Cohomogeneity-one Einstein–Weyl structures: a local approach

We analyse in a systematic way the (non-) compact n-dimensional Einstein–Weyl spaces equipped with a cohomogeneity-one metric. In that context, with no compactness hypothesis for the manifold on which lives the Einstein–Weyl structure, we prove that, as soon as the (n−1)-dimensional space is a homogeneous reductive Riemannian space with a unimodular group of left-acting isometries G:

• •there exists a Gauduchon gauge such that the Weyl-form is co-closed and its dual is a Killing vector for the metric;
• •in that gauge, a simple constraint on the conformal scalar curvature holds;
• •a non-exact Einstein–Weyl structure may exist only if the (n−1)-dimensional homogeneous space G/H contains a non-trivial subgroup H′ that commutes with the isotropy subgroup H;
• •the extra isometry due to this Killing vector corresponds to the right-action of one of the generators of the algebra of the subgroup H′.

The first two results are well known when the Einstein–Weyl structure lives on a compact manifold, but our analysis gives the first hints on the enlargement of the symmetry due to the Einstein–Weyl constraint.We also prove that the subclass with G compact, a one-dimensional subgroup H′ and the (n−2)-dimensional space G/(H×H′) being an arbitrary compact symmetric Kähler coset space, corresponds to n-dimensional Riemannian locally conformally Kähler metrics. The explicit family of structures of cohomogeneity-one under SU(n/2) being, thanks to our results, invariant under U(1)×SU(n/2), it coincides with the one first studied by Madsen; our analysis allows us to prove most of his conjectures.     

Geometrical approach to the reduction of gauge theories with spontaneously broken symmetry

The reduction of a theory with gauge group G to a theory which is gauge invariant with respect to a subgroup H of G is formulated in a geometrical language. It is assumed that among the physical fields considered as cross-sections of fibre bundles with structure group G there exists a section of the fibre bundle with fibre isomorphic to G/H — a Higgs field. The investigation of the broken gauge symmetry is based on the reduction theorem for structure groups of principal fibre bundles. The reduction of fields and their covariant derivatives is studied.     

Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories

We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal G-bundle with connection and a class in H⁴(BG, ℤ) for a compact semi-simple Lie group G. The Chern-Simons bundle 2-gerbe realises differential geometrically the Cheeger-Simons invariant. We apply these notions to refine the Dijkgraaf-Witten correspondence between three dimensional Chern-Simons functionals and Wess-Zumino-Witten models associated to the group G. We do this by introducing a lifting to the level of bundle gerbes of the natural map from H⁴(BG, ℤ) to H³(G, ℤ). The notion of a multiplicative bundle gerbe accounts geometrically for the subtleties in this correspondence for non-simply connected Lie groups. The implications for Wess-Zumino-Witten models are also discussed.The authors acknowledge the support of the Australian Research Council. ALC thanks MPI für Mathematik in Bonn and ESI in Vienna and BLW thanks CMA of Australian National University for their hospitality during part of the writing of this paper.     

Symmetric space scalar field theory

Some quantum field theories, such as the chiral SU(2) ? SU(2) theory, can have a dynamics invariant under a group $\text{G}$ that is realized on a vacuum which is invariant only under a subgroup $\text{H}$ of $\text{G}$. These theories may be defined by scalar fields which are coordinates for the coset manifold $\text{G}$/$\text{H}$. They are thus non-polynomial theories on a symmetric space, with the group motions in this space described by a set of Killing vectors. We show how the Lagrange function may be constructed entirely from the Killing vectors. In particular, all physical quantities may be expressed in terms of the currents formed out of the Killing vectors. The current correlation functions do not exhibit the spurious wave function renormalizations which are encountered if ordinary Green's functions are computed. We illustrate the general method by calculating one-loop counter terms in a completely invariant fashion. An Appendix describes in simple terms the general theory of symmetric spaces, which should prove useful in other contexts.     

BFV quantization on hermitian symmetric spaces

Gauge-invariant BFV approach to geometric quantization is applied to the case of hermitian symmetric spaces G/H. In particular, gauge invariant quantization on the Lobachevski plane and sphere is carried out. Due to the presence of symmetry, master equations for the first-class constraints, quantum observables and physical quantum states are exactly solvable. BFV-BRST operator defines a flat G-connection in the Fock bundle over G/H. Physical quantum states are covariantly constant sections with respect to this connection and are shown to coincide with the generalized coherent states for the group G. Vacuum expectation values of the quantum observables commuting with the quantum first-class constraints reduce to the covariant symbols of Berezin. The gauge-invariant approach to quantization on symplectic manifolds synthesizes geometric, deformation and Berezin quantization approaches.     

<Emphasis Type="Italic">SU</Emphasis>(3)-Instantons and <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>,<Emphasis Type="Italic">Spin</Emphasis>(7)-Heterotic String Solitons

Necessary and sufficient conditions to the existence of a hermitian connection with totally skew-symmetric torsion and holonomy contained in SU(3) are given. A formula for the Riemannian scalar curvature is obtained. Non-compact solution to the supergravity-type I equations of motion with non-zero flux and non-constant dilaton is found in dimension 6. Non-conformally flat non-compact solutions to the supergravity-type I equations of motion with non-zero flux and non-constant dilaton are found in dimensions 7 and 8. A Riemannian metric with holonomy contained in G₂ arises from our considerations and Hitchin’s flow equations, which seems to be new. Compact examples of SU(3),G₂ and Spin(7) instanton satisfying the anomaly cancellation conditions are presented.     

Projective and volume-preserving bundle structures involved in the formulation ofA(4) gauge theories

The bundle structures required by volume-preserving and related projective properties are developed and discussed in the context ofA(4) gauge theories which may be taken as the proper framework for Poincaré gauge theories. The results of this paper include methods for extending both tensors and connections to a principal fiber bundle havingG1(4,R)xG1(4,R) as its structure group. This bundle structure is shown to be a natural arena for the generalized (±) covariant differentiation utilized by Einstein for his extended gravitational theories involving nonsymmetric connections. In particular, it is shown that this generalized (±) covariant differentiation is actually a special case of ordinary covariant differentiation with respect to a connection on theG1(4,R) xG1(4,R) bundle. These results are discussed in relation to certain properties of generalized gravitational theories based on a nonsymmetric connection which include the metric affine theories of Hehl et al. and the general requirement that it should be possible to formulate well-defined local conservation laws. In terms of the extended bundle structure considered in this paper, it is found that physically distinct particle number type conservation expressions could exist for certain given types of matter currents.     

Group actions on quantum bundles

Suppose we are given a group G acting through canonical transformations on a symplectic manifold (M, ω). If there is a quantum bundle over (M, ω), a carrier for wave functions in the geometric quantization theory, then G acts infinitesimally on the bundle in a natural way. We give a necessary and sufficient condition for the infinitesimal G-action to integrate up to a global G-action. This is used for an investigation how the choice of the quantum bundle over (M, ω) influences the integrability of the corresponding infinitesimal G-action. The relationship to group representations is briefly mentioned.     

Semi-classical spectrum of the homogeneous sine-Gordon theories

The semi-classical spectrum of the homogeneous sine-Gordon theories associated with an arbitrary compact simple Lie group G is obtained and shown to be entirely given by solitons. These theories describe quantum integrable massive perturbations of Gepner's G-parafermions whose classical equations-of-motion are non-abelian affine Toda equations. One-soliton solutions are constructed by embeddings of the SU(2) complex sine-Gordon soliton in the regular SU(2) subgroups of G. The resulting spectrum exhibits both stable and unstable particles, which is a peculiar feature shared with the spectrum of monopoles and dyons in N = 2 and N = 4 supersymmetric gauge theories.     

Quantum groups as generalized gauge symmetries in WZNW models. Part II. The quantized model

This is the second part of a paper dealing with the “internal” (gauge) symmetry of the Wess–Zumino–Novikov–Witten (WZNW) model on a compact Lie group G. It contains a systematic exposition, for G = SU(n), of the canonical quantization based on the study of the classical model (performed in the first part) following the quantum group symmetric approach first advocated by L.D. Faddeev and collaborators. The internal symmetry of the quantized model is carried by the chiral WZNW zero modes satisfying quadratic exchange relations and an n-linear determinant condition. For generic values of the deformation parameter the Fock representation of the zero modes’ algebra gives rise to a model space of U _q (sl(n)). The relevant root of unity case is studied in detail for n = 2 when a “restricted” (finite dimensional) quotient quantum group is shown to appear in a natural way. The module structure of the zero modes’ Fock space provides a specific duality with the solutions of the Knizhnik–Zamolodchikov equation for the four point functions of primary fields suggesting the existence of an extended state space of logarithmic CFT type. Combining left and right zero modes (i.e., returning to the 2D model), the rational CFT structure shows up in a setting reminiscent to covariant quantization of gauge theories in which the restricted quantum group plays the role of a generalized gauge symmetry.     

A note on coherent states

Given a connected Lie groupG with an Abelian invariant Lie subgroup and a continuous unitary representation ofG on the Hilbert space ?, we investigate a relationship between the first cohomology groupH ¹(G, ?) and classes of sectors, determined by coherent states with a projectivelyG-covariant Weyl system. This result is applied to calculateH ¹(G, ?), if the groupG has in addition a compact subgroup with certain properties.     

Gauge theories with non-unitary parallel transporters: a soluble Higgs model<Superscript>*</Superscript>

It has been proposed to abandon the requirement that parallel transporters in gauge theories are unitary (or pseudo-orthogonal). This leads to a geometric interpretation of Vierbein fields as parts of gauge fields, and non-unitary parallel transport in extra directions yields Higgs fields. In such theories, the holonomy group H is larger than the gauge group G. Here we study a one-dimensional model with fermions which retains only the extra dimension, and which is soluble in the sense that its renormalization group flow may be exactly computed, with G = SU(2) and non-compact , or G = U(2), H = GL(2,C). In all cases the asymptotic behavior of the Higgs potential is computed, and with one possible exception for G = SU(2), H = GL(2,C), there is a flow of the action from a UV fixed point which describes a SU(2)-gauge theory with unitary parallel transporters, to a IR fixed point. We explain how exponential mass ratios of fermions of different flavor can arise through spontaneous symmetry breaking, within the general framework.Received: 2 June 2003, Revised: 14 September 2004, Published online: 21 January 2005PACS: 11.10.Hi, 11.10.Kk, 11.15.Ex, 11.15.Tk, 12.15.Ff, 12.15.HhWork supported by Deutsche Forschungsgemeinschaft.     

Generalized Voigt functions and their derivatives

This paper deals with a special class of functions called generalized Voigt functions H⁽ⁿ⁾(x,a) and G⁽ⁿ⁾(x,a) and their partial derivatives, which are useful in the theory of polarized spectral line formation in stochastic media. For n=0 they reduce to the usual Voigt and Faraday-Voigt functions H(x,a) and G(x,a). A detailed study is made of these new functions. Simple recurrence relations are established and employed for the calculation of the functions themselves and of their partial derivatives. Asymptotic expansions are given for large values of x and a. They are used to examine the range of applicability of the recurrence relations and to construct a numerical algorithm for the calculation of the generalized Voigt functions and of their derivatives valid in a large (x,a) domain. It is also shown that the partial derivatives of the usual H(x,a) and G(x,a) can be expressed in terms of H⁽ⁿ⁾(x,a) and G⁽ⁿ⁾(x,a).     

On a microscopic representation of space-time

We start from a noncompact Lie algebra isomorphic to the Dirac algebra and relate this Lie algebra in a brief review to low-energy hadron physics described by the compact group SU(4). This step permits an overall physical identification of the operator actions. Then we discuss the geometrical origin of this noncompact Lie algebra and ??reduce?? the geometry in order to introduce in each of these steps coordinate definitions which can be related to an algebraic representation in terms of the spontaneous symmetry breakdown along the Lie algebra chain su*(4) ?? usp(4) ?? su(2) × u(1). Standard techniques of Lie algebra decomposition(s) as well as the (physical) operator identification give rise to interesting physical aspects and lead to a rank-1 Riemannian space which provides an analytic representation and leads to a 5-dimensional hyperbolic space H ₅ with SO(5, 1) isometries. The action of the (compact) symplectic group decomposes this (globally) hyperbolic space into H ₂ ?? H ₃ with SO(2, 1) and SO(3, 1) isometries, respectively, which we relate to electromagnetic (dynamically broken SU(2) isospin) and Lorentz transformations. Last not least, we attribute this symmetry pattern to the algebraic representation of a projective geometry over the division algebra H and subsequent coordinate restrictions.     

Non-Perturbative Heat Kernel Asymptotics on Homogeneous Abelian Bundles

We study the heat kernel for a Laplace type partial differential operator acting on smooth sections of a complex vector bundle with the structure group G × U(1) over a Riemannian manifold M without boundary. The total connection on the vector bundle naturally splits into a G-connection and a U(1)-connection, which is assumed to have a parallel curvature F. We find a new local short time asymptotic expansion of the off-diagonal heat kernel U(t|x, x′) close to the diagonal of M × M assuming the curvature F to be of order t ⁻¹. The coefficients of this expansion are polynomial functions in the Riemann curvature tensor (and the curvature of the G-connection) and its derivatives with universal coefficients depending in a non-polynomial but analytic way on the curvature F, more precisely, on tF. These functions generate all terms quadratic and linear in the Riemann curvature and of arbitrary order in F in the usual heat kernel coefficients. In that sense, we effectively sum up the usual short time heat kernel asymptotic expansion to all orders of the curvature F. We compute the first three coefficients (both diagonal and off-diagonal) of this new asymptotic expansion.     

From chiral to two-dimensional Wess-Zumino-Novikov-Witten model via quantum gauge group

We study the canonical quantization of the SU(n) WZNW model. Decoupling the chiral dynamics requires an extended state space including left and right monodromies as independent variables. In the simplest (n = 2) case we explicitly show that the zero modes of the monodromy extended SU(2) WZNW model give rise to a quantum group gauge theory in a finite-dimensional Fock space. We define the subspace of U_q(sl(2)) ⊗ U_q(sl(2))-invariant vectors on which the monodromy invariance is also restored and construct the physical space applying a generalized cohomology condition.     

Octonionic Non-Abelian Gauge Theory

We have made an attempt to describe the octonion formulation of Abelian and non-Abelian gauge theory of dyons in terms of 2×2 Zorn vector matrix realization. As such, we have discussed the U(1) _e ×U(1) _m Abelian gauge theory and U(1)×SU(2) electroweak gauge theory and also the SU(2) _e ×SU(2) _m non-Abelian gauge theory in term of 2×2 Zorn vector matrix realization of split octonions. It is shown that SU(2) _e characterizes the usual theory of the Yang Mill’s field (isospin or weak interactions) due to presence of electric charge while the gauge group SU(2) _m may be related to the existence of ’t Hooft-Polyakov monopole in non-Abelian Gauge theory. Accordingly, we have obtained the manifestly covariant field equations and equations of motion.     

Extensions of representations and cohomology

In this article we study the extensions of Banach space representations of a Lie group G. We introduce different spaces of 1-cohomology on G, or on its Lie algebra $\text{G}$, and make the connection between these spaces and the equivalence (or weak equivalence) classes of extensions.We characterize, from the properties of the 1-cohomology groups, the spaces of differentiable and analytic vectors of an extension and prove a kind of Whitehead's lemma.For Lie groups with a large compact subgroup K, we specialize to K-finite representations, and introduce and study Naimark equivalence of extensions.The results are applied to classify the extensions of the irreducible representations of $\text{G =}\text{SL}\text{(2,}\text{R}\text{)}$.     

Construction of grand unified models under maximal subalgebras

A construction of grand unified models of the strong, weak and electromagnetic interactions is described based on the transformation properties of the group generators under a maximal subgroup decomposition without recourse to large representation matrices or to the specific algebraic structures of some classical Lie-groups, such as the Clifford algebra associated with the orthogonal groups or the octonionic structure of the exceptional groups. To illustrate the procedure an explicit construction is given of the SU(5) model useful in the discussion of higher rank groups, of SO(10) under the maximal subalgebras SU(2)_L × SU(2)_R × SU(4)_c and SU(5) × U(1)_r and of the exceptional group E₆ under SU(3)_L × SU(3)_R × SU(3)_c and SO(10) × U(1)_t. The construction procedure can be used as well with any classical Lie-group.