Confinement and Graded Partition Functions for $\mathcal{N}=4$ SYM

Gauge theories with confining phases at low temperatures tend to deconfine at high temperatures. In some cases, for example in supersymmetric theories, confinement can persist for all temperatures provided the partition function includes a grading by $(-1)^F$. When it is possible to define partition functions which smoothly interpolate between no grading and $(-1)^F$ grading, it is natural to ask if there are other choices of grading that have the same effect as $(-1)^F$ on confinement. We explore how this works for $\mathcal{N}=4$ SYM on $S^1\times S^3$ in the large $N$ limit at both small and large coupling. We find evidence for a continuous range of grading parameters that preserve confinement for all temperatures at large coupling, while at small coupling only a discrete set of gradings preserves confinement.

Introduction. Gauge theories tend to deconfine at high temperatures. This statement can be made precise in theories that have a center symmetry, such as SU (N ) Yang-Mills theory, see e.g. [1,2]. Studying the properties of a 4d theory on a spatial manifold M 3 at finite temperature is equivalent to asking about the behavior of the Euclidean path integral on M 3 × S 1 β with periodic boundary conditions for bosons and anti-periodic boundary conditions for fermions. This path integral gives the thermodynamic partition function Z(β) = tr e −βH (1) where β = 1/T . Wilson loops that wind around the thermal circle, tr Ω = Pe i S 1 A transform by a Z N phase under center symmetry, tr Ω → e 2πi/N tr Ω. In pure SU (N ) YM theory, it is known that tr Ω = 0 at low temperatures, see e.g. [3], but it is non-zero at high temperatures [1]. This implies that the high and low temperature regimes are separated by a phase transition in the thermodynamic limit. Note however that there are actually two physically distinct ways to take the thermodynamic limit for SU (N ) YM theory. The standard way is take the volume of M 3 to infinity. Alternatively, one can take N → ∞ with the 't Hooft coupling λ = g 2 YM N and other parameters such as the spatial volume held fixed [4][5][6]. There is a temperature-driven deconfinement transition in both of these two thermodynamic limits. In this paper we will discuss the deconfinement phase transition in the the large N thermodynamic limit.
The deconfinement phase transition is rather difficult to evade by, e.g., varying parameters controlling the matter content. Indeed, in the large N limit, the deconfinement transition is forced by the fact that the confining phase features a Hagedorn density of states. This means that the density of states, ρ(E), grows exponentially with the energy ρ(E) ∼ e +β H E , where it is assumed that E ∼ N 0 is large compared to the characteristic mass scale of the theory, and β H > 0, the Hagedorn temperature, depends on the matter content of the theory, but can be estimated as β H ∼ min(Λ −1 , R) where Λ is the large scale and R is the characteristic size of the spatial box. Given this density of states, the thermodynamic partition function cannot be analytic at β = β H . Thus the theory must have deconfinement transition at some β c ≥ β H at large N .
However, the deconfinement phase transition can be evaded if one considers some graded partition functions. The most famous example is where F is fermion parity. This partition function is calculated by a path integral with periodic boundary conditions for all fields on S 1 β . If the distribution of bosonic and fermionic states are uncorrelated, a (−1) F -graded partition function will also exhibit phase transitions as a function of β in the thermodynamic limit. However, if the spectra of bosonic and fermionic states are related in a sufficiently precise way, there will be large cancellations and the graded density of statesρ(E) can have sub-exponential growth at large E at arbitrary β. Theñ Z(β) can avoid the Hagedorn instability and the deconfinement phase transition.
This naturally occurs in supersymmetric gauge theories, where the bosonic and fermionic states with energies E > 0 necessarily come in degenerate multiplets in the infinite-volume limit. Surprisingly, deconfinement phase transitions of (2) can sometimes be avoided even without supersymmetry. Indeed, this is the case for adjoint QCD, YM theory coupled to N f > 1 adjoint fermions [7][8][9][10][11], and even in some theories with fundamental fermions, which require more complicated gradings [12]. In such theories,Z(β) is smooth, and hence the theory remains confining at all temperatures.
In this paper, we want to study what happens to deconfinement transition in theories where it is possible to smoothly interpolate between (1) and (2), both at large and small 't Hooft coupling. This is most easily done for the large N limit of 4d N = 4 super Yang-Mills theory, formulated in Euclidean signature on S 3 R × S 1 β , where R is the radius of S 3 . Indeed, it is known that without any grading, large N , N = 4 SYM has a deconfinement as a function of β/R, and the critical value of β/R is known both at small [4][5][6] and large [13][14][15][16] 't Hooft coupling 1 due to the AdS/CFT correspondence. Supersymmetry suggests that the theory should be in a confined phase for all β/R if the partition function includes a grading by (−1) F , and indeed this has been verified at large coupling in Ref. [16]. Finally, N = 4 SYM has a continuous internal global symmetry, SO(6) R , which, as we will show, can be used to define a grading that smoothly interpolates between (1) and (2). We define this grading below and study the resulting phase structure at small and large 't Hooft coupling.
Graded partition functions. In a conformal field theory (such as N = 4 SYM) on S 3 R × S 1 β , the partition function can be viewed as Z(β/R) = ∆ e −∆β/R , where the sum runs over the scaling dimensions of local operators, so that the Hamiltonian is related to the dilation operator H = D/R. In theories with global symmetries it is also possible to turn on chemical potentials µ i associated to conserved charges Q i , yielding In the specific case of N = 4 SYM, there is an SO(6) R-symmetry, and so one can turn on three independent chemical potentials associated with the three Cartan generators of SO(6) R . We can consider N = 4 SYM as N = 1 SYM coupled to three adjoint chiral supermultiplets, then the three complex scalar fields φ 1 , φ 2 , φ 3 in N = 4 SYM are the lowest components of the three chiral supermultiplets. The three scalars transform in the 6 representation of SO(6) R , and following the conventions of Ref. [19], we assemble the three scalars into a six-dimensional vector (φ 1 , φ * 1 , φ 2 , φ * 2 , φ 3 , φ * 3 ) T , and write the Cartan generators in the 6 representation as Then 2Q 6 i acts with charge 2 on φ i , and the superconformal R-charge r can be written as r = 2 3 (Q 1 + Q 2 + Q 3 ), see e.g. [20]. We will set µ 1 = µ 2 = µ 3 = 2iθ/β and study the partition function This graded partition function is periodic in θ with period 2π. It implements a simple interpolation from (1) at θ = 0 to (2) at θ = π, thanks to the fact that in N = 4 SYM there is a spin-charge relation e 3πir = (−1) F , see e.g. [21]. Charge conjugation symmetry further relates r to −r, so that the θ peridicity is reduced to θ θ + π. Zero coupling. Let us determine the β and θ dependence of Z in (7) in the zero coupling limit λ → 0, at large 1 More generally, the critical value of β/R is bounded from below by the Hagedorn temperature, which is known for all λ [17,18].
for λ → 0 as a function of inverse temperature β and grading θ. The theory is in a confined phase in the white region, and is partially confined in the colored regions. In between θ = π/3 and θ = 5π/3 center symmetry is only partially broken, with the unbroken subgroup of ZN indicated by the color scheme in the legend. The theory confines at all β/R when θ = π/3, π, 5π/6.
N . The partition function is given by a matrix integral which can be interpreted as an integral over the holonomy Ω [5], which implements the Gauss law constraint [1]. The matrix integral can be reduced to an integral over the eigenvalues of the holonomy where λ k are the eigenvalues, {µ i } are the three independent chemical potentials for mutually commuting charges in SO(6) R , and S eff is an effective action. The holonomy eigenvalues take values on a circle, and the effective action can be written in terms of Fourier coefficients ρ n of the eigenvalue distribution ρ where ρ n ≡ π −π dαρ(α) cos(nα) and the V n coefficients are given by The parameter x = e −β/R , and the functions z B = z V + z S , z F are related to the single-particle partition functions for the massless vector, scalar, and fermion fields of N = 4 SYM on S 3 [19]: We now set µ i = 2iθ β , so that So long as all of the coefficients V n are positive, the matrix integral is dominated by the center-symmetric ρ n = 0 minimum. When there is value of n for which V n becomes negative there is a phase transition. We can interpret these phase transitions as center-breaking phase transitions where Z N is spontaneously broken to an (approximate) subgroup Z n . (For an early discussion of partial deconfinement see e.g. [22].) For large β/R, corresponding to x 1, it is easy to verify that V n > 0 for all n, but for small β/R some V n generically become negative. Figure 1 is a plot of the phase diagram as a function of θ and β/R. Center symmetry is preserved above the black curve, and is broken either completely or to a subgroup of Z N below the black curve.
There are two interesting features in Fig. 1. First, the lack of deconfinement for any β at θ = π is consistent with our original expectations: it amounts to working with a (−1) F graded partition function. However, the theory also remains confining for all β/R if θ = π/3, 5π/3. In the region (π/3, π) ∪ (π, 5π/3) the system is in a (partially) center-broken phase for sufficiently small values of β/R. The dependence of the deconfinement temperature on θ is highly non-monotonic.
What should we expect as λ is increased from zero? At a heuristic level, increasing the coupling λ should increase the fluctuations of the Polyakov loop eigenvalues, and it is natural to expect such fluctuations to increase the range of β/R values where all traces of powers of the Polyakov loop vanish. It would be interesting but challenging to directly compute corrections to our results in powers of λ, see e.g. [6,23] for the challenges that arise already in pure YM theory. However, one of the special features of N = 4 SYM is that the λ → ∞, N → ∞ limit is just as calculable as the λ → 0, N → ∞ limit thanks to the AdS/CFT correspondence [13][14][15][16]. In what follows, we take advantage of AdS/CFT and study the θ dependence of the phase diagram at large 't Hooft coupling.
Infinite coupling. At large N and λ, N = 4 SU (N ) SYM is believed to have a dual gravitational description [13][14][15][16]. The dependence of the theory on R-charge chemical potentials has been extensively explored in the literature, see e.g. [15,16,19,24,25]. The AdS/CFT dictionary for conserved charges implies that to study N = 4 SYM with chemical potentials for R symmetry, we should consider the truncation of Type IIB supergravity on AdS 5 × S 5 to a 5d Einstein gravity theory coupled to three U (1) gauge fields (A I ), I = 1, 2, 3, associated with the U (1) 3 Cartan subgroup of SO(6) R [26][27][28][29][30]: where (F I ) M N = ∂ M (A I ) N − ∂ N (A I ) M are the field strengths of the three gauge fields, M, N = 1, . . . , 5, κ is related to the 5d Newton constant G 5 via κ = 8πG 5 , and the cosmological constant is Λ = −6/ 2 . The boundary values of bulk gauge fields act as sources for conserved currents in the boundary theory, and our goal is to turn on equal chemical potentials for each of the three Cartan charges. For this purpose we can simply set A I = A above, and then the bulk action reduces to an Einstein-Maxwell-AdS 5 theory The Einstein-Maxwell-AdS 5 system has been extensively explored in [24,25], and has Reissner-Nordstrom-AdS 5 black hole solutions 2 with parameters (L m , L q ) which are related to the ADM mass and the charge density respectively. The charged black hole solution reads This non-supersymmetric family of non-rotating solutions is expected to correspond to spatially-homogeneous equilibrium states of the dual field theory. We briefly review how the thermodynamic parameters of the bulk solution, i.e., the inverse temperature and chemical potential, map to the corresponding field theory parameters. First, we recall that can be identified with the radius of the S 3 spatial manifold in the dual field theory [19]. To identify β, we pass to Euclidean signature t → iτ , and recall that the metric is free of conical singularities provided τ is periodic with period where r + is the outer horizon corresponding to the largest positive root of f (r + ) = 0. The Hawking temperature T = 1/β is then identified with the temperature of the dual field theory. Next, the chemical potential is determined by the asymptotic value of the field strength along the τ − r disk, D 2 : 2 Our normalization convention is different from [24,25], so that The gauge field must be regular at r + , meaning that A τ (r + ) = 0, which yields the identification The confinement/deconfinement phase transition is mapped onto a gravitational phase transition in the following way. The solution with L m = L q = 0 should be thought of as thermal AdS 5 , and is dual to the confining phase of the dual gauge theory, whereas solutions with L 2 m ∈ R + are dual to the deconfined phase [16]. The difference of the free energies of these two phases is [24,25,31] When this quantity changes sign, there is a deconfinement phase transition from the point of view of the field theory [15,16], which is realized as a Hawking-page phase transition in the gravitational theory [32].
To map out the dependence of the phase transition temperature on the parameter θ in (7), we study how (23) behaves as a function of an appropriate imaginary chemical potential. We take µ → iμ = 2iθ β while keeping r + , which requires sending L 2 q → iL 2 q . Then A τ is real and continues to satisfy A τ (r + ) = 0. The on-shell Euclidean action becomes This expression is manifestly invariant under θ → −θ.
In addition, we should understand the θ dependence in this formula and the ones that follow mod 2π. In field theory, this basic fact follows from charge quantization. To see how this is matched in the bulk, we recall that Type IIB string theory has an NS-NS two-form gauge field B. One can turn on a flat B potential (that is, obeying dB = 0) in the solutions discussed above at no cost in energy, with an arbitrary value of the 2π periodic parameter α [16,33,34]: The periodicity of the chemical potential under shifts of 2πi/β is ensured by the shift freedom of α → α + 2π. It is now straightforward to determine the phase transition temperature as a function of θ. The critical temperature, T c , is determined by the solution of F deconfined − F confined = 0, which reduces to Using (20), we may trade r + for β c , replace by R, and find the transition temperature in field theory variables

FIG. 2.
Large N phase diagram of N = 4 SYM theory on S 3 R as a function of inverse temperature β and grading θ at zero and infinite 't Hooft coupling λ. The grading θ interpolates between the thermal and (−1) F graded partition functionsf. At λ = 0 the theory has a deconfinement phase transition along the black curve, and has at least partially broken center symmetry everywhere in the colored regions. The analogous phase transition curve due to known bulk solutions at large coupling λ → ∞ is shown in blue, with center symmetry broken everywhere in the hatched region. Note that the deconfined region at large-coupling is a subset of the deconfined region at zero coupling.
When θ = 0, corresponding to conventional thermal field theory we recover the standard Hawking-Page phase transition for uncharged SAdS 5 black holes β c = 2πR/3 [32]. As θ is increased the deconfinement temperature decreases. At θ = π/3 mod 2π the critical temperature becomes infinite, and there are no sensible Reissner-Nordstrom-AdS 5 black hole solutions because β c becomes complex. The fact that N = 4 SYM is confined for all β when θ = π, corresponding to (−1) F graded partition function, has been known since Ref. [16]. But our analysis here suggests that there is actually a whole window of θ values, namely θ ∈ π 3 , 5π 3 (28) where N = 4 SU (N ) SYM apparently remains confined for all values of β/R at large λ. We compare the zero and large coupling phase diagrams in Fig. 2. We should emphasize that to reach (27) we have only studied the known (analytically-continued) solutions that correspond to homogeneous equilibrium states in the dual field theory. It would be very interesting to see whether there might be some sort of previously unknown (perhaps multi-center) black objects which would be associated to the partially-confined phases 3 we saw at small coupling, see e.g. [42] for a discussion in the context of the superconformal index.
Conclusions. In N = 4 SYM theory, it is possible to smoothly interpolate from the standard thermodynamic partition function to a (−1) F -graded partition function by taking advantage of the SO(6) R-symmetry. We studied the simplest such generalized partition function, where the grading factor is e 3iθ r uses the superconformal R-charge r. We found that the large N theory is in the confined phase on S 3 R × S 1 β when θ = π/3, π, 5π/3 both at small and large coupling. The result for e 3πir = (−1) F was already known. The other two points with complete confinement for all temperatures at both small and large 't Hooft coupling correspond to the more complicated grading (−1) F e ±2πir . The small coupling and large coupling corners of parameter space differ in the behavior for θ ∈ (π/3, π) ∪ (π, 5π/3). At small coupling, the phase transition temperature depends non-monotonically on θ in the region θ ∈ (π/3, π)∪(π, 5π/3), with center symmetry partially broken for high enough temperatures, with the critical temperature diverging as θ → π. At large coupling, on the other hand, we found no evidence of such rich behavior: the theory appears to be in the confined phase for all temperatures for θ ∈ [π/3, 5π/3].
It is interesting to connect our results to some recent studies of deconfinement in the superconformal index of N = 4 SYM. At the outset, however, we should stress that our graded partition function is quite different from the superconformal index of N = 4 SYM [43], see e.g. Refs. [44,45] for recent reviews. The partition function we focus on here receives contributions from all of the states in the theory and depends non-trivially on the 't Hooft coupling. On the other hand, the superconformal index is a special S 3 R × S 1 β partition function which is designed to only receive contributions from 1/4 BPS states (in N = 1 language). It is independent of the 't Hooft coupling, so that its exact form for all values of λ and N can be determined in terms of a matrix integral. The superconformal index involves two chemical potentials (and corresponding fugacities p, q) built out of appropriate combinations of ∆ R-charge r and the left and right moving angular momenta j 1 , j 2 . When the fu-gacities are real, the superconformal index I is always in the 'confining phase', meaning that log I ∼ O(1) [43]. It was recently discovered that log I can become O(N 2 ) if the fugacities are rotated into the complex plane, so that the superconformal index can be used to study deconfinement and black holes, see Ref. [20,21,42,[46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65]. While the superconformal index and our partition functions are distinct, they do share an interesting parallel: the grading (−1) F e ±2πir plays the same role in both cases as being on the boundary between deconfinement for some β/R and confinement for all β/R. This was recently shown in Ref. [21], which studied the superconformal index as a function of p = q = ye iψ , finding a deconfined region for ψ ∈ (−4π, −3π) ∪ (−3π, −2π), with e.g. ψ = −2π corresponding to a grading by (−1) F e −2πir .
Our large-coupling results are less complete than our results at small coupling, because we do not know how to exclude the possibility that there are some presentlyunknown black objects which correspond to partially confined phases at large coupling. One might guess that increasing the 't Hooft coupling should increase eigenvalue fluctuations, and heuristically this could be expected to increase the deconfinement temperature and (naively) wipe out partially confined phases. This simple picture is consistent with our results. However, Ref. [42] recently also found evidence for partially-confined phases within the superconformal index, which is independent of the 't Hooft coupling. So from the field theory point of view, partially confined phases appear to be fairly ubiquitous. It would be very interesting to find new black objects dual to partially-confined phases, or to prove that they do not exist at least within the supergravity approximation.