Lectures , pages : Geometry of gauge fields notes on this are kind of sketchy , abelian Higgs model and vortices, local discrete symmetry, anyons , abelian Chern -Simons theory, fractional quantum Hall effect. Lectures , pages : Magnetic Cheshire charge, Strings ending on monopoles, walls bounded by strings, topological classification of gauge fields, cohomology with integer coefficients, U 1 bundles, first Chern class, torsion classes, flat connections on nonorientable manifolds, G-bundles.

Lectures , pages : Yang-Mills theory and its quantization, theta vacua and CP nonconservation , theta-dependent dyon charge, nonabelian monopoles and global gauge transformations. Part IV: Anomalies. Lecture 32, pages : Chiral anomaly in two and four dimensions as chiral pair production in electromagnetic fields, vector and axial Ward identities, anomalies and massless particles. Chapters Chapter 1, Quantum Mechanics: states, observables, measurements, dynamics, spectra. Chapter 2, Time-Dependent Scattering Formalism: asymptotic states, wave operators, S-matrix, cross section, optical theorem.

Chapter 3, Analytic functions: derivative, integrals, power series, residue theorem, analytic continuation, Riemann surfaces. This was achieved by considering more general groups SU N , where the number of gauge fields is N 2 Even though considered a very nice theoretical idea, the first successful application of the Yang-Mills fields just occurred almost fifteen years later , in the consistent description of the weak interaction in a unified theory also involving QED. It is opportune to mention that the experimental observation of weak gauge fields took fifteen years more, after the construction of powerful accelerators.

Later on, the strong interaction was also formulated as a Yang-Mills theory, whose symmetry group is the SU 3 and the basic ingredients are not protons and neutrons, as in the old Nuclear Physics, but quarks and gluons. The standard model and problems related with confinement, vacuum QCD etc. We also mention that problems related to the Casimir Effect have increased of interest in recent years [5,6,7,8,9,10,11,12].

The path integral formalism in Field Theory has a very high resemblance with the partition function in Statistical Physics. The intersection of these initially distinct subjects leads to a fruitful line of research because the knowledge of each one could be used into the other. The so called Quantum Field Theory at Finite Temperature , or Thermal Field Theory , has always been an interesting research subject [13,14,15,16,17,18,19,20,21,22,23,24]. In the meeting of this year, it was devoted a parallel talk on this subject, involving the Derivative expansion technique and Chern-Simons theories [25].

There are many interesting research areas at this part, involving Chern-Simons theories [26,27,28,29,30,31,32,33,34,35,36,37,38,39], Anyons and Fractons [18,40], Nonlinear Sigma-model [41,42,43,44], Schwinger and Chiral-Schwinger models [45,46] etc. Further, they have also connection with the real world, in a research area involving condensed matter with many interesting applications [72,28,73,74,75,76,77,78,79].

It was also devoted a parallel talk on this subject [80]. After the success of the field quantization method in the weak, electromagnetic and strong interactions, the most natural step would be to use the same quantization rules into the Einstein theory of gravity. This did not work! The main reason is that quantum field theories deal with many infinite quantities which are either simple discarded, as the vacuum energy, or are intelligent circumvented the renormalization program. Both these procedures cannot be applied to the gravity theory. First because sources of energy cannot be simple discarded in presence of gravity and second because the renormalization program simply does not work.

A first attempt to circumvent this problem was to follow the same idea before the advent of QED, that is, just quantum matter fields were quantized and interacting with a classical electromagnetic background. The corresponding research area of quantum matter and quantum gauge fields propagating in a classical gravitational background, and the problems related with geometrization, lead to a very interesting developments and constitute a very fruitful research area [81,82,83,84,85,86,87,88,89,90,91]. The current idea on this subject is that there must exist some more general formulation for the gravitational theory where the quantization procedures can be applied.

The first consistent attempt was based on the supersymmetry , that is an extension of the Lorentz symmetry where the spacetime also has fermionic degrees of freedom. The corresponding supersymmetric theory of gravity, called supergravity , was then formulated and it contains the Einstein gravitational theory as a particular case.

Both supersymmetry and supergravity always deal with interesting research problems to be solved [92,93,94,95,96,97,98,30,90,,,]. However, the problems related with the infinities of quantum gravity were not completely solved with supergravity. An important step was done based on the idea that fields could not depend on points considered to be a mathematical idealization , but on extended objects.

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## Effective hydrodynamic field theory and condensation picture of topological insulators

Strings are the simplest extended objects. However, a field theory where fields are functions or functionals of strings is very difficult to be handled. What remained is the string idea itself, where elementary particles are not points, but vibrational states of strings. More general extended objects, the branes, were also considered, but it is opportune to say that they are not exactly the same of the modern branes.

These are related to boundary surfaces described by strings. Anyway, string and branes with their supersymmetric version are a very fruitful research area [,,,,,87,,,,,,,]. There was a plenary talk involving string theory and noncommutative fields []. The mathematical structures of strings and branes are much more involved and their quantization led to a great development in the quantization methods and in the study of anomalies [,,,,,,,,,,,,,,,].

The treatment of constrained systems Dirac, symplectic, Senjanovic etc. Mathematical structures and their algebras, as well as problems related to Lax pairs, KP hierarchy, integrable models acquired much interest [,,,,,34,,,]. This might means that there exists some compactification procedure which leads to our spacetime dimension [28,]. There is another interesting aspect in super string theories. At first, it appeared to exist five independent theories for them. It is important to emphasize that supergravity is again a very interesting research area. In this line of research, problems related to duality [,,,,39,] and topology [,98,,,,,,,,] have a great deal of interest.

After a brilliant work due to J. Maldacena, published in , the old idea and dream of a total unification and to have a unique theory able to describe everything is again in evidence. As it was said, there were problems to include gravitation in the family of quantum gauge theories.

On the other hand, after the advent of strings, the problem has changed in a reversed way. String theories naturally contain gravity, but the difficulty was to include Maxwell and Yang-Mills theories into this formalism. This line of research, including the recent works of Lisa Randall and Raman Sundrum and the possibility of noncommutative fields, is the most recent subjects in quantum field theory [,,,,,]. Arcuri and M. Srivastava and S. Brodsky, Standard model of electro-weak interactions in light-cone gauge: a renormalizable, unitary, and ghost-free theory.

Alves, C. Farina, P. Maia Neto, and A. Tort, Dynamical Casimir effect with Neumann condition at finite temperature. Matos Neto, A. Santana, and F. Khanna, Generalized Bogoliubov transformation in thermofield dynamics and the Casimir effect. Rodrigues and N. Svaiter, Vacuum stress tensor of a scalar field in a retangular waveguide.

Nogueira, On the effects of homogeneous Dirichlet's boundary conditions in the spontaneous symmetry breaking of a real scalar field. Mendes and A. Cougo-Pinto, C. Farina, and J. Mendes, Casimir effect in kappa deformed electrodynamics. Farina, A. Tort, F.

Farias, and M. Ribeiro, Casimir energy of Dirac field in external magnetic field by the method of generalized zeta function. Silva, Alternative dimensional reduction. Cabral and E. Ramos, Testing quantum field theory methods in experiments of Bose-Einstein condensation. Leite and L. Ferreira Jr. Del Cima, and J. Borges, H. Boschi Filho, and M. Machado and F. Moutinho, W. Moura-Neto, S. Thomaz, Modelo de Thirring a altas temperaturas. Caldas, The damping rate of a fermion within the linear sigma model at finite temperature. Moura-Melo, S. Rojas, and M. Pacheco and R. Queiroz and J.

Belich, M. Moura-Melo, Some consequences of the quantum violation of the Huyghens principle on the behavior of radiation in Maxwell and Maxwell-Chern-Simons models in three dimensions. Barcelos-Neto and E. Oxman and S. Sorella, Vortex correlation functions in nonlocal Maxwell-Chern-Simons models. Nogueira, J. Boldo, O. Piguet, and A. Boldo, L. Delpupo, J. Boldo, and O. Fernandes and D. Franco, Asymptotic scale invariance of a non-Abelian Chern-Simons matter model. Paschoal, Quantum mechanical aspects of charged particles in Maxwell-Chern-Simons electrodynamics.

Scarpelli and J. Santos, and P. Trajtenberg, Probing the symplectic projector method of quantization in planar gauge models. Crispim, R. Landim, and C. Mendes, R. Bazeia, A. Ilha, J. An algebraic technique of separation of gauge modes in Abelian gauge theories on homogeneous spaces is proposed. An effective potential for the Maxwell- Chern-Simons theory on S 3 is calculated. Starting from gravity as a Chern-Simons action for the AdS algebra in five dimensions, it is possible to modify the theory through an expansion of the Lie algebra that leads to a system consisting of the Einstein-Hilbert action plus non-minimally coupled matter.

The modified system is gauge invariant under the Poincare group enlarged by an Abelian ideal. Although the resulting action naively looks like general relativity plus corrections due to matter sources, it is shown that the non-minimal couplings produce a radical departure from GR. Indeed, the dynamics is not continuously connected to the one obtained from Einstein-Hilbert action. In a matter-free configuration and in the torsionless sector, the field equations are too strong a restriction on the geometry as the metric must satisfy both the Einstein and pure Gauss-Bonnet equations.

In particular, the five-dimensional Schwarzschild geometry fails to be a solution; however, configurations corresponding to a brane-world with positive cosmological constant on the worldsheet are admissible when one of the matter fields is switched on. These results can be extended to higher odd dimensions. Dynamics of Chern-Simons vortices.

We study vortex dynamics in three-dimensional theories with Chern-Simons interactions. The dynamics is governed by motion on the moduli space M in the presence of a magnetic field.

For Abelian vortices, the magnetic field is shown to be the Ricci form over M; for non- Abelian vortices, it is the first Chern character of a suitable index bundle. We derive these results by integrating out massive fermions and following the fate of their zero modes. Kaehler- Chern-Simons theory and symmetries of anti-self-dual gauge fields. Kaehler- Chern-Simons theory , which was proposed as a generalization of ordinary Chern-Simons theory , is explored in more detail.

The theory describes anti-self-dual instantons on a four-dimensional Kaehler manifold. The phase space is the space of gauge potentials, whose symplectic reduction by the constraints of anti-self-duality leads to the moduli space of instantons. We show that infinitesimal Baecklund transformations, previously related to 'hidden symmetries' of instantons, are canonical transformations generated by the anti-self-duality constraints.

## Fractional quantum Hall effect via holography: Chern-Simons, edge states and hierarchy

The quantum wave functions naturally lead to a generalized Wess-Zumino-Witten action, which in turn has associated chiral current algebras. The dimensional reduction of the anti-self-duality equations leading to integrable two-dimensional theories is briefly discussed in this framework. Physically meaningful and not so meaningful symmetries in Chern-Simons theory. We explicitly show that the Landau gauge supersymmetry of Chern-Simons theory does not have any physical significance. In fact, the difference between an effective action both BRS invariant and Landau supersymmetric and an effective action only BRS invariant is a finite field redefinition.

Finally, to convince ourselves that the shift above is not an accident of our regularization method, we comment on the fact that all BRS invariant regulators used as yet yield the same value for the shift. Existence of local degrees of freedom for higher dimensional pure Chern-Simons theories. The canonical structure of higher dimensional pure Chern-Simons theories is analyzed.

It is shown that these theories have generically a nonvanishing number of local degrees of freedom, even though they are obtained by means of a topological construction. This number of local degrees of freedom is computed as a function of the spacetime dimension and the dimension of the gauge group. Perturbative expansion of Chern-Simons theory with non-compact gauge group. Naive imitation of the usual formulas for compact gauge group in quantizing three dimensional Chern-Simons gauge theory with non-compact gauge group leads to formulas that are wrong or unilluminating.

In this paper, an appropriate modification is described, which puts the perturbative expansion in a standard manifestly 'unitary' format. The one loop contributions which differ from naive extrapolation from the case of compact gauge group are computed, and their topological invariance is verified. Skein relations and Wilson loops in Chern-Simons gauge theory. These relations can be used recursively to calculate the expectation values of Wilson lines in three-dimensional Chern-Simons gauge theory with these gauge groups. A combination of braiding and tying of Wilson lines completely describes the skein relations.

On the phase of Chern-Simons theory with complex gauge group. We compute the eta function for Chern-Simons quantum field theory with complex gauge group. The calculation is performed using the Schwinger expansion technique. We discuss, in particular, the role of the metric on the field configuration space, and demonstrate that for a certain class of acceptable metrics the one-loop phase contribution to the effective action can be calculated explicitly.

The result is found to be proportional to a gauge invariant part of the action. Exact solubility of Chern-Simons theory with compact simple gauge group. Interestingly enough, vacuum expectation values for unknotted Wilson loop operators in any representation of any compact and simple group are exactly computed by solving the equations. So-called 'skein relations', which give us algebraic equations among vacuum expectation values of different Wilson loop operators, are constructed. In our formalism, quantum group symmetry appears naturally.

Supersymmetry, quantum gauge anomalies and generalized Chern-Simons terms in chiral gauge theory. The purpose of this thesis is to investigate the interplay of anomaly cancellation and generalized Chern-Simons terms in four-dimensional chiral gauge theory. We start with a detailed discussion of generalized Chern-Simons terms with the canellation of anomalies via the Green-Schwarz mechanism.

Two simple consistency conditions are shown to encode strong constraints on the allowed anomalies for different types of gauge groups. In one major part of this thesis we are going to display to what extent one has to modify the existing formalism in order to allow for the cancellation of quantum gauge anomalies via the Green-Schwarz mechanism. At the end of this thesis we comment on a puzzle in the literature on supersymmetric field theories with massive tensor fields. The potential contains a term that does not arise from eliminating an auxiliary field.

We clarify the origin of this term and display the relation to standard D-term potential. In an appendix it is explicitly shown how these low energy effective actions might be connected to the formulation of four-dimensional gauge theories discussed at earlier stages of this thesis. We have studied the existence of self-dual solitonic solutions in a generalization of the Abelian Chern-Simons -Higgs model. Also, we have shown that by properly choosing the generalizing functions it is possible to reproduce the Bogomolnyi equations of the Abelian Maxwell-Higgs and Chern-Simons -Higgs models.

Multi-boundary entanglement in Chern-Simons theory and link invariants. The entanglement entropies of the resulting states define framing-independent link invariants which are sensitive to the topology of the chosen link. The formula involves the number of solutions to certain Diophantine equations with coefficients related to the Gauss linking numbers mod k between the two sublinks.

This formula connects simple concepts in quantum information theory , knot theory , and number theory , and shows that entanglement entropy between sublinks vanishes if and only if they have zero Gauss linking mod k. We show that the 2-component Hopf link is maximally entangled, and hence analogous to a Bell pair, and that the Whitehead link, which has zero Gauss linking, nevertheless has entanglement entropy.

Self-dual Maxwell- Chern-Simons theory on a cylinder. In this paper, we study the relativistic Maxwell- Chern-Simons vortices on an asymptotically flat cylinder. A topological multivortex solution is constructed by variational methods, and the Maxwell and the Chern-Simons limits are verified. The expectation value at strong coupling is found by constructing the string theory duals of these operators. For low dimensional representations these are fundamental strings, for high dimensional representations these are D2-branes and D6-branes.

In support of this identification we demonstrate that these string theory solutions match the symmetries, charges and the preserved supersymmetries of their Chern-Simons theory counterparts.

### Bubbling Solutions for the SU(3) Chern‐Simons Model on a Torus

A Chern-Simons -like action for closed-string field theory. A Chern-Simons -like action is proposed for closed-string field theory. Two-dimensional Lorentz-Weyl anomaly and gravitational Chern-Simons theory. Two-dimensional chiral fermions and bosons, more generally conformal blocks of two-dimensional conformal field theories , exhibit Weyl-, Lorentz- and mixed Lorentz-Weyl anomalies. A novel way of computing these anomalies for a system of chiral bosons of arbitrary conformal spin j is sketched. It is shown that the Lorentz- and mixed Lorentz-Weyl anomalies of these theories can be cancelled by the anomalies of a three-dimensional classical Chern-Simons action for the spin connection, expressed in terms of the dreibein field.

Some tentative applications of this result to string theory are indicated. Perturbed Chern-Simons theory , fractional statistics, and Yang-Baxter algebra. Topological Chern-Simons theory coupled to matter fields is analysed in the framework of Dirac's method of quantising constrained systems in a general class of linear, non-local gauges. We show that in the weak coupling limit gauge invariant operators in the theory transform under an exchange according to a higher dimensional representation of the braid group which is built out of the fundamental representation matrices of the gauge group and thus behave like anyons.

We also discover new solutions of the Yang-Baxter equation which emerges as a consistency condition on the structure functions of the operator algebra of the matter fields. Time dependent solitons of noncommutative Chern-Simons theory coupled to scalar fields. We study one- and two-soliton solutions of noncommutative Chern-Simons theory coupled to a nonrelativistic or a relativistic scalar field. In the nonrelativistic case, we find a tower of new stationary time-dependent solutions, all with the same charge density, but with increasing energies.

The dynamics of these solitons cannot be studied using traditional moduli space techniques, but we do find a nontrivial symplectic form on the phase space indicating that the moduli space is not flat. In the relativistic case we find the metric on the two soliton moduli space. Observables, skein relations, and tetrahedra in Chern-Simons gauge theory. The observables in three-dimensional Chern-Simons gauge theory are Wilson lines and Wilson graphs. Skein relations are non-trivial identities between expectation values of distinct Wilson graphs.

We discuss various kinds of skein relations and the relationships between them. By comparing different kinds of skein relations, we show how to calculate the expectation value of a general tetrahedral Wilson graph. This is shown to be the last and most difficult step in a systematic procedure for calculating the expectation values of arbitrary Wilson graphs in arbitrary representations of arbitrary gauge groups. Effective actions for gauge theories with Chern-Simons terms - I. The effective Lagrangian for a three-dimensional gauge theory with a Chern-Simons term is evaluated upto one-loop effects.

It is shown to be completely finite. It also does not exhibit any imaginary part. The calculation is carried out in a background field analogue of the Feynman gauge and gauge invariance is maintained throughout the calculation. In an appendix an argument is presented as to why this Feynman gauge may be a 'good' gauge for our results to be applied to high temperature QCD and in particular to the quark-gluon plasma. Despite the technical complexity of the Lovelock Lagrangian we obtain a remarkably simple expression for the variation of the charges ensuing from the diffeomorphism covariance of the theory.

The viability of the result is tested in specific applications and the formal e Self-duality in Maxwell- Chern-Simons theories with non minimal coupling with matter field. We consider a general class of non-local MCS models whose usual minimal coupling to a conserved current is supplemented with a non-minimal magnetic Pauli-type coupling. We find that the considered models exhibit a self-duality whenever the magnetic coupling constant reaches a special value: the partition function is invariant under a set of transformations among the parameter space the duality transformations while the original action and its dual counterpart have the same form.

We present a classical ISO 2,1 Chern-Simons gauge theory for planar gravity coupled to point-like sources. The theory is defined in terms of flat coordinates whose relation with the space-time coordinates is established. Though flat, the theory is equivalent to Einstein's as we show explicitly in two examples. The relationship between the 3-algebras, Lie superalgebras, Lie algebras, and embedding tensors proposed in [E.

Bergshoeff, O. Hohm, D. Roest, H. Samtleben, and E. Sezgin, J.

## Vortex Condensates¶for the SU(3) Chern–Simons Theory

High Energy Phys. Periodic electromagnetic vacuum in the two-dimensional Yang-Mills theory with the Chern-Simons mass. The periodic vacuum structure formed from magnetic and electric fields is derived in the two-dimensional Yang-Mills theory with the Chern-Simons term. It is shown that both the magnetic flux quantization in the fundamental sell and conductivity quantization inherent to the vacuum.

Hence, the quantum Hall effect gets its natural explanation. New reparametrization invariant p-brane actions and their Moyal star product deformations follows. It is conjectured that topological Chern-Simons brane actions associated with higher-dimensional "knots" have a one-to-one correspondence with topological Chern-Simons Matrix models in the large N limit. The corresponding large N limit of Topological BF Matrix models leads to Kalb-Ramond couplings of antisymmetric-tensor fields to p-branes.

We conclude by arguing why this interplay between condensed matter models, higher-dimensional extensions of the Quantum Hall effect, Chern-Simons Matrix mod Our analysis is based on the supersymmetry localization and the Fermi-gas formalism. The resulting matrix model can be regarded as a two-parameter deformation of the ABJM matrix model, and has richer non-perturbative structures.

Based on a systematic semi-classical analysis, we find analytic expressions of membrane instanton corrections. We also exactly compute the partition function for various cases and find some exact forms of worldsheet instanton corrections, which appear as quantum mechanical non-perturbative corrections in the Fermi-gas system.

In the cases with no vanishing Chern-Simons levels, we find a pair of Wilson loops for each pair of adjacent nodes on the quiver connected by a hypermultiplet nodes connected by twisted hypermultiplets have Wilson loops preserving another set of supercharges.

We expect this classical pairwise degeneracy to be lifted by quantum corrections. When the nodes with vanishing Chern-Simons levels are connected by untwisted hypermultiplets, we do not find any Wilson loops coupling to those nodes which are classically invariant. Rather, we find several loops whose supersymmetry variation, while non zero, vanishes in any correlation function, so is weakly zero.

We expect only one linear combination of those Wilson loops to remain BPS when quantum corrections are included. We analyze the M- theory duals of those Wilson loops and comment on their degeneracy. When the gauge group is SU 2 x SU 2 our theory has extra symmetries and becomes identical to the Bagger-Lambert theory. We then consider orbifold projections of this theory that give non-chiral and chiral U N x U N n superconformal quiver gauge theories.

Warner 25 years ago and whose uplifting to 11 dimensions was found more recently. Knot invariants and universal R-matrices from perturbative Chern-Simon theory in the almost axial gauge. Using perturbative Chern-Simons theory in the almost axial gauge on the euclidean manifold S 1 xR 2 , we give a prescription for the computation of knot invariants. The method gives the correct expectation value of the unknot to all orders in perturbation theory and gives the correct answer for the spectral-parameter-dependent universal R-matrix to second order.

All results are derived for a general semi-simple Lie algebra. A series of studies in two and three dimensional theories is presented. The two dimensional problems are considered in the framework of String Theory. The first one determines the region of integration in the space of inequivalent tori of a tadpole diagram in Closed String Field Theory , using the naive Witten three-string vertex. It is shown that every surface is counted an infinite number of times and the source of this behavior is identified.

The second study analyzes the behavior of the discrete matrix model of two dimensional gravity without matter using a mathematically well-defined construction, confirming several conjectures and partial results from the literature. The studies in three dimensions are based on Chern Simons pure gauge theory. The first one deals with the projection of the theory onto a two-dimensional surface of constant time, whereas the second analyzes the large N behavior of the SU N theory and makes evident a duality symmetry between the only two parameters of the theory.

Chern-Simons field theory of two-dimensional electrons in the lowest Landau level. We propose a fermion Chern-Simons field theory describing two-dimensional electrons in the lowest Landau level. Unlike the field theory constructed directly with the states in the lowest Landau level, this theory allows one, utilizing the physical picture of open-quote open-quote composite fermion,close-quote close-quote to study the fractional quantum Hall states by mapping them onto certain integer quantum Hall states; but, unlike its application in the unconstrained theory , such a mapping is sensible only when interactions between electrons are present.

An open-quote open-quote effective mass,close-quote close-quote which characterizes the scale of low energy excitations in the fractional quantum Hall systems, emerges naturally from our theory. Fre, P. From the structural view-point, we use the new formalism of integral forms in superspace that utilizes the rheonomic Lagrangians and the Picture Changing Operators, as an algorithmic tool providing the connection between different approaches to supersymmetric theories. KGaA, Weinheim. We first revisit the Yang-Baxter equation for a spin chain system associated with the single trace operators.

We show that the integrability by itself does not preclude parity symmetry breaking. We construct two-parameter family of parity non-invariant, alternating spin chain Hamiltonian involving three-site interactions between 4 and 4-bar of SU 4 R. At weak 't Hooft coupling, we study the Chern-Simons theory perturbatively and calculate anomalous dimension of single trace operators up to two loops. We find that resulting spin chain Hamiltonian matches with the Hamiltonian derived from Yang-Baxter equation, but to the one preserving parity symmetry.

We give several intuitive explanations why the parity symmetry breaking is not detected in the Chern-Simons spin chain Hamiltonian at perturbative level. We suggest that open spin chain, associated with open string excitations on giant gravitons or dibaryons, can detect discrete flat holonomy and hence parity symmetry breaking through boundary field. Non-minimal Maxwell- Chern-Simons theory and the composite Fermion model. The magnetic field redefinition in Jain's composite fermion model for the fractional quantum Hall effect is shown to be effective described by a mean-field approximation of a model containing a Maxwell- Chern-Simons gauge field nominally coupled to matter.

Further implications will be considered elsewhere. We compute explicitly the effect for the case of a symmetric configuration where the two external bound states, each of A and B particles, have the same momentum p and spin J 2. We compare this with the classical string theory result which we computed by reducing it to the Neumann-Rosochatius system.

The two results match perfectly. We use consistency of Coulomb gauge condition to naturally deduce a new gauge condition. Furthermore, we obtain the generating functional of Green function in phase space, deduce the angular momentum based on the global canonical Noether theorem at quantum level, obtain the fractional spin of this supersymmetric system, and show that the total angular momentum is the sum of the orbital angular momentum and spin angular momentum of the non- Abelian gauge field. Finally, we obtain the anomalous fractional spin and discover that the fractional spin has the contributions of both the group superscript components and A 0 s x charge.

It is found that in a sample with only one boundary -a half-plane- a total Meissner effect takes place, while in a sample with two boundaries -an infinite strip- the external magnetic field partially penetrates the material. Wilson loops in superconformal Chern-Simons theory and fundamental strings in Anti-de Sitter supergravity dual.

Novelty of Wilson loop operators in this theory is that, for a given contour, there are two linear combinations of Wilson loop transforming oppositely under time-reversal transformation. We show that one combination is holographically dual to IIA fundamental string, while orthogonal combination is set to zero.

We gather supporting evidences from detailed comparative study of generalized time-reversal transformations in both D2-brane worldvolume and ABJM theories. We next study Wilson loop expectation value in planar perturbation theory. First, all odd loop diagrams vanish identically and even loops contribute nontrivial contributions.

Combining these results, we propose that expectation value of circular Wilson loop is given by Wilson loop expectation value in pure Chern-Simons theory times zero-dimensional Gaussian matrix model whose variance is specified by an interpolating function of 't Hooft coupling. We analyze holographic field theory dual to Lovelock Chern-Simons anti-de Sitter AdS gravity in higher dimensions using first order formalism. We first find asymptotic symmetries in the AdS sector showing that they consist of local translations, local Lorentz rotations, dilatations and non- Abelian gauge transformations.

Then, we compute 1-point functions of energy-momentum and spin currents in a dual conformal field theory and write Ward identities. We find that the holographic theory possesses Weyl anomaly and also breaks non- Abelian gauge symmetry at the quantum level. Holographic Chern-Simons defects. We study SU N Yang-Mills- Chern-Simons theory in the presence of defects that shift the Chern-Simons level from a holographic point of view by embedding the system in string theory.

We holographically renormalize the free energy of the defect system with sources, from which we obtain the correlation functions for certain operators naturally associated to these defects. We find interesting phase transitions when the separation of the defects as well as the temperature are varied. U 1 x SU 2 Chern-Simons gauge theory of underdoped cuprate superconductors. After neglecting the feedback of holon fluctuations on the U 1 field B and spinon fluctuations on the SU 2 field V, the holon field is a fermion and the spinon field is a hard-core boson.

Moreover, we derive a low-energy effective action in terms of spinons holons and a self-generated U 1 gauge field. The gauge fluctuations are not confining due to coupling to holons, but nevertheless yield an attractive interaction between spinons and holons leading to a bound state with electron quantum numbers. The renormalisation effects due to gauge fluctuations give rise to non-Fermi liquid behaviour for the composite electron, in certain temperature range showing the linear in T resistivity.

This formalism provides a new interpretation of the spin gap in the underdoped superconductors. The problem of uniform magnetic fields passing perpendicularly through a 2-torus, Abelian and Non- Abelian , is considered. Focus is on dynamical effects of non-integrable phases on the torus at non zero B and from magnetic fields themselves in the vacuum. The spectrum is computed and is shown to be always independent of the non-integrable phases on the torus.

The special case of an electromagnetically uncharged anyon gas in noted and shown to be a system whose spectrum can depend on the non-integrable phases in the two torus directions, subject to a consistency requirement. In three and four dimensions, dynamical symmetry breaking of non- Abelian fields and associated condensate formation is possible by radiative corrections.

The classification on non- Abelian magnetic fields in terms of ''flux integers'' is discussed, and a method for obtaining such integers for an arbitrary gauge algebra is presented. This provides a rigorous generalisation of Hooft's su 2 classification. We present a detailed analysis of the quantum field theory of a Chern-Simons field coupled minimally to massive charged bosonic matter.

This analysis is carried out in the Coulomb and covariant gauges. Some aspects concerning the transformation law of the fields under Poincare transformations are clarified. Emphasis is placed on gauge-invariant operators. The order and disorder operators are constructed from their dual algebra. The order operator is shown to obey anyonic statistics. The correlator of the disorder operator is computed in the large boson-mass limit, and the corresponding cluster properties are discussed. In the absence of a symmetry-breaking Higgs potential, there is no evidence for the ground state being anyonic.

Maxwell- Chern-Simons Casimir effect. Circular boundary conditions. In odd-dimensional spaces, gauge invariance permits a Chern-Simons mass term for the gauge fields in addition to the usual Maxwell-Yang-Mills kinetic energy term. The case of parallel conducting lines was considered by us in a previous paper. Here we discuss the Casimir effect for a circle and examine the effect of finite temperature. The Casimir stress is found to be attractive at both low and high temperatures. Two ansatze are constructed which yield respectively analytic Bessel function solutions and elliptic function solutions.

The Jacobi elliptic function solutions are string-like, have finite energy and magnetic flux concentrated along a line in the x 1 - x 2 plane. The Chern-Simons diffusion rate is a crucial. Superfiled formulation of Chern-Simons supersymmetry. We discuss an extra supersymmetry present in the covariantly quantized Chern-Simons action within the superfield formalism.

By introducing scalar superfields we show how the component transformations are naturally reproduced from the superfield transformation. When the superspace is extended to include an additional odd coordinate for the BRST symmetry, the entire theory is described by a single odd scalar superfield. The implications of this supersymmetry for the renormalized theory are also discussed.

We discuss the Casimir effect between parallel lines in such a theory. The effect of finite temperature is also considered. In principle, our results provide a way to measure the topological mass of the photon. Remarks on Chern-Simons Invariants. The perturbative Chern-Simons theory is studied in a finite-dimensional version or assuming that the propagator satisfies certain properties as is the case, e. It turns out that the effective BV action is a function on cohomology with shifted degrees that solves the quantum master equation and is defined modulo certain canonical transformations that can be characterized completely.

Out of it one obtains invariants. Fractional exclusion and braid statistics in one dimension: a study via dimensional reduction of Chern-Simons theory. The relation between braid and exclusion statistics is examined in one-dimensional systems, within the framework of Chern-Simons statistical transmutation in gauge invariant form with an appropriate dimensional reduction. If the matter action is anomalous, as for chiral fermions, a relation between braid and exclusion statistics can be established explicitly for both mutual and nonmutual cases.

However, if it is not anomalous, the exclusion statistics of emergent low energy excitations is not necessarily connected to the braid statistics of the physical charged fields of the system. Finally, we also discuss the bosonization of one-dimensional anyonic systems through T-duality.

Dedicated to the memory of Mario Tonin. Chern-Simons terms and cocycles in physics and mathematics. Contemporary topological research in Yang-Mills theory is reviewed, emphasizing the Chern-Simons terms and their relatives. Three applications of the Chern-Simons terms in physical theory are described: to help understanding gauge theories in even dimensional space-time; gauge field dynamics in odd dimensional space-time; and mathematically coherent description of even-dimensional gauge theories with chiral fermions that are apparently inconsistent due to chiral anomalies.

Discussion of these applications is preceded by explanation of the mathematical preliminaries and examples in simple quantum mechanical settings. Chern-Simons gravity in four dimensions. Five-dimensional Chern-Simons theory with anti- de Sitter SO 1,5 or SO 2,4 gauge invariance presents an alternative to general relativity with cosmological constant. We consider the zero modes of its Kaluza-Klein compactification to four dimensions. We also check that vanishing torsion is a stable feature of the solutions. We construct quantum-mechanical models that are analogs of three-dimensional, topologically massive as well as Chern-Simons gauge-field theories , and we study the phase-space reductive limiting procedure that takes the former to the latter.

The zero-point spectra of operators behave discontinuously in the limit, as a consequence of a nonperturbative quantum-mechanical anomaly. The nature of the limit for wave functions depends on the representation, but is always such that normalization is preserved. Even-dimensional topological gravity from Chern-Simons gravity. Yang-Mills- Chern-Simons supergravity. There is in addition a triplet of matter vectors.

After diagonalization, these fields describe two triplets of topologically-massive vector fields of opposite helicities. It provides the first example of a three-dimensional gauged supergravity that can be obtained by a consistent reduction of string theory or M- theory and that admits AdS 3 as a vacuum solution. There are unusual features in the reduction from six-dimensional supergravity, owing to the self-duality condition on the 3-form field. Equivalence of several Chern-Simons matter models.

Chern-Simons CS coupling characterizes not only statistics, but also spin and scaling dimension of matter fields. We demonstrate spin transmutation in relativistic CS matter theory , and moreover show equivalence of several models. We study the CS vector model in some detail, which provides a consistent check to the assertion of the equivalence.

Gauge fixing of Chern-Simons N-extended supergravity. We fix the gauge of this theory within the Batalin-Vilkovisky scheme. We pointed out that there exists a critical frequency of oscillation for the vortex-like solution above which the system switches to the fields of an alternating current inside a solenoid. We also show the existence a non- abelian gauge for which the fields can alternate periodically in space between the abelian fields and another abelian or non- abelian field pointing in a different isospin space direction.

The solution discussed here are real, zero-action Minkowski space configurations. Stochastic quantization of topological field theory : generalized Langevin equation with memory kernel. We use the method of stochastic quantization in a topological field theory defined in an Euclidean space, assuming a Langevin equation with a memory kernel. We show that our procedure for the Abelian Chern-Simons theory converges regardless of the nature of the Chern-Simons coefficient. Absence of higher order corrections to noncommutative Chern-Simons coupling.

We analyze the structure of noncommutative pure Chern-Simons theory systematically in the axial gauge. In fact, we show, using the usual BRST identities as well as the identities following from vector supersymmetry, that this is a free theory. As a result, the tree level Chern-Simons coefficient is not renormalized. It also holds that the Chern-Simons coefficient is not modified at finite temperature.

Chern-Simons expectation values and quantum horizons from loop quantum gravity and the Duflo map.