Lower-dimensional Theory Research Articles

Duality in the Low Dimensional Field Theory

A strong-weak coupling duality appears in the quantum mechanical many body system with the interacting potential proportional to the pair-wise inverse-squared distance. In the large N limit the field theory formulation possess the soliton solution in non-perturbative “Bogomolny-Prasad-Sommerfeld” limit (BPS). In this limit we establish duality between original and soliton fields and the dual relation among coupling constants.

Duality in the Low Dimensional Field Theory

A strong-weak coupling duality appears in the quantum mechanical many body system with the interacting potential proportional to the pair-wise inverse-squared distance. In the large N limit the field theory formulation possess the soliton solution in non-perturbative “Bogomolny-Prasad-Sommerfeld” limit (BPS). In this limit we establish duality between original and soliton fields and the dual relation among coupling constants.

Extra dimensions as an alternative to Higgs mechanism?

We show that a pure gauge theory in higher dimensions may lead to an effective lower-dimensional theory with massive vector field, broken gauge symmetry and no fundamental Higgs boson. The mechanism we propose employs the localization of a vector field on a lower-dimensional defect. No non-zero expectation values of the vector field components along extra dimensions are required. New possibilities for the solution to the gauge hierarchy problem are discussed.

Knot concordance and torsion

Abstract Algebra, 2nd ed. , Prentice Hall, New Jersey, 1999.[F] R. Fox, A quick trip through knot theory , 1962 Topology of 3–manifolds and relatedtopics (Proc. The Univ. of Georgia Institute, 1961) 120–167 Prentice-Hall, EnglewoodCliffs, N.J.[FM] R. Fox and J. Milnor, Singularities of 2 –spheres in 4 –space and cobordism of knots ,Osaka J. Math. 3 (1966) 257–267.[G] C. Gordon, Problems , in Knot Theory, ed. J.-C. Hausmann, Springer Lecture Notesno. 685, 1977.[Gi] P. Gilmer, Slice knots in S 3 , Quart. J. Math. Oxford Ser. (2) 34 (1983), no. 135,305–322.[J] B. Jiang, A simple proof that the concordance group of algebraically slice knots isinfinitely generated , Proc. Amer. Math. Soc. 83 (1981), 189–192.[K1] R. Kirby, Problems in low dimensional manifold theory , in Algebraic and GeometricTopology (Stanford, 1976), vol 32, part II of Proc. Sympos. Pure Math., 273–312.[K2] R. Kirby, Problems in low dimensional manifold theory , Geometric Topology, AMS/IPStudies in Advanced Mathematics, ed. W. Kazez, 1997.[L1] J. Levine,

On the nonlinear KK reductions on spheres of supergravity theories

We address some issues related to the construction of general Kaluza-Klein (KK) ans\"atze for the compactification of a supergravity (sugra) theory on a sphere $S_m$. We first reproduce various ans\"atze for compactification to 7d from the ansatz for the full nonlinear KK reduction of 11d sugra on $AdS_7\times S_4$. As a side result, we obtain a lagrangian formulation of 7d ${\cal N}=2$ gauged sugra, which so far had only a on-shell formulation, through field equations and constraints. The $AdS_7\times S_4$ ansatz generalizes therefore all previous sphere compactifications to 7d. Then we consider the case when the scalars in the lower dimensional theory are in a coset $Sl(m+1)/SO(m+1)$, and we keep the maximal gauge group $SO(m+1)$. The 11-dimensional sugra truncated on $S_4$ fits precisely the case under consideration, and serves as a model for our construction. We find that the metric ansatz has a universal expression, with the internal space deformed by the scalar fluctuations to a conformally rescaled ellipsoid. We also find the ansatz for the dependence of the antisymmetric tensor on the scalars. We comment on the fermionic ansatz, which will contain a matrix $U$ interpolating between the spinorial $SO(m+1)$ indices of the spherical harmonics and the $R$-symmetry indices of the fermionic fields in the lower dimensional sugra theory. We derive general conditions which the matrix $U$ has to satisfy and we give a formula for the vielbein in terms of $U$. As an application of our methods we obtain the full ansatz for the metric and vielbein for 10d sugra on $AdS_5\times S_5$ (with no restriction on any fields).

Holography based on noncommutative geometry and the AdS/CFT correspondence

Exponential regularization of orthogonal and Anti-de Sitter (AdS) space is presented based on noncommutative geometry. We show that an adequately adopted noncommutative deformation of geometry makes the holography of higher dimensional quantum theory of gravity and low dimensional theory possible. Detail counting for observable degrees of freedom of quantum system of gravity in the bulk of noncommutative space SO_q(3) and the classical limit of its boundary surface S^2 is discussed. Taking noncommutivity effect into account, we get the desired form of entropy for our world, which is consistent with the physical phenomena associated with gravitational collapse. Conformally invariant symmetry is obtained for the equivalent theory of the qunatum gravity living on the classical limit of boundary of the noncommutative AdS space. This is the basis of the AdS/CFT correspondence in string theory.

Domain-wall supergravities from sphere reduction

Kaluza-Klein sphere reductions of supergravities that admit Ads × Sphere vacuum solutions are believed to be consistent. The examples include the S 4 and S 7 reductions of eleven-dimensional supergravity, and the S 5 reduction of ten-dimensional type IIB supergravity . In this paper we provide evidence that sphere reductions of supergravities that admit instead Domain-wallxSphere vacuum solutions are also consistent, where the background can be viewed as the near-horizon structure of a dilatonic p -brane of the theory. The resulting lower-dimensional theory is a gauged supergravity that admits a domain wall, rather than AdS, as a vacuum solution. We illustrate this consistency by taking the singular limits of certain modulus parameters, for which the original S n compactifying spheres ( n = 4, 5 or 7) becomes S p × R q , with p = n − q < n . The consistency of the S 4 , S 7 reductions then implies the consistency of the S p reductions of the lower-dimensional supergravities. In particular, we obtain explicit non-linear ansätze for the S 3 reduction of type IIA and heterotic supergravities, restricting to the U (1) 2 subgroup of the SO (4) gauge group of S 3 . We also study the black-hole solutions in the lower-dimensional gauged supergravities with domain-wall backgrounds. We find new domain-wall black holes which are not the singular-modulus limits of the AdS black holes of the original theories, and we obtain their Killing spinors.

Brane baldness versus superselection sectors

The search for intersecting brane solutions in supergravity is a large and profitable industry. Recently, attention has focused on finding localized forms of known `delocalized' solutions. However, in some cases, a localized version of the delocalized solution simply does not exist. Instead, localized separated branes necessarily delocalize as the separation is removed. This phenomenon is related to black hole no-hair theorems, i.e. `baldness.' We continue the discussion of this effect and describe how it can be understood, in the case of Dirichlet branes, in terms of the corresponding intersection field theory. When it occurs, it is associated with the quantum mixing of phases and lack of superselection sectors in low dimensional field theories. We find surprisingly wide agreement between the field theory and supergravity both with respect to which examples delocalize and with respect to the rate at which this occurs.

Integrability in supersymmetric gauge theories. I

This is the first part of a review devoted to the exact, nonperturbative solutions of supersymmetric gauge theories and their formulation in terms of integrable systems. The general phenomenon of integrability as it appears in the formulation of effective actions for various models of topological, low-dimensional string theories and almost realistic supersymmetric gauge field theories is discussed. First the basic features of the string-theory path integral are discussed in order to understand better the nonperturbative properties of the theory. Then a formulation of the exact effective actions based on systems of nonlinear differential equations is proposed. It is demonstrated that the resulting nonlinear differential equations belong to the class of integrable models of the Kadomtsev–Petviashvili and Toda type. Particular models of this class are discussed, with special focus on the integrable systems appearing in the context of multidimensional supersymmetric gauge theories. Their Lax representations and spectral curves are studied in detail, and a classification of the exact solutions of N=2 supersymmetric gauge theories along these lines is proposed.

An Sl(2, Z) multiplet of nine-dimensional type II supergravity theories

We show that only by performing generalized dimensional reductions all possible brane configurations are taken into account and one gets the complete lower-dimensional theory. We apply this idea to the reduction of type IIB supergravity in an SL(2, Z)-covariant way and establish T-duality for the type II superstring effective action in the context of generalized dimensional reduction giving the corresponding generalized Buscher's T-duality rules. The full (generalized) dimensional reduction involves all the S-duals of D7-branes: Q7-branes and a sort of composite 7-branes. The three species constitute an SL(2, Z) triplet. Their presence induces the appearance of the triplet of masses of the 9-dimensional theory. The T-duals, including a “KK-8A-brane”, which must have a compact transverse dimension have to be considered in the type IIA side. Compactification of 11-dimensional KK-9M-branes (a.k.a. M9-branes) on the compact transverse dimension give D8-branes while compactification on a world-volume dimension gives KK-8A-branes. The presence of these KK-monopole-type objects breaks translation invariance and two of them given rise to an SL(2, Z)-covariant massive 11-dimensional supergravity whose reduction gives the massive 9-dimensional type II theories.

Superalgebraic Truncations from D=10, N=2 Chiral Supergravity

We study 10-dimensional N=2 maximal chiral supergravity in the context of Lie superalgebra SU(8/1). The possible successive superalgebraic truncations from 10-dimensional N=2 chiral theory to the lower-dimensional supergravity theories are systematically realized as sub-superalgebraic chains of SU(8/1) by using the Kac–Dynkin weight techniques.

Z-Symmetry and Polyakov Loops in Hot Abelian Gauge Theories

We discuss the Z symmetry of finite temperature QED, which is related to the confinement properties of the theory. We show the correlators of Polyakov loop operators with incommensurate charges can be used as order parameters for this symmetry. The screening or confining properties of lower dimensional abelian gauge theories are analyzed. In the Schwinger model, the Z-symmetry is broken and the system is in a screening phase; with a non-vanishing fermion mass the symmetry is recovered and the system confines. In parity invariant 2+1-dimensional QED, there is a phase transition between phases with unbroken and broken realizations of this symmetry. This confinement–deconfinement transition is of the Berezinskii–Kosterlitz–Thouless (BKT) type. When there is a topological mass the model exhibits a screening phase. However, if the topological mass is much smaller than the other dimensional parameters there is a vestige of the BKT transition separating regions with screening and confining behavior.

Domain walls in massive supergravities

We show how toroidally compactified eleven-dimensional supergravity can be consistently truncated to yield a variety of maximally supersymmetric massive supergravities in space-time dimensions D ⩽ 8. The mass terms arise as a consequence of making a more general ansatz than that in usual Kaluza-Klein dimensional reduction, in which one or more axions are given an additional linear dependence on one of the compactification coordinates. The lower-dimensional theories are nevertheless consistent truncations of eleven-dimensional supergravity. Owing to the fact that the generalised reduction commutes neither with U-duality nor with ordinary dimensional reduction, many different massive theories can result. The simplest examples arise when just a single axion has the additional linear coordinate dependence. We find five inequivalent such theories in D = 7, and 71 inequivalent ones in D = 4. The massive theories admit no maximally symmetric vacuum solution, but they do admit (D - 2)-brane solutions, i.e. domain walls, which preserve half the supersymmetry. We present examples of these solutions, and their oxidations to D = 11. Some of the latter are new solutions of D = 11 supergravity.

D-dimensional gravity from (D + 1) dimensions

We generalize Wesson's procedure, whereby vacuum (4 + 1)-dimensional field equations give rise to (3 + 1)-dimensional equations with sources, to arbitrary dimensions. We then employ this generalization to relate the usual (3 + 1)-dimensional vacuum field equations to (2 + 1)-dimensional field equations with sources and derive the analogues of the classes of solutions obtained by Ponce de Leon. This way of viewing lower dimensional gravity theories can be of importance in establishing a relationship between such theories and the usual four-dimensional general relativity, as well as giving a way of producing exact solutions in (2 + 1) dimensions that are naturally related to the vacuum (3 + 1)-dimensional solutions. An outcome of this correspondence, regarding the nature of lower dimensional gravity, is that the intuitions obtained in (3 + 1) dimensions may not be automatically transportable to lower dimensions. We also extend a number of physically motivated solutions studied by Wesson and Ponce de Leon to (D + 1) dimensions and employ the equivalence between the (D + 1) Kaluza - Klein theories with empty D-dimensional Brans - Dicke theories (with ) to throw some light on the solutions derived by these authors.

Compactification of D = 11 supergravity on spaces of exceptional holonomy

We investigate the compactification of D = 11 supergravity to D = 5,4,3, on compact manifolds of holonomy SU(3) (Calabi-Yau), G 2, and Spin(7), respectively, making use of examples of the latter two cases found recently by Joyce. In each case the lower dimensional theory is a Maxwell/Einstein supergravity theory. We find evidence for an equivalence, in certain cases, with heterotic string compactifications from D = 10 to D = 5,4,3, on compact manifolds of holonomy SU(2) ( K 3 × S 1), SU(3), and G 2, respectively. Calabi-Yau manifolds with Hodge numbers h 1,1 = h 1,2 = 19 play a significant role in the proposed equivalences.

Lower-dimensional theories of gravity and gravitational waves

We demonstrate that there are no gravitational waves in vacuum in 1+1 gravity, with or without a cosmological constant, and a wave-like field in regions of non-vanishing Tμv is quite unlike a conventional gravitational wave in 3+1 gravity. This, and other dissimilarities, lend a word of caution to the extraction of information from lower-dimensional theories for the understanding of 3+1 gravity.

Classical Electromagnetic Radiation in Multidimensional Space-Times

The dependence of various features of electromagnetic radiation on the space-time dimension is investigated. The dimension is d ≥ 4, the lower dimensional theory is not considered. Some features known in four dimensions are carried over (possibly with slight modifications) to extra dimensions, whereas other ones exist only for d = 4. The first group contains, e.g., the notion of null characteristic surfaces and null electromagnetic fields, the geometrical optics approximation ( nearly plane waves), and Fermat′s principle. It turns out that the null rays of the null field are geodesic lines for any dimension (generalization of Mariot′s theorem). The independent functional (algebraic) invariants for a generic electromagnetic field are explicitly constructed, there are [ d/2] of such invariants for d dimensions. A generalized Newman-Penrose formalism (complex Ricci rotation coefficients) is developed in d = 6 and it is used to show that four dimensions are exceptional: only for d = 4 are the rays of a null field shear-free (Robinson′s theorem); in extra dimensions Maxwell′s equations are less restrictive. This formalism is also applied for proving a generalization of Sachs′ optical theorem: for any d, the shadow cast by a small object in a light beam is rotated, expanded, and sheared, but it is not Lorentz contracted.

TWO-DIMENSIONAL QUANTUM GRAVITY, MATRIX MODELS AND STRING THEORY

We review some results of the recent progress in understanding two-dimensional quantum gravity and low-dimensional string theories based on the lattice approach. The possibility to solve the lattice models exactly comes from their equivalence to large N matrix models. We describe various matrix models and their continuum limits, and discuss in some detail the phase structure of Hermitian one-matrix models. For the one-dimensional matrix model we discuss its field theoretic formulation through a collective field method and summarize some perturbative results. We compare the results obtained from matrix models to the results in the continuum approach to string theory.

Differential renormalization of lower dimensional gauge theories

We apply the method of differential renormalization to two and three dimensional abelian gauge theories. The method is especially well suited for these theories as the problems of defining the antisymmetric tensor are avoided and the calculus involved is impressively simple. The topological and dynamical photon masses are obtained.

Gravitation as gauge theory of the diffeomorphism group

The ( m+ n)-dimensional Einstein theory of gravitation is identified with an m-dimensional generally invariant gauge theory of DiffN, where N is an n-dimensional manifold. This means that the four-dimensional Einstein gravity can be identified as a lower dimensional gauge theory of an infinite dimensional group of diffeomorphism. We discuss the physical implications of the results.