Narrow Results

We generalize several results on Chern–Simons models on Σ × S¹ in the so-called "torus gauge" which were obtained in [A. Hahn, An analytic approach to Turaev's shadow invariant, J. Knot Theory Ramifications17(11) (2008) 1327–1385] (= arXiv:math-ph/0507040) to the case of general (simply-connected simple compact) structure groups and general link colorings. In particular, we give a non-perturbative evaluation of the Wilson loop observables corresponding to a special class of simple but non-trivial links and show that their values are given by Turaev's shadow invariant. As a byproduct, we obtain a heuristic path integral derivation of the quantum Racah formula.

We give a detailed introduction to the classical Chern–Simons gauge theory, including the mathematical preliminaries. We then explain the perturbative quantization of gauge theories via the Batalin–Vilkovisky (BV) formalism. We then define the perturbative Chern–Simons partition function at any (possibly non-acylic) reference flat connection using the BV formalism, using a Riemannian metric for gauge fixing. We show that it exhibits an anomaly known as the “framing anomaly” when the Riemannian metric is changed, that is, it fails to be gauge invariant. We explain how one can deal with this anomaly to obtain a topological invariant of framed manifolds.

We examine Chern–Simons theory written on a noncommutative plane with a "hole", and show that the algebra of observables is a nonlinear deformation of the w_∞ algebra. The deformation depends on the level (the coefficient in the Chern–Simons action), and the noncommutativity parameter, which were identified, respectively, with the inverse filling fraction (minus one) and the inverse density in a recent description of the fractional quantum Hall effect. We remark on the quantization of our algebra. The results are sensitive to the choice of ordering in the Gauss law.

We illustrate how boundary states are recovered when going from a noncommutative manifold to a commutative one with a boundary. Our example is the noncommutative plane with a defect, whose commutative limit was found to be a punctured plane - so here the boundary is one point. Defects were introduced by removing states from the standard harmonic oscillator Hilbert space. For Chern-Simons theory, the defect acts as a source, which was found to be associated with a nonlinear deformation of the w_∞ algebra. The undeformed w_∞ algebra is recovered in the commutative limit, and here we show that its spatial support is in a tiny region near the puncture.

We study the properties of matrix models with soft confining potentials. Their precise mathematical characterization is that their weight function is not determined by its moments. We mainly rely on simple considerations based on orthogonal polynomials and the moment problem. In addition, some of these models are equivalent, by a simple mapping, to matrix models that appear in Chern–Simons theory. The models can be solved with q deformed orthogonal polynomials (Stieltjes–Wigert polynomials), and the deformation parameter turns out to be the usual q parameter in Chern–Simons theory. In this way, we give a matrix model computation of the Chern–Simons partition function on S³ and show that there are infinitely many matrix models with this partition function.

We study a topological Abelian gauge theory that generalizes the Abelian Chern–Simons one, and that leads in a natural way to the Milnor's link invariant when the classical action on-shell is calculated.

We show that matrix models in Chern–Simons theory admit an interpretation as 1D exactly solvable models, paralleling the relationship between the Gaussian model and the Calogero model. We compute the corresponding Hamiltonians, ground-state wave functions and ground-state energies and point out that the models can be interpreted as quasi-1D Coulomb plasmas. We also study the relationship between Chern–Simons theory on S³ and a system of N one-dimensional fermions at finite temperature with harmonic confinement. In particular, we show that the Chern–Simons partition function can be described by the density matrix of the free fermions in a very particular, crystalline, configuration. For this, we both use the Brownian motion and the matrix model description of Chern–Simons theory and find several common features with c = 1 theory at finite temperature. Finally, using the exactly solvable model result, we show that the finite temperature effect can be described with a specific two-body interaction term in the Hamiltonian, with 1D Coulombic behavior at large separations.

We study supersymmetric D-brane configurations in AdS₄×ℂℙ³ background, which is dual to the D = 3, Chern–Simons-matter theory discovered by Aharony et al. In particular, we consider D2 and D6-branes with fundamental strings attached to them. D2-branes are wrapped on ℂℙ¹ ⊂ ℂℙ³ and dual to a symmetric product of Wilson lines, while D6-branes wrapping the entire ℂℙ³ correspond to baryonic vertices. We perform the κ-symmetry analysis to construct the BPS equations and find explicit solutions of the world-volume D-brane action. When one turns on the electric field, D2 and D6-brane configurations preserve 4 and 2 supercharges respectively.

U(1) Chern–Simons theory is quantized canonically on manifolds of the form , where Σ is a closed orientable surface. In particular, we investigate the role of mapping class group of Σ in the process of quantization. We show that, by requiring the quantum states to form representation of the holonomy group and the large gauge transformation group, both of which are deformed by quantum effect, the mapping class group can be consistently represented, provided the Chern–Simons parameter k satisfies an interesting quantization condition. The representations of all the discrete groups are unique, up to an arbitrary sub-representation of the mapping class group. Also, we find a k↔1/k duality of the representations.

We consider general supersymmetric Wilson loops in ABJM model, which is Chern–Simons-matter theory in (2+1) dimensions with 𝒩 = 6 supersymmetry. The Wilson loops of our interest are so-called Zarembo-type: they have generic contours in spacetime, but the scalar field coupling is arranged accordingly so that there are unbroken supersymmetries. Following the supermatrix construction of Wilson loops by Drukker and Trancanelli and the generalization by Griguolo et al., we study 1/6-BPS Wilson loops and check that their expectation value is protected using perturbation up to two loops. We also study the dual string configuration in AdS₄×ℂℙ³ background and check the supersymmetry.

We implement the metric-independent Fock–Schwinger gauge in the quantum Chern–Simons (CS) field theory defined in a three-manifold M which is homeomorphic with ℝ³. The expressions of various components of the propagator are determined. Although the gauge field propagator differs from the Gauss linking density, we prove that its integral along two oriented knots is equal to the linking number.

Recent results of Gu and Jockers provide the lacking initial conditions for the evolution method in the case of the first nontrivially colored HOMFLY polynomials H_[21] for the family of twist knots. We describe this application of the evolution method, which finally allows one to penetrate through the wall between (anti)symmetric and non-rectangular representations for a whole family. We reveal the necessary deformation of the differential expansion, what, together with the recently suggested matrix model approach gives new opportunities to guess what it could be for a generic representation, at least for the family of twist knots.

In this paper, we consider topological gauge theories in three dimensions which are defined by metric independent lagrangians. It has been claimed that the functional integration necessarily depends nontrivially on the gauge-fixing metric. We demonstrate that the partition function and the mean values of the gauge invariant observables do not really depend on the gauge-fixing metric.

A deformation of the dimensional-regularization technique that is useful for theories where the common dimensional regularization does not apply is formulated. The Dirac algebra is not dimensionally continued, to avoid inconsistencies with the trace of an odd product of gamma matrices in odd dimensions. The regularization is completed with an evanescent higher-derivative deformation, which proves to be efficient in practical computations. This technique is particularly convenient in three dimensions for Chern–Simons gauge fields, two-component fermions and four-fermion models in the large N limit, eventually coupled with quantum gravity. Differently from even dimensions, in odd dimensions it is not always possible to have propagators with fully Lorentz invariant denominators. The main features of the deformed technique are illustrated in a set of sample calculations. The regularization is universal, local, manifestly gauge-invariant and Lorentz invariant in the physical sector of space–time. In flat space power-like divergences are set to zero by default. Infinitely many evanescent operators are automatically dropped.

The equations of motion for a theory described by a Chern–Simons type of action in two dimensions are obtained and investigated. The equation for the classical, continuous Heisenberg model is used as a form of gauge constraint to obtain a result which provides a completely integrable dynamics and which partially fixes the gauge degrees of freedom. Under a particular form of the spin connection, an integrable equation which can be analytically extended to a form of the nonlinear Schrödinger equation is obtained. Some explicit solutions are presented, and in particular a soliton solution is shown to lead to an integrable two-dimensional model of gravity.

We evaluate the ground state degeneracy of noncommutative Chern–Simons models on the two-torus, a quantity that is interpreted as the "topological order" of associated phases of Hall fluids. We define the noncommutative theory via T-duality from an ordinary Chern–Simons model with non-Abelian 't Hooft magnetic fluxes. Motivated by this T-duality, we propose a discrete family of noncommutative, non-Abelian fluid models, arising as a natural generalization of the standard noncommutative Chern–Simons effective models. We compute the topological order for these universality classes, and comment on their possible microscopic interpretation.

We study several types of classical rotating membrane solutions in AdS₄ ×Q^{1, 1, 1} and discuss their field theory duals. Q^{1, 1, 1} is a seven-dimensional Sasaki–Einstein manifold given as a nontrivial U(1) fibration over S²×S²×S², equipped with SU(2)³ ×U(1) isometry. It is recently suggested that there exist quiver Chern–Simons theories which are dual to M-theory in certain orbifolds of Q^{1, 1, 1}. The membrane solutions we consider have in general nonvanishing angular momenta both in AdS₄ and Q^{1, 1, 1} spaces. We present solutions for folded and wrapped membranes. According to the AdS/CFT correspondence, such classical solutions are dual to long operators of the dual conformal field theories in the large coupling limit. We analyze the asymptotic behaviour of the dispersion relation between energy (conformal dimension) and angular momenta (global charges).

We continue the program of systematic study of extended HOMFLY polynomials, suggested in [A. Mironov, A. Morozov and And. Morozov, arXiv:1112.5754] and [A. Mironov, A. Morozov and And. Morozov, J. High Energy Phys. 03, 034 (2012), arXiv:1112.2654]. Extended polynomials depend on infinitely many time-variables, are close relatives of integrable τ-functions, and depend on the choice of the braid representation of the knot. They possess natural character decompositions, with coefficients which can be defined by exhaustively general formula for any particular number m of strands in the braid and any particular representation R of the Lie algebra GL(∞). Being restricted to "the topological locus" in the space of time-variables, the extended HOMFLY polynomials reproduce the ordinary knot invariants. We derive such a general formula, for m = 3, when the braid is parametrized by a sequence of integers (a₁, b₁, a₂, b₂, …) and for the first nonfundamental representation R = [2]. Instead of calculating the mixing matrices directly, as suggested [A. Mironov, A. Morozov and And. Morozov, J. High Energy Phys. 03, 034 (2012), arXiv:1112.2654], we deduce them from comparison with the known answers for torus and composite knots. A simple reflection symmetry converts the answer for the symmetric representation [2] into that for the antisymmetric one [1, 1]. The result applies, in particular, to the figure eight knot 4₁, and was further extended to superpolynomials in arbitrary symmetric and antisymmetric representations in H. Itoyama, A. Mironov, A. Morozov and And. Morozov, arXiv:1203.5978.

We review quantum field theory approach to the knot theory. Using holomorphic gauge, we obtain the Kontsevich integral. It is explained how to calculate Vassiliev invariants and coefficients in Kontsevich integral in a combinatorial way which can be programmed on a computer. We discuss experimental results and temporal gauge considerations which lead to representation of Vassiliev invariants in terms of arrow diagrams. Explicit examples and computational results are presented.

Character expansion expresses extended HOMFLY polynomials through traces of products of finite-dimensional - and Racah mixing matrices. We conjecture that the mixing matrices are expressed entirely in terms of the eigenvalues of the corresponding -matrices. Even a weaker (and, perhaps, more reliable) version of this conjecture is sufficient to explicitly calculate HOMFLY polynomials for all the 3-strand braids in arbitrary (anti)symmetric representations. We list the examples of so obtained polynomials for R = [3] and R = [4], and they are in accordance with the known answers for torus and figure-eight knots, as well as for the colored special and Jones polynomials. This provides an indirect evidence in support of our conjecture.