Conformal symmetry of relativistic and nonrelativistic systems and AdS/CFT correspondence

The nonlinear realization of conformal so(2,d) symmetry for relativistic systems and the dynamical conformal so(2,1) symmetry of nonrelativistic systems are investigated in the context of AdS/CFT correspondence. We show that the massless particle in d-dimensional Minkowski space can be treated as the system confined to the border of the AdS_d+1 of infinite radius, while various nonrelativistic systems may be canonically related to a relativistic (massless, massive, or tachyon) particle on the AdS₂ × S^d−1. The list of nonrelativistic systems “unified” by such a correspondence comprises the conformal mechanics model, the planar charge-vortex and three-dimensional charge-monopole systems, the particle in a planar gravitational field of a point massive source, and the conformal model associated with the charged particle propagating near the horizon of the extreme Reissner-Nordström black hole.     

Integral dimension in the string theory based on a group manifold

We study string models on a group manifold with Kac-Moody symmetry where the critical dimensiond is integer. In particular the possibility of fourdimensional models is investigated. We find that only nine group manifolds with a relevant level can have four as the critical dimension among an infinite number of compact Lee groups. They re all listed. The models with minimal conformal sectors adding to the Kac-Moody sector are investigated. In the cases with one minimal conformal sector, there are only two groups,SU (5) andSO (43), that can gived=4. Among the cases with some tensoring products of minimal conformal sectors we discuss a few special cases withk=0 andk=1. The cases based onN=1 super Kac-Moody algebra are also studied. Finally we discuss the possibility of the enlargement of gauge symmetry.     

Structure constants for the osp(1|2) current algebra

We study the free field realization of the two-dimensional osp(1|2) current algebra. We consider the case in which the level of the affine osp(1|2) symmetry is a positive integer. Using the Coulomb gas technique we obtain integral representations for the conformal blocks of the model. In particular, from the behaviour of the four-point function, we extract the structure constants for the product of two arbitrary primary operators of the theory. From this result we derive the fusion rules of the osp(1|2) conformal field theory and we explore the connections between the osp(1|2) affine symmetry and the N = 1 superconformal field theories.     

Hidden conformal symmetry of rotating black holes in the five-dimensional Gödel universe

We have investigated the hidden conformal symmetry of generic non-extremal rotating black holes in the five-dimensional Gödel universe. In a range of parameters, the low-frequency massless scalar wave equation in the “near region” can be described by an SL(2, R) _L × SL(2, R) _R conformal symmetry. We further found that the microscopic entropy via Cardy formula matches the macroscopic Bekenstein-Hawking entropy and the absorption cross section for the massless scalar also agrees with the one for the two dimensional finite temperature conformal field theory (CFT). All these evidences support the conjecture that the generic non-extremal rotating black hole immersed in the Gödel universe can be dual to a two dimensional finite temperature CFT. In addition, we have reformulated the first laws of thermodynamics associated with the inner and outer horizons of the rotating Gödel-type black holes into the forms of conformal thermodynamics.     

Notes on the hidden conformal symmetry in the near horizon geometry of the Kerr black hole

Toward the Kerr/CFT correspondence for the generic non-extremal Kerr black hole, the analysis of scattering amplitudes by near extremal Kerr provides a clue. This pursuit reveals a hidden conformal symmetry in the low frequency wave equation for a scalar field in a certain spacetime region referred to as the near region. For extremal case, the near region is expected to be the near horizon region in which the correspondence via the asymptotic symmetry is studied. We investigate the hidden conformal symmetry in the near horizon limit and consider the relation between the hidden conformal symmetry and the asymptotic symmetry in the near horizon limit. By using an appropriate definition of the quasi-local charge, we obtain the deviation of the entropy from the extremality.     

Mean field at finite temperature and symmetry breaking

For an infinite system of nucleons interacting through a central spin-isospin schematic force we discuss how the Hartree-Fock theory at finite temperature T yields back, in the T=0 limit, the standard zero-temperature Feynman theory when there is no symmetry breaking. The attention is focused on the mechanism of cancellation of the higher order Hartree-Fock diagrams and on the dependence of this cancellation upon the range of the interaction. When a symmetry breaking takes place it turns out that more iterations are required to reach the self-consistent Hartree-Fock solution, because the cancellation of the Hartree-Fock diagrams of order higher than one no longer occurs. We explore in particular the case of an explicit symmetry breaking induced by a constant, uniform magnetic field B acting on a system of neutrons. Here we compare calculations performed using either the single-particle Matsubara propagator or the zero-temperature polarization propagator, discussing under which perturbative scheme they lead to identical results (if B is not too large). We finally address the issue of the spontaneous symmetry breaking for a system of neutrons using the technique of the anomalous propagator: in this framework we recover the Stoner equation and the critical values of the interaction corresponding to a transition to a ferromagnetic phase.     

An xp model on AdS2 spacetime

In this paper we formulate the xp model on the AdS₂ spacetime. We find that the spectrum of the Hamiltonian has positive and negative eigenvalues, whose absolute values are given by a harmonic oscillator spectrum, which in turn coincides with that of a massive Dirac fermion in AdS₂. We extend this result to generic xp models which are shown to be equivalent to a massive Dirac fermion on spacetimes whose metric depend of the xp   Hamiltonian. Finally, we construct the generators of the isometry group SO(2,1)$\mathit{SO}(2,1)$ of the AdS₂ spacetime, and discuss the relation with conformal quantum mechanics.     

Elliptic Hypergeometry of Supersymmetric Dualities II. Orthogonal Groups, Knots, and Vortices

We consider Seiberg electric-magnetic dualities for 4d ${\mathcal{N} = 1}$ SYM theories with SO(N) gauge group. For all such known theories we construct superconformal indices (SCIs) in terms of elliptic hypergeometric integrals. Equalities of these indices for dual theories lead both to proven earlier special function identities and new conjectural relations for integrals. In particular, we describe a number of new elliptic beta integrals associated with the s-confining theories with the spinor matter fields. Reductions of some dualities from SP(2N) to SO(2N) or SO(2N + 1) gauge groups are described. Interrelation of SCIs and the Witten anomaly is briefly discussed. Possible applications of the elliptic hypergeometric integrals to a two-parameter deformation of 2d conformal field theory and related matrix models are indicated. Connections of the reduced SCIs with the state integrals of knot theory, generalized AGT duality for (3 + 3)d theories, and a 2d vortex partition function are described.     

Hidden symmetry in the presence of fluxes

We derive the most general first-order symmetry operator for the Dirac equation coupled to arbitrary fluxes. Such an operator is given in terms of an inhomogeneous form ω   which is a solution to a coupled system of first-order partial differential equations which we call the generalized conformal Killing–Yano system. Except trivial fluxes, solutions of this system are subject to additional constraints. We discuss various special cases of physical interest. In particular, we demonstrate that in the case of a Dirac operator coupled to the skew symmetric torsion and U(1)$U(1)$ field, the system of generalized conformal Killing–Yano equations decouples into the homogeneous conformal Killing–Yano equations with torsion introduced in D. Kubiznak et al. (2009) [8] and the symmetry operator is essentially the one derived in T. Houri et al. (2010) [9]. We also discuss the Dirac field coupled to a scalar potential and in the presence of 5-form and 7-form fluxes.     

Orthosymplectic unification of the Galilei group and BRST

The inclusion of BRST generators into the Poincarè group in D dimensions is known to lead IOsp[D,2|2]. Similarly, conformal symmetry gets extended into Osp[D×1,3|2]. For the non-relativistic case we find that the Galilei symmetry gets extended, by inclusion of the BRST generators, into an orthosymplectic symmetry possessing Osp[D,1|2] as a subgroup. All such extensions express the possibility of formulating the classical theories in reparametrization invariant ways. They include besides the generators of the initial kinematical symmetry (Poincarè, or conformal, or Galilei), the generators of Parisi-Sourlas transformations. The extended symmetries follow directly through BRST quantization.     

On second-quantized open superstring theory

The SO(32) theory, in the limit where it is an open superstring theory, is completely specified in the light-cone gauge as a second-quantized string theory in terms of a “matrix string” model. The theory is defined by the neighborhood of a 1 + 1-dimensional fixed point theory, characterized by an Abelian gauge theory with type IB Green-Schwarz form. Non-orientability and SO(32) gauge symmetry arise naturally, and the theory effectively constructs an orientifold projection of the (weakly coupled) matrix type IIB theory (also discussed herein). The fixed point theory is a conformal field theory with boundary, defining the free string theory. Interactions involving the interior of open and closed strings are governed by a twist operator in the bulk, while string endpoints are created and destroyed by a boundary twist operator.     

Flavored surface defects in 4d $$\mathcal{N}=1$$ N = 1 SCFTs

We discuss supersymmetric surface defects in compactifications of six-dimensional minimal conformal matter of types SU(3) and SO(8) to four dimensions. The relevant field theories in four dimensions are \(\mathcal{N}=1\) quiver gauge theories with SU(3) and SU(4) gauge groups, respectively. The defects are engineered by giving space-time-dependent vacuum expectation values to baryonic operators. We find evidence that in the case of SU(3) minimal conformal matter, the defects carry SU(2) flavor symmetry which is not a symmetry of the four-dimensional model. The simplest case of a model in this class is SU(3) SQCD with nine flavors, and thus the results suggest that this admits natural surface defects with SU(2) flavor symmetry. We analyze the defects using the superconformal index and derive analytic difference operators introducing the defects into the index computation. The duality properties of the four-dimensional theories imply that the index of the models is a kernel function for such difference operators. In turn, checking the kernel property constitutes an independent check of the dualities and the dictionary between six- dimensional compactifications and four-dimensional models.

     

A gravitating SO(3, 1) gauge field

In this article, we postulate SO(3, 1) as a local symmetry of any relativistic theory. This is equivalent to assuming the existence of a gauge field associated with this noncompact group. This SO(3, 1) gauge field is the spinorial affinity which usually appears when we deal with weighting spinors, which, as is well known, cannot be coupled to the metric tensor field. Furthermore, according to the integral approach to gauge fields proposed by Yang, it is also recognized that in order to obtain models of gravity we have to introduce ordinary affinities as the gauge field associated with GL(4) (the local symmetry determined by the parallel transport). Thus if we assume both L(4) and SO(3, 1) as local independent symmetries we are led to analyze the dynamical gauge system constituted by the Einstein field interacting with the SO(3, 1) Weyl-Yang gauge field. We think this system is a possible model of strong gravity. Once we give the first-order action for this Einstein-Weyl-Yang system we study whether the SO(3, 1) gauge field could have a tetrad associated with it. It is also shown that both fields propagate along a unique characteristic cone. Algebraic and differential constraints are solved when the system evolves along a null coordinate. The unconstrained expression for the action of the system is found working in the Bondi gauge. That allows us to exhibit an explicit expression of the dynamical generator of the system. Its signature turns out to be nondefinite, due to the nondefinite contribution of the Weyl-Yang field, which has the typical spinorial behavior. A conjecture is made that such an unpleasant feature could be overcome in the quantized version of this model.     

Multi-instantons and exact results II: specific cases, higher-order effects, and numerical calculations

In this second part of the treatment of instantons in quantum mechanics, the focus is on specific calculations related to a number of quantum mechanical potentials with degenerate minima. We calculate the leading multi-instanton contributions to the partition function, using the formalism introduced in the first part of the treatise [Ann. Phys. (N. Y.) (previous issue) (2004)]. The following potentials are considered: (i) asymmetric potentials with degenerate minima, (ii) the periodic cosine potential, (iii) anharmonic oscillators with radial symmetry, and (iv) a specific potential which bears an analogy with the Fokker-Planck equation. The latter potential has the peculiar property that the perturbation series for the ground-state energy vanishes to all orders and is thus formally convergent (the ground-state energy, however, is non-zero and positive). For the potentials (ii), (iii), and (iv), we calculate the perturbative B-function as well as the instanton A-function to fourth order in g. We also consider the double-well potential in detail, and present some higher-order analytic as well as numerical calculations to verify explicitly the related conjectures up to the order of three instantons. Strategies analogous to those outlined here could result in new conjectures for problems where our present understanding is more limited.     

Conformal anomaly in 4D gravity-matter theories non-minimally coupled with a dilaton

The conformal anomaly for 4D gravity-matter theories, which are non-minimally coupled with the dilaton, is systematically studied. Special care is taken of rescaling of fields, treatment of total derivatives, hermiticity of the system operator and the choice of measure. Scalar, spinor and vector fields are taken as the matter quantum fields and their explicit conformal anomalies in the gravity-dilaton background are found. The cohomology analysis is carried out and some new conformal invariants and trivial terms, involving the dilaton, are obtained. The symmetry of the constant shift of the dilaton field plays an important role. The general structure of the conformal anomaly is examined. It is shown that the dilaton affects the conformal anomaly characteristically for each case: (1) [Scalar] The dilaton changes the conformal anomaly only by a new conformal invariant, I₄; (2) [Spinor] The dilaton does not change the conformal anomaly; (3) [Vector] The dilaton changes the conformal anomaly by three new (generalized) conformal invariants, I₄, I₂, I₁. We present some new anomaly formulae which are useful for practical calculations. Finally, the anomaly induced action is calculated for the dilatonic Wess-Zumino model. We point out that the coefficient of the total derivative term in the conformal anomaly for the 2D scalar coupled to a dilaton is ambiguous. This resolves the disagreement between earlier calculations and the result of Hawking and Bousso.     

Maximal symmetry and mass generation of Dirac fermions and gravitational gauge field theory in six-dimensional spacetime

The relativistic Dirac equation in four-dimensional spacetime reveals a coherent relation between the dimensions of spacetime and the degrees of freedom of fermionic spinors. A massless Dirac fermion generates new symmetries corresponding to chirality spin and charge spin as well as conformal scaling transformations. With the introduction of intrinsic W-parity, a massless Dirac fermion can be treated as a Majorana-type or Weyl-type spinor in a six-dimensional spacetime that reflects the intrinsic quantum numbers of chirality spin. A generalized Dirac equation is obtained in the six-dimensional spacetime with a maximal symmetry. Based on the framework of gravitational quantum field theory proposed in Ref. [1] with the postulate of gauge invariance and coordinate independence, we arrive at a maximally symmetric gravitational gauge field theory for the massless Dirac fermion in six-dimensional spacetime. Such a theory is governed by the local spin gauge symmetry SP(1,5) and the global Poincar′e symmetry P(1,5)= SO(1,5) P~(1,5) as well as the charge spin gauge symmetry SU(2). The theory leads to the prediction of doubly electrically charged bosons. A scalar field and conformal scaling gauge field are introduced to maintain both global and local conformal scaling symmetries. A generalized gravitational Dirac equation for the massless Dirac fermion is derived in the six-dimensional spacetime. The equations of motion for gauge fields are obtained with conserved currents in the presence of gravitational effects. The dynamics of the gauge-type gravifield as a Goldstone-like boson is shown to be governed by a conserved energy-momentum tensor, and its symmetric part provides a generalized Einstein equation of gravity. An alternative geometrical symmetry breaking mechanism for the mass generation of Dirac fermions is demonstrated.     

Liouville field theory revisited: Zero modes andS-matrix element

A new analysis of the intrinsic structure of Liouville field theory (LFT) is presented. We prove that LFT displays a zero mode if its Laplacian is defined in terms of the square of the corresponding Dirac operator. Further, by interpreting the spacetime asSO(2,1)/SO(1, 1) (analogous toSO(3, 2)/SO(3, 1)) we present, arguments which support the nontrivial approximativeS-matrix element we derived in [2]. Some connected questions are also discussed.     

On the renormalization of the bosonized multi-flavor Schwinger model

The phase structure of the bosonized multi-flavor Schwinger model is investigated by means of the differential renormalization group (RG) method. In the limit of small fermion mass the linearized RG flow is sufficient to determine the low-energy behavior of the N  -flavor model, if it has been rotated by a suitable rotation in the internal space. For large fermion mass, the exact RG flow has been solved numerically. The low-energy behavior of the multi-flavor model is rather different depending on whether N=1$N=1$ or N>1$N>1$, where N   is the number of flavors. For N>1$N>1$ the reflection symmetry always suffers breakdown in both the weak and strong coupling regimes, in contrary to the N=1$N=1$ case, where it remains unbroken in the strong coupling phase.     

Phase diagram of the Gross-Neveu model: exact results and condensed matter precursors

Recently the revised phase diagram of the (large N) Gross-Neveu model in 1 + 1 dimensions with discrete chiral symmetry has been determined numerically. It features three phases, a massless and a massive Fermi gas and a kink-antikink crystal. Here we investigate the phase diagram by analytical means, mapping the Dirac-Hartree-Fock equation onto the non-relativistic Schrödinger equation with the (single gap) Lamé potential. It is pointed out that mathematically identical phase diagrams appeared in the condensed matter literature some time ago in the context of the Peierls-Fröhlich model and ferromagnetic superconductors.