Duality and an exact Landau-Ginzburg potential for quasi-bosonic Chern-Simons-Matter theories (original) (raw)
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Bose-Fermi Chern-Simons Dualities in the Higgsed Phase
JHEP 1811 (2018) 177
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.
Chern Simons duality with a fundamental boson and fermion
Journal of High Energy Physics, 2013
We compute the thermal free energy for all renormalizable Chern Simon theories coupled to a single fundamental bosonic and fermionic field in the 't Hooft large N limit. We use our results to conjecture a strong weak coupling duality invariance for this class of theories. Our conjectured duality reduces to Giveon Kutasov duality when restricted to N = 2 supersymmetric theories and to an earlier conjectured bosonization duality in an appropriate decoupling limit. Consequently the bosonization duality may be regarded as a deformation of Giveon Kutasov duality, suggesting that it is true even at large but finite N. B. Bose Condensation? 29 B.1 Space of solutions of (B.1) 30 B.2 Dualization of the fermionic gap equation 30 B.3 Dualization of the bosonic gap equation 33
Physical Review D
While the phase structure of the U (1) × U (1)-symmetric Higgs theory is still under debate, a version of this theory with an additional Chern-Simons term was recently shown to undergo a second-order phase transition [V. Shyta, J. van den Brink, and F. S. Nogueira, Phys. Rev. Lett. 127, 045701 (2021)]. This theory is dual to a topological field theory of massless fermions featuring two gauge fields. Here we elaborate on several aspects of this duality, focusing on the critical current correlators and on the nature of the critical point as reflected by the bosonization duality. The current correlators associated to the U (1) × U (1) symmetry and the topological current are shown to coincide up to a universal prefactor, which we find to be the same for both U (1) and U (1) × U (1) topological Higgs theories. The established duality offers in addition another way to substantiate the claim about the existence of a critical point in the bosonic Chern-Simons U (1) × U (1) Higgs model: a Schwinger-Dyson analysis of the fermionic dual model shows that no dynamical mass generation occurs. The same cannot be said for the theory without the Chern-Simons term in the action.
Phase structure of Chern-Simons gauge theories
arXiv (Cornell University), 2003
We study the effect of a Chern-Simons (CS) term in the phase structure of two different Abelian gauge theories. For the compact Maxwell-Chern-Simons theory, with the CSterm properly defined, we obtain that for values g = n/2π of the CS coupling with n = ±1, ±2, the theory is equivalent to a gas of closed loops with contact interaction, exhibiting a phase transition in the 3dXY universality class. We also employ Monte Carlo simulations to study the noncompact U (1) Abelian Higgs model with a CS term. Finite size scaling of the third moment of the action yields critical exponents α and ν that vary continuously with the strength of the CS term, and a comparison with available analytical results is made.
Physical Review D, 2005
We use AdS/CFT duality to study the thermodynamics of a strongly coupled N = 2 supersymmetric large N c SU(N c ) gauge theory with N f = 2 fundamental hypermultiplets. At finite temperature T and isospin chemical potential µ, a potential on the Higgs branch is generated, corresponding to a potential on the moduli space of instantons in the AdS description. For µ = 0, there is a known first order phase transition around a critical temperature T c . We find that the Higgs VEV is a suitable order parameter for this transition; for T > T c , the theory is driven to a non-trivial point on the Higgs branch. For µ = 0 and T = 0, the Higgs potential is unbounded from below, leading to an instability of the field theory due to Bose-Einstein In its original form, AdS/CFT duality [1, 2, 3] relates theories of closed strings in asymptotically AdS spaces to large N c gauge theories with matter in the adjoint representation. Fields in the fundamental representation may be added by including an open string sector through the introduction of branes probing the supergravity background. Much effort has gone into studying dualities of this type, motivated largely by the goal of finding a supergravity background dual to QCD.
The Hilbert Space of large NNN Chern-Simons matter theories
arXiv (Cornell University), 2022
The impact of the quantum singlet constraint in the limit V 2 → ∞ 4.1 Contrasting the 'classical' and 'quantum' Gauss laws at large volume 4.2 Simplification at large V 2 4.3 Some q-number identities 4.4 Interpolation between free bosons and free fermions 4.5 A note on quantum dimensions 4.6 Universal distribution of representations 5 Energy renormalization 5.1 Only forward scattering interactions contribute to the free energy at large N 5.2 The forward scattering truncated effective Hamiltonian 5.2.1 Effective Hamiltonian from the Lagrangian written in terms of c 2 B and σ B 5.2.2 Effective Hamiltonian from the Lagrangian written only in terms of φ 5.3 The partition function of the RB theory as a trace over H eff 5.4 Effect of F int the expectation value of charge 6 Thermodynamics from an entropy functional 6.1 Thermodynamics of generalized 'free' theories 6.2 Specialization to Chern-Simons matter theories at large volume 6.3 The entropy functional 6.4 The entropy functional for large volume Chern-Simons matter theories including interactions 7 Counting quantum singlets 7.1 Chern-Simons theories and WZW theories 7.2 The methods employed to evaluate the number of singlets 7.2.1 The Verlinde formula 7.2.2 Evaluation of the Chern-Simons path integral in the presence of Wilson lines 7.2.3 Evaluation of the path integral using supersymmetric localization 7.3 SU (N) k Chern-Simons theory 7.3.1 A simple expression for the Verlinde S-matrix 7.3.2 An orthonormal basis for su(N) weights and a lightning review of the su(N) Lie algebra 7.3.3 Discretized SU (N) eigenvalues 7.3.4 The Vandermonde factor 7.3.5 The final formula 7.3.6 A special eigenvalue configuration 7.3.7 Interpretation in terms of path integrals 7.3.8 Verification using supersymmetric localization 7.4 U (N) k,k theories 7.4.1 u(N) representation theory-ii-7.4.2 Integrable representations for the U (N) k,k WZW model 7.4.3 Verlinde S-matrices 7.4.4 Discretized U (N) eigenvalues 7.4.5 The Vandermonde 7.4.6 The final formula 7.4.7 Eigenvalues corresponding to the trivial representation 7.4.8 A path integral derivation 7.5 Supersymmetric localization and the path integral 7.5.1 N=2 Chern-Simons matter theories 7.5.2 Relationship between supersymmetric and non-supersymmetric Chern-Simons levels 7.5.3 The superconformal index 7.5.4 Supersymmetric Wilson loops on S 2 × S 1 7.5.5 The fugacity x in the index of pure N = 2 Chern-Simons theory with Wilson loops 7.5.6 Partition function from index at x = −1 7.5.7 The index at x = 1 and the Bose-Fermi nature of states on S 2 7.6 A Check of level-rank duality in Type I U (N) theory 7.6.1 Type I-Type I duality: map between representations 7.6.2 Implications 7.7 Check of Level-Rank duality between SU (N) k and Type II U (k) −N,−N theory 7.7.1 SU (N)-Type II duality: map between representations 8 Discussion A Enumerating classical SU (N) and U (N) singlets J One-loop determinants in U (N) k,k Chern-Simons theories 149 K Transformation of the Verlinde formula measure under duality 154 L More about level-rank duality and the character formula 156 L.1 Map of characters under level rank duality for general representations in Type I theory at even κ 156 L.1.1 Representations with less than κ boxes 156 L.2 Total number of boxes in tableau less than κ 157 L.2.1 Comments on representations with more than κ boxes 158 M Conventions on S 2 × S 1 and N = 2 supersymmetry transformations 159 N Some explicit results for the index in the presence of Wilson loops 159 N.1 SU (2) κ index 160 N.2 U (2) κ,k index 160 1 More explicitly, the Hilbert space H Fock is a free Fock space over the one-particle Hilbert space, i.e. over the space of solutions of the linearized Yang-Mills equations on S 2 × S 1. 2 The subscript 'cl' in I cl signifies that we are looking at a 'classical' situation that arises in the limit k → ∞ of the Chern-Simons matter theories that we study in this paper, where k is the level of the Chern-Simons theory. The meaning of this notational point will become clearer in later sections (say, e.g. in subsection 1.4) when we discuss the corresponding claim in the finite k Chern-Simons coupled matter theories. 3 Even though H is originally defined as a Hamiltonian on H Fock , as H is gauge invariant, it commutes with the projector onto singlets and its restriction to the space H cl is unambiguous and Hermitian. 4 This seemingly innocuous projection can have a dramatic impact on the value of the partition function e.g. the large N limit. In the large N limit, the projection is responsible for the partition function (1.4) to undergo phase transitions as a function of temperature, from a low temperature 'glueball' phase to a high temperature gluonic phase. 18 Once again the expression on the last two lines of the off-shell free energies for the critical and regular boson theories (2.10) and (2.11) are identical and equal to the partition function of N B free bosons of mass c B twisted by the holonomy U , but corrected with the strange looking term proportional to Θ(|µ| − c B). Again, we find it convenient to give a new symbol 17 The variable σB in the regular boson free energy (2.11) has a simple physical interpretation. It is related to the expectation value of the lightest gauge-invariant operator,φφ, of the regular theory as σB = 2π N B φ φ. 18 The equation of motion that results from varying w.r.t.S in (2.11) is quite simple. One solution to this equation of motion is given by settingS = −cB. This solution puts us in the unHiggsed phase [71]. After making this choice, the terms in the first and second line of (2.11) reduce to the function Fint(c 2 B , σ) of (1.13).
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.
Phases of large N vector Chern-Simons theories on S 2 × S 1
Journal of High Energy Physics, 2013
We study the thermal partition function of level k U (N) Chern-Simons theories on S 2 interacting with matter in the fundamental representation. We work in the 't Hooft limit, N, k → ∞, with λ = N/k and T 2 V 2 N held fixed where T is the temperature and V 2 the volume of the sphere. An effective action proposed in arXiv:1211.4843 relates the partition function to the expectation value of a 'potential' function of the S 1 holonomy in pure Chern-Simons theory; in several examples we compute the holonomy potential as a function of λ. We use level rank duality of pure Chern-Simons theory to demonstrate the equality of thermal partition functions of previously conjectured dual pairs of theories as a function of the temperature. We reduce the partition function to a matrix integral over holonomies. The summation over flux sectors quantizes the eigenvalues of this matrix in units of 2π k and the eigenvalue density of the holonomy matrix is bounded from above by 1 2πλ. The corresponding matrix integrals generically undergo two phase transitions as a function of temperature. For several Chern-Simons matter theories we are able to exactly solve the relevant matrix models in the low temperature phase, and determine the phase transition temperature as a function of λ. At low temperatures our partition function smoothly matches onto the N and λ independent free energy of a gas of non renormalized multi trace operators. We also find an exact solution to a simple toy matrix model; the large N Gross-Witten-Wadia matrix integral subject to an upper bound on eigenvalue density. 6.4.1 Low temperature expansion 6.5 Critical fermion 6.5.1 Low temperature expansion 7. Exact solution of the Large N capped GWW model 7.1 No gap solution 7.2 Single lower gap solution 7.3 Single upper gap solution 7.4 One lower gap and one upper gap solution 7.5 Level Rank Duality 7.6 Summary 7.6.1 λ < 1 2 7.6.2 λ > 1 2 8. Discussion A. Interpretation of the Fadeev-Popov Determinant B. Solution to the saddle point equations of 'capped' unitary matrix models B.1 Review of standard Unitary matrix integrals at large N B.1.1 General solution of (B.1) B.1.2 The GWW problem B.2 The capped unitary matrix model C. Solution of the Capped GWW model C.1 One upper gap solution of the capped GWW model C.2 Two cut solution of the capped GWW model C.3 Special Limits of the capped GWW model C.3.1 ρ(α) as b → π C.3.2 ρ(α) as a → 0 C.3.3 The large ζ limit C.3.4 Eigenvalue density near the end points of cuts C.4 Level rank duality of the solution to the capped GWW model C.5 Behavior of (λ, ζ) with respect to (a, b) in the one lower gap and one upper gap solution D. Level-rank duality of the saddle point equation in the multi trace potential E. High temperature limit of the partition function of a gas of non renormalized multitrace operators 8 In the case that the base manifold is an S 2 , as studied in our paper, the exclusion of such configurations follows from the fact that the measure factor in (1.1) eliminates their contributions. The generalization of (1.6) to the partition function of Chern-Simons theory on Σg × S 1 where Σg is a genus g manifold of arbitrary metric is given by the formula (4.6). In these cases the measure factors is either constant (in the case of g = 1) or diverges when two eigenvalues are equal. Nonetheless the correct prescription (the one that agrees with Chern-Simons computations using other techniques) appears to be to omit the contribution of such sectors. The justification for this prescription does not appear to be clearly understood from first principle path integral reasoning. We hope to clear up this point in the future. We thank O. Aharony, S. Giombi and J. Maldacena for extensive discussions on this point. 9 The same upper bound for the density function appears in two dimensional Yang-Mills theory (p = 1), in which case the situation becomes similar due to the fact that there is no propagating degrees of freedom for the gauge field. A phase transition relevant to this upper bound was studied in 2d (q-deformed) Yang-Mills theory on S 2 [33, 34, 35, 36], which we will see from Chern-Simons theory below. We noticed these relevant papers when we were completing this paper.
On thermodynamics of 𝒩 = 6 superconformal Chern-Simons theories at strong coupling
Journal of High Energy Physics, 2008
Recently it has been conjectured that N = 6, U (N ) k × U (N ) −k Chern-Simons theory is dual to M-theory on AdS 4 × S 7 /Z k . By studying one-loop correction to the M-theory effective action, we calculate the correction to the entropy of thermal field theory at strong coupling. For large k level, we have also found the α ′ correction to the entropy from the string correction of the type IIA effective action. The structure of these two corrections at strong 't Hooft coupling are different. 1
Flows, fixed points and duality in Chern-Simons-matter theories
Journal of High Energy Physics, 2018
It has been conjectured that 3d fermions minimally coupled to Chern-Simons gauge fields are dual to 3d critical scalars, also minimally coupled to Chern-Simons gauge fields. The large N arguments for this duality can formally be used to show that Chern-Simons-gauged critical (Gross-Neveu) fermions are also dual to gauged ‘regular ’ scalars at every order in a 1/N expansion, provided both theories are well-defined (when one fine-tunes the two relevant parameters of each of these theories to zero). In the strict large N limit these ‘quasi-bosonic’ theories appear as fixed lines parameterized by x 6, the coefficient of a sextic term in the potential. While x 6 is an exactly marginal deformation at leading order in large N, it develops a non-trivial β function at first subleading order in 1/N. We demonstrate that the beta function is a cubic polynomial in x 6 at this order in 1/N, and compute the coefficients of the cubic and quadratic terms as a function of the ’t Hooft coupling. We co...