Higher Dimensional Field Theories Research Articles

Running with the radius in RS1

We derive a renormalization group formalism for the Randall-Sundrum scenario, where the renormalization scale is set by a floating compactification radius. While inspired by the AdS-CFT conjecture, our results are derived concretely within higher-dimensional effective field theory. Matching theories with different radii leads to running hidden brane couplings. The hidden brane Lagrangian consists of four-dimensional local operators constructed from the induced value of the bulk fields on the brane. We find hidden Lagrangians which are nontrivial fixed points of the RG flow. Calculations in RS1 can be greatly simplified by ``running down'' the effective theory to a small radius. We demonstrate these simplifications by studying the Goldberger-Wise stabilization mechanism. In this paper, we focus on the classical and tree-level quantum field theory of bulk scalar fields, which demonstrates the essential features of the RG in the simplest context.

Localized anomalies in heterotic orbifolds

Recently spatially localized anomalies have been considered in higher-dimensional field theories. The question of the quantum consistency and stability of these theories needs further discussion. Here we would like to investigate what string theory might teach us about theories with localized anomalies. We consider the Z 3 orbifold of the heterotic E 8×E′ 8 theory, and compute the anomaly of the gaugino in the presence of Wilson lines. We find an anomaly localized at the fixed points, which depends crucially on the local untwisted spectra at those points. We show that non-Abelian anomalies cancel locally at the fixed points for all Z 3 models with or without additional Wilson lines. At various fixed points different anomalous U(1)'s may be present, but at most one at a given fixed point. It is in general not possible to construct one generator which is the sole source of the anomalous U(1)'s at the various fixed points.

Gauge-Higgs unification in higher dimensions

The electroweak Higgs doublets are identified as components of a vector multiplet in a higher-dimensional supersymmetric field theory. We construct a minimal model in 6D where the electroweak SU(2)⊗ U(1) gauge group is extended to SU(3), and unified 6D models with the unified SU(5) gauge symmetry extended to SU(6). In these realistic theories the extended gauge group is broken by orbifold boundary conditions, leaving Higgs doublet zero modes which have Yukawa couplings to quarks and leptons on the orbifold fixed points. In one SU(6) model the weak mixing angle receives power law corrections, while in another the fixed point structure forbids such corrections. A 5D model is also constructed in which the Higgs doublet contains the fifth component of the gauge field. In this case Yukawa couplings are introduced as nonlocal operators involving the Wilson line of this gauge field.

Modular origin of chiral diffeomorphisms and their fuzzy analogs in higher-dimensional quantum field theories

The infinite-dimensional conformal symmetries beyond the Moebius-group (the vacuum symmetry) of chiral theories are shown to allow a natural derivation and interpretation in the operator-algebraic modular setting of algebraic QFT. In this way one learns that the diffeomorphisms of the circle beyond the Moebius group are bona fide Wigner symmetries with respect to suitably chosen non-vacuum state vector. The separation of chiral diffeomorphism symmetry from the issue of the structure of the energy–momentum tensor of 2-dimensional conformal theories permits the use of these symmetries in more general chiral theories which one encounters in the algebraic lightfront holography of massive d⩾1+1 dimensional QFTs. They only act locally on the lightfront projection but correspond to a new class of “fuzzy” symmetries in the original theory before the projection. These fuzzy symmetries remained hidden to quantization methods since they have a pure quantum origin; they do not require any spacetime noncommutativity and are in complete harmony with the causality and locality of the real time (non-commutative) operator algebra structure. Their use makes recent attempts to explain the desired universal structure of a possible future quantum Bekenstein-like area law in terms of Virasoro algebra structures on black hole horizons more palatable.

Strongly coupled grand unification in higher dimensions

We consider the scenario where all the couplings in the theory are strong at the cutoff scale, in the context of higher dimensional grand unified field theories where the unified gauge symmetry is broken by an orbifold compactification. In this scenario, the non-calculable correction to gauge unification from unknown ultraviolet physics is naturally suppressed by the large volume of the extra dimension, and the threshold correction is dominated by a calculable contribution from Kaluza-Klein towers that gives the values for ${\mathrm{sin}}^{2}{\ensuremath{\theta}}_{w}$ and ${\ensuremath{\alpha}}_{s}$ in good agreement with low-energy data. The threshold correction is reliably estimated despite the fact that the theory is strongly coupled at the cutoff scale. A realistic 5D supersymmetric $\mathrm{SU}(5)$ model is presented as an example, where rapid $d=6$ proton decay is avoided by putting the first generation matter in the 5D bulk.

Studying quantum field theories numerically on the continuum and a new look at perturbation theory

The Source Galerkin method finds approximate solutions to the functional differential equations of field theories in the presence of external sources. While developing this process, it was recognized that approximations of the spectral representations of the Green's functions by Sinc function expansions are an extremely powerful calculative tool. Specifically, this understanding makes it not only possible to apply the Source Galerkin method to higher dimensional field theories, but also leads to a new approach to perturbation theory calculations in scalar and fermionic field theories. This report summarizes the methodologies for solving quantum field theories with the Source Galerkin method and for performing perturbation theory calculations using Sinc approximations.

Brane-bulk duality and non-conformal gauge theories

We discuss non-conformal gauge theories from type IIB D3-branes embedded in orbifolded space-times. Such theories can be obtained by allowing some non-vanishing logarithmic twisted tadpoles. In certain cases with N=0,1 supersymmetry correlation functions in the planar limit are the same as in the parent N=2 supersymmetric theories. In particular, the effective action in such theories perturbatively is not renormalized beyond one loop in the planar limit. In the N=2 as well as such N=0,1 theories quantum corrections in the D3-brane gauge theories are encoded in the corresponding classical higher dimensional field theories whose actions contain the twisted fields with non-vanishing tadpoles. We argue that this duality can be extended to the non-perturbative level in the N=2 theories. We give some evidence that this might also be the case for N=0,1 theories as well.

Modal expansions and nonperturbative quantum field theory in Minkowski space

We introduce a spectral approach to non-perturbative field theory within the periodic field formalism. As an example we calculate the real and imaginary parts of the propagator in 1+1 dimensional phi^4 theory, identifying both one-particle and multi-particle contributions. We discuss the computational limits of existing diagonalization algorithms and suggest new quasi-sparse eigenvector methods to handle very large Fock spaces and higher dimensional field theories.

Class Formations and Higher Dimensional Local Class Field Theory

The reciprocity law of higher dimensional local class field theory is proved with the help of class formations.

Renormalization group and spontaneous compactification of a higher-dimensional scalar field theory in curved spacetime.

The renormalization group (RG) is used to study the asymptotically free $\phi_6^3$-theory in curved spacetime. Several forms of the RG equations for the effective potential are formulated. By solving these equations we obtain the one-loop effective potential as well as its explicit forms in the case of strong gravitational fields and strong scalar fields. Using zeta function techniques, the one-loop and corresponding RG improved vacuum energies are found for the Kaluza-Klein backgrounds $R^4\times S^1\times S^1$ and $R^4\times S^2$. They are given in terms of exponentially convergent series, appropriate for numerical calculations. A study of these vacuum energies as a function of compactification lengths and other couplings shows that spontaneous compactification can be qualitatively different when the RG improved energy is used.

Characteristic classes and multidimensional reciprocity laws

In this note, we formulate and prove a non-commutative generalization of the reciprocity laws of Parshin and Kato in higher dimensional class field theory [Kat][P1][P4]. This can be regarded as a geometric realization of the reciprocity laws implicit in the Gersten-Quillen complex in algebraic K-theory [Q]. We expect this work to be related to the “higher dimensional Langlands program”, which has yet to be formulated precisely (although see [Kap]). Let X be an k-dimensional complex analytic space (possibly singular). Let F := (C0 ⊂ C1 ⊂ · · · ⊂ Ck = X) be a flag of irreducible subspaces in X, with dim Ci = i (we often omit Ck from the notation). For any analytic open set V ⊂ X, we constructed in [Br-M1] a homology class κF,V ∈ Hk(V ;Z) and showed that it satisfies the homological reciprocity law described in Theorem 2.1. Let ĉp denote the characteristic class of Deligne-Beilinson [Be] which refines the usual pth Chern class cp. In section 3, we define the non-commutative symbol F,V at the “place” F , by pulling back ĉk+1 via the evaluation map BGL(n,O(V )) × V → BGL(n,C) and taking the “slant-product” of the resulting class with κF,V . Here O(V ) denotes the algebra of holomorphic functions on V and we are viewing GL(n,O(V )) as a discrete group. It is important to note that the non-commutative symbol is not just a number, rather it is the cohomology class in Hk+1(BGL(n,O(V )),C∗) of a degree k + 1 group cocycle on GL(n,O(V )). Since the homology class κF,V satisfies a reciprocity law, we obtain

GAUGE-COVARIANT WESS-ZUMINO ACTIONS FOR SUPER p-BRANES IN SUPERSPACE

We present a general method of constructing manifestly world-volume supersymmetric actions of super p-branes, starting from some higher-dimensional field theories in which the partial breaking of global supersymmetry (PBGS) comes about. Our approach is based upon a substitution of the superspace coordinates which relates linear and nonlinear realizations of PBGS. As instructive examples we consider the massive d=2 superparticle (p=0) and d=4 superstring (p=1). The relevant world-line and world-sheet superfied actions naturally appear as a long-wavelength limit of the superfield actions of some two- and four-dimensional supersymmetric field theories possessing topologically nontrivial soliton solutions. The corresponding topological charges prove to enter the super p-brane actions as the coupling constants. We also give a new general Wess-Zumino-type representation of the d=2 superparticle action via the world-line superfields. It respects invariances both under the target space Poincaré supersymmetry and the gauge group of general reparametrizations of the world-line superspace. In one gauge, it is reduced to the standard PBGS form while in another, it gives rise to a very simple action which displays manifest world-line superconformal symmetry and is directly related to the familiar component action of the superparticle. The fermionic κ-symmetry of the latter can be identified with the odd sector of superconformal symmetry. This identity suggests a simple recipe for building higher-order κ-invariant corrections to the minimal superparticle action.

The effective action for wormholes

There is a strong similarity between space-time wormholes and the multiply connected topology that occurs in one- and two-dimensional quantum field theories, like Feynman diagrams and string theory. Integrating out such topological fluctuations gives rise to a quantum field theory on a topologically trivial manifold with effective interactions whose couplings are given by a-parameters. It is shown that these α-parameters are the moments of a single quantum field α on the superspace of the original theory. In the one-dimensional case, this superspace is space-time, but in higher-dimensional field theories, it will be infinite dimensional. However, the superspace for the string contains space-time as a mini-superspace. The single α-field on superspace can be dimensionally reduced to an infinite tower of fields on space-time. If the tachyon is projected out, the lowest members of this tower are the dilaton, graviton and anti-symmetric tensor fields. In the case of space-time wormholes, the α-field is defined on the superspace of General Relativity. There is a discussion of the implications of this for the suggestion that wormholes uniquely fix the values of the physical constants.

Cosmological impact of winding strings

The thermodynamics and cosmology of torus-compactified heterotic strings are studied. The authors emphasise qualitatively new effects due to compactification. New topologically stable states appear which correspond to strings winding around the non-simply-connected compact manifold. Under reasonable assumptions they avoid the blowing up of the compactification scale when the universe becomes matter dominated. For a higher-dimensional point field theory with a scale-invariant ground state this blowing up would be unavoidable.

On the splitting problem in higher-dimensional field theories

Much recent work on higher-dimensional field theories is based on the assumption that the universe (or its ground state) may be represented as a product M4*N, where N is a compact Riemannian manifold. The author argues that, just as it is necessary to justify the assumption of compactness by means of a physical mechanism (spontaneous compactification), so also does one need to explain the splitting between internal and external spaces. Various possible mathematical interpretations of the splitting are discussed, as well as some results which may be relevant to the solution of the 'splitting problem'.

Inhomogeneous Einstein metrics on complex line bundles

Recent developments in higher-dimensional unified field theories have led to a great deal of interest in compact spaces admitting Einstein metrics. Almost all the physics literature on such spaces has been concerned with the very atypical case in which the space is homogeneous. The authors present a very simple construction of a wide class of inhomogeneous compact Einstein spaces with positive Ricci curvature found earlier by Berard-Bergery, (1982) which arise as certain 2-sphere bundles over an arbitrary Einstein-Kahler base space of positive Ricci curvature. Solutions on complete non-compact manifolds also exist, with negative or zero Ricci curvature.

The generalized Kerr-Schild transform in eleven-dimensional supergravity

The Kerr-Schild transformation in general relativity is proposed as a technique to generate vacuum solutions of higher-dimensional field theories coupled to gravity. In particular, it is shown that, in eleven-dimensional supergravity the superposition of the Freund-Rubin and the pp-wave metric yields a new class of exact solutions.

Geometric symmetry breaking.

The spontaneous symmetry breaking in higher-dimensional unified field theories is discussed in a general setting. A purely geometric symmetry-breaking mechanism involving no free parameters is presented, in which the symmetry is broken by the higher-dimensional metric through Einstein's equation.

Depletion of Fractional Fermion Number of a Soliton at Finite Chemical Potential and Temperature

We find an expression for the fractional fermion number of a soliton at finite fermion chemical potential ..mu.. and finite temperature T in the (1+1)-dimensional chiral field theory. We discuss the physical origin of the depletion of the soliton charge by finite ..mu.. and the relevance to lower-dimensional charge-density-wave systems and to topological solitons in higher-dimensional field theories.

Scalars coupled to fermions in 0+1 dimensions

A model of a Bose field coupled to a Fermi field is presented which shows interesting features that might arise in higher dimensional field theories. The model is partly analytically soluble. The remaining, part is investigated using a computer solution and some analytic approximations.