# Ammon, Gutperle, Kraus, Perlmutter ## Spacetime Geometry in Higher Spin Gravity \[Links: [arXiv](https://arxiv.org/abs/1106.4788), [PDF](https://arxiv.org/pdf/1106.4788)\] \[Abstract: [[0421 Higher-spin gravity|Higher spin gravity]] is an interesting toy model of stringy geometry. Particularly intriguing is the presence of higher spin gauge transformations that redefine notions of invariance in gravity: the existence of event horizons and singularities in the metric become gauge dependent. In previous work, solutions of spin-3 gravity in the $SL(3,R) \times SL(3,R)$ [[0089 Chern-Simons theory|Chern-Simons]] formulation were found, and were proposed to play the role of black holes. However, in the gauge employed there, the spacetime metric describes a [[0083 Traversable wormhole|traversable wormhole]] connecting two asymptotic regions, rather than a black hole. In this paper, we show explicitly that under a higher spin gauge transformation these solutions can be transformed to describe black holes with manifestly smooth event horizons, thereby changing the spacetime causal structure. A related aspect is that the Chern-Simons theory admits two distinct AdS$_3$ vacua with different asymptotic W-algebra symmetries and central charges. We show that these vacua are connected by an explicit, Lorentz symmetry-breaking RG flow, of which our solutions represent finite temperature generalizations. These features will be present in any $SL(N,R) \times SL(N,R)$ Chern-Simons theory of higher spins.\] # Andersen, Kashaev ## A TQFT from quantum Teichmüller theory \[Links: [arXiv](https://arxiv.org/abs/1109.6295), [PDF](https://arxiv.org/pdf/1109.6295.pdf)\] \[Abstract: By using quantum Teichmüller theory, we construct a one parameter family of TQFT's on the categroid of admissible leveled shaped 3-manifolds.\] # Bautista, Bawane ## Boundary timelike Liouville theory: bulk 1-point & boundary 2-point functions \[Links: [arXiv](https://arxiv.org/abs/2111.04715), [PDF](https://arxiv.org/pdf/2111.04715)\] \[Abstract: We consider [[0622 Timelike Liouville|timelike Liouville theory]] with [[0658 FZZT brane|FZZT]]-like boundary conditions. The bulk one-point and boundary two-point structure constants on a disk are derived using bootstrap. We find that these structure constants are not the analytic continuations of their spacelike counterparts.\] # Blackman. McDermott, van Raamsdonk ## Acceleration-Induced Deconfinement Transitions in dS \[Links: [arXiv](https://arxiv.org/abs/1105.0440), [PDF](https://arxiv.org/pdf/1105.0440.pdf)\] \[Abstract: \] ## Summary - extends [[2010#Marolf, Rangamani, van Raamsdonk]] to $S^2\times dS_n$ and $T^2\times dS_n$ # Boucher-Veronneau, Larkoski ## Constructing Amplitudes from Their Soft Limits \[Links: [arXiv](https://arxiv.org/abs/1108.5385), [PDF](https://arxiv.org/pdf/1108.5385.pdf)\] \[Abstract: The existence of universal soft limits for gauge-theory and gravity amplitudes has been known for a long time. The properties of the soft limits have been exploited in numerous ways; in particular for relating an $n$-point amplitude to an $(n-1)$-point amplitude by removing a soft particle. Recently, a procedure called [[0515 Inverse soft construction|inverse soft]] was developed by which "soft" particles can be systematically added to an amplitude to construct a higher-point amplitude for generic kinematics. We review this procedure and relate it to [[0058 BCFW|Britto-Cachazo-Feng-Witten recursion]]. We show that all tree-level amplitudes in gauge theory and gravity up through seven points can be constructed in this way, as well as certain classes of NMHV gauge-theory amplitudes with any number of external legs. This provides us with a systematic procedure for constructing amplitudes solely from their soft limits.\] # Chiodaroli, D'Hoker, Gutperle ## Simple holographic duals to boundary CFTs \[Links: [arXiv](https://arxiv.org/abs/1111.6912), [PDF](https://arxiv.org/pdf/1111.6912.pdf)\] \[Abstract: By relaxing the regularity conditions imposed in [arXiv:1107.1722](https://arxiv.org/abs/1107.1722) on half-[[0178 BPS|BPS]] solutions to six-dimensional Type 4b supergravity, we enlarge the space of solutions to include two new half-BPS configurations, which we refer to as the $AdS_2$-cap and the $AdS_2$-funnel. We give evidence that the $AdS_2$-cap and $AdS_2$-funnel can be interpreted as fully back-reacted brane solutions with respectively $AdS_2$ and $AdS_2\times S^2$ world volumes. $AdS_2$-cap and $AdS_2$-funnel solutions with a single asymptotic $AdS_3 \times S^3$ region are constructed analytically. We argue that $AdS_2$-cap solutions provide simple examples of holographic duals to boundary CFTs in two dimensions and present calculations of their holographic boundary entropy to support the [[0548 Boundary CFT|BCFT]] dual picture.\] # Cremonini (Review) ## The Shear Viscosity to Entropy Ratio: A Status Report \[Links: [arXiv](https://arxiv.org/abs/1108.0677), [PDF](https://arxiv.org/pdf/1108.0677.pdf)\] \[Abstract: This review highlights some of the lessons that the [[0001 AdS-CFT|holographic gauge/gravity duality]] has taught us regarding the behavior of the shear viscosity to entropy density in strongly coupled field theories. The viscosity to entropy ratio has been shown to take on a very simple universal value in all gauge theories with an [[0554 Einstein gravity|Einstein gravity]] dual. Here we describe the origin of this universal ratio, and focus on how it is modified by generic [[0006 Higher-derivative gravity|higher derivative corrections]] corresponding to curvature corrections on the gravity side of the duality. In particular, certain curvature corrections are known to push the [[0430 Holographic shear viscosity|viscosity to entropy ratio]] below its universal value. This disproves a longstanding conjecture that such a universal value represents a strict lower bound for any fluid in nature. We discuss the main developments that have led to insight into the violation of this bound, and consider whether the consistency of the theory is responsible for setting a fundamental lower bound on the viscosity to entropy ratio.\] # Fujita, Takayanagi, Tonni ## Aspects of AdS/BCFT \[Links: [arXiv](https://arxiv.org/abs/1108.5152), [PDF](https://arxiv.org/pdf/1108.5152.pdf)\] \[Abstract: We expand the results of [arXiv:1105.5165](https://arxiv.org/abs/1105.5165), where a holographic description of a conformal field theory defined on a manifold with boundaries (so called [[0548 Boundary CFT|BCFT]]) was proposed, based on AdS/CFT. We construct gravity duals of conformal field theories on strips, balls and also time-dependent boundaries. We show a holographic g-theorem in any dimension. As a special example, we can define a 'boundary central charge' in three dimensional conformal field theories and our holographic $g$-theorem argues that it decreases under RG flows. We also computed holographic one-point functions and confirmed that their scaling property agrees with field theory calculations. Finally, we give an example of string theory embedding of this holography by inserting orientifold 8-planes in AdS(4)xCP(3).\] # Giombi, Prakash, Yin ## A Note on CFT Correlators in Three Dimensions \[Links: [arXiv](https://arxiv.org/abs/1104.4317), [PDF](https://arxiv.org/pdf/1104.4317)\] \[Abstract: In this note we present a simple method of constructing general conformally invariant [[0633 CFT correlators|three point functions]] of operators of various spins in three dimensions. Upon further imposing current conservation conditions, we find new parity violating structures for the three point functions involving either the stress-energy tensor, spin one currents, or higher spin currents. We find that all parity preserving structures for conformally invariant three point functions of higher spin conserved currents can be realized by free fields, whereas there is at most one parity violating structure for three point functions for each set of spins, which is not realized by free fields.\] # Gopakumar ## What is the Simplest Gauge-String Duality? \[Links: [arXiv](https://arxiv.org/abs/1104.2386), [PDF](https://arxiv.org/pdf/1104.2386)\] \[Abstract: We make a proposal for the string dual to the simplest large $N$ theory, the Gaussian [[0197 Matrix model|matrix integral]] in the 't Hooft limit, and how this dual description emerges from double line graphs. This is a specific realisation of the general approach to gauge-string duality which associates worldsheet Riemann surfaces to the Feynman-'t Hooft diagrams of a large $N$ gauge theory. We show that a particular version (proposed by Razamat) of this connection, involving integer Strebel differentials, naturally explains the combinatorics of Gaussian matrix correlators. We find that the correlators can be explicitly realised as a sum over a special class of holomorphic maps (Belyi maps) from the worldsheet to a *target space* ${\mathbb P}^1$. We are led to identify this target space with the Riemann surface associated with the (eigenvalues of the) matrix model. In the process, an [[0001 AdS-CFT|AdS/CFT]] like dictionary, for arbitrary correlators of single trace operators, also emerges in which the holomorphic maps play the role of stringy Witten diagrams. Finally, we provide some evidence that the above string dual is the conventional A-model topological string theory on ${\mathbb P}^1$.\] # Gutperle, Kraus ## Higher Spin Black Holes \[Links: [arXiv](https://arxiv.org/abs/1103.4304), [PDF](https://arxiv.org/pdf/1103.4304)\] \[Abstract: We study classical solutions of three dimensional [[0421 Higher-spin gravity|higher spin gravity]] in the Chern-Simons formulation. We find solutions that generalize the [[0086 Banados-Teitelboim-Zanelli black hole|BTZ]] black hole and carry spin-3 charge. The [[0004 Black hole entropy|black hole entropy]] formula yields a result for the asymptotic growth of the partition function at finite spin-3 chemical potential. Along the way, we develop technology for computing [[0001 AdS-CFT|AdS/CFT]] correlation functions involving higher spin operators.\] # Harlow, Maltz, Witten ## Analytic Continuation of Liouville Theory \[Links: [arXiv](https://arxiv.org/abs/1108.4417), [PDF](https://arxiv.org/pdf/1108.4417)\] \[Abstract: Correlation functions in [[0562 Liouville theory|Liouville theory]] are meromorphic functions of the Liouville momenta, as is shown explicitly by the [[0598 DOZZ formula|DOZZ formula]] for the three-point function on the sphere. In a certain physical region, where a real classical solution exists, the semiclassical limit of the DOZZ formula is known to agree with what one would expect from the action of the classical solution. In this paper, we ask what happens outside of this physical region. Perhaps surprisingly we find that, while in some range of the Liouville momenta the semiclassical limit is associated to complex saddle points, in general Liouville's equations do not have enough complex-valued solutions to account for the semiclassical behavior. For a full picture, we either must include "solutions" of Liouville's equations in which the Liouville field is multivalued (as well as being complex-valued), or else we can reformulate Liouville theory as a [[0089 Chern-Simons theory|Chern-Simons theory]] in three dimensions, in which the requisite solutions exist in a more conventional sense. We also study the case of [[0622 Timelike Liouville|"timelike" Liouville theory]], where we show that a proposal of Al. B. Zamolodchikov for the exact three-point function on the sphere can be computed by the original Liouville path integral evaluated on a new integration cycle.\] # Harlow, Stanford ## Operator dictionaries and wave functions in AdS/CFT and dS/CFT \[Links: [arXiv](https://arxiv.org/abs/1104.2621), [PDF](https://arxiv.org/pdf/1104.2621.pdf)\] \[Abstract: Dual AdS/CFT correlators can be computed in two ways: [[0591 Differentiate dictionary|differentiate]] the bulk partition function with respect to boundary conditions, or [[0590 Extrapolate dictionary|extrapolate]] bulk correlation functions to the boundary. These dictionaries were conjectured to be equivalent by Banks, Douglas, Horowitz, and Martinec. We revisit this question at the level of bulk path integrals, showing that agreement in the presence of interactions requires careful treatment of the renormalization of bulk composite operators. By contrast, we emphasize that proposed dS/CFT analogues of the two dictionaries are inequivalent. Next, we show quite generally that the wave function for Euclidean AdS analytically continues to the dS wave function with Euclidean initial conditions. Most of our arguments consider interacting fields on a fixed background, but in a final section we discuss the inclusion of bulk dynamical gravity.\] ## Refs - [[1999#Giddings (a)]] - shows equivalence in the case of free fields - justification for interacting fields: interactions turn off at infinity - here we will deal with interacting fields more carefully ## Summary - show two [[0001 AdS-CFT]] are the same - at the level of bulk PI - in the presence of interacting fields, require careful treatment of renormalisation of bulk composite operators - Not true for dS/CFT - generically WF for Euclidean AdS analytically continue to dS WF with Euclidean initial conditions - discuss inclusion of bulk **dynamical gravity** ## Issues with interacting fields - a scheme-dependent factor arising from renormalisation of **composite** fields in the bulk <-> in CFT, this corresponds to an **arbitrariness** in normalising the dual CFT operators ## GKPW - [[1998#Gubser, Klebanov, Polyakov]] and [[1998#Witten (Feb)]] - $Z_\text{bulk}[\phi_0]=Z_\text{CFT}[\phi_0]$ - bulk side: $\phi_0$ = BC - CFT side: $\phi_0$ = coefficients of operator deformation of the CFT Lagrangian ## BDHM - [[BanksDouglasHorowitzMartinec1998]] - $\left\langle\mathcal{O}\left(x_{1}\right) \ldots \mathcal{O}\left(x_{n}\right)\right\rangle_\text{CFT}=\lim _{z \rightarrow 0} z^{-n \Delta}\left\langle\phi\left(x_{1}, z\right) \ldots \phi\left(x_{n}, z\right)\right\rangle_\text{bulk}$ ## Equivalence - want to show $\left[\frac{\delta}{\delta \beta\left(x_{1}\right)} \cdots \frac{\delta}{\delta \beta\left(x_{n}\right)} Z_\text{bulk}[\beta]\right]_{\beta=0} \sim \lim _{z \rightarrow 0} z^{-n \Delta}\left\langle\phi\left(x_{1}, z\right) \ldots \phi\left(x_{n}, z\right)\right\rangle_\text{bulk}$ - i.e. both bulk operations compute the same quantities ## Relation between dS and AdS - at the level of wavefunction, but not expectations values - some Euclidean initial data will evolve to arbitrary fluctuations in the future, so will not match an analytic continuation from AdS # Hristov, Toldo, Vandoren ## On BPS bounds in D=4 N=2 gauged supergravity \[Links: [arXiv](https://arxiv.org/abs/1110.2688), [PDF](https://arxiv.org/pdf/1110.2688.pdf)\] \[Abstract: We determine the [[0178 BPS|BPS]] bounds in minimal gauged supergravity in four spacetime dimensions. We concentrate on asymptotically anti-de Sitter (AdS) spacetimes, and find that there exist two disconnected BPS ground states of the theory, depending on the presence of magnetic charge. Each of these ground states comes with a different superalgebra and a different BPS bound, which we derive. As a byproduct, we also demonstrate how the [[0359 Supersymmetry|supersymmetry]] algebra has a built-in [[0209 Holographic renormalisation|holographic renormalization]] method to define finite conserved charges.\] # Hung, Myers, Smolkin ## On HEE and HDG \[Links: [arXiv](https://arxiv.org/abs/1101.5813), [PDF](https://arxiv.org/pdf/1101.5813.pdf)\] \[Abstract: \] ## Remarks - this is pre-[[2013#Lewkowycz, Maldacena]] ## Summary - points out that Wald entropy is the wrong entanglement entropy but JM entropy is correct for Lovelock - (according to #xidong) relations between [[0145 Generalised area|HEE]] and [[0306 Weyl anomaly]] - a new ambiguity for Lovelock but can be fixed by considering variational principle ## Relation to trace anomaly - trace anomaly - $\left\langle T^{i}{ }_{i}\right\rangle=\sum_{n} B_{n} I_{n}-2(-)^{d / 2} A E_{d}+B^{\prime} \nabla_{i} J^{i}$ # Hung, Myers, Smolkin, Yale ## Holographic Calculations of Renyi Entropy \[Links: [arXiv](https://arxiv.org/abs/1110.1084), [PDF](https://arxiv.org/pdf/1110.1084.pdf)\] \[Abstract: We extend the approach of Casini, Huerta and Myers to a new calculation of the [[0293 Renyi entropy|Renyi entropy]] of a general CFT in $d$ dimensions with a spherical entangling surface, in terms of certain thermal partition functions. We apply this approach to calculate the Renyi entropy in various holographic models. Our results indicate that in general, the Renyi entropy will be a complicated nonlinear function of the [[0033 Central charge|central charges]] and other parameters which characterize the CFT. We also exhibit the relation between this new thermal calculation and a conventional calculation of the Renyi entropy where a twist operator is inserted on the spherical entangling surface. The latter insight also allows us to calculate the scaling dimension of the twist operators in the holographic models.\] # Kol, Smolkin ## Black hole stereotyping: Induced gravito-static polarization \[Links: [arXiv](https://arxiv.org/abs/1110.3764), [PDF](https://arxiv.org/pdf/1110.3764.pdf)\] \[Abstract: We discuss the black hole effective action and define its static subsector. We determine the induced gravito-static polarization constants (electric [[0581 Tidal Love numbers|Love numbers]]) of static black holes (Schwarzschild) in an arbitrary dimension, namely the induced mass multipole as a result of an external gravitational field. We demonstrate that in 4d these constants vanish thereby settling a disagreement in the literature. Yet in higher dimensions these constants are non-vanishing, thereby disproving (at least in $d>4$) speculations that black holes have no effective couplings beyond the point particle action. In particular, when $l/(d-3)$ is half integral these constants demonstrate a (classical) renormalization flow consistent with the divergences of the effective field theory. In some other cases the constants are negative indicating a novel non-spherical instability. The theory of hypergeometric functions plays a central role.\] # Komargodski, Schwimmer ## On Renormalization Group Flows in Four Dimensions \[Links: [arXiv](https://arxiv.org/abs/), [PDF](https://arxiv.org/pdf/.pdf)\] \[Abstract: We discuss some general aspects of renormalization group flows in four dimensions. Every such flow can be reinterpreted in terms of a spontaneously broken conformal symmetry. We analyze in detail the consequences of trace anomalies for the effective action of the Nambu-Goldstone boson of broken conformal symmetry. While the c-anomaly is algebraically trivial, the a-anomaly is "non-Abelian," and leads to a positive-definite universal contribution to the S-matrix of 2->2 dilaton scattering. Unitarity of the S-matrix results in a monotonically decreasing function that interpolates between the Euler anomalies in the ultraviolet and the infrared, thereby establishing the [[0351 Irreversibility theorems|a-theorem]].\] ## Remarks - derives the [[0351 Irreversibility theorems|a-theorem]] # Maldacena, Pimentel ## On graviton non-Gaussianities during inflation \[Links: [arXiv](https://arxiv.org/abs/1104.2846), [PDF](https://arxiv.org/pdf/1104.2846)\] \[Abstract: We consider the most general three point function for gravitational waves produced during a period of exactly de Sitter expansion. The de Sitter isometries constrain the possible shapes to only three: two preserving parity and one violating parity. These isometries imply that these correlation functions should be conformal invariant. One of the shapes is produced by the ordinary gravity action. The other shape is produced by a higher derivative correction and could be as large as the gravity contribution. The parity violating shape does not contribute to the bispectrum \[[1106.3228](https://arxiv.org/abs/1106.3228), [1108.0175](https://arxiv.org/abs/1108.0175)\], even though it is present in the wavefunction. We also introduce a spinor helicity formalism to describe de Sitter gravitational waves with circular polarization. These results also apply to [[0105 AdS amplitudes|correlation functions]] in Anti-de Sitter space. They also describe the general form of stress tensor [[0633 CFT correlators|correlation functions]], in momentum space, in a three dimensional conformal field theory. Here all three shapes can arise, including the parity violating one.\] # Maldacena, Zhiboedov ## Constraining conformal field theories with a higher spin symmetry \[Links: [arXiv](https://arxiv.org/abs/1112.1016), [PDF](https://arxiv.org/pdf/1112.1016.pdf)\] \[Abstract: We study the constraints imposed by the existence of a single higher spin conserved current on a three dimensional conformal field theory. A single higher spin conserved current implies the existence of an infinite number of higher spin conserved currents. The correlation functions of the stress tensor and the conserved currents are then shown to be equal to those of a free field theory. Namely a theory of $N$ free bosons or free fermions. This is an extension of the Coleman-Mandula theorem to CFT's, which do not have a conventional $S$ matrix. We also briefly discuss the case where the higher spin symmetries are "slightly" broken.\] # Monteiro, O'Connell ## The kinematic algebra from the self-dual sector \[Links: [arXiv](https://arxiv.org/abs/1105.2565), [PDF](https://arxiv.org/pdf/1105.2565.pdf)\] \[Abstract: \] ## Refs - [[0136 Self-dual Yang-Mills]] - [[0152 Colour-kinematics duality]] ## Summary - in ==4d==, for ==MHV== amplitudes, duality-satisfying numerators can be obtained from Feynman rules of some Lagrangian # Padmanabhan ## Some aspects of field equations in generalised theories of gravity \[Links: [arXiv](https://arxiv.org/abs/1109.3846), [PDF](https://arxiv.org/pdf/1109.3846.pdf)\] \[Abstract: A class of theories of gravity based on a Lagrangian which depends on the curvature and metric - but not on the derivatives of the curvature tensor - is of interest in several contexts including in the development of the paradigm that treats gravity as an emergent phenomenon. This class of models contains, as an important subset, all Lanczos-Lovelock models of gravity. I derive several identities and properties which are useful in the study of these models and clarify some of the issues that seem to have received insufficient attention in the past literature.\] # Solodukhin (Review) ## Entanglement entropy of black holes \[Links: [arXiv](https://arxiv.org/abs/1104.3712), [PDF](https://arxiv.org/pdf/1104.3712.pdf)\] \[Abstract: \] <!-- ## Comments - #ziyuewang : this does the [[hayward]] UV to IR idea with loop corrections for the matter field --> # Takayanagi ## Holographic Dual of BCFT \[Links: [arXiv](https://arxiv.org/abs/1105.5165), [PDF](https://arxiv.org/pdf/1105.5165.pdf)\] \[Abstract: We propose a holographic dual of a conformal field theory defined on a manifold with boundaries, i.e. boundary conformal field theory ([[0181 AdS-BCFT|BCFT]]). Our new holography, which may be called AdS/BCFT, successfully calculates the boundary entropy or g-function in two dimensional BCFTs and it agrees with the finite part of the [[0145 Generalised area|holographic entanglement entropy]]. Moreover, we can naturally derive a holographic g-theorem. We also analyze the holographic dual of an interval at finite temperature and show that there is a first order phase transition.\] ## Summary - OG for [[0181 AdS-BCFT|AdS/BCFT]] ## Proposal - Dirichlet BC on BCFT and boundary of BCFT, Neumann on bulk brane ## Comments - [[2017#Chu, Miao, Guo (Long)]] claims that the proposal here is too restrictive and always makes some boundary central charges vanish # Terashima, Yamazaki ## SL(2,R) Chern-Simons, Liouville, and Gauge Theory on Duality Walls \[Links: [arXiv](https://arxiv.org/abs/1103.5748), [PDF](https://arxiv.org/pdf/1103.5748.pdf)\] \[Abstract: We propose an equivalence of the partition functions of two different 3d gauge theories. On one side of the correspondence we consider the partition function of 3d $SL(2,R)$ [[0089 Chern-Simons theory|Chern-Simons theory]] on a 3-manifold, obtained as a punctured Riemann surface times an interval. On the other side we have a partition function of a 3d $N=2$ superconformal field theory on $S^3$, which is realized as a duality domain wall in a 4d gauge theory on $S^4$. We sketch the proof of this conjecture using connections with quantum [[0562 Liouville theory|Liouville theory]] and quantum Teichmuller theory, and study in detail the example of the once-punctured torus. Motivated by these results we advocate a direct Chern-Simons interpretation of the ingredients of (a generalization of) the Alday-Gaiotto-Tachikawa relation. We also comment on M5-brane realizations as well as on possible generalizations of our proposals.\] # Wall ## A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices \[Links: [arXiv](https://arxiv.org/abs/1105.3445), [PDF](https://arxiv.org/pdf/1105.3445.pdf)\] \[Abstract: The [[0082 Generalised second law|GSL]] is proven for semiclassical quantum fields falling across a causal horizon, minimally coupled to general relativity. The proof is much more general than previous proofs in that it permits the quantum fields to be rapidly changing with time, and shows that entropy increases when comparing any slice of the horizon to any earlier slice. The proof requires the existence of an algebra of observables restricted to the horizon, satisfying certain axioms (Determinism, Ultralocality, Local Lorentz Invariance, and Stability). These axioms are explicitly verified in the case of free fields of various spins, as well as 1+1 conformal field theories. The validity of the axioms for other interacting theories is discussed.\] ## Summary - OG: proves [[0082 Generalised second law|GSL]] for quantum fields minimally coupled to GR assuming some axioms - axioms checked explicitly for free fields and 1+1 CFT