0-extended supergravity and Chern-Simons theories

We give generalizations of extended Poincaré supergravity with arbitrarily many supersymmetries in the absence of central charges in three dimensions by gauging its intrinsic global SO(N) symmetry. We call these ₀ (Aleph-null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to ₀ supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond N = 4. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as N = 1 self-dual supersymmetric Yang-Mills theory or N = 1 supergravity theory that can generate ₀ globally and locally supersymmetric theories in three dimensions. As an interesting application, we present ₀ supergravity Liouville theory in two dimensions after appropriate dimensional reduction from three dimensions.     

ℵ0-extended supergravity and Chern-Simons theories

We give generalizations of extended Poincaré supergravity with arbitrarily many supersymmetries in the absence of central charges in three dimensions by gauging its intrinsic global SO(N) symmetry. We call these ℵ₀ (Aleph-null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to ℵ₀ supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond N = 4. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as N = 1 self-dual supersymmetric Yang-Mills theory or N = 1 supergravity theory that can generate ℵ₀ globally and locally supersymmetric theories in three dimensions. As an interesting application, we present ℵ₀ supergravity Liouville theory in two dimensions after appropriate dimensional reduction from three dimensions.     

0-extended supergravity and Chern-Simons theories

We give generalizations of extended Poincaré supergravity with arbitrarily many supersymmetries in the absence of central charges in three dimensions by gauging its intrinsic global SO(N) symmetry. We call these ₀ (Aleph-null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to ₀ supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond N = 4. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as N = 1 self-dual supersymmetric Yang-Mills theory or N = 1 supergravity theory that can generate ₀ globally and locally supersymmetric theories in three dimensions. As an interesting application, we present ₀ supergravity Liouville theory in two dimensions after appropriate dimensional reduction from three dimensions.     

On the low energy spectra of the non-supersymmetric heterotic string theories

The ten dimensional string theories as well as eleven dimensional supergravity are conjectured to arise as limits of a more basic theory, traditionally dubbed M-theory. This notion is confined to the ten dimensional supersymmetric theories. String theory, however, also contains ten dimensional non-supersymmetric theories that have not been incorporated into this picture. In this note we explore the possibility of generating the low energy spectra of various non-supersymmetric heterotic string vacua from the Horava–Witten model. We argue that this can be achieved by imposing on the Horava–Witten model an invariance with respect to some extra operators which identify the orbifold fixed planes in a non-trivial way, and we demonstrate it for the E₈ and SO(16)×SO(16) heterotic string vacua in ten dimensions.     

Black hole attractors and the entropy function in four‐ and five‐dimensional N = 2 supergravity

In this overview of selected aspects of the black hole attractor mechanism, after introducing the necessary foundations, we examine the relationship between two ways to describe the attractor phenomenon in four‐dimensional N = 2 supergravity: the entropy function and the black hole potential. We also exemplify their practical application to finding solutions to the attractor equations for a conifold prepotential. Next we describe an extension of the original definition of the entropy function to a class of rotating black holes in five‐dimensional N = 2 supergravity based on cubic polynomials, exploiting a connection between four‐ and five‐dimensional black holes. This link allows further the derivation of five‐dimensional first‐order differential flow equations governing the profile of the fields from infinity to the event horizon and construction of non‐supersymmetric interpolating solutions in four dimensions by dimensional reduction. Finally, since four‐dimensional extremal black holes in N = 2 supergravity can be viewed as certain two‐dimensional string compactifications with fluxes, we discuss implications of the conifold example in the context of the entropic principle, which postulates as a probability measure on the space of these string compactifications the exponentiated entropy of the corresponding black holes.     

Dynamical inflation and unification scale on quantum moduli spaces

We show that simple strongly coupled supersymmetric gauge theories with quantum moduli spaces can naturally lead to hybrid inflation. These theories contain no input dimensionful or small parameters. The effective superpotential is linear in the inflaton field; this ensures that supergravity corrections do not spoil the slow roll conditions for inflation. We construct a simple theory in which the classical moduli space exhibits neither GUT-symmetry-breaking nor inflation whereas its quantum modification exhibits both. As a result, the dynamical origin and scales of inflation and grand unification coincide.     

New Gaugings and Non‐Geometry

We discuss the possible realisation in string/M theory of the recently discovered family of four‐dimensional maximal gauged supergravities, and of an analogous family of seven‐dimensional half‐maximal gauged supergravities. We first prove a no‐go theorem that neither class of gaugings can be realised via a compactification that is locally described by ten‐ or eleven‐dimensional supergravity. In the language of Double Field Theory and its M theory analogue, this implies that the section condition must be violated. Introducing the minimal number of additional coordinates possible, we then show that the standard S ³ and S ⁷ compactifications of ten‐ and eleven‐dimensional supergravity admit a new class of section‐violating generalised frames with a generalised Lie derivative algebra that reproduces the embedding tensor of the and gaugings respectively. The physical meaning, if any, of these constructions is unclear. They highlight a number of the issues that arise when attempting to apply the formalism of Double Field Theory to non‐toroidal backgrounds. Using a naive brane charge quantisation to determine the periodicities of the additional coordinates restricts the gaugings to an infinite discrete set and excludes all the gaugings other than the standard one.     

Superpotentials for Gauge and Conformal Supergravity Backgrounds

Effective superpotentials obtained by integrating out matter in super Yang-Mills and conformal supergravity backgrounds in SUSY theories are considered. The pure gauge and supergravity contributions (generalizing Veneziano-Yankielowicz terms) are derived by considering the case with matter fields in the fundamental representation of the gauge group. These contributions represent quantum corrections to the tree-level Yang-Mills and conformal supergravity actions. The classical equations of motion following from the conformal supergravity action require the background to be (super)conformally flat. This condition is unchanged by quantum corrections to the effective superpotential, irrespective of the matter content of the theory.     

Koopman‐von Neumann formulation of classical Yang‐Mills theories: I

In this paper we present the Koopman‐von Neumann (KvN) formulation of classical non‐Abelian gauge field theories. In particular we shall explore the functional (or classical path integral) counterpart of the KvN method. In the quantum path integral quantization of Yang‐Mills theories concepts like gauge‐fixing and Faddeev‐Popov determinant appear in a quite natural way. We will prove that these same objects are needed also in this classical path integral formulation for Yang‐Mills theories. We shall also explore the classical path integral counterpart of the BFV formalism and build all the associated universal and gauge charges. These last are quite different from the analog quantum ones and we shall show the relation between the two. This paper lays the foundation of this formalism which, due to the many auxiliary fields present, is rather heavy. Applications to specific topics outlined in the paper will appear in later publications.     

Gauged supergravities in various spacetime dimensions

In this review article we study the gaugings of extended supergravity theories in various space‐time dimensions. These theories describe the low‐energy limit of non‐trivial string compactifications. For each theory under consideration we review all possible gaugings that are compatible with supersymmetry. They are parameterized by the so‐called embedding tensor which is a group theoretical object that has to satisfy certain representation constraints. This embedding tensor determines all couplings in the gauged theory that are necessary to preserve gauge invariance and supersymmetry. The concept of the embedding tensor and the general structure of the gauged supergravities are explained in detail. The methods are then applied to the half‐maximal (N = 4) supergravities in d = 4 and d = 5 and to the maximal supergravities in d = 2 and d = 7. Examples of particular gaugings are given. Whenever possible, the higher‐dimensional origin of these theories is identified and it is shown how the compactification parameters like fluxes and torsion are contained in the embedding tensor.     

Topological order and conformal quantum critical points

We discuss a certain class of two-dimensional quantum systems which exhibit conventional order and topological order, as well as quantum critical points separating these phases. All of the ground-state equal-time correlators of these theories are equal to correlation functions of a local two-dimensional classical model. The critical points therefore exhibit a time-independent form of conformal invariance. These theories characterize the universality classes of two-dimensional quantum dimer models and of quantum generalizations of the eight-vertex model, as well as and non-abelian gauge theories. The conformal quantum critical points are relatives of the Lifshitz points of three-dimensional anisotropic classical systems such as smectic liquid crystals. In particular, the ground-state wave functional of these quantum Lifshitz points is just the statistical (Gibbs) weight of the ordinary two-dimensional free boson, the two-dimensional Gaussian model. The full phase diagram for the quantum eight-vertex model exhibits quantum critical lines with continuously varying critical exponents separating phases with long-range order from a deconfined topologically ordered liquid phase. We show how similar ideas also apply to a well-known field theory with non-Abelian symmetry, the strong-coupling limit of 2+1-dimensional Yang–Mills gauge theory with a Chern–Simons term. The ground state of this theory is relevant for recent theories of topological quantum computation.     

The dimensional reduction of eleven dimensional supergravity into a product of Robertson-Walker spacetime and the seven sphere

The dimensional reduction of eleven dimensional supergravity is discussed. It is shown that there is no dimensional reduction onto Robertson-Walker space with the asymmetric tensorF giving a realistic fluid. Furthermore it is shown that the ansatz’s for the scale factorR:R=at ⁿ, R=a exp (bt ⁿ), andR=aZ ⁿ, there is no dimensional reduction except the known example of the Freund-Rubin-Englert solution.     

M‐theory and gauged supergravities

We present a pedagogical discussion of the emergence of gauged supergravities from M‐theory. First, a review of maximal supergravity and its global symmetries and supersymmetric solutions is given. Next, different procedures of dimensional reduction are explained: reductions over a torus, a group manifold and a coset manifold and reductions with a twist. Emphasis is placed on the consistency of the truncations, the resulting gaugings and the possibility to generate field equations without an action. Using these techniques, we construct a number of gauged maximal supergravities in diverse dimensions with a string or M‐theory origin. One class consists of the CSO gaugings, which comprise the analytic continuations and group contractions of SO(n) gaugings. We construct the corresponding half‐supersymmetric domain walls and discuss their uplift to D‐ and M‐brane distributions. Furthermore, a number of gauged maximal supergravities are constructed that do not have an action.     

Supersymmetry algebras that include topological charges

We show that in supersymmetric theories with solitons, the usual supersymmetry algebra is not valid; the algebra is modified to include the topological quantum numbers as central charges. Using the corrected algebra, we are able to show that in certain four dimensional gauge theories, there are no quantum corrections to the classical mass spectrum. These are theories for which Bogomolny has derived a classical bound; the argument involves showing that Bogomolny's bound is valid quantum mechanically and that it is saturated.     

The Parametrization Invariant and Gauge Invariant Effective Actions in Quantum Field Theory

The review of formulation and methods of calculation of the parametrization and gauge invariant effective actions in quantum field theory is given. As an example the Vilkovisky-De Witt Effective action (EA) is studied (this EA is a natural representative of gauge and parametrization invariant EA's). The examples where the use of the standard EA leads to the ambiguity are demonstrated. This happens as the result of dependence of the standard EA upon the choice of gauge condition. These examples are as follows: Coleman-Weinberg potential in the finite theories and symmetry breaking, EA in quantum gravity with matter and d = 5 gauged supergravity, the possibility of spontaneous supersymmetry breaking in N = 1 supergravity and the spontaneous compactification in the multidimensional R²-gravity. In all these cases the one-loop Vilkovisky-De Witt EA is found and therefore the problem of gauge dependence of EA is solved. The dependence of standard EA of composite fields upon the gauge is studied for the general gauge theories. The class of gauge and parametrization invariant EA's of the composite fields is offered.     

Lectures on attractors and black holes

A basic survey on some aspects of four‐dimensional black holes (BHs) is given in these lectures. It covers thermodynamical properties as well as the Attractor Mechanism for extremal BHs in an environment of scalar field background. Some relevant formulæ for the critical points of the BH “effective potential” are discussed, and the simplest example uncovering the attractor behavior, the Maxwell‐Einstein‐dilaton supergravity, is analyzed in detail. Observations on similarities between BH entropy (as given by the Bekenstein‐Hawking entropy‐area formula) and multipartite entanglement of qubits in quantum information theory are reported, as well. Finally, among the latest developments, the moduli space of attractor points for 𝒩 ≥ 2 supergravities is also considered. Based on lectures given by S. Ferrara at the International School of Subnuclear Physics, 45th Course: Search for the “Totally Unexpected” in the LHC era, Erice, Italy, 29 August – 7 September 2007 (Directors: G. 't Hooft – A. Zichichi), and at the III Avogadro Meeting on Theoretical Physics, Alessandria, Italy, 19 – 21 December 2007.     

周期轨道的量子化与量子能谱中的长程关联

用二维可积系统的半经典量子化方案和二维无关联振子系统的量子能级与周期轨道之间的对应关系,讨论了一组量子能级之间具有长程关联的内在机制,在二维无关联振子系统中,发现了具有相同拓扑M(M1,M2)的周期轨道相对应的量子能级之间存在着长程关联,并以二维4次无关联振子系统为例做了具体说明.