Classical and quantum Chern-Simons with gauge group $\text{U}(1)^N$ were classified by Belov and Moore in \cite{belov_moore}. They studied both ordinary topological quantum field theories as well as spin theories. On the other hand a correspondence is well known between ordinary $(2+1)$-dimensional TQFTs and modular tensor categories. We study group categories and extend them slightly to produce modular tensor categories that correspond to toral Chern-Simons. Group categories have been widely studied in other contexts in the literature \cite{frolich_kerler},\cite{quinn},\cite{joyal_street},\cite{eno},\cite{dgno}. The main result is a proof that the associated projective representation of the mapping class group is isomorphic to the one from toral Chern-Simons. We also remark on an algebraic theorem of Nikulin that is used in this paper.
In this paper, we completely characterize time-reversal invariant three-dimensional Chern-Simons gauge theories with torus gauge group. At the level of the Lagrangian, toral Chern-Simons theory is defined by an integral lattice, while at the quantum level, it is entirely determined by a quadratic function on a finite Abelian group and an integer mod 24. We find that quantum time-reversally symmetric theories can be defined by classical Lagrangians defined by integral lattices which have self-perpendicular embeddings into a unimodular lattice. We find that the quantum toral Chern-Simons theory …
We perform a resurgence analysis of the $SU(2)$ Chern-Simons partition function on a Brieksorn homology sphere $Σ(2,5,7)$. Starting from an exact Chern-Simons partition function, we study the Borel resummation of its perturbative expansion.
We use some BRS techniques to construct Chern-Simons forms generalizing the Chern character of K_1 groups in the Cuntz-Quillen description of cyclic homology.
It has been a long-standing problem how to relate Chern-Simons theory to the quantum groups. In this paper we recover the classical $r$-matrix directly from a 3-dimensional Chern-Simons theory with boundary conditions, thus creating a direct link to the quantum groups. It is known that the Jones polynomials can be constructed using an $R$-matrix. We show how these constructions can be seen to arise directly from 3-dimensional Chern-Simons theory.
We derive discrete and oscillatory Chern-Simons matrix models. The method is based on fundamental properties of the associated orthogonal polynomials. As an application, we show that the discrete model allows to prove and extend the recently found relationship between Chern-Simons theory and q-deformed 2dYM. In addition, the equivalence of the Chern-Simons matrix models gives a complementary view on the equivalence of effective superpotentials in N=1 gauge theories.
The dimensional reduction of the three-dimensional fermion-Chern-Simons model (related to Hopf maps) of Adam et el. is shown to be equivalent to (i) either the static, fixed--chirality sector of our non-relativistic spinor-Chern-Simons model in 2+1 dimensions, (ii) or a particular Heisenberg ferromagnet in the plane.
We find a Chern-Simons propagator on the ball with the chiral boundary condition. We use it to study perturbatively Chern-Simons boundary conditions related to 2-dim $σ$-models and to Poisson-Lie T-duality. In particular, we find their renormalization group flow, given by the generalized Ricci tensor. Finally we briefly discuss what happens when the Chern-Simons theory is replaced by a Courant $σ$-model or possibly by a more general AKSZ model.
Being gauge non-invariant, a Chern-Simons (2k-1)-form seen as a Lagrangian of gauge theory on a (2k-1)-dimensional manifold leads to the gauge conservation law of a modified Noether current.
Though a Chern-Simons (2k-1)-form is not gauge-invariant and it depends on a background connection, this form seen as a Lagrangian of gauge theory on a (2k-1)-dimensional manifold leads to the energy-momentum conservation law.
An existence theorem is established for the solutions to the non-Abelian relativistic Chern--Simons--Higgs vortex equations over a doubly periodic domain when the gauge group $G$ assumes the most general and important prototype form, $G=SU(N)$
We comment on the calculation of the Chern-Simons coefficient in (2+1)-dimensional gauge theories at finite chemical potential made by A.N.Sissakian, O.Yu.Shevchenko and S.B.Solganik (hep-th/9608159 and hep-th/9612140).
Chern-Simons gravities and gravitational Chern-Simons densities are constructed using the non-Abelian Yang-Mills Chern-Simons densities. As such, they are defined only in odd dimensions. We propose instead an analogous construction employing what we term Higgs--Chern-Simons (HCS) densities, which are defined in $all\ dimensions$. This enables the definition of extended versions of Chern-Simons gravities in all dimensions. Employing the same prescription, the definition of gravitational Chern-Simons densities is extended to all even dimensions, but only to $4p-1$ $odd$ dimensions. All our consi…
By taking into account the effect of the would be Chern-Simons term, we calculate the quantum correction to the Chern-Simons coefficient in supersymmetric Chern-Simons Higgs theories with matter fields in the fundamental representation of SU(n). Because of supersymmetry, the corrections in the symmetric and Higgs phases are identical. In particular, the correction is vanishing for N=3 supersymmetric Chern-Simons Higgs theories. The result should be quite general, and have important implication for the more interesting case when the Higgs is in the adjoint representation.
In this article it is studied the 3-brane world in the context of five-dimensional Einstein-Chern-Simons gravity. We started by considering Israel's junction condition for AdS-Chern-Simons gravity. Using the S-expansion procedure, we mapped the AdS-Chern-Simons junction conditions to Einstein-Chern-Simons gravity, allowing us to derive effective four-dimensional Einstein-Chern-Simons field equations.
We study a one-dimensional toy version of the Chern-Simons theory. We construct its simplicial version which comprises features of a low-energy effective gauge theory and of a topological quantum field theory in the sense of Atiyah.
We define the notion of spectral network on manifolds of dimension $\le 3$. For a manifold $X$ equipped with a spectral network, we construct equivalences between Chern-Simons invariants of flat ${\mathrm {SL}}(2,{\mathbb C})$-bundles over $X$ and Chern-Simons invariants of flat ${\mathbb C}^\times$-bundles over ramified double covers $\widetilde X$. Applications include a new viewpoint on dilogarithmic formulas for Chern-Simons invariants of flat ${\mathrm {SL}}(2,{\mathbb C})$-bundles over triangulated 3-manifolds, and an explicit description of Chern-Simons lines of flat ${\mathrm {SL}}(2,{…
We study a three-dimensional symmetric Chern-Simons field theory with a general covariance and it turns out that the original Chern-Simons theory is just a gauge fixed action of the symmetric Chern-Simons theory whose constraint algebra belongs to fully first class constraint system. The Abelian Chern-Simons theory with matter coupling is studied for the construction of anyon operators without any ordering ambiguity with the help of this symmetric Chern-Simons action. Finally we shall discuss some connections between the present symmetric formulation of Chern-Simons theory and the Stükelberg m…
By analyzing the odd parity part of the gauge field two and three point vertex functions, the one-loop radiative correction to the Chern-Simons coefficient is computed in noncommutative Chern-Simons-Higgs model at zero and at high temperature. At high temperature, we show that the static limit of this correction is proportional to $T$ but the first noncommutative correction increases as $T\log T$. Our results are analytic functions of the noncommutative parameter.
Lectures at the 1998 Les Houches Summer School: Topological Aspects of Low Dimensional Systems. These lectures contain an introduction to various aspects of Chern-Simons gauge theory: (i) basics of planar field theory, (ii) canonical quantization of Chern-Simons theory, (iii) Chern-Simons vortices, and (iv) radiatively induced Chern-Simons terms.
A perturbative analysis of a massive Chern-Simons is presented. The mass term is introduced in a BRST invariant way. With these method we prove that the number of independent renormalisation of massive Chern-Simons is equal to topologically massive Yang-Mills and different from pure Chern-Simons.
Noncommutative Maxwell-Chern-Simons theory in 3-dimensions is defined in terms of star product and noncommutative fields. Seiberg-Witten map is employed to write it in terms of ordinary fields. A parent action is introduced and the dual action is derived. For spatial noncommutativity it is studied up to second order in the noncommutativity parameter θ. A new noncommutative Chern-Simons action is defined in terms of ordinary fields, inspired by the dual action. Moreover, a transformation between noncommuting and ordinary fields is proposed.
We study the manifestly covariant three-dimensional symmetric Chern-Simons action in terms of the Batalin-Vilkovisky quantization method. We find that the Lorentz covariant gauge fixed version of this action is reduced to the usual Chern-Simons type action after a proper field redefinition. Furthermore, the renormalizability of the symmetric Chern-Simons theory turns out to be the same as that of the original Chern-Simons theory.
Some of supersymmetric Chern-Simons theories are known to exhibit supersymmetry breaking when the Chern-Simons level is less than a certain number. The mechanism of the supersymmetry breaking is, however, not clear from the field theory viewpoint. In this paper, we discuss vacuum states of ${\cal N}=2$ pure Chern-Simons theory and ${\cal N}=2$ Chern-Simons-matter theories of quiver type using related theories in which Chern-Simons terms are replaced with (anti-)fundamental chiral multiplets. In the latter theories, supersymmetry breaking can be shown to occur by examining that the vacuum energ…
Some general features of locally supersymmetric theories (N=1 in four dimensions) involving Chern-Simons forms and antisymmetric tensors are sketched out. The relevance of the three-form multiplet both for the description of Chern-Simons forms and the supersymmetry properties of the gaugino condensate is pointed out.
We construct the (enhanced Rogers) dilogarithm function from the spin Chern-Simons invariant of C*-connections. This leads to geometric proofs of basic dilogarithm identities and a geometric context for other properties, such as the branching structure.
We construct parity and time reversal invariant Maxwell-Chern-Simons gauge theory coupled to fermions with adding the parity partner to the matter and the gauge fields, which can give nontopological vortex solutions depending on the sign of the Chern-Simons coupling constant.
I present two calculations of the holographic Weyl anomalies induced by Chern-Simons gravity theories alternative to the ones presented in the literature. The calculations presented here rest on the extension from Chern-Simons to Transgression forms as lagrangians, motivated by gauge invariance, which automatically yields the boundary terms suitable to regularize the theory. The procedure followed here sheds light in the structure of Chern-Simons gravities and their regularization.
We derive the noncommutative Chern-Simons action induced by Dirac fermions coupled to a background gauge field, for the fundamental, antifundamental, and the adjoint representation. We discuss properties of the noncommutative Chern-Simons action showing in particular that the Seiberg-Witten formula maps it into the standard commutative Chern-Simons action.
We develop a supersymmetric extension of Chern-Simons theory and Chern-Simons-Landau-Ginzburg theory for supersymmetric quantum Hall liquid. Supersymmetric counterparts of topological and gauge structures peculiar to the Chern-Simons theory are inspected in the supersymmetric Chern-Simons theory. We also explore an effective field theoretical description for the supersymmetric quantum Hall liquid. The key observation is the the charge-flux duality. Based on the duality, we derive a dual supersymmetric Chern-Simons-Landau-Ginzburg theory, and discuss physical properties of the topological excit…
The relation between the Dirac quantization condition of magnetic charge and the quantization of the Chern-Simons coefficient is obtained. It implies that in a (2+1)-dimensional QED with the Chern-Simons topological mass term and the existence of a magnetic monopole with magnetic charge $g$, the Chern-Simons coefficient must be also quantized, just as in the non-Abelian case.
The relation between open topological strings and Chern-Simons theory was discovered by E. Witten. He proved that A-model on T*M where M is a three-dimensional manifold is equivalent to Chern-Simons theory on M and that A-model on arbitrary Calabi-Yau 3-fold is related to Chern-Simons theory with instanton corrections. In present paper we discuss multidimensional generalization of these results.
The BRST quantization of matrix Chern-Simons theory is carried out, the symmetries of the theory are analysed and used to constrain the form of the effective action.
We explain how static multi-vortex solutions arise in non-linear field theories, by taking the non-linear Schrödinger equation coupled to Chern-Simons field (Jackiw-Pi model) and a fermion Chern-Simons theory as simple examples. We then construct a fermion Maxwell-Chern-Simons theory which has consistent static field equations, and show that it has the same vortex solutions as the Jackiw-Pi model, but gives rise to quite different vortex dynamics.
We show explicitly that the question of gauge invariance of the effective potential in standard scalar electrodynamics remains unchanged despite the introduction of the Chern-Simons term. The result does not depend on the presence of the Maxwell term in the Chern-Simons territory.
We study one-loop correction to the Chern-Simons coefficient $κ=k/4π$ in $N=1,2,3$ supersymmetric Yang-Mills Chern-Simons systems. In the pure bosonic case, the shift of the parameter $k$ is known to be $k\rightarrow k + c_v$, where $c_v$ is the quadratic Casimir of the gauge group. In the $N=1$ case, the fermionic contribution cancels the bosonic contribution by half and the shift is $k \rightarrow k+ c_v/2$, making the theory anomalous if $c_v$ is odd. In the $N=2,3$ cases, the fermionic contribution cancels the bosonic contribution completely and there is no correction. We also calculate th…
The use of the physical variables in the fashion of Dirac in the three-dimensional Chern-Simons theories is presented. Our previous results are reinterpreted in a new aspect.
The renormalization of the Chern-Simons parameter is investigated by using an exact and manifestly gauge invariant evolution equation for the scale-dependent effective average action.
We propose a new nonrelativistic Chern-Simons theory based on a simple modification of the standard Lagrangian. This admits asymptotically nonvanishing field configurations and is applicable to the description of systems of repulsive bosons. The new model supports topological vortices and has a self-dual limit, both in the pure Chern-Simons and in the mixed Chern-Simons-Maxwell cases. The analysis is based on a new formulation of the Chern-Simons theories as constrained Hamiltonian systems.
The large-mass behaviour of loop variables in Maxwell-Chern-Simons theory is analysed by means of a gauge-field transformation which allows to reset the Maxwell-Chern-Simons action to pure Chern-Simons.
We argue that N=2 supersymmetric Chern-Simons theories exhibit a strong-weak coupling Seiberg-type duality. We also discuss supersymmetry breaking in these theories.
We study the effect of noncommutative Chern-Simons term on fundamental fermions. In particular the one-loop contribution to the magnetic moment to order $þ$ is calculated.
Strong/weak coupling duality in Chern-Simons supergravity is studied. It is argued that this duality can be regarded as an example of superduality. The use of supergroup techniques for the description of Chern-Simons supergravity greatly facilitates the analysis.
We consider here a generalization of the Abelian Higgs model in curved space, by adding a Chern--Simons term. The static equations are self-dual provided we choose a suitable potential. The solutions give a self-dual Maxwell--Chern--Simons soliton that possesses a mass and a spin.
We quantize the Maxwell Chern Simons theory in a geometric representation that generalizes the Abelian Loop Representation of Maxwell theory. We find that in the physical sector, the model can be seen as the theory of a massles scalar field with a topological interaction that enforces the wave functional to be multivalued. This feature allows to relate the Maxwell Chern Simons theory with the quantum mechanics of particles interacting through a Chern Simons field
We study the relation between the partition function of refined SU(N) and SO(2N) Chern-Simons on the 3-sphere and the universal Chern-Simons partition function in the sense of Mkrtchyan and Veselov. We find a four-parameter generalization of the integral representation of universal Chern-Simons that includes refined SU(N) and SO(2N) Chern-Simons for special values of parameters. The large N expansion of the integral representation of refined SU(N) Chern-Simons explicitly shows the replacement of the virtual Euler characteristic of the moduli space of complex curves with a refined Euler charact…
The Chern-Simons topological term coefficient is derived at arbitrary finite density. As it occures that $μ^2 = m^2$ is the crucial point for Chern-Simons. So when $μ^2 < m^2 μ$--influence disappears and we get the usual Chern-Simons term. On the other hand when $μ^2 > m^2$ the Chern-Simons term vanishes because of non-zero density of background fermions. In particular for massless case parity anomaly is absent at any finite density. This result holds in any odd dimension as in abelian so as in nonabelian cases.
We search for vortices in a generalized Abelian Chern-Simons model with a nonstandard kinetic term. We illustrate our results, plotting and comparing several features of the vortex solution of the generalized model with those of the vortex solution found in the standard Chern-Simons model.
Three dimensional Yang-Mills gauge theories in the presence of the Chern-Simons action are seen as being generated by the pure topological Chern-Simons term through nonlinear covariant redefinitions of the gauge field
