Showing papers in "International Journal of Geometric Methods in Modern Physics in 2004"
TL;DR: In this article, the authors identify a deformation of the N = 2 supersymmetric sigma model on a Calabi-Yau manifold X which has the same effect on B-branes as a non-commutative deformation on X and show that for hyperkahler X such deformations allow one to interpolate continuously between the A-model and the B-model.
Abstract: We identify a deformation of the N=2 supersymmetric sigma model on a Calabi–Yau manifold X which has the same effect on B-branes as a noncommutative deformation of X. We show that for hyperkahler X such deformations allow one to interpolate continuously between the A-model and the B-model. For generic values of the noncommutativity and the B-field, properties of the topologically twisted sigma-models can be described in terms of generalized complex structures introduced by N. Hitchin. For example, we show that the path integral for the deformed sigma-model is localized on generalized holomorphic maps, whereas for the A-model and the B-model it is localized on holomorphic and constant maps, respectively. The geometry of topological D-branes is also best described using generalized complex structures. We also derive a constraint on the Chern character of topological D-branes, which includes A-branes and B-branes as special cases.
198 citations
TL;DR: A survey of geometric techniques in quantum models can be found in this article, where the main peculiarities of geometric quantization techniques are discussed. But the main point why geometry plays a prominent role in classical field theory lies in the fact that it enables one to deal with invariantly defined objects.
Abstract: In this scientific preface to the first issue of International Journal of Geometric Methods in Modern Physics1, we briefly survey some peculiarities of geometric techniques in quantummodels. Contemporary quantum theory meets an explosion of different types of quantization. Some of them (geometric quantization, deformation quantization, noncommutative geometry, topological field theory etc.) speak the language of geometry, algebraic and differential topology. We do not pretend for any comprehensive analysis of these quantization techniques, but aims to formulate and illustrate their main peculiarities. As in any survey, a selection of topics has to be done, and we apologize in advance if some relevant works are omitted. Geometry of classical mechanics and field theory is mainly differential geometry of finitedimensional smooth manifolds, fiber bundles and Lie groups. The key point why geometry plays a prominent role in classical field theory lies in the fact that it enables one to deal with invariantly defined objects. Gauge theory has shown clearly that this is a basic physical principle. At first, a pseudo-Riemannian metric has been identified to a gravitational field in the framework of Einstein’s General Relativity. In 60-70th, one has observed that connections on a principal bundle provide the mathematical model of classical gauge potentials [1-3]. Furthermore, since the characteristic classes of principal bundles are expressed in terms of the gauge strengths, one can also describe the topological phenomena in classical gauge models [4]. Spontaneous symmetry breaking and Higgs fields have been explained in terms of reduced G-structures [5]. A gravitational field seen as a pseudo-Riemannian metric exemplifies such a Higgs field [6]. In a general setting, differential geometry of smooth fiber bundles gives the adequate mathematical formulation of classical field theory, where fields are represented by sections of fiber bundles and their dynamics is phrased in terms of jet manifolds [7]. Autonomous classical mechanics speaks the geometric language of symplectic and Poisson Web: http://www.worldscinet.com/ijgmmp/ijgmmp.shtml
91 citations
TL;DR: In this article, the multisymplectic description of classical field theories is revisited, including its relation with the presymplectic formalism on the space of Cauchy data.
Abstract: The multisymplectic description of Classical Field Theories is revisited, including its relation with the presymplectic formalism on the space of Cauchy data. Both descriptions allow us to give a complete scheme of classification of infinitesimal symmetries, and to obtain the corresponding conservation laws.
58 citations
TL;DR: In this paper, the coefficients of Tian-Yau-Zelditch asymptotic expansion of a compact Kahler manifold were computed using quantization techniques alternative to Lu's computations.
Abstract: Let M be a compact Kahler manifold endowed with a real analytic and polarized Kahler metric g and let Tmω(x) be the corresponding Kempf's distortion function. In this paper we compute the coefficients of Tian–Yau–Zelditch asymptotic expansion of Tmω(x) using quantization techniques alternative to Lu's computations in [10].
52 citations
TL;DR: In this paper, the authors extend Vorobiev's theory of coupling Poisson structures from fiber bundles to foliated manifolds and give simpler proofs of the existence and equivalence theorems of coupling structures on duals of kernels of transitive Lie algebroids over symplectic manifolds.
Abstract: Let M be a differentiable manifold endowed with a foliation ℱ. A Poisson structure P on M is ℱ-coupling if ♯P(ann(Tℱ)) is a normal bundle of the foliation. This notion extends Sternberg's coupling symplectic form of a particle in a Yang–Mills field [11]. In the present paper we extend Vorobiev's theory of coupling Poisson structures [16] from fiber bundles to foliated manifolds and give simpler proofs of Vorobiev's existence and equivalence theorems of coupling Poisson structures on duals of kernels of transitive Lie algebroids over symplectic manifolds. We then discuss the extension of the coupling condition to Jacobi structures on foliated manifolds.
27 citations
TL;DR: In this paper, the authors give a modern account of the construction and structure of the space of generalized connections, an extension of the spaces of connections that plays a central role in loop quantum gravity.
Abstract: We give a modern account of the construction and structure of the space of generalized connections, an extension of the space of connections that plays a central role in loop quantum gravity.
25 citations
TL;DR: In this article, the functional integral quantization of non-Abelian gauge theories is affected by the Gribov problem at non-perturbative level, where the requirement of preserving the supplementary conditions under gauge transformations leads to a nonlinear differential equation, and the various solutions of such a non-linear equation represent different gauge configurations known as gribov copies.
Abstract: The functional-integral quantization of non-Abelian gauge theories is affected by the Gribov problem at non-perturbative level: the requirement of preserving the supplementary conditions under gauge transformations leads to a nonlinear differential equation, and the various solutions of such a nonlinear equation represent different gauge configurations known as Gribov copies. Their occurrence (lack of global cross-sections from the point of view of differential geometry) is called Gribov ambiguity, and is here presented within the framework of a global approach to quantum field theory. We first give a simple (standard) example for the SU(2) group and spherically symmetric potentials, then we discuss this phenomenon in general relativity, and recent developments, including lattice calculations.
24 citations
TL;DR: In this paper, the global aspects of Abelian and center projection of a SU(2) gauge theory on an arbitrary manifold are naturally described in terms of smooth Deligne cohomology.
Abstract: We show that the global aspects of Abelian and center projection of a SU(2) gauge theory on an arbitrary manifold are naturally described in terms of smooth Deligne cohomology. This is achieved through the introduction of a novel type of differential topological structure, called Cho structure. Half integral monopole charges appear naturally in this framework.
22 citations
TL;DR: In this paper, the geometrical origin and interpretation for the nilpotent (anti-)BRST charges, (anti)-co-BRST charge and a non-nilpotent bosonic charge are provided.
Abstract: In the framework of augmented superfield approach, we provide the geometrical origin and interpretation for the nilpotent (anti-)BRST charges, (anti-)co-BRST charges and a non-nilpotent bosonic charge. Together, these local and conserved charges turn out to be responsible for a clear and cogent definition of the Hodge decomposition theorem in the quantum Hilbert space of states. The above charges owe their origin to the de Rham cohomological operators of differential geometry which are found to be at the heart of some of the key concepts associated with the interacting gauge theories. For our present review, we choose the two (1+1)-dimensional (2D) quantum electrodynamics (QED) as a prototype field theoretical model to derive all the nilpotent symmetries for all the fields present in this interacting gauge theory in the framework of augmented superfield formulation and show that this theory is a unique example of an interacting gauge theory which provides a tractable field theoretical model for the Hodge theory.
20 citations
TL;DR: In this article, the authors constructed a de Rham cohomology class on the space Imb(S1,ℝn) of imbeddings of the circle into ℝ n by means of Feynman diagrams.
Abstract: We construct nontrivial cohomology classes of the space Imb(S1,ℝn) of imbeddings of the circle into ℝn by means of Feynman diagrams. More precisely, starting from a suitable linear combination of nontrivalent diagrams, we construct, for every even number n≥4, a de Rham cohomology class on Imb(S1,ℝn). We prove nontriviality of these classes by evaluation on the dual cycles.
19 citations
TL;DR: In this paper, the authors characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor.
Abstract: We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex space forms if the dimension is at least 8. We also study when the Jacobi operator associated to the Weyl conformal curvature tensor of a Riemannian manifold has constant eigenvalues on the bundle of unit tangent vectors and classify such manifolds which are not conformally flat in dimensions congruent to 2 mod 4.
TL;DR: In this article, the concept of a gauge natural theory is discussed and its role in implementing the very basic principles of any reasonable fundamental physical interaction is discussed, and conservation laws are investigated by means of Nother theorem together with the theory of superpotentials for conserved currents.
Abstract: We review the notion of a gauge natural theory and discuss in particular its role in implementing the very basic principles of any reasonable fundamental physical interaction. Conservation laws are investigated in this general framework by means of Nother theorem together with the theory of superpotentials for conserved currents. Tetrad gravity and Wess–Zumino model interacting with a gravitational field are considered as pedagogical applications of the gauge natural framework.
TL;DR: In this article, the explicit form of the evolution operator of the Tavis-Cummings model with three and four atoms is given, which is an important progress in quantum optics or mathematical physics.
Abstract: In this letter the explicit form of the evolution operator of the Tavis–Cummings model with three and four atoms is given. This is an important progress in quantum optics or mathematical physics.
TL;DR: In this paper, it is shown that the introduction of suitable "symmetryadapted" variables for the study of differential equations can be efficient and useful even if the problem does not admit symmetries.
Abstract: It is shown that the introduction of suitable "symmetry-adapted" variables for the study of differential equations can be efficient and useful even if the problem does not admit symmetries. This method not only provides new solutions but also leads to the introduction of weaker notions of symmetry, and allows a natural classification of the possible types of symmetry, each of which is characterized by a specific form of the equation when written in the appropriate variables. Some simple examples are briefly proposed.
TL;DR: In this article, the Nash embedding theorem is used to construct generators for the space of algebraic covariant derivative curvature tensors, which are then used as generators for algebraic derivative curvatures.
Abstract: We use the Nash embedding theorem to construct generators for the space of algebraic covariant derivative curvature tensors.
TL;DR: In this article, a fractal supersymmetric quantum mechanical (SUSY-QM) model implementing the Hilbert-Polya proposal was proposed to prove the Riemann's hypothesis.
Abstract: The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn=1/2+iλn. Earlier work on the RH based on supersymmetric QM, whose potential was related to the Gauss–Jacobi theta series, allows us to provide the proper framework to construct the well-defined algorithm to compute the density of zeros in the critical line, which would complement the existing formulas in the literature for the density of zeros in the critical strip. Geometric probability theory furnishes the answer to the difficult question whether the probability that the RH is true is indeed equal to unity or not. To test the validity of this geometric probabilistic framework to compute the probability if the RH is true, we apply it directly to the the hyperbolic sine function sinh(s) case which obeys a trivial analog of the RH (the HSRH). Its zeros are equally spaced in the imaginary axis sn=0+inπ. The geometric probability to find a zero (and an infinity of zeros) in the imaginary axis is exactly unity. We proceed with a fractal supersymmetric quantum mechanical (SUSY-QM) model implementing the Hilbert–Polya proposal to prove the RH by postulating a Hermitian operator that reproduces all the λn for its spectrum. Quantum inverse scattering methods related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) are applied to the fractal analog of the Comtet–Bandrauk–Campbell (CBC) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter β is one-half the fractal dimension (D=1.5) of the Weierstrass function. An ordinary SUSY-QM oscillator is also constructed whose eigenvalues are of the form λn=nπ and which coincide with the imaginary parts of the zeros of the function sinh(s). Finally, we discuss the relationship to the theory of 1/f noise.
TL;DR: In this article, a method of regularization in infinite dimensional calculus, based on spectral zeta function and zeta regularization, is proposed, and mathematical justification of appearance of Ray-Singer determinant in Gaussian Path integral, regularized volume form of a Hilbert space with the determinant bundle, eigenvalue problems of regularized Laplacian, are investigated.
Abstract: A method of regularization in infinite dimensional calculus, based on spectral zeta function and zeta regularization is proposed. As applications, a mathematical justification of appearance of Ray–Singer determinant in Gaussian Path integral, regularized volume form of the sphere of a Hilbert space with the determinant bundle, eigenvalue problems of regularized Laplacian, are investigated. Geometric counterparts of regularization procedure are also discussed applying arguments from noncommutative geometry.
TL;DR: In this article, the basic BRST cohomological properties of a free, massless tensor field with the mixed symmetry of the Riemann tensor are studied in detail.
Abstract: The basic BRST cohomological properties of a free, massless tensor field with the mixed symmetry of the Riemann tensor are studied in detail. It is shown that any non-trivial co-cycle from the local BRST cohomology group can be taken to stop at antighost number three, its last component belonging to the cohomology of the exterior longitudinal derivative and containing non-trivial elements from the (invariant) characteristic cohomology.
TL;DR: In this paper, it was shown that the de Rham cohomology of an algebra with the differentiable coaction of a cosemisimple Hopf algebra with trivial 0-th cohomologies group, reduces to the (co)invariant forms of Hopf algebras.
Abstract: Various aspects of the de Rham cohomology of Hopf algebras are discussed. In particular, it is shown that the de Rham cohomology of an algebra with the differentiable coaction of a cosemisimple Hopf algebra with trivial 0-th cohomology group, reduces to the de Rham cohomology of (co)invariant forms. Spectral sequences are discussed and the van Est spectral sequence for Hopf algebras is introduced. A definition of Hopf–Lie algebra cohomology is also given.
TL;DR: In this paper, the authors give an expository account of the physical relevance of K-theory in string theory and show how the inclusion of the B-field modifies the general structure leading to the twisted K-groups.
Abstract: In this review we show how K-theory classifies RR-charges in type II string theory and how the inclusion of the B-field modifies the general structure leading to the twisted K-groups. Our main purpose is to give an expository account of the physical relevance of K-theory. To do that, we consider different points of view: processes of tachyon condensation, cancellation of global anomalies and gauge fixings. As a field to test the proposals of K-theory, we concentrate on the study of the D6-brane, now seen as a non-abelian monopole.
TL;DR: In this article, Batalin-Vilkoviski (BV) quantization of polysymplectic Hamiltonian field theory is compared in the case of almost-regular quadratic Lagrangians.
Abstract: Covariant (polysymplectic) Hamiltonian field theory is the Hamiltonian counterpart of classical Lagrangian field theory. They are quasi-equivalent in the case of almost-regular Lagrangians. This work addresses Batalin–Vilkoviski (BV) quantization of polysymplectic Hamiltonian field theory. We compare BV quantizations of associated Lagrangian and polysymplectic Hamiltonian field systems in the case of almost-regular quadratic Lagrangians.
TL;DR: In this article, three kinds of systems of differential equations which are relevant in physics, control theory and other applications in engineering and applied mathematics are considered: Hamilton equations, singular differential equations, and partial differential equations in field theories.
Abstract: In this review paper, we consider three kinds of systems of differential equations, which are relevant in physics, control theory and other applications in engineering and applied mathematics; namely: Hamilton equations, singular differential equations, and partial differential equations in field theories. The geometric structures underlying these systems are presented and commented on. The main results concerning these structures are stated and discussed, as well as their influence on the study of the differential equations with which they are related. In addition, research to be developed in these areas is also commented on.
TL;DR: In this article, two different ways of constructing surface holonomies, one by using a pair of one-and two-form connections, and another by using an one-form connection and a two-and-one-connections, were considered.
Abstract: Just as point objects are parallel transported along curves, giving holonomies, string-like objects are parallel transported along surfaces, giving surface holonomies. The composition of these surfaces correspond to products in a category theoretic generalization of the gauge group, called a 2-group. I consider two different ways of constructing surface holonomies, one by using a pair of one- and two-form connections, and another by using a pair of one-form connections. Both procedures result in the structure of a 2-group.
TL;DR: In this paper, action-angle coordinates are shown to exist around an instantly compact invariant submanifold of a time-dependent completely integrable Hamiltonian system, and a comparison is made with other possible approaches.
Abstract: Action-angle coordinates are shown to exist around an instantly compact invariant submanifold of a time-dependent completely integrable Hamiltonian system. Partially integrable Hamiltonian systems are also considered in the noncompact case; a comparison is made with other possible approaches. Results on symplectically complete foliations contained in the Appendix A can be used to give alternative proofs of some propositions.
TL;DR: In this article, a formulation of NT=1, D=8 Euclidean super Yang-Mills theory with generalized self-duality and reduced Spin(7)-invariance is given which avoids the peculiar extra constraints introduced by Nishino and Rajpoot.
Abstract: A formulation of NT=1, D=8 Euclidean super Yang–Mills theory with generalized self-duality and reduced Spin(7)-invariance is given which avoids the peculiar extra constraints introduced by Nishino and Rajpoot. Its reduction to seven dimensions leads to the G2-invariant NT=2, D=7 super Yang–Mills theory which may be regarded as a higher-dimensional analogue of the NT=2, D=3 super-BF theory. When further reducing that G2-invariant theory to three dimensions one gets the NT=2 super-BF theory coupled to a spinorial hypermultiplet.
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TL;DR: In this paper, on-shell symmetries for Lagrangian systems are defined and characterized, and the behavior with respect to solution dragging is investigated in order to discuss relations with the theory of internal symmetry of a PDE.
Abstract: We define on-shell symmetries and characterize them for Lagrangian systems. The terms appearing in the variation of the Poincare–Cartan form, which vanish because of field equations, are found to be strongly constrained if the space of solutions has to be preserved. The behavior with respect to solution dragging is also investigated in order to discuss relations with the theory of internal symmetries of a PDE.
TL;DR: A new aproach of the noncommutative geometry of the matrix algebra by defining two different differential calculi and introducing linear connections on using the framewok of ρ-algebras.
Abstract: In this paper we present a new aproach of the noncommutative geometry of the matrix algebra . We define two different differential calculi, and we introduce linear connections on using the framewok of ρ-algebras.
TL;DR: In this article, the determining equations for deformation terms are shown to have an elegant formulation using Lie derivatives in the jet space associated with the gauge field variables, and the obstructions (integrability conditions) that must be satisfied by lowest-order deformations terms for existence of a deformation to higher orders are explicitly identified.
Abstract: A basic problem of classical field theory, which has attracted growing attention over the past decade, is to find and classify all nonlinear deformations of linear abelian gauge theories. The physical interest in studying deformations is to address uniqueness of known nonlinear interactions of gauge fields and to look systematically for theoretical possibilities for new interactions. Mathematically, the study of deformations aims to understand the rigidity of the nonlinear structure of gauge field theories and to uncover new types of nonlinear geometrical structures. The first part of this paper summarizes and significantly elaborates a field-theoretic deformation method developed in earlier work. Some key contributions presented here are, firstly, that the determining equations for deformation terms are shown to have an elegant formulation using Lie derivatives in the jet space associated with the gauge field variables. Secondly, the obstructions (integrability conditions) that must be satisfied by lowest-order deformations terms for existence of a deformation to higher orders are explicitly identified. Most importantly, a universal geometrical structure common to a large class of nonlinear gauge theory examples is uncovered. This structure is derived geometrically from the deformed gauge symmetry and is characterized by a covariant derivative operator plus a nonlinear field strength, related through the curvature of the covariant derivative. The scope of these results encompasses Yang–Mills theory, Freedman–Townsend theory, and Einstein gravity theory, in addition to their many interesting types of novel generalizations that have been found in the past several years. The second part of the paper presents a new geometrical type of Yang–Mills generalization in three dimensions motivated from considering torsion in the context of nonlinear sigma models with Lie group targets (chiral theories). The generalization is derived by a deformation analysis of linear abelian Yang–Mills Chern–Simons gauge theory. Torsion is introduced geometrically through a duality with chiral models obtained from the chiral field form of self-dual (2+2) dimensional Yang–Mills theory under reduction to (2+1) dimensions. Field-theoretic and geometric features of the resulting nonlinear gauge theories with torsion are discussed.
TL;DR: In this article, the authors constructed a solution with Lp-estimates, 1≤p≤∞, to the strongly q-convex domain of Kahler manifold.
Abstract: The purpose of this paper is to construct a solution with Lp-estimates, 1≤p≤∞, to the equation on strongly q-convex domain of Kahler manifold. This is done for forms of type (n,s), s≥ max(q,k), with values in a holomorphic vector bundle which is Nakano semi-positive of type k and for forms of type (0,s), q≤s≤n-k, with values in a holomorphic vector bundle which is Nakano semi-negative of type k.
TL;DR: In this article, basic results on gauge invariance of differential forms on the bundle of connections of an arbitrary principal U(1)-bundle and its associated bundles are reviewed in terms of the underlying geometry to such bundles, within the framework of classical electromagnetism.
Abstract: Basic results on gauge invariance of differential forms on the bundle of connections of an arbitrary principal U(1)-bundle and its associated bundles, are reviewed in terms of the underlying geometry to such bundles, within the framework of classical electromagnetism.