Heisenberg groups and noncommutative fluxes

We develop a group-theoretical approach to the formulation of generalized abelian gauge theories, such as those appearing in string theory and M-theory. We explore several applications of this approach. First, we show that there is an uncertainty relation which obstructs simultaneous measurement of electric and magnetic flux when torsion fluxes are included. Next, we show how to define the Hilbert space of a self-dual field. The Hilbert space is Z₂-graded and we show that, in general, self-dual theories (including the RR fields of string theory) have fermionic sectors. We indicate how rational conformal field theories associated to the two-dimensional Gaussian model generalize to (4k + 2)-dimensional conformal field theories. When our ideas are applied to the RR fields of string theory we learn that it is impossible to measure the K-theory class of a RR field. Only the reduction modulo torsion can be measured.     

Statistics dependence of the entanglement entropy

The entanglement entropy of a distinguished region of a quantum many-body system reflects the entanglement in its pure ground state. Here we establish scaling laws for this entanglement in critical quasifree fermionic and bosonic lattice systems, without resorting to numerical means. We consider the setting of D-dimensional half-spaces which allows us to exploit a connection to the one-dimensional case. Intriguingly, we find a difference in the scaling properties depending on whether the system is bosonic-where an area law is proven to hold-or fermionic where we determine a logarithmic correction to the area law, which depends on the topology of the Fermi surface. We find Lifshitz quantum phase transitions accompanied with a nonanalyticity in the prefactor of the leading order term.     

Linear bosonic and fermionic quantum gauge theories on curved spacetimes

We develop a general setting for the quantization of linear bosonic and fermionic field theories subject to local gauge invariance and show how standard examples such as linearised Yang-Mills theory and linearised general relativity fit into this framework. Our construction always leads to a well-defined and gauge-invariant quantum field algebra, the centre and representations of this algebra, however, have to be analysed on a case-by-case basis. We discuss an example of a fermionic gauge field theory where the necessary conditions for the existence of Hilbert space representations are not met on any spacetime. On the other hand, we prove that these conditions are met for the Rarita-Schwinger gauge field in linearised pure $N=1$ supergravity on certain spacetimes, including asymptotically flat spacetimes and classes of spacetimes with compact Cauchy surfaces. We also present an explicit example of a supergravity background on which the Rarita-Schwinger gauge field can not be consistently quantized.     

Light composite fermions

In this paper we consider the possibility that QCD-like theories can lead to massless or near-massless composite fermions. The method of analysis relies on a conjectured equivalence between the confined and Higgs phases of certain non-abelian gauge theories. This “complementarity” principle allows us to analyze a theory as if the Higgs phenomenon occurred and then reinterpret the results in the language of composite gauge singlets. Those fermions which remain massless in the Higgs picture may then be interpreted as massless fermionic composites.The principle of complementarity, when applied to a class of extended technicolor models, implies that quarks and leptons are composites bound at a scale of order 1–100 TeV.     

More on axial anomalies of Lifshitz fermions

We show that the gauge and metric field contribution to the axial anomaly of a four‐dimensional massless Lifshitz fermion theory with anisotropy scaling exponent z is identical to the relativistic case, hereby extending the results found in arXiv:1103.5693 to arbitrary values of z. This is in accordance with the fact that the axial anomaly is an infra‐red phenomenon in disguise. We also provide some new models that realize baryon and lepton number violation in non‐relativistic theories of gravity. In all cases, the volume of space exhibits a lower bound that is fixed by the gravitational coupling parameters.     

Gravitational effective action and entanglement entropy in UV modified theories with and without Lorentz symmetry

We calculate parameters in the low energy gravitational effective action and the entanglement entropy in a wide class of theories characterized by improved ultraviolet (UV) behavior. These include (i) local and non-local Lorentz invariant theories in which inverse propagator is modified by higher-derivative terms and (ii) theories described by non-Lorentz invariant Lifshitz type field operators. We demonstrate that the induced cosmological constant, gravitational couplings and the entropy are sensitive to the way the theory is modified in UV. For non-Lorentz invariant theories the induced gravitational effective action is of the Horava–Lifshitz type. We show that under certain conditions imposed on the dimension of the Lifshitz operator the couplings of the extrinsic curvature terms in the effective action are UV finite. Throughout the paper we systematically exploit the heat kernel method appropriately generalized for the class of theories under consideration.     

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.     

Event-Symmetry for Superstrings

I apply the principle of event-symmetry tosimple string models and discuss how these lead to theconviction that multiple quantization is linked todimension. It may be that string theory has to beformulated in the absence of space-time, which will thenemerge as a derived property of the dynamics. Anotherinterpretation of the event-symmetric approach whichembodies this is that instantons are fundamental. Just as solitons may be dual to fundamentalparticles, instantons may be dual to space-time events.Event-symmetry is then dual to instanton statistics. Inthat case a unification between particle statistics and gauge symmetry follows on naturally fromthe principle of event-symmetry. I build algebras whichrepresent symmetries of superstring theories extendingevent-symmetry, but which are also isomorphic to an algebra of creation and annihilationoperators for strings of fermionic partons.     

Casimir effect with dynamical matter on thin mirrors

We calculate the Casimir energy for scalar and gauge fields in interaction with zero-width mirrors, including quantum effects due to the matter fields inside the mirrors. We consider models where those fields are either scalar or fermionic, obtaining general expressions for the energy as a function of the vacuum field 1PI function. We also study, within the frame of a concrete model, the role of the dissipation induced by those degrees of freedom, showing that, after integration of the matter fields, the effective theory for the electromagnetic field contains modes with complex energies. As for the case of Lifshitz formula, we show that the formal result obtained by neglecting dissipation coincides with the correct result that comes from the quantum fluctuations of both bulk and matter fields.     

Anomalous gauge theories as constrained Hamiltonian systems

Anomalous gauge theories considered as constrained systems are investigated. The effects of chiral anomaly on the canonical structure are examined first for nonlinear σ-model and later for fermionic theory. The breakdown of the Gauss law constraints and the anomalous commutators among them are studied in a systematic way. An intrinsic mass term for gauge fields makes it possible to solve the Gauss law relations as second class constraints. Dirac brackets between the time components of gauge fields are shown to involve anomalous terms. Based upon the Ward-Takahashi identities for gauge symmetry, we investigate anomalous fermionic theory within the framework of path integral approach.     

Axial anomalies of Lifshitz fermions

We compute the axial anomaly of a Lifshitz fermion theory with anisotropic scaling z = 3 which is minimally coupled to geometry in 3+1 space‐time dimensions. We find that the result is identical to the relativistic case using path integral methods. An independent verification is provided by showing with spectral methods that the η‐invariant of the Dirac and Lifshitz fermion operators in three dimensions are equal. Thus, by the integrated form of the anomaly, the index of the Dirac operator still accounts for the possible breakdown of chiral symmetry in non‐relativistic theories of gravity. We apply this framework to the recently constructed gravitational instanton backgrounds of Hořava–Lifshitz theory and find that the index is non‐zero provided that the space‐time foliation admits leaves with harmonic spinors. Using Hitchin's construction of harmonic spinors on Berger spheres, we obtain explicit results for the index of the fermion operator on all such gravitational instanton backgrounds with SU(2) × U(1) isometry. In contrast to the instantons of Einstein gravity, chiral symmetry breaking becomes possible in the unimodular phase of Hořava–Lifshitz theory arising at λ = 1/3 provided that the volume of space is bounded from below by the ratio of the Ricci to Cotton tensor couplings raised to the third power. Some other aspects of the anomalies in non‐relativistic quantum field theories are also discussed.     

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.     

Gauge Model with Massive Gravitons

Gauge theory of gravity is formulated based on principle of local gauge invariance. Because the model hasstrict local gravitational gauge symmetry, and gauge theory of gravity is a perturbatively renormalizable quantum model.However, in the original model, all gauge gravitons are massless. We want to ask whether there exist massive gravitonsin Nature. In this paper, we will propose a gauge model with massive gravitons. The mass term of gravitational gaugefield is introduced into the theory without violating the strict local gravitational gauge symmetry. Massive gravitons canbe considered to be possible origin of dark energy and dark matter in the Universe.     

Properties of the covariant formulation of superstring theories

The covariant two-dimensional action principle that describes the dynamics of free superstrings in a Minkowski background is reviewed. Covariant gauge conditions are formulated, which simplify the equations of motion of the superspace coordinates to free equations. In this gauge there are bosonic and fermionic constraints whose generators give a supersymmetric generalization of the Virasoro algebra. As in certain supersymmetric field theories, closure of the algebra requires using the equations of motion. Covariant constrained bracket relations are obtained for the classical theory, but it is very difficult to extend them to quantum mechanical commutation relations. Interaction vertices satisfying supersymmetry and the necessary gauge conditions are constructed. They reduce in a special frame to ones found in earlier work in the light-cone gauge, and then can be interpreted quantum mechanically.     

Fermion condensation in the field strength approach to non-abelian yang-mills theories

Within the field strength approach to Yang-Mills theories the fermionic sectors of gauge theories are bosonized for the SU(2) and SU(3) gauge group. The emerging effective meson theories are studied in the tree approximation. In this approximation the original minimal gauge coupling of the quarks to gluons is rendered into an effective local four-fermion interaction with non-trivial Lorentz and gauge structure. The Schwinger-Dyson equation is solved in the strong coupling limit and the quark condensates and constituent masses are evaluated.     

Higher-Order Inhomogeneous Generalized Heisenberg Supermagnetic Model

We construct the fourth-order inhomogeneous generalized HS model and investigate the integrability property of the supersymmetric integrable system. Moreover, in terms of the gauge transformation, we investigate the corresponding gauge equivalent counterparts under two constraints, i.e., the super inhomogeneous generalized nonlinear Schr?dinger equation and the fermionic inhomogeneous generalized nonlinear Schr?dinger equation.     

Gauged Lifshitz scalar field theories in two dimensions

We present two-dimensional gauged Lifshitz scalar field theories by considering the duality relation between the source current and the Noether current. Requiring the duality partially, we obtain a gauged model which recovers the bosonized Schwinger model for the IR limit. For the exact duality, however, the source current is not conserved, which means that the resulting theory is anomalous, so that the number of degrees of freedom is increased. The second model is consistently formulated by adding the Wess–Zumino type action to maintain the gauge invariance.     

Particle picture in external fields,quantum holonomy,and gauge anomaly

We formulate a path integral of chiral gauge theories by means of the canonical quantization of fermions in time-dependent background gauge fields. The expression of the path integral is composed of two parts. One is due to the nontrivial holonomy of the fermionic Fock vacua and the other is the conventional form which is used in the perturbation theory. The nontrivial holonomy part is expected to be a nonlocal counter term. We show a possibility of the perturbative calculation     

On the True Nature of Renormalizability in Horava-Lifshitz Gravity

We argue that the true nature of the renormalizability of Horava-Lifshitz gravity lies in the presence of higher order spatial derivatives and not in the anisotropic Lifshitz scaling of space and time. We discuss the possibility of constructing a higher order spatial derivatives model that has the same renormalization properties of Horava-Lifshitz gravity but that does not make use of the Lifshitz scaling. In addition, the state-of-the-art of the Lorentz symmetry restoration in Horava-Lifshitz-type theories of gravitation is reviewed.     

Gauge invariant theories of the generalized chiral Schwinger model

The gauge invariant theories of the generalized chiral Schwinger model are constructed in terms of two schemes with and without Wess-Zumino terms, respectively. Following the former scheme, we calculate the Wess-Zumino term which cancels the gauge anomaly, and then constitute the gauge invariant theory by adding the Wess-Zumino term to the original Lagrangian of the model. According to the latter, we modify the original Hamiltonian by adding a term composed of constraints of the model. It is so designed that the theory described by the modified Hamiltonian and its corresponding first-order Lagrangian maintains gauge invariance. We show by the canonical Dirac method that each of the two gauge invariant theories has the same physical spectrum as that of the original gauge noninvariant formulation.