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Register a new userFermi seas from Bose condensates in Chern-Simons matter theories and a bosonic exclusion principle
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We generalize previously obtained results for the (all orders in the t Hooft coupling) thermal free energy of bosonic and fermionic large $N$ Chern-Simons theories with fundamental matter, to values of the chemical potential larger than quasiparticle thermal masses. Building on an analysis by Geracie, Goykhman and Son, we present a simple explicit formula for the occupation number for a quasiparticle state of any given energy and charge as a function of the temperature and chemical potential. This formula is a generalization to finite t Hooft coupling of the famous occupation number formula of Bose-Einstein statistics, and implies an exclusion principle for Chern-Simons coupled bosons: the total number of bosons occupying any particular state cannot exceed the Chern-Simons level. Specializing our results to zero temperature we construct the phase diagrams of these theories as a function of chemical potential and the UV parameters. At large enough chemical potential, all the bosonic theories we study transit into a compressible Bose condensed phase in which the runaway instability of free Bose condensates is stabilized by the bosonic exclusion principle. This novel Bose condensate is dual to - and reproduces the thermodynamics of - the fermionic Fermi sea.
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It has been conjectured that Chern-Simons (CS) gauged `regular bosons in the fundamental representation are `level-rank dual to CS gauged critical fermions also in the fundamental representation. Generic relevant deformations of these conformal field theories lead to one of two distinct massive phases. In previous work, the large $N$ thermal free energy for the bosonic theory in the unHiggsed phase has been demonstrated to match the corresponding fermionic results under duality. In this note we evaluate the large $N$ thermal free energy of the bosonic theory in the Higgsed phase and demonstrate that our results, again, perfectly match the predictions of duality. Our computation is performed in a unitary gauge by integrating out the physical excitations of the theory - i.e. W bosons - at all orders in the t Hooft coupling. Our results allow us to construct an exact quantum effective potential for ${bar phi} phi$, the lightest gauge invariant scalar operator in the theory. In the zero temperature limit this exact Landau-Ginzburg potential is non-analytic at ${bar phi phi}=0$. The extrema of this effective potential at positive ${bar phi}phi$ solve the gap equations in the Higgsed phase while the extrema at negative ${bar phi} phi$ solve the gap equations in the unHiggsed phase. Our effective potential is bounded from below only for a certain range of $x_6$ (the parameter that governs sextic interactions of $phi$). This observation suggests that the regular boson theory has a stable vacuum only when $x_6$ lies in this range.
We study a certain class of supersymmetric (SUSY) observables in 3d $mathcal{N}=2$ SUSY Chern-Simons (CS) matter theories and investigate how their exact results are related to the perturbative series with respect to coupling constants given by inverse CS levels. We show that the observables have nontrivial resurgent structures by expressing the exact results as a full transseries consisting of perturbative and non-perturbative parts. As real mass parameters are varied, we encounter Stokes phenomena at an infinite number of points, where the perturbative series becomes non-Borel-summable due to singularities on the positive real axis of the Borel plane. We also investigate the Stokes phenomena when the phase of the coupling constant is varied. For these cases, we find that the Borel ambiguities in the perturbative sector are canceled by those in nonperturbative sectors and end up with an unambiguous result which agrees with the exact result even on the Stokes lines. We also decompose the Coulomb branch localization formula, which is an integral representation for the exact results, into Lefschetz thimble contributions and study how they are related to the resurgent transseries. We interpret the non-perturbative effects appearing in the transseries as contributions of complexified SUSY solutions which formally satisfy the SUSY conditions but are not on the original path integral contour.
We study large $N$ 2+1 dimensional fermions in the fundamental representation of an $SU(N)_k$ Chern Simons gauge group in the presence of a uniform background magnetic field for the $U(1)$ global symmetry of this theory. The magnetic field modifies the Schwinger Dyson equation for the propagator in an interesting way; the product between the self energy and the Greens function is replaced by a Moyal star product. Employing a basis of functions previously used in the study of non-commutative solitons, we are able to exactly solve the Schwinger Dyson equation and so determine the fermion propagator. The propagator has a series of poles (and no other singularities) whose locations yield a spectrum of single particle energies at arbitrary t Hooft coupling and chemical potential. The usual free fermion Landau levels spectrum is shifted and broadened out; we compute the shifts and widths of these levels at arbitrary tHooft coupling. As a check on our results we independently solve for the propagators of the conjecturally dual theory of Chern Simons gauged large $N$ fundamental Wilson Fisher bosons also in a background magnetic field but this time only at zero chemical potential. The spectrum of single particle states of the bosonic theory precisely agrees with those of the fermionic theory under Bose-Fermi duality.
Continuing the work arXiv:1603.06207, we study perturbative series in general 3d $mathcal{N}=2$ supersymmetric Chern-Simons matter theory with $U(1)_R$ symmetry, which is given by a power series expansion of inverse Chern-Simons levels. We find that the perturbative series are usually non-Borel summable along positive real axis for various observables. Alternatively we prove that the perturbative series are always Borel summable along negative (positive) imaginary axis for positive (negative) Chern-Simons levels. It turns out that the Borel resummations along this direction are the same as exact results and therefore correct ways of resumming the perturbative series.
It has been conjectured that fermions minimally coupled to a Chern-Simons gauge field define a conformal field theory (CFT) that is level-rank dual to Chern-Simons gauged Wilson-Fisher Bosons. The CFTs in question admit relevant deformations parametrized by a real mass. When the mass deformation is positive, the duality of the two deformed theories has previously been checked in detail in the large $N$ limit by comparing explicit all orders results on both sides of the duality. In this paper we perform a similar check for the case of negative mass deformations. In this case the bosonic field condenses triggering the Higgs mechanism. The effective excitations in this phase are massive $W$ bosons. By summing all leading large $N$ graphs involving these $W$ bosons we find an all orders (in the t Hooft coupling) result for the thermal free energy of the bosonic theory in the condensed phase. Our final answer perfectly matches the previously obtained fermionic free energy under the conjectured duality map.
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