Topological interactions in gauge theories

Topological defects

Much of the relevant physics described by gauge theories is non-perturbative in nature; for example the spectrum, the study of confinement and the study of topological defects and their physical properties. The research in the group of Bais has in recent years focused on some - up to now - poorly understood features of topological defects in gauge theories such as their interactions. In particular, the interplay of topological and non-topological quantum numbers on fluxes and monopoles has been studied, also in view of the search for truly non-abelian dualities. In this context a lot has been learned from the study of non-abelian discrete gauge theories [18]. These feature topological (magnetic flux) sectors which are characterized by certain non-abelian quantum numbers (conjugacy classes to be specific), whereas the allowed dyonic sectors carry electric charges falling into representations of the stabilizer group of the conjugacy class. This yields an intricate interdependence of admissible electric and magnetic quantum numbers in the semi-classical excitation spectrum of the theory. Important was the discovery that this interplay is a manifestation of an underlying quantum group structure, whose representation theory exactly accounts for the structure just mentioned [16], [13].

It should be added that for the monopoles in phases with residual non-abelian continuous symmetries interesting results have been obtained [14], [11] which show that a similar picture emerges, but the precise mathematical structure has not yet been fully determined. In other words the algebraic and physical structure reflecting the rather well studied geometry of the moduli space is still an important topic.

Important was the realization that these features are quite generic and may lead to unusual physical phenomena.

Topological Interactions

Defects in phases with non-abelian residual symmetries, often exhibit nontrivial topological interactions with the other (fundamental and/or topological) degrees of freedom in the theory. These interactions are a consequence of the nontrivial connectivity properties of the solution space of a single defect.

From a physical point of view these topological interactions manifest themselves through the possibility of nontrivial entanglements and consequently of various scattering phenomena which can be described as non-abelian generalizations of the Ahoronov-Bohm effect [6], [13]. Another consequence is the possibility of exotic, non-abelian quantum statistics for collective excitations in certain phases of the theory, as occurring for example in the fractional quantum Hall effect [7]. Also the peculiar situation of general relativity in (2+1) dimensions - which as was shown by Witten, is a topological field theory - was studied from this perspective and the quantum symmetry underlying its structure was constructed [12], [10]. The Hopf algebra in question turned out to be the quantum double of the group SU(2). Recently we have shown how gravitational scattering of particles, with and without spin, can be treated consistently within this framework [2].

Broken Hopf-symmetry and confinement

We have mentioned the appearance of quantum symmetries, generally speaking Hopf-algebra's, which have the important feature that they allow one to treat topological (say magnetic) and ordinary (say color-electric) quantum-numbers on same footing. Having states labeled in this universal way it is interesting to study the question of duality and conceivable condensation phenomena.

We are at present rather excited about the phase classification of the non-abelian (topological) field theories and the many distinct but allowed types of confinement one may have. These phases are precisely characterized by the the breaking of the Hopf symmetry through the condensation of certain well defined bosonic order-parameter fields (electric, magnetic or both) in the theory [3],[4]. These findings may be linked to certain duality properties of the Hopf symmetries we have studied [9]. This phenomenon of the breaking of quantum-symmetries appears to be important and generic in 2 dimensional physics. We therefore plan to further explore this topic in the near future.

Core instabilities of monopoles

Interesting in the above context are phases of theories where different types of topological excitations can coexist, in which case the topological interactions between these will lead to rather exotic physical properties. As we showed long ago, topological excitations (monopoles an instantons) in these theories may exhibit core deformations and instabilities[17].

A simple but interesting theory of this type is Alice electrodynamics, a theory whose gauge group is the usual U(1) enhanced with a local realization of charge conjugation symmetry. One of the exotic properties of the theory is the emergence of the topological concept of "cheshire charge", a nonlocalizable manifestation of electric and/or magnetic charge. We have shown that this elusive concept for certain parameter ranges in the theory manifests itself through a core instability of magnetic monopoles, where the point defect decays into a ring-shaped object carrying non-localisable magnetic charge [5]. This is a line of research which we certainly intent to pursue, as there may have been phases in the early universe featuring these phenomena (SO(10) breaking to SU(5) for example). At the moment we are involved in doing lattice simulations to determine the quantum properties of these models, such as the phase diagram and its unusual screening properties in certain phases [1].

Future plans

Bais and collaborators will continue exploring the aspects of gauge theories in 2 and 3 dimensions, in which the interplay between topological and non-topological features leads to unusual physics. As these features are only beginning to be uncovered but appear to be generic, we consider it worthwhile to continue our research efforts along these lines. We think of lifting the discussion of quantumsymmetries and confinement to (3+1)-dimensions. Also applications of the breaking idea to (2+1)-dimensional gravity may lead to new insights. We want to fully understand the algebraic structure of moonopoles in phases with a nonabelian residual symmetry. We 'll study the charge instabilities that may occur on the quantum level due to the creation of alice strings.

Bibliography

1

J. Striet and F. A. Bais, "Simulations of Alice electrodynamics on a lattice" Nucl.Phys.B647:215-234,2002 [arXiv:hep-lat/0210009]

F. A. Bais and J. Striet, "Charge Instabilities due to local Charge Conjugation Symmetry in (2+1)-dimensions" ITFA-2003-20 (Apr 2003) 25p. [arXiv:hep-th/0304186]

2

F. A. Bais, N. M. Muller and B. J. Schroers, "Quantum group symmetry and particle scattering in (2+1)-dimensional quantum gravity" Nucl.Phys.B640:3-45,2002 [arXiv:hep-th/0205021]

3

F. A. Bais, J. K. Slingerland and B. J. Schroers, "Broken quantum symmetry and confimnement in planar physics" Phys.Rev.Lett.89:181601,2002. [arXiv:hep-th/0205117]

4

F. A. Bais, J. K. Slingerland and B. J. Schroers, "Hopf symmetry breaking and confinement in (2+1)-dimensionsal gauge theory", JHEP (to be published) [arXiv:hep-th/0205114]

5

F. A. Bais and J. Striet "On a core instability of magnetic monopoles", Phys.Lett.B540:319-323,2002 [arXiv:hep-th/0205152]

J. Striet and F. A. Bais, "More on core instabilities of Magnetic Monopoles" ITFA-2003-19 (Apr 2003) 15p. [arXiv:hep-th/0304189]

6

B. J. Overbosch and F. A. Bais, "Inequivalent classes of interference experiments with non-abelian anyons", Phys. Rev. A 64 (2001) 062107 [arXiv:quant-ph/0105015].

7

J. K. Slingerland and F. A. Bais, "Quantum groups and nonabelian braiding in quantum Hall systems," Nucl. Phys. B 612 (2001) 229 [arXiv:cond-mat/0104035].

8

J. Striet and F. A. Bais, "Simple models with Alice fluxes," Phys. Lett. B 497 (2000) 172 [arXiv:hep-th/0010236].

9

T. H. Koornwinder, B. J. Schroers, J. K. Slingerland and F. A. Bais, "Fourier transform and the Verlinde formula for the quantum double of a finite group," J. Phys. A 32 (1999) 8539 [arXiv:math.qa/9904029].

10

T. H. Koornwinder, F. A. Bais and N. M. Muller, "Tensor product representations of the quantum double of a compact group," Commun. Math. Phys. 198 (1998) 157 [arXiv:q-alg/9712042].

11

B. J. Schroers and F. A. Bais, "S-duality in Yang-Mills theory with non-abelian unbroken gauge group," Nucl. Phys. B 535 (1998) 197 [arXiv:hep-th/9805163].

12

F. A. Bais and N. M. Muller, "Topological field theory and the quantum double of SU(2)," Nucl. Phys. B 530 (1998) 349 [arXiv:hep-th/9804130].

13

M. De Wild Propitius and F. A. Bais, "Discrete Gauge Theories ," Published in Particles and Fields. Edited by G.W. Semenoff. Berlin, Germany, Springer Verlag, 1998, (CRM Series in Math. Physics), pp. 353-440.

14

F. A. Bais and B. J. Schroers, "Quantisation of monopoles with non-abelian magnetic charge," Nucl. Phys. B 512 (1998) 250 [arXiv:hep-th/9708004].

15

F. A. Bais, P. van Driel and M. de Wild Propitius, "Anyons in discrete gauge theories with Chern-Simons terms," Nucl. Phys. B 393 (1993) 547 [arXiv:hep-th/9203047].

16

F. A. Bais, P. van Driel and M. de Wild Propitius, "Quantum symmetries in discrete gauge theories," Phys. Lett. B 280 (1992) 63 [arXiv:hep-th/9203046].

17

F. A. Bais and P. John, "Core deformations of topological defects," Int. J. Mod. Phys. A 10 (1995) 3241.

J. Striet and F. A. Bais, "Dynamical Vacuum Selection in Field Theories with Flat Directions in their Potential" JHEP 0301:032,2003. [arXiv:hep-th/0211265]

18

F. A. Bais, "Flux Metamorphosis," Nucl. Phys. B 170 (1980) 32.