Matter Chern Simons theories in a background magnetic field (original) (raw)
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Chern–Simons theory with vector fermion matter
The European Physical Journal C, 2012
We study three dimensional conformal field theories described by U (N) Chern-Simons theory at level k coupled to massless fermions in the fundamental representation. By solving a Schwinger-Dyson equation in lightcone gauge, we compute the exact planar free energy of the theory at finite temperature on R 2 as a function of the 't Hooft coupling λ = N/k. Employing a dimensional reduction regularization scheme, we find that the free energy vanishes at |λ| = 1; the conformal theory does not exist for |λ| > 1. We analyze the operator spectrum via the anomalous conservation relation for higher spin currents, and in particular show that the higher spin currents do not develop anomalous dimensions at leading order in 1/N. We present an integral equation whose solution in principle determines all correlators of these currents at leading order in 1/N and present explicit perturbative results for all three point functions up to two loops. We also discuss a lightcone Hamiltonian formulation of this theory where a W ∞ algebra arises. The maximally supersymmetric version of our theory is ABJ model with one gauge group taken to be U (1), demonstrating that a pure higher spin gauge theory arises as a limit of string theory.
Non-Abelian Chern-Simons particles in an external magnetic field
Nuclear Physics B, 1999
The quantum mechanics and thermodynamics of SU(2) non-Abelian Chern-Simons particles (non-Abelian anyons) in an external magnetic field are addressed. We derive the N-body Hamiltonian in the (anti-)holomorphic gauge when the Hilbert space is projected onto the lowest Landau level of the magnetic field. In the presence of an additional harmonic potential, the N-body spectrum depends linearly on the coupling (statistics) parameter. We calculate the second virial coefficient and find that in the strong magnetic field limit it develops a step-wise behavior as a function of the statistics parameter, in contrast to the linear dependence in the case of Abelian anyons. For small enough values of the statistics parameter we relate the N-body partition functions in the lowest Landau level to those of SU(2) bosons and find that the cluster (and virial) coefficients dependence on the statistics parameter cancels.
Journal of High Energy Physics, 2015
We present explicit computations and conjectures for 2 → 2 scattering matrices in large N U (N) Chern-Simons theories coupled to fundamental bosonic or fermionic matter to all orders in the 't Hooft coupling expansion. The bosonic and fermionic S-matrices map to each other under the recently conjectured Bose-Fermi duality after a level-rank transposition. The S-matrices presented in this paper may be regarded as relativistic generalization of Aharonov-Bohm scattering. They have unusual structural features: they include a non-analytic piece localized on forward scattering, and obey modified crossing symmetry rules. We conjecture that these unusual features are properties of S-matrices in all Chern-Simons matter theories. The S-matrix in one of the exchange channels in our paper has an anyonic character; the parameter map of the conjectured Bose-Fermi duality may be derived by equating the anyonic phase in the bosonic and fermionic theories. 4.8 The onshell limit 4.8.1 An infrared 'ambiguity' and its resolution 4.8.2 Covariantization of the amplitude 4.9 The S-matrix in the adjoint channel 4.10 The S-matrix for particle-particle scattering 5. The onshell one loop amplitude in Landau Gauge 6. Scattering in the fermionic theory 6.1 The offshell four point amplitude 6.2 The onshell limit 6.3 S-matrices 6.3.1 S-matrix for adjoint exchange in particle-antiparticle scattering 6.3.2 S-matrix for particle-particle scattering 7. Scattering in the identity channel and crossing symmetry 7.1 Crossing symmetry 7.2 A conjecture for the S-matrix in the singlet channel 7.3 Bose-Fermi duality in the S-channel 7.4 A heuristic explanation for modified crossing symmetry 7.5 Direct evaluation of the S-matrix in the identity channel 7.5.1 Double analytic continuation 7.5.2 Schrodinger equation in lightfront quantization? 8. Discussion A. The identity S-matrix as a function of s, t, u B. Tree level S-matrix B.1 Particle-particle scattering B.2 Particle-antiparticle scattering B.3 Explicit tree level computation C. Aharonov-Bohm scattering C.1 Derivation of the scattering wave function C.2 The scattering amplitude C.3 Physical interpretation of the δ function at forward scattering D. Details of the computation of the scalar S-matrix D.1 Computation of the effective one particle exchange interaction D.2 Euclidean rotation D.3 Solution of the Euclidean integral equations D.4 The one loop box diagram computed directly in Minkowski space D.4.1 Scalar poles D.4.2 Contributions of the gauge boson poles off shell-2-D.4.3 The onshell contribution of the gauge boson poles 88 E. Details of the one loop Landau gauge computation 91 E.1 Simplification of the integrand of the box graph E.2 Simplification of the remaining integrands 93 E.3 Absence of IR divergences 94 E.4 Absence of gauge boson cuts 95 E.5 Potential subtlety at special values of external momenta 98 F. Details of scattering in the fermionic theory 98 F.1 Off shell four point function 98 G. Preliminary analysis of the double analytic continuation 103 G.1 Analysis of the scalar integral equation after double analytic continuation 103 G.2 The oneloop box diagram after double analytic continuation 103 G.2.1 Setting up the computation 103 G.2.2 The contribution of the pole at zero 105 G.2.3 The contribution of the remaining four poles 105 G.3 Solutions of the Dirac equation at q ± = 0 after double analytic continuation. 107 G.4 Aharonov-Bohm in the non-relativistic limit 108 5 Readers familiar with the relationship between Chern-Simons theory and WZW theory may recognize this formula in another guise. C 2 (R) k is the holomorphic scaling dimension of a primary operator in the integrable representation R, and e 2πiνm is the monodromy of the four point function < R1, R2,R1,R2 > in the conformal block corresponding to the OPE R1R2 → Rm. 6 The additional-1 in the fermionic theory comes from Fermi statistics. We have used −1 = e ±iπ = e −iπsgn(λ F) .
Four-fermion field theories and the Chern-Simons field: A renormalization group study
Physical Review D, 1999
In (2+1) dimensions, we consider the model of a N flavor, two-component fermionic field interacting through a Chern-Simons field besides a four fermion self-interaction which consists of a linear combination of the Gross-Neveu and Thirring like terms. The four fermion interaction is not perturbatively renormalizable and the model is taken as an effective field theory in the region of low momenta. Using Zimmerman procedure for reducing coupling constants, it is verified that, for small values of the Chern-Simons parameter, the origin is an infrared stable fixed point but changes to ultraviolet stable as α becomes bigger than a critical α c . Composite operators are also analyzed and it is shown that a specific four fermion interaction has an improved ultraviolet behavior as N increases. Fermionic quartic interactions have been very important for the clarification of conceptual aspects as well as for the applications of Quantum Field Theory. Illustrative examples of such dual role are provided by the Thirring and Nambu-Jona Lasinio models. However, perturbative studies of the models have been hampered by the fact that only in two dimensions they are renormalizable. If the number of flavors is high enough, a better ultraviolet behavior is achieved in the context of the 1/N expansion which turns out to be renormalizable up
Fermionic matter coupled to higher derivative Chern–Simons theories. II
Journal of Mathematical Physics, 1996
The diagrammatic and the Feynman rules for the higher derivative Chern-Simons theories in ͑2ϩ1͒ dimensions coupled to fermionic matter are constructed. This is done by starting from the path-integral quantization. Once the diagrammatic and the Feynman rules are given, the regularization and renormalization problem of this higher derivative model is analysed in the framework of the perturbation theory. The unitarity problem related with the possible appearance of ghost states with negative norm is also discussed. Finally, the BRST formalism for the model is constructed and some interesting differences with respect to the formalism applied to usual Chern-Simons models are presented.
Poles in the S-matrix of relativistic Chern-Simons matter theories from quantum mechanics
Journal of High Energy Physics, 2015
An all orders formula for the S-matrix for 2 → 2 scattering in large N Chern-Simons theory coupled to a fundamental scalar has recently been conjectured. We find a scaling limit of the theory in which the pole in this S-matrix is near threshold. We argue that the theory must be well described by non-relativistic quantum mechanics in this limit, and determine the relevant Schroedinger equation. We demonstrate that the S-matrix obtained from this Schroedinger equation agrees perfectly with this scaling limit of the relativistic S-matrix; in particular the pole structures match exactly. We view this matching as a nontrivial consistency check of the conjectured field theory S-matrix.
Low energy limit of the Chern-Simons theory coupled to fermions
Physical Review D
We study the nonrelativistic limit of the theory of a quantum Chern-Simons field minimally coupled to Dirac fermions. To get the nonrelativistic effective Lagrangian one has to incorporate vacuum polarization and anomalous magnetic moment effects. Besides that, an unsuspected quartic fermionic interaction may also be induced. As a by product, the method we use to calculate loop diagrams, separating low and high loop momenta contributions, allows to identify how a quantum nonrelativistic theory nests in a relativistic one.
Nuclear Physics B, 1993
We explicitly carry Out the renormalization of non-abelian Chern-Simons field theories with matter in (2+1) dimensions. All the renormalization constants are calculated to the leading two-loop order both in terms of component fields and N = 1 superfields for the fundamental representation of the SU(n), Sp(n) and SO(n) groups. Renormalization-group fixed points are found, and their stability properties are examined. It is shown that the N 2 supersymmetry is realized as an infrared fixed-point solution, where the ultraviolet divergencies cancel.
Integrable Chern-Simons Gauge Field Theory in 2+1 Dimensions
1995
The classical spin model in planar condensed media is represented as the U(1) Chern-Simons gauge field theory. When the vorticity of the continuous flow of the media coincides with the statistical magnetic field, which is necessary for the model's integrability, the theory admits zero curvature connection. This allows me to formulate the model in terms of gauge - invariant fields whose evolution is described by the Davey-Stewartson (DS) equations. The Self-dual Chern-Simons solitons described by the Liouville equation are subjected to corresponding integrable dynamics. As a by-product the 2+1-dimensional zero-curvature representation for the DS equation is obtained as well as the new reduction conditions related to the DS-I case. Some possible applications for the statistical transmutation in the anyon superfluid and TQFT are briefly discussed.