Strong Coupling Universality at Large N for Pure CFT Thermodynamics in 2+1 dimensions

Pure CFTs have vanishing $\beta$-function at any value of the coupling. One example of a pure CFT is the O(N) Wess-Zumino model in 2+1 dimensions in the large N limit. This model can be analytically solved at finite temperature for any value of the coupling, and we find that its entropy density at strong coupling is exactly equal to 31/35 of the non-interacting Stefan-Boltzmann result. We show that a large class of theories with equal numbers of N-component fermions and bosons, supersymmetric or not, for a large class of interactions, exhibit the same universal ratio. For unequal numbers of fermions and bosons we find that the strong-weak thermodynamic ratio is bounded to lie in between 4/5 and 1.


Introduction
Among quantum field theories, conformal field theories (CFTs) are special because their β-function vanishes and no mass scales are present. Typically, CFTs can be obtained from a quantum field theory parent by tuning the coupling to a critical value. However, there is a class of special CFTs, sometimes referred to as "pure" CFTs, which are CFTs for all values of the coupling, not just at a finite number of critical points. Pure CFTs are not common, because they are special examples of an already special set of quantum field theories. One example is N = 4 Super-Yang-Mills theory in 3+1 dimensions, which is a pure CFT for any number of colors N . However, one may also have theories that are pure CFTs only in the limit N → ∞, such as the interacting O(N) model in 2+1 dimensions.
Because pure CFTs are so special, it is interesting to study some of their properties. A famous property of N = 4 SYM in 3+1 dimensions is that its free energy at finite temperature evaluated at infinite coupling (calculated via its conjectured gravity dual [1]) is exactly 3/4 of the Stefan-Boltzmann value expected for a free theory [2]. Using the strong constraints of CFTs in 1+1 dimensions [3], it is possible to construct pure CFTs with a strong-weak thermodynamic ratio given by the ratio of central charges, finding a strong-weak thermodynamic ratio of 1 2 for scalar φ 4 theory in 1+1 dimensions with a mass tuned to the Ising point [4]. Finally, for scalar theory with quartic interaction in 2+1 dimensions, a strong-weak thermodynamic ratio of 4/5 has been found in Ref. [5] (cf. Ref. [6]), and the ratio has recently been shown to be universal in a large class of pure CFTs with only bosonic degrees of freedom [7].
How general is the ratio of 4/5 for pure CFTs in 2+1 dimensions? In order to answer this question, we consider generalizing Ref. [7] to include fermionic degrees of freedom in this work. We will start by considering a supersymmetric theory in 2+1 dimensions, the O(N) Wess-Zumino model, but then start modifying the Lagrangian, first by breaking supersymmetry, then by changing the interactions, and finally by considering unequal numbers of fermionic and bosonic degrees of freedom.

Calculation
Let us consider the O(N) supersymmetric Wess-Zumino model [8] in 2+1 dimensions given by the superspace action where the superfield Φ a and its spinorial differentiation can be written in components as Here φ a , F a are N-component real scalar fields and ψ a is an N-component Majorana spinor in 3 dimensions. Performing the integration over supervariablesθ, θ and integrating out the scalars F a in the action leads to (2.4) The Yukawa-type terms break the parity symmetry x 2 → −x 2 , ψ a → iγ 2 ψ a , but this indicates the physics of the model is invariant under λ → −λ up to a parity transformation. The termψ a ψ b φ a φ b does not contribute to leading order in the large N limit and hence will be neglected in the following.
The theory can be heated up to temperature T by analytically continuing time to be imaginary and compactified on a circle of radius β = T −1 . The partition function is then given by with S E the Euclidean action Introducing an auxiliary field σ = φ c φ c /N and its Lagrange multiplier ζ as 1 = DσDζe i ζ(σ−φcφc/N ) , only the zero modes of σ, ζ contribute to the leading order large N result of the partition function. As a result, the Gaussian integral over the bosonic and fermionic fields may be performed as where bosonic and fermionic sum-integrals in 3 − 2 dimensions are written as K = T n 2 ) (see e.g. Ref. [9]). Rescaling ζ → N ζ/2 and σ → σ/(4λ) this leads to the partition function given by where the bosonic and fermionic thermal sum-integrals J B , J F in 2+1 dimensions are finite in the → 0 limit. For large N, the partition function may be evaluated exactly using the saddle points located at iζ = z * , σ = σ * given by the solution of the nonperturbative coupled gap equations It is tempting to interpret σ * as the in-medium fermion mass and √ z * as the in-medium boson mass. Since the model does not break chiral symmetry, these 1 Note that the results for I F,B (m) can be obtained directly by writing I B,F (m) = P,{P } 1 P 2 +m 2 and performing the thermal sums, finding 1 e x/T ∓1 for bosons and fermions, respectively, cf. Ref. [9]. in-medium masses vanish in the zero-temperature limit. Hence we may set z * = m 2 B T 2 , σ * = m F T such that the above gap equations become It is straightforward to verify that in the weak coupling limit λ → 0, the solution to these equations is m F = m B = 0, indicating vanishing thermal masses for both bosons and fermions. At finite interaction strength, bosons and fermions develop non-vanishing thermal masses. At weak coupling λ 1, these may be calculated in perturbation theory, but the gap equations (2.12) can be solved numerically for any value of the interaction strength (see Fig. 1). Curiously, we find that in the strong coupling limit, the masses tend to

Thermodynamics
The pressure P (minus the free energy density) for the Wess-Zumino model in 2+1 dimensions can be calculated as (2.14) The functions J B (m), J F (m) may be evaluated by direct integration of the known forms of I B (m), I F (m) in Eq. (2.11). One finds Plugging in the expressions for the bosonic in-medium mass √ z * = m B T and fermionic in-medium mass σ * = m F T allows evaluation of the pressure for all values of the interaction. However, the entropy density s = ∂P ∂T is a somewhat more convenient thermodynamic variable because the temperature derivatives of the in-medium masses √ z * , σ * cancel as a result of the saddle point conditions (2.10). One finds In the limit where in-medium masses vanish, one has where the factor 3/4 is the known ratio of fermionic to bosonic degrees of freedom for a free theory in 2+1 dimensions. Thus, for a free theory, For finite values of the interaction parameter, Eqns. (2.16) may be evaluated numerically using the solutions for m B , m F from the non-perturbative gap equations (2.12) at any value of the coupling λ. The resulting dependence of the entropy density (scaled by s free ) is shown in Fig. 2.
In the strong coupling limit, m B → 2 ln 1+ √ 5 2 and the properties of the polylogarithms may be used to show [5,6] Since in the strong coupling limit the fermionic in-medium mass vanishes m F → 0, the entropy density in the strong coupling limit becomes

Strong Coupling Universality
Taken by itself, the result that the large N Wess-Zumino model in 2+1 dimensions is a pure CFT with a strong-weak thermodynamic ratio of 31 35 is perhaps not that relevant or enlightening. For this reason, it is interesting to study if the ratio of 31 35 is valid more generally for a larger class of theories.
First, while the Wess-Zumino realizes N = 1 supersymmetry at zero temperature, note that the ratio 31 35 does not depend on the theory to be supersymmetric (this is not expected given that finite temperature breaks supersymmetry anyway). To see this, simply change the coefficient of the bosonic interaction potential in (2.6) from we can consider arbitrary interaction potentials U including the possibility of adding relevant operators. The only conditions on U (x) are that there is no spontaneous mass generation which would break the CFT, and that its argument is non-negative (since x = φaφa N ). The former condition requires that the potential only has one trivial minimum located at x = 0 or U (0) = 0, U (0) > 0.
For any such potential U , the relevant gap equation gets modified to as before, such that the lhs of this equation tends to zero. Since U (0) > 0 and the only minimum of the potential is located at x = 0, this implies that the rhs of the above equation is positive definite for all m F > 0, leading to m F = 0 as the only possible solution.
As a consequence, the strong coupling results (2.13), (2.20) are universally true for a large class of interaction potentials U (x) in the large N limit.

Changing the balance between fermions and bosons
As is easy to see, the above argument trivially extends to cases where the field content of the theory is modified to contain multiple copies of the scalars and fermions as long as they come in equal number. This establishes the universality of the strong-weak thermodynamic ratio of 31 35 for a large class of 2+1 dimensional pure CFTs with equal number of bosonic and fermionic degrees of freedom in the large N limit.
But what about changing the balance between fermionic and bosonic degrees of freedom? For this reason, let us consider B N-component scalars coupled to F Ncomponent Majorana spinors in 2+1 dimensions with a Euclidean action given by (3.4) Proceeding as above leads to a partition function for this theory which is given by 5) and the gap equations become which is a function that monotonically increases with the ratio of fermionic to bosonic degrees of freedom F/B. For F = B, one recovers 31 35 , as above. Interesting limits of the ratio s s free are no fermions (or, equivalently, an infinite number of bosons with only a finite number of fermions), for which s s free = 4 5 , whereas for the opposite limit of no bosons (or an infinite number of fermions), s s free = 1. Thus we find that for a large class of theories with different field content and different interactions the thermodynamic ratio is bounded by in the large N limit in 2+1 dimensions.

Summary and Conclusions
In this work, we have considered the thermodynamics of pure CFTs in 2+1 dimensions with interacting fermions and bosons in the large N limit. We solved the supersymmetric O(N) Wess-Zumino model analytically at finite temperature for all coupling values in the large N limit. Our results show that the entropy density monotonically decreases from its Stefan-Boltzmann value s free at weak coupling to 31 35 × s free at infinite coupling. At intermediate values of the coupling, the thermodynamic ratio s s free depends on the details of the interaction. However, we found that the infinitely strong to weak coupling thermodynamic ratio of 31 35 is universal for a large class of interaction potentials with equal number of fermionic and bosonic degrees of freedom in 2+1 dimensions, with and without supersymmetry. Furthermore, we found that for unequal numbers of fermionic and bosonic degrees of freedom the corresponding weak-strong thermodynamic ratio is bounded by the purely bosonic result 4/5 from Ref. [7] from below, and unity above.
Many open questions remain, such as finite N corrections to the ratios 4/5 and 31/35 within the class of theories considered here, or if pure CFTs with gauge fields exist that can be solved for all interaction strengths. Extensions of this work to higher odd dimensions is feasible and could be considered along the lines of Ref. [10]. Examples of solvable field theories with simple known gravity duals are rare, cf. Refs. [11,12]. Therefore, another direction to consider could be studies of possible gravitational duals to the CFTs considered here, such as the gravity dual to the O(N) Wess-Zumino model in 2+1 dimensions along the lines of Ref. [13]. We leave these questions to future work.