebras can be used to construct string compactifications with reduced rank of the gauge group.
TL;DR: In this paper, the Konishi anomaly equations were used to construct the superpotential of the glueball superfields in various N = 1 supersymmetric gauge theories, including chiral and non-chiral SU(N) models, the exceptional gauge group G_2 and models that break supersymmetry dynamically.
Abstract: We use the Konishi anomaly equations to construct the exact effective superpotential of the glueball superfields in various N = 1 supersymmetric gauge theories. We use the superpotentials to study in detail the structure of the spaces of vacua of these theories. We consider chiral and non-chiral SU(N) models, the exceptional gauge group G_2 and models that break supersymmetry dynamically.
TL;DR: The covariant superstring is a twisted version of another formulation with an equivariant cohomology as discussed by the authors, and the vertex operators correspond in one-to-one fashion to the vertex operator in Berkovits' approach based on pure spinors.
Abstract: We replace our earlier condition that physical states of the superstring have non-negative grading by the requirement that they are analytic in a new real commuting constant t which we associate with the central charge of the underlying Kac-Moody superalgebra. The analogy with the twisted N=2 SYM theory suggests that our covariant superstring is a twisted version of another formulation with an equivariant cohomology. We prove that our vertex operators correspond in one-to-one fashion to the vertex operators in Berkovits' approach based on pure spinors. Also the zero-momentum cohomology is equal in both cases. Finally, we apply the methods of equivariant cohomology to the superstring, and obtain the same BRST charge as obtained earlier by relaxing the pure spinor constraints.
TL;DR: In this article, the glueball superpotential for four-dimensional super-ymmetric gauge theories with arbitrary gauge groups and massive matter representations is computed, by perturbatively integrating massive charged fields.
Abstract: We compute glueball superpotentials for four-dimensional, ${\cal N}=1$
supersymmetric gauge theories, with arbitrary gauge groups and massive
matter representations. This is done by perturbatively integrating
out massive charged fields. The Feynman diagram computations
simplify, and are related to the corresponding matrix model. This
leads to a natural notion of ``projection to planar diagrams'' for
arbitrary gauge groups and representations. We discuss a general
ambiguity in the glueball superpotential $W(S)$ for terms, $S^n$,
whose order, $n$ is greater than the dual Coxeter number. This
ambiguity can be resolved for all classical gauge groups $(A,B,C,D)$,
via a natural embedding in an infinite rank supergroup. We use this to
resolve some recently raised puzzles. For exceptional groups, we
compute the superpotential terms for low powers of the glueball field
and propose an all-order completion for some examples including ${\cal
N}=1^*$ for all simply-laced groups. We also comment on
compactification of these theories to lower dimensions.
TL;DR: In this article, the Seiberg-Witten curve for the E-string was reduced to the known E-layer curve of four-and five-dimensional gauge theories by setting some of the Wilson line parameters to special values and also a curve by Donagi and Witten describing the dynamics of a perturbed N = 4 theory.
Abstract: We discuss various properties of the Seiberg–Witten curve for the E-string theory which we have obtained recently in hep-th/0203025. Seiberg–Witten curve for the E-string describes the low-energy dynamics of a six-dimensional (1,0) SUSY theory when compactified on R 4 × T 2 . It has a manifest affineE8 global symmetry with modulus τ and E8 Wilson line parameters {mi}, i = 1,2, . . . ,8 which are associated with the geometry of the rational elliptic surface. When the radii R5, R6 of the torus T 2 degenerate R5, R6 → 0, E-string curve is reduced to the known Seiberg–Witten curves of four- and five-dimensional gauge theories. In this paper we first study the geometry of rational elliptic surface and identify the geometrical significance of the Wilson line parameters. By fine tuning these parameters we also study degenerations of our curve corresponding to various unbroken symmetry groups. We also find a new way of reduction to four-dimensional theories without taking a degenerate limit of T 2 so that the SL(2, Z) symmetry is left intact. By setting some of the Wilson line parameters to special values we obtain the four-dimensional SU(2) Seiberg–Witten theory with 4 flavors and also a curve by Donagi and Witten describing the dynamics of a perturbed N = 4 theory.
TL;DR: In this article, the authors give a general framework for constructing supersymmetric solutions in the presence of non-trivial fluxes of tensor gauge fields, by making a general Ansatz for the metric and defining the Killing spinors in terms of very simple projectors on the spinor fields.
Abstract: We give a general framework for constructing supersymmetric solutions in the presence of non-trivial fluxes of tensor gauge fields. This technique involves making a general Ansatz for the metric and then defining the Killing spinors in terms of very simple projectors on the spinor fields. These projectors and, through them, the spinors, are determined algebraically in terms of the metric Ansatz. The Killing spinor equations then fix the tensor gauge fields algebraically, and, with the Bianchi identities, provide a system of equations for all the metric functions. We illustrate this by constructing an infinite family of massive flows that preserve eight supersymmetries in M-theory. This family constitutes all the radially symmetric Coulomb branch flows of the softly broken, large N scalar-fermion theory on M2-branes. We reduce the problem to the solution of a single, non-linear partial differential equation in two variables. This equation governs the flow of the fermion mass, and the function that solves it then generates the entire M-theory solution algebraically in terms of the function and its first derivatives. While the governing equation is non-linear, it has a very simple perturbation theory from which one can see how the Coulomb branch is encoded.
TL;DR: In this paper, the massless boundary Ramond sector spectra of open strings were extended to include cases with nonzero flat $B$ fields, where D-branes are no longer well modelled precisely by sheaves, but rather they are replaced by twisted sheaves.
Abstract: In this paper we extend previous work on calculating massless boundary Ramond sector spectra of open strings to include cases with nonzero flat $B$ fields. In such cases, D-branes are no longer well-modeled precisely by sheaves, but rather they are replaced by `twisted' sheaves, reflecting the fact that gauge transformations of the $B$ field act as affine translations of the Chan-Paton factors. As in previous work, we find that the massless boundary Ramond sector states are counted by Ext groups -- this time, Ext groups of twisted sheaves. As before, the computation of BRST cohomology relies on physically realizing some spectral sequences. Subtleties that cropped up in previous work also appear here.
TL;DR: In this article, the authors analyzed three aspects of N = 1 heterotic string compactications on elliptically bounded Calabi-Yau three-folds: stability of vector bundles, v e-brane instanton transitions and chiral matter.
Abstract: We analyze three aspects of N = 1 heterotic string compactications on elliptically b ered Calabi-Yau threefolds: stability of vector bundles, v e-brane instanton transitions and chiral matter. First we show that relative Fourier-Mukai transformation preserves absolute stability. This is relevant for vector bundles whose spectral cover is reducible. Then we derive an explicit formula for the number of moduli which occur in (vertical) v e-brane instanton transitions provided a certain vanishing argument applies. Such transitions increase the holonomy of the heterotic vector bundle and cause gauge changing phase transitions. In a M-theory description the transitions are associated with collisions of bulk v e-branes with one of the boundary xed planes. In F-theory they correspond to three-brane instanton transitions. Our derivation relies on an index computation with data localized along the curve which is related to the existence of chiral matter in this class of heterotic vacua. Finally, we show how to compute the number of chiral matter multiplets for this class of vacua allowing to discuss associated Yukawa couplings.
TL;DR: In this paper, an analogue of the group that is used in the canonical quantisation of a system whose configuration space is a manifold Q is isomorphic to G/H where G and H are Lie groups.
Abstract: A new approach is suggested to the problem of quantising causal sets, or topologies, or other such models for space-time (or space). The starting point is the observation that entities of this type can be regarded as objects in a category whose arrows are structure-preserving maps. This motivates investigating the general problem of quantising a system whose `configuration space' (or history-theory analogue) can be regarded as the set of objects Ob(Q) in a category Q. In this first of a series of papers, we study this question in general and develop a scheme based on constructing an analogue of the group that is used in the canonical quantisation of a system whose configuration space is a manifold Q is isomorphic to G/H where G and H are Lie groups. In particular, we choose as the analogue of G the monoid of 'arrow fields' on Q. Physically, this means that an arrow between two objects in the category is viewed as some sort of analogue of momentum. After finding the 'category quantisation monoid', we show how suitable representations can be constructed using a bundle of Hilbert spaces over Ob(Q).
TL;DR: In this article, the authors show how super BF theory in any dimension can be quantized as a spin foam model, generalizing the situation for ordinary BF theory. And they obtain a super-Ponzano-Regge model with gauge group OSp$(1|2) with fermionic degrees of freedom.
Abstract: We show how super BF theory in any dimension can be quantized as a spin foam model, generalizing the situation for ordinary BF theory. This is a first step toward quantizing supergravity theories. We investigate in particular 3-dimensional $(p=1,q=1)$ supergravity which we quantize exactly. We obtain a super-Ponzano-Regge model with gauge group OSp$(1|2)$. A main motivation for our approach is the implementation of fermionic degrees of freedom in spin foam models. Indeed, we propose a description of the fermionic degrees of freedom in our model. Complementing the path integral approach we also discuss aspects of a canonical quantization in the spirit of loop quantum gravity. Finally, we comment on 2+1-dimensional quantum supergravity and the inclusion of a cosmological constant.
TL;DR: In this paper, the authors construct the Calabi-Yau threefold for a large class of M-matrix models, and explain how the geometry encodes the correlators.
Abstract: Large N geometric transitions and the Dijkgraaf-Vafa conjecture suggest a deep relationship between the sum over planar diagrams and Calabi-Yau threefolds. We explore this correspondence in details, explaining how to construct the Calabi-Yau for a large class of Mmatrix models, and how the geometry encodes the correlators. We engineer in particular two-matrix theories with potentials W (X; Y ) that reduce to arbitrary functions in the commutative limit. We apply the method to calculate all correlators htr X p i and htr Y p i in models of the form W (X; Y )= V (X )+ U(Y ) XY and W (X; Y )= V (X )+ YU (Y 2 )+ XY 2 . The solution of the latter example was not known, but when U is a constant we are able to solve the loop equations, nding a precise match with the geometric approach. We also discuss special geometry in multi-matrix models, and we derive an important property, the entanglement of eigenvalues, governing the expansion around classical vacua for which the matrices do not commute.
TL;DR: In this paper, a procedure for quantifying a system whose configuration space is the set of objects in a small category Q is presented, where Q is defined as a category of sets.
Abstract: In [1], a new approach was suggested for finding quantum structures that might, in particular, be used in potential approaches to quantum gravity that involve non-manifold models for space and/or space-time (for example, causal sets). This involved developing a procedure for quantising a system whose configuration space--or history-theory analogue--is the set of objects in a (small) category Q. In the present paper, we show how this theory can be applied to the special case when Q is a category of sets.
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TL;DR: In this paper, the authors studied BPS states in type IIA string compactification on a local Calabi-Yau 3-fold which are related to the BPS state of the E-string.
Abstract: We study BPS states in type IIA string compactification on a local Calabi-Yau 3-fold which are related to the BPS states of the E-string. Using Picard-Lefshetz transformations of the 3-cycles on the mirror manifold we determine automorphisms of the K-theory of the compact divisor of the Calabi-Yau which maps certain D-brane configurations to a bound state of single D4-brane with multiple D0-branes. This map allows us to write down the generating functions for the multiplicity of these BPS states.