The thermal partition function, Z Z , of a CFT_d C F T d on S^{d-1} S d − 1 is parameterized by the inverse temperature \beta β along with \lfloor d/2\rfloor ⌊ d / 2 ⌋ angular velocities \omega_i ω i . In this paper, we investigate the behaviour of this partition function when n n of the \omega_i ω i are scaled to unity (the largest allowed value) at fixed values of the other (\lfloor d/2\rfloor-n) ( ⌊ d / 2 ⌋ − n ) angular velocities. We argue that \ln Z ln ⁡ Z develops a simple pole in (1-\omega_i) ( 1 − ω i ) for each \omega_i ω i that is scaled to unity. The residue of this product of poles is a theory-dependent (so non-universal) function of \beta β and the fixed angular velocities. The inverse Laplace transformation of this partition function constrains the functional form of the field theory entropy as a function of charges in a limit in which angular momenta and the twist are scaled as follows. While n n special angular momenta J_1\ldots J_n J 1 … J n are scaled to infinity, the twist and the other angular momenta – collectively denoted x_i x i – are also taken to infinity but at the slower rate that ensures that the scaled charges x_i/(J_1 J_2 \ldots J_n)^{\frac{1}{n+1}} x i / ( J 1 J 2 … J n ) 1 n + 1 are held fixed. In this limit, we demonstrate that the scaled entropy S/(J_1 J_2 \ldots J_n)^{\frac{1}{n+1}} S / ( J 1 J 2 … J n ) 1 n + 1 depends only on the \lfloor d/2\rfloor-n+1 ⌊ d / 2 ⌋ − n + 1 scaled charges defined above (the precise form of this dependence is non-universal). We verify our predictions (and compute all non-universal functions) in the case of free scalar theories (which show surprisingly rich behaviour) as well as large N N , strongly coupled {\cal N}=4 Yang-Mills theory. The last theory is analyzed in the bulk via the AdS/CFT correspondence. In the scaling limit described above, its phase diagram displays sharp phase transitions between black hole, grey galaxy, and thermal gas phases.

We consider the path integral of a quantum field theory in Minkowski spacetime with fixed boundary values (for the elementary fields) on asymptotic boundaries. We define and study the corresponding boundary correlation functions obtained by taking derivatives of this path integral with respect to the boundary values. The S-matrix of the QFT can be extracted directly from these boundary correlation functions after smearing. We interpret this relation in terms of coherent state quantization and derive the constraints on the path-integral as a function of boundary values that follow from the unitarity of the S-matrix. We then study the locality structure of boundary correlation functions. In the massive case, we find that the boundary correlation functions for generic locations of boundary points are dominated by a saddle point which has the interpretation of particles scattering in a small elevator in the bulk, where the location of the elevator is determined dynamically, and the S-matrix can be recovered after stripping off some dynamically determined but non-local “renormalization” factors. In the massless case, we find that while the boundary correlation functions are generically analytic as a function on the whole manifold of locations of boundary points, they have special singularities on a sub-manifold, points on which correspond to light-like scattering in the bulk. We completely characterize this singular scattering sub-manifold, and find that the corresponding residues of the boundary correlations at these singularities are precisely given by S-matrices. This analysis parallels the analysis of bulk-point singularities in AdS/CFT and generalizes it to the case of multi-bulk point singularities.

The free energy of any CFT, ln Z(β; ω_i), admits two expansions: high temperature (β→ 0) and fast rotation (ω_i → 1). We demonstrate that locality of the thermal effective action forces ln Z to take a simple analytic form at all orders in the high temperature expansion, and further imposes an infinite number of sharp relations on the coefficients in this expansion. All are homogeneous, except at order β^1 due to the Weyl anomaly. From this, the a-anomaly can be extracted from the partition function. The relations resum in the fast-spinning expansion into differential equations in β obeyed by the semi-universal limit and its corrections. We verify the relations in a variety of CFTs. We generalize to any even d, but find no similar relations at odd d.

While correlators of a CFT are single valued in Euclidean Space, they are multi-valued - and have a complicated sheet structure - in Lorentzian space. Correlators on R1,1 are well known to access a finite number of these sheets. In this paper, we demonstrate the spiral nature of lightcones on S1x time, which allows the time-ordered correlators of a C FT2 on this spacetime - the Lorentzian cylinder - to access an infinite number of sheets of the correlator. We present a complete classification, both of the sheets accessed as well as of the various distinct causal configurations that lie on a particular sheet. Our construction provides a physical interpretation for an infinite number of sheets of the correlator, while, however, leaving a larger infinity of these sheets uninterpreted.

Most of the computational evidence for the Bose$\small-$Fermi duality of fundamental fields coupled to $U(N)$ Chern$\small-$Simons theories originates in the large-$N$ calculations performed in the light-cone gauge. In this paper, we use another gauge, the 'temporal' gauge, to evaluate the finite temperature partition function of $U(N)$ coupled regular and critical fermions on $\mathbb{R}^2$ at large $N$. We first set up the finite temperature gap equations, and then use tricks explored in arXiv:1410.0558 to solve these equations and evaluate the partition function. Our final results are in perfect agreement with earlier light-cone gauge results. The success of our 'temporal' gauge calculation potentially opens a path to computations that are awkward in light-cone gauge but more natural in the 'temporal' gauge, e.g. the evaluation of the thermal free energy on a finite-sized sphere.

Motivated by the recent construction of grey galaxy and dual dressed black hole solutions in AdS5 x S5, we present two conjectures relating to the large N entropy of supersymmetric states in N = 4 Yang-Mills theory. Our first conjecture asserts the existence of a large number of supersymmetric states which can be thought of as a non interacting mix of supersymmetric black holes and supersymmetric "gravitons". It predicts a microcanonical phase diagram of supersymmetric states with eleven distinct phases, and makes a sharp prediction for the supersymmetric entropy (as a function of 5 charges) in each of these phases. The microcanonical version of the superconformal index involves a sum over states-with alternating signs-over a line in 5 parameter charge space. Our second (and more tentative) conjecture asserts that this sum is dominated by the point on the line that has the largest supersymmetric entropy. This conjecture predicts a large N formula for the superconformal index as a function of indicial charges, and predicts a microcanonical indicial phase diagram with nine distinct phases. It predicts agreement between the superconformal index and black hole entropy in one phase (so over one range of charges), but disagreement in other phases (and so at other values of charges). We compare our predictions against numerically evaluated superconformal index at N <= 10, and find qualitative agreement.

Charged Black holes in AdS_5 × S^5 AdS5×S5 suffer from superradiant instabilities over a range of energies. Hairy black hole solutions (constructed within gauged supergravity) have previously been proposed as endpoints to this instability. We demonstrate that these hairy black holes are themselves unstable to the emission of large dual giant gravitons. We propose that the endpoint to this instability is given by Dual Dressed Black Holes (DDBH)s; configurations consisting of one, two, or three very large dual giant gravitons surrounding a core AdS AdS black hole with one, two, or three SO(6) SO(6) chemical potentials equal to unity. The dual giants each live at AdS AdS radial coordinates of order \sqrt{N} N and each carry charge of order N^2 N2 . The large separation makes DDBHs a very weakly interacting mix of their components and allows for a simple computation of their thermodynamics. We conjecture that DDBHs dominate the phase diagram of \mathcal{N}=4 𝒩=4 Yang-Mills over a range of energies around the BPS plane, and provide an explicit construction of this phase diagram, briefly discussing the interplay with supersymmetry. We develop the quantum description of dual giants around black hole backgrounds and explicitly verify that DDBHs are stable to potential tunneling instabilities, precisely when the chemical potentials of the core black holes equal unity. We also construct the 10-dimensional DDBH bulk solutions.

It has recently been conjectured that the end point of the rotational superradiant instability of black holes in AdS_4 is a Grey Galaxy: an ω=1 black hole sitting at the centre of AdS_4, surrounded by a large disk of rapidly rotating gravitons and other bulk fields. In this paper we study Grey Galaxies in AdS_5. In this case, the rotational group is of rank 2, and so has two distinct angular velocities ω_1 and ω_2. We demonstrate that AdS_5 hosts two qualitatively distinct Grey Galaxy phases: the first with either ω_1≈ 1 or ω_2≈ 1, and the second with both angular velocities ≈ 1. We use these results to present a conjecture for a part of the phase diagram of N=4 Yang-Mills (as a function of energy and the two angular momenta) that displays several phase transitions between regular black holes and various Grey Galaxy phases. We present an explicit gravitational construction of the phases in which ω_1 and ω_2 are both parametrically close to unity, and demonstrate that the corresponding boundary stress tensor is the sum of two pieces. The first is the stress tensor of the central black hole. The second - the contribution of the bulk gas - takes the form of the stress tensor of an equilibrated boundary conformal fluid, rotating at the given angular speeds ω_i. We also briefly comment on the structure of Grey Galaxies in AdS_D for D > 5.

We present a conjecture for the crossing symmetry rules for Chern-Simons gauge theories interacting with massive matter in $2+1$ dimensions. Our crossing rules are given in terms of the expectation values of particular tangles of Wilson lines, and reduce to the standard rules at large Chern-Simons level. We present completely explicit results for the special case of two fundamental and two antifundamental insertions in $SU(N)_k$ and $U(N)_k$ theories. These formulae are consistent with the conjectured level-rank, Bose-Fermi duality between these theories and take the form of a $q=e^{\frac{ 2 \pi i }{\kappa}}$ deformation of their large $k$ counterparts. In the 't Hooft large $N$ limit our results reduce to standard rules with one twist: the $S$-matrix in the singlet channel is reduced by the factor $\frac{\sin \pi \lambda}{\pi \lambda} $ (where $\lambda$ is the 't Hooft coupling), explaining `anomalous' crossing properties observed in earlier direct large $N$ computations.

Kerr-AdSd+1 black holes for d ≥ 3 suffer from classical superradiant instabilities over a range of masses above extremality. We conjecture that these instabilities settle down into Grey Galaxies (GGs) — a new class of coarse-grained solutions to Einstein’s equations which we construct in d = 3. Grey Galaxies are made up of a black hole with critical angular velocity ω = 1 in the ‘centre’ of AdS, surrounded by a large flat disk of thermal bulk gas that revolves around the centre of AdS at the speed of light. The gas carries a finite fraction of the total energy, as its parametrically low energy density and large radius are inversely related. GGs exist at masses that extend all the way down to the unitarity bound. Their thermodynamics is that of a weakly interacting mix of Kerr-AdS black holes and the bulk gas. Their boundary stress tensor is the sum of a smooth ‘black hole’ contribution and a peaked gas contribution that is delta function localized around the equator of the boundary sphere in the large N limit. We also construct another class of solutions with the same charges; ‘Revolving Black Holes (RBHs)’. RBHs are macroscopically charged SO(d, 2) descendants of AdS-Kerr solutions, and consist of ω = 1 black holes revolving around the centre of AdS at a fixed radial location but in a quantum wave function in the angular directions. RBH solutions are marginally entropically subdominant to GG solutions and do not constitute the endpoint of the superradiant instability. Nonetheless, we argue that supersymmetric versions of these solutions have interesting implications for the spectrum of supersymmetric states in, e.g. 𝒩 = 4 Yang-Mills theory.

Abstract We demonstrate that the known expressions for the thermal partition function of large N Chern-Simons matter theories admit a simple Hilbert space interpretation as the partition function of an associated ungauged large N matter theory with one additional condition: the Fock space of this associated theory is projected down to the subspace of its quantum singlets i.e. singlets under the Gauss law for Chern-Simons gauge theory. Via the Chern-Simons / WZW correspondence, the space of quantum singlets are equivalent to the space of WZW conformal blocks. One step in our demonstration involves recasting the Verlinde formula for the dimension of the space of conformal blocks in SU(N) k and U(N) k,k′ WZW theories into a simple and physically transparent form, which we also rederive by evaluating the partition function and superconformal index of pure Chern-Simons theory in the presence of Wilson lines. A particular consequence of the projection of the Fock space of Chern-Simons matter theories to quantum (or WZW) singlets is the ‘Bosonic Exclusion Principle’: the number of bosons occupying any single particle state is bounded above by the Chern-Simons level. The quantum singlet condition (unlike its Yang-Mills Gauss Law counterpart) has a nontrivial impact on thermodynamics even in the infinite volume limit. In this limit the projected Fock space partition function reduces to a product of partition functions, one for each single particle state. These single particle state partition functions are q-deformations of their free boson and free fermion counterparts and interpolate between these two special cases. We also propose a formula for the large N partition function that is valid for arbitrary finite volume of the spatial S 2 and not only at large volume.

We study four-point functions of scalars, conserved currents, and stress tensors in a conformal field theory, generated by a local contact term in the bulk dual description, in two different causal configurations. The first of these is the standard Regge configuration in which the chaos bound applies. The second is the `causally scattering configuration' in which the correlator develops a bulk point singularity. We find an expression for the coefficient of the bulk point singularity in terms of the bulk S matrix of the bulk dual metric, gauge fields and scalars, and use it to determine the Regge scaling of the correlator on the causally scattering sheet in terms of the Regge growth of this S matrix. We then demonstrate that the Regge scaling on this sheet is governed by the same power as in the standard Regge configuration, and so is constrained by the chaos bound, which turns out to be violated unless the bulk flat space S matrix grows no faster than $s^2$ in the Regge limit. It follows that in the context of the AdS/CFT correspondence, the chaos bound applied to the boundary field theory implies that the S matrices of the dual bulk scalars, gauge fields, and gravitons obey the Classical Regge Growth (CRG) conjecture.

A bstract We study the space of all kinematically allowed four photon and four graviton S-matrices, polynomial in scattering momenta. We demonstrate that this space is the permutation invariant sector of a module over the ring of polynomials of the Mandelstam invariants s , t and u . We construct these modules for every value of the spacetime dimension D , and so explicitly count and parameterize the most general four photon and four graviton S-matrix at any given derivative order. We also explicitly list the local Lagrangians that give rise to these S-matrices. We then conjecture that the Regge growth of S-matrices in all physically acceptable classical theories is bounded by s 2 at fixed t . A four parameter subset of the polynomial photon S-matrices constructed above satisfies this Regge criterion. For gravitons, on the other hand, no polynomial addition to the Einstein S-matrix obeys this bound for D ≤ 6. For D ≥ 7 there is a single six derivative polynomial Lagrangian consistent with our conjectured Regge growth bound. Our conjecture thus implies that the Einstein four graviton S-matrix does not admit any physically acceptable polynomial modifications for D ≤ 6. A preliminary analysis also suggests that every finite sum of pole exchange contributions to four graviton scattering also violates our conjectured Regge growth bound, at least when D ≤ 6, even when the exchanged particles have low spin.

We explicitly construct every kinematically allowed three particle graviton- graviton-P and photon-photon-P S-matrix in every dimension and for every choice of the little group representation of the massive particle P. We also explicitly construct the spacetime Lagrangian that generates each of these couplings. In the case of gravitons we demonstrate that this Lagrangian always involves (derivatives of ) two factors of the Riemann tensor, and so is always of fourth or higher order in derivatives. This result verifies one of the assumptions made in the recent preprint [1] while attempting to establish the rigidity of the Einstein tree level S-matrix within the space of local classical theories coupled to a collection of particles of bounded spin.

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.

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 t’Hooft 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.

We study the theory of a single fundamental fermion and boson coupled to Chern-Simons theory at leading order in the large N limit. Utilizing recent progress in understanding the Higgsed phase in Chern-Simons-Matter theories, we compute the quantum effective potential that is exact to all orders in the ’t Hooft coupling for the lightest scalar operator of this theory at finite temperature. Specializing to the zero temperature limit we use this potential to determine the phase diagram of the large N $$ \mathcal{N} $$ = 2 supersymmetric theory with this field content. This intricate two dimensional phase diagram has four topological phases that are separated by lines of first and second order phase transitions and includes special conformal points at which the infrared dynamics is governed by Chern-Simons theory coupled respectively to free bosons, Gross-Neveu fermions, and to a theory of Wilson-Fisher bosons plus free fermions. We also describe the vacuum structure of the most general $$ \mathcal{N} $$ = 1 supersymmetric theory with one fundamental boson and one fundamental fermion coupled to an SU(N ) Chern-Simons gauge field, at arbitrary values of the ’t Hooft coupling.

It has recently been demonstrated that the large N limit of a model of fermions charged under the global/gauge symmetry group O(N)q−1 agrees with the large N limit of the SYK model. In these notes we investigate aspects of the dynamics of the O(N)q−1 theories that differ from their SYK counterparts. We argue that the spectrum of fluctuations about the finite temperature saddle point in these theories has \( \left(q-1\right)\frac{N^2}{2} \) new light modes in addition to the light Schwarzian mode that exists even in the SYK model, suggesting that the bulk dual description of theories differ significantly if they both exist. We also study the thermal partition function of a mass deformed version of the SYK model. At large mass we show that the effective entropy of this theory grows with energy like E ln E (i.e. faster than Hagedorn) up to energies of order N2. The canonical partition function of the model displays a deconfinement or Hawking Page type phase transition at temperatures of order 1/ln N. We derive these results in the large mass limit but argue that they are qualitatively robust to small corrections in J/m.

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 x6, the coefficient of a sextic term in the potential. While x6 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 x6 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 conjecture that flows governed by this leading large N beta function have three fixed points for x6 at every non-zero value of the ’t Hooft coupling, implying the existence of three distinct regular bosonic and three distinct dual critical fermionic conformal fixed points, at every value of the ’t Hooft coupling. We analyze the phase structure of these fixed point theories at zero temperature. We also construct dual pairs of large N fine-tuned renormalization group flows from supersymmetric \( \mathcal{N}=2 \) Chern-Simons-matter theories, such that one of the flows ends up in the IR at a regular boson theory while its dual partner flows to a critical fermion theory. This construction suggests that the duality between these theories persists at finite N, at least when N is large.