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.     

Exactly soluble diffeomorphism invariant theories

A class of diffeomorphism invariant theories is described for which the Hilbert space of quantum states can be explicitly constructed. These theories can be formulated in any dimension and include Witten's solution to 2+1 dimensional gravity as a special case. Higher dimensional generalizations exist which start with an action similar to the Einstein action inn dimensions. Many of these theories do not involve a spacetime metric and provide examples of topological quantum field theories. One is a version of Yang-Mills theory in which the only quantum states onS ³×R are the vacua. Finally it is shown that the three dimensional Chern-Simons theory (which Witten has shown is intimately connected with knot theory) arises naturally from a four dimensional topological gauge theory.On leave from the Department of Physics, University of California, Santa Barbara, CA, USA     

Chiral anomalies in even and odd dimensions

Odd dimensional Yang-Mills theories with an extra topological mass term, defined by the Chern-Simons secondary characteristic, are discussed. It is shown in detail how the topological mass affects the equal time charge commutation relations and how the modified commutation relations are related to non-abelian chiral anomalies in even dimensions. We also study the SU(3) chiral model (Wess-Zumino model) in four dimensions and we show how a gauge invariant interaction with an external SU(3) vector potential can be defined with the help of the Chern-Simons characteristic in five dimensions.     

Elliptic operators in the functional quantisation for gauge field theories

Given a gauge theory with gauge groupG acting on a path spaceX,G andX being both infinite dimensional manifolds modelled on spaces of sections of vector bundles on a compact riemannian manifold without boundary, it is shown that when the action ofG onX is smooth, free and proper, the same ellipticity condition on an operator naturally given by the geometry of the problem yields both the existence of a principal fibre bundle structure induced by the canonical projection :XX/G and the existence of the Faddeev-Popov determinant arising in the functional quantisation of the gauge theory. This holds for certain gauge theories with anomalies like bosonic closed string theory in non-critical dimension and also holds for a class of gauge theories which includes Yang-Mills theory.     

Unitary positive energy representations of the gauge group

We investigate the positive energy representations (also called highest weight representations) of the gauge groupC (T ^v,G ₀),G ₀ being a compact simple Lie group, and discuss their unitarity, using the technique of Verma modules constructed from generalized loop algebras (a simple generalization of Kac-Moody affine Lie algebras). We show that the unitarity of the representation imposes severa restrictions in it. In particular, we show, as a part of a more general result, that the gauge group does not admit faithful unitary positive energy representations.Allocataire du MRT.     

Local BRST cohomology in the antifield formalism: I. General theorems

We establish general theorems on the cohomologyH ^* (s/d) of the BRST differential modulo the spacetime exterior derivative, acting in the algebra of localp-forms depending on the fields and the antifields (=sources for the BRST variations). It is shown thatH ^–k (s/d) is isomorphic toH _k (/d) in negative ghost degree–k (k>0), where is the Koszul-Tate differential associated with the stationary surface. The cohomology groupH ₁ (/d) in form degreen is proved to be isomorphic to the space of constants of the motion, thereby providing a cohomological reformulation of Noether's theorem. More generally, the groupH _k (/d) in form degreen is isomorphic to the space ofn–k forms that are closed when the equations of motion hold. The groupsH _k (/d)(k>2) are shown to vanish for standard irreducible gauge theories. The groupH ₂ (/d) is then calculated explicitly for electromagnetism, Yang-Mills models and Einstein gravity. The invariance of the groupsH ^k (s/d) under the introduction of non-minimal variables and of auxiliary fields is also demonstrated. In a companion paper, the general formalism is applied to the calculation ofH ^k (s/d) in Yang-Mills theory, which is carried out in detail for an arbitrary compact gauge group.Supported by Deutsche Forschungsgemeinschaft     

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.     

The Faddeev-Popov procedure and application to bosonic strings: An infinite dimensional point of view

A generalisation of the finite dimensional presentation of the Faddeev-Popov perocedure is derived, in an infinite dimensional framework for gauge theories with finite dimensional moduli space using heat-kernel regularised determinants. It is shown that the infinite dimensional Faddeev-Popov determinant is-up to a finite dimensional determinant determined by a choice of a slice-canonically determined by the geometrical data defining the gauge theory, namely a fibre bundlePP/G with structure groupG and the invariance group of a metric structure given on the total spaceP. The case of (closed) bosonic string theory is discussed.     

Group duality and the Kubo-Martin-Schwinger condition

We consider clusteringG-invariant states of aC*-algebraU endowed with an action of a locally compact abelian groupG. Denoting as usual byF _AB,G _AB, the corresponding two-point functions, we give criteria for the fulfillment of the KMS condition (w.r.t. some one-parameter subgroup ofG) based upon the existence of a closable mapT such thatTF _AB =G _AB for allA,BU. Closability is either inL (G),B(G), orC (G), according to clustering assumptions. Our criteria originate from the combination of duality results for the groupG (phrased in terms of functions systems), with density results for the two-point functions.Supported in part by the National Science Foundation     

Quantum groupSU(1, 1) ⋊ Z2 and “super-tensor” products

A quantum analogue of the groupSU(1,1)Z ₂—the normalizer ofSU(1, 1) inSL ₂(C)—is introduced and studied. Although there isno correctly defined tensor product in the category of *-representations of the quantum algebraC[SU(1, 1)] _q of regular functions, some categories of *-representations ofC[SU(1, 1)Z ₂] _q turn out to be endowed with a certainZ ₂-graded structure which can be considered as a super-generalization of the monoidal category structure. This quantum effect may be considered as a step to understanding the concept of quantum topological locally compact group.In fact, there seems to be afamily of quantum groupsSU(1, 1)Z ₂ parameterized by unitary characters T ¹ of the fundamental group of the two-dimensional symplectic leaf ofSU(1, 1)/T, whereT is the subgroup of diagonal matrices.It is shown that thequasi-classical analogues of the results of the paper are connected with the decomposition of Schubert cells of the flag manifoldSL ₂(C)_R/B (whereB is the Borel subgroup of upper-triangular matrices) into symplectic leaves.Supported by the Rosenbaum Fellowship.     

Integration on then th power of a hyperbolic space in terms of invariants under diagonal action of isometries (Lorentz transformations)

The integral of a function over then'th power of hyperbolicd-dimensional spaceH is decomposed into integration along each orbit under diagonal action onH ⁿ of the isometry groupG onH, followed by integration over the orbit space, parametrized in terms of a complete set of invariants. The Jacobian entering in this last integral is expressed explicitly in terms of certain determinants. When viewingH as a half-hyperboloid in _d+1 ,G is induced by the homogeneous Lorentz groupO (1,d) acting on _d+1 .     

Primitive ideals of C q [SL(3)]

The primitive ideals of the Hopf algebraC _q [SL(3)] are classified. In particular it is shown that the orbits in PrimC _q [SL(3)] under the action of the representation groupH C ^*×C ^* are parameterized naturally byW×W, whereW is the associated Weyl group. It is shown that there is a natural one-to-one correspondence between primitive ideals ofC _q [SL(3)] and symplectic leaves of the associated Poisson algebraic groupSL(3,C).Partially supported by a grant from the N.S.A.     

Monopole charges for arbitrary compact gauge groups and Higgs fields in any representation

The topological invariants of monopoles are described for an arbitrary compact gauge groupG and Higgs field in any representation. The results generalize those obtained recently for compact and simply connectedG and in the adjoint representation. The cases when the residual symmetry group isH=U(1) orH=U(3) are worked out explicitly. This latter is needed to accommodate fractional electric charge with monopoles having one Dirac unit magnetic charge.The general theory is illustrated on the SU(5) monopole.     

Torsion Structure in Riemann-Cartan Manifold and Dislocation

The U(1) gauge structure of torsion and dislocation in three dimensional Riemann-Cartan manifold have been studied. The local topological structure of dislocation have been presented by so-called topological method in which the quantum number is by Hopf indices and Brouwer degree. Furthermore, the relationship between the dislocation lines and Wilson lines of the U(1) gauge theory is discussed by using the Chern-Simons theory.     

Chern-Simons-Witten invariants of lens spaces and torus bundles,and the semiclassical approximation

We derive explicit formulas for the Chern-Simons-Witten invariants of lens spaces and torus bundles overS ¹, for arbitrary values of the levelk. Most of our results are for the groupG=SU(2), though some are for more general compact groups. We explicitly exhibit agreement of the limiting values of these formulas ask with the semiclassical approximation predicted by the Chern-Simons path integral.Partially supported by an NSF Graduate FellowshipAddress as of September 1, 1991: School of Natural Science, Institute for Advanced Study, Princeton, NJ 08540; USA     

A Gauge Theoretical View of the Charge Concept in Einstein Gravity

We will discuss some analogies between internal gauge theories and gravity in order to better understand the charge concept in gravity. A dimensional analysis of gauge theories in general and a strict definition of elementary, monopole, and topological charges are applied to electromagnetism and to teleparallelism, a gauge theoretical formulation of Einstein gravity. As a result we inevitably find that the gravitational coupling constant has dimension /l ², the mass parameter of a particle dimension /l, and the Schwarzschild mass parameter dimension l (where l means length). These dimensions confirm the meaning of mass as elementary and as monopole charge of the translation group, respectively. In detail, we find that the Schwarzschild mass parameter is a quasi–electric monopole charge of the time translation whereas the NUT parameter is a quasi–magnetic monopole charge of the time translation as well as a topological charge. The Kerr parameter and the electric and magnetic charges are interpreted similarly. We conclude that each elementary charge of a Casimir operator of the gauge group is the source of a (quasi-electric) monopole charge of the respective Killing vector.     

The geometry of gauge fields

Principal fibre bundles with connections provide geometrical models of gauge theories. Bundles allow for a global formulation of gauge theories: the potentials used in physics are pull-backs, by means of local sections, of the connection form defined on the total spaceP of the bundle. Given a representationP of the structure (gauge) groupG in a vector spaceV, one defines a (generalized) Higgs field as a map fromP toV, equivariant under the action ofG inP. If the image of is an orbitW V ofG, then a breaks (spontaneously) the symmetry: the isotropy (little) group ofw ₀ W is the unbroken groupH. The principal bundleP is then reduced to a subbundleQ with structure groupH. Gravitation corresponds to a linear connection, i.e. to a connection on the bundle of frames. This bundle has more structure than an abstract principal bundle: it is soldered to the base. Soldering results in the occurrence of torsion. The metric tensor is a Higgs field breaking the symmetry fromGL (4,R) to the Lorentz group.Invited talk at the Symposium on Mathematical Methods in the Theory of Elementary Particles, Liblice castle, Czechoslovakia, June 18–23, 1978.Work on this paper was supported in part by the Polish Research Programme MR. I. 7.This paper is based in part on the research done in 1976–77 when I was Visiting Professor at the State University of New York at Stony Brook. I thankChen Ning Yang for encouragement, discussions and hospitality at the Institute for Theoretical Physics, SUSB. I have also learned much from conversations with D. Z.Freedman, A. S.Goldhaber, P.van Nieuwenhuizen, J.Smith, P. K.Townsend, W. I.Weisberger, and D.Wilkinson.     

Higher algebraic structures and quantization

We derive (quasi-)quantum groups in 2+1 dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the Chern-Simons theory with finite gauge group. The principles behind our computations are presumably more general. We extend the classical action in ad+1 dimensional topological theory to manifolds of dimension less thand+1. We then construct a generalized path integral which ind+1 dimensions reduces to the standard one and ind dimensions reproduces the quantum Hilbert space. In a 2+1 dimensional topological theory the path integral over the circle is the category of representations of a quasi-quantum group. In this paper we only consider finite theories, in which the generalized path integral reduces to a finite sum. New ideas are needed to extend beyond the finite theories treated here.The author is supported by NSF grant DMS-8805684, a Presidential Young Investigators award DMS-9057144, and by the O'Donnell Foundation. He warmly thanks the Geometry Center at the University of Minnesota for their hospitality while this work was undertaken     

Exact Results for Topological Strings on Resolved <Emphasis Type="Italic">Y</Emphasis><Superscript><Emphasis Type="Italic"> p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Superscript> Singularities

We obtain exact results in α′ for open and closed A-model topological string amplitudes on a large class of toric Calabi-Yau threefolds by using their correspondence with five dimensional gauge theories. The toric Calabi-Yaus that we analyze are obtained as minimal resolution of cones over Y ^p,q manifolds and give rise via M-theory compactification to SU(p) gauge theories on . As an application we present a detailed study of the local case and compute open and closed genus zero Gromov-Witten invariants of the orbifold. We also display the modular structure of the topological wave function and give predictions for higher genus amplitudes. The mirror curve in this case is the spectral curve of the relativistic A ₁ Toda chain. Our results also indicate the existence of a wider class of relativistic integrable systems associated to generic Y ^p,q geometries.