# Bern, Chalmers, Dixon, Kosower ## One-loop N gluon amplitudes with maximal helicity violation via collinear limits \[Links: [arXiv](https://arxiv.org/abs/hep-ph/9312333), [PDF](https://arxiv.org/pdf/hep-ph/9312333.pdf), [PRL](https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.72.2134)\] \[Abstract: We present a conjecture for the $n$-gluon one-loop amplitudes with [[0061 Maximally helicity violating amplitudes|maximal helicity violation]]. The conjecture emerges from the powerful requirement that the amplitudes have the correct behavior in the [[0078 Collinear limit|collinear limits]] of external momenta. One implication is that the corresponding amplitudes where three or more gluon legs are replaced by photons vanish for $n>4$.\] ## Refs - [[0078 Collinear limit]] - [[0260 Celestial loops]] ## Summary - *conjectures* 1-loop [[0061 Maximally helicity violating amplitudes|MHV]] gluon amplitudes - using constraints from the [[0078 Collinear limit|collinear limit]] - $A_{n ; 1}^{\text {loop }} \stackrel{a \| b}{\rightarrow} \sum_{\lambda=\pm}\left\{ \text{Split}^{\text {tree}}_{-\lambda}\left(a^{\lambda_{a}}, b^{\lambda_{b}}\right) A_{n-1 ; 1}^{\text {loop }}\left[\cdots(a+b)^{\lambda} \cdots\right]+ \text{Split}_{-\lambda}^{\text {loop }}\left(a^{\lambda_{a}}, b^{\lambda_{b}}\right) A_{n-1}^{\text {tree }}\left[\cdots(a+b)^{\lambda} \cdots\right]\right\}$ - $n;1$ means 1-loop and $n$ external gluons ## Restrictions - explicit expression only given for scalars in the loop - because SUSY Ward identities relate the gluon and fermion contributions to the scalar one - but to find explicit loop splitting function, can use known gluon 4-point and 5-point functions - 4-point: [[BernKosower1992]][](https://www.sciencedirect.com/science/article/pii/055032139290134W) - 5-point: [[BernDixonKosower1993]][](https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.70.2677) # Blau, Thompson ## Derivation of the Verlinde Formula from Chern-Simons Theory and the $G/G$ model \[Links: [arXiv](https://arxiv.org/abs/hep-th/9305010), [PDF](https://arxiv.org/pdf/hep-th/9305010.pdf)\] \[Abstract: We give a derivation of the Verlinde formula for the $G_{k}$ [[0601 Weiss-Zumino-Witten models|WZW model]] from [[0089 Chern-Simons theory|Chern-Simons theory]], without taking recourse to CFT, by calculating explicitly the partition function $Z_{\Sigma\times S^{1}}$ of $\Sigma\times S^{1}$ with an arbitrary number of labelled punctures. By a suitable gauge choice, $Z_{\Sigma\times S^{1}}$ is reduced to the partition function of an Abelian topological field theory on $\Sigma$ (a deformation of non-Abelian BF and Yang-Mills theory) whose evaluation is straightforward. This relates the Verlinde formula to the Ray-Singer torsion of $\Sigma\times S^{1}$. We derive the $G_{k}/G_{k}$ model from Chern-Simons theory, proving their equivalence, and give an alternative derivation of the Verlinde formula by calculating the $G_{k}/G_{k}$ path integral via a functional version of the Weyl integral formula. From this point of view the Verlinde formula arises from the corresponding Jacobian, the Weyl determinant. Also, a novel derivation of the shift $k\rightarrow k+h$ is given, based on the index of the twisted Dolbeault complex.\] # Deser, Schwimmer ## Geometric Classification of Conformal Anomalies in Arbitrary dimensions \[Links: [arXiv](https://arxiv.org/abs/hep-th/9302047), [PDF](https://arxiv.org/pdf/hep-th/9302047.pdf)\] \[Abstract: We give a complete geometric description of [[0306 Weyl anomaly|conformal anomalies]] in arbitrary, (necessarily even) dimension. They fall into two distinct classes: the first, based on Weyl invariants that vanish at integer dimensions, arises from finite -- and hence scale-free -- contributions to the effective gravitational action through a mechanism analogous to that of the (gauge field) chiral anomaly. Like the latter, it is unique and proportional to a topological term, the Euler density of the dimension, thereby preserving scale invariance. The contributions of the second class, requiring introduction of a scale through regularization, are correlated to all local conformal scalar polynomials involving powers of the Weyl tensor and its derivatives; their number increases rapidly with dimension. Explicit illustrations in dimensions 2, 4 and 6 are provided.\] ## Summary - OG of [[0306 Weyl anomaly]] categorisation - gives explicit expressions in 2, 4, 6 dimensions # Di Francesco, Ginsparg, Zinn-Justin (Review) ## 2D Gravity and Random Matrices \[Links: [arXiv](https://arxiv.org/abs/hep-th/9306153), [PDF](https://arxiv.org/pdf/hep-th/9306153)\] \[Abstract: We review recent progress in 2D gravity coupled to $d<1$ conformal matter, based on a representation of discrete gravity in terms of random [[0197 Matrix model|matrices]]. We discuss the saddle point approximation for these models, including a class of related $O(n)$ matrix models. For $d<1$ matter, the matrix problem can be completely solved in many cases by the introduction of suitable orthogonal polynomials. Alternatively, in the continuum limit the orthogonal polynomial method can be shown to be equivalent to the construction of representations of the canonical commutation relations in terms of differential operators. In the case of pure gravity or discrete Ising-like matter, the sum over topologies is reduced to the solution of non-linear differential equations (the Painlevé equation in the pure gravity case) which can be shown to follow from an action principle. In the case of pure gravity and more generally all unitary models, the perturbation theory is not Borel summable and therefore alone does not define a unique solution. In the non-Borel summable case, the matrix model does not define the sum over topologies beyond perturbation theory. We also review the computation of correlation functions directly in the continuum formulation of matter coupled to 2D gravity, and compare with the matrix model results. Finally, we review the relation between matrix models and topological gravity, and as well the relation to intersection theory of the moduli space of punctured Riemann surfaces.\] # Fioravanti, Pradisi, Sagnotti ## Sewing Constraints and Non-Orientable Open Strings \[Links: [arXiv](https://arxiv.org/abs/hep-th/9311183), [PDF](https://arxiv.org/pdf/hep-th/9311183.pdf), [DOI](https://doi.org/10.1016/0370-2693(94)90255-0)\] \[n.b. arXiv version does not have figures.\] \[Abstract: We extend to [[0620 Non-orientable CFT|non-orientable]] surfaces previous work on sewing constraints in Conformal Field Theory. A new constraint, related to the real projective plane, is described and is used to illustrate the correspondence with a previous construction of open-string spectra.\] # Friedman, Schleich, Witt ## Topological Censorship \[Links: [arXiv](https://arxiv.org/abs/gr-qc/9305017), [PDF](https://arxiv.org/pdf/gr-qc/9305017.pdf)\] \[Abstract: All three-manifolds are known to occur as Cauchy surfaces of asymptotically flat vacuum spacetimes and of spacetimes with positive-energy sources. We prove here the conjecture that [[0554 Einstein gravity|general relativity]] does not allow an observer to probe the topology of spacetime: any topological structure collapses too quickly to allow light to traverse it. More precisely, in a globally hyperbolic, asymptotically flat spacetime satisfying the null energy condition, every causal curve from $\mathscr{I}^-$ to $\mathscr{I}^+$ is homotopic to a topologically trivial curve from $\mathscr{I}^-$ to $\mathscr{I}^+$. (If the Poincaré conjecture is false, the theorem does not prevent one from probing fake 3-spheres).\] # Garfinkle, Giddings, Strominger ## Entropy in BH pair production \[Links: [arXiv](https://arxiv.org/abs/gr-qc/9306023), [PDF](https://arxiv.org/pdf/gr-qc/9306023.pdf)\] \[Abstract: \] ## Comments - in [[1995#Iyer, Wald]], this method is compared to Noether charge method for computing [[0004 Black hole entropy|black hole entropy]] # Ginsparg, Moore (Lectures) ## Lectures on 2D gravity and 2D string theory (TASI 1992) \[Links: [arXiv](https://arxiv.org/abs/hep-th/9304011), [PDF](https://arxiv.org/pdf/hep-th/9304011)\] # Gurarie ## Logarithmic Operators in Conformal Field Theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/9303160), [PDF](https://arxiv.org/pdf/hep-th/9303160.pdf)\] \[Abstract: Conformal field theories with correlation functions which have logarithmic singularities are considered. It is shown that those singularities imply the existence of additional operators in the theory which together with ordinary primary operators form the basis of the Jordan cell for the operator $L_{0}$. An example of the field theory possessing such correlation functions is given.\] ## Comments - original work on [[0563 Log CFT|Log CFT]] # Hayward (Geoff) ## Gravitational action for spacetimes with nonsmooth boundaries \[Links: [PRD](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.47.3275)\] \[Abstract: In this paper, I examine the gravitational action for spacetimes with nonsmooth boundaries. By two independent techniques, I derive the contribution to the gravitational action of spacelike and timelike two-surfaces on the boundary at which the unit normal changes discontinuously. I discuss the relationship between constraints imposed at such two-surfaces and their contribution to the gravitational action. I derive the form of the action and the juncture conditions appropriate to cases in which a spacetime includes a singular matter distribution whose world history corresponds to a timelike two-dimensional surface.\] ## Refs - [[0102 Hayward term|Corner term]] - used in [[2019#Takayanagi, Tamaoka]] # Hayward (Sean) ## Quasi-Local Gravitational Energy \[Links: [arXiv](https://arxiv.org/abs/gr-qc/9303030), [PDF](https://arxiv.org/pdf/gr-qc/9303030.pdf)\] \[Abstract: A dynamically preferred [[0595 Quasi-local energy|quasi-local]] definition of [[0592 Gravitational energy|gravitational energy]] is given in terms of the Hamiltonian of a '2+2' formulation of general relativity. The energy is well-defined for any compact orientable spatial 2-surface, and depends on the fundamental forms only. The energy is zero for any surface in flat spacetime, and reduces to the Hawking mass in the absence of shear and twist. For asymptotically flat spacetimes, the energy tends to the Bondi mass at null infinity and the ADM mass at spatial infinity, taking the limit along a foliation parametrised by area radius. The energy is calculated for the Schwarzschild, Reissner-Nordström and Robertson-Walker solutions, and for plane waves and colliding plane waves. Energy inequalities are discussed, and for static black holes the irreducible mass is obtained on the horizon. Criteria for an adequate definition of quasi-local energy are discussed.\] ## Summary - proposes the [[0595 Quasi-local energy#Hayward energy|Hayward energy]] # Jacobson, Kang, Myers ## On black hole entropy \[Links: [arXiv](https://arxiv.org/abs/gr-qc/9503020), [PDF](https://arxiv.org/pdf/gr-qc/9503020.pdf)\] \[Abstract: \] ## Summary - Develop a method for calculating [[0004 Black hole entropy|the gravitational entropy]]: field redefinition - [[1993#Wald]] result extended to work for arbitrary cross-section of event horizon - Wald formula has three types of ambiguities that all vanish for stationary black holes - talks about [[0018 JKM ambiguity]] ## Field redefinition method - **key property** - $\bar g_{ab} = g_{ab} + \Delta_{ab}$, where $\Delta_{ab}$ is constructed out of stationary fields - => then Killing horizon and surface gravity of a stationary black hole are invariant ## Results - entropy formula $S=-2 \pi \oint\left(Y^{a b c d}-\nabla_{e} Z^{e: a b c d}\right) \hat{\epsilon}_{a b} \hat{\epsilon}_{c d} \bar{\epsilon}$ - $Y^{a b c d} \equiv \partial \widetilde{L} / \partial R_{a b c d}$ - $Z^{e: a b c d}=\partial \widetilde{L} / \partial \nabla_{e} R_{a b c d}$ - $\widetilde{L}\left(\psi_{m}, \nabla_{a} \psi_{m}, g_{a b}, R_{a b c d}, \nabla_{e} R_{a b c d}\right)$ - where $\mathbf{L}=\widetilde{L} \epsilon$ i.e. $\widetilde{L}$ is a scalar function ## Caveats - results here works for black holes with a *bifurcate* Killing horizon # Jacobson, Myers ## Entropy of Lovelock black holes \[Links: [arXiv](https://arxiv.org/abs/hep-th/9305016), [PDF](https://arxiv.org/pdf/hep-th/9305016.pdf)\] \[Abstract: \] Calculate [[0004 Black hole entropy]] using *Hamiltonian methods*. - follows a method by [[1992#Sudarsky, Wald]] - also closely related method in [[BrownMartinezYorkJr1991]] Integrate the 1st law to get an entropy. # Osborn, Petkos ## Implications of Conformal Invariance in Field Theories for General Dimensions \[Links: [arXiv](https://arxiv.org/abs/hep-th/9307010), [PDF](https://arxiv.org/pdf/hep-th/9307010)\] \[Abstract: The requirements of [[0028 Conformal symmetry|conformal invariance]] for [[0633 CFT correlators|two and three point functions]] for general dimension $d$ on flat space are investigated. A compact group theoretic construction of the three point function for arbitrary spin fields is presented and it is applied to various cases involving conserved vector operators and the energy momentum tensor. The restrictions arising from the associated conservation equations are investigated. It is shown that there are, for general $d$, three linearly independent conformal invariant forms for the three point function of the energy momentum tensor, although for $d=3$ there are two and for $d=2$ only one. The form of the three point function is also demonstrated to simplify considerably when all three points lie on a straight line. Using this the coefficients of the conformal invariant point functions are calculated for free scalar and fermion theories in general dimensions and for abelian vector fields when $d=4$. [[0106 Ward identity|Ward identities]] relating three and two point functions are also discussed. This requires careful analysis of the singularities in the short distance expansion and the method of differential regularisation is found convenient. For $d=4$ the coefficients appearing in the energy momentum tensor three point function are related to the coefficients of the two possible terms in the trace anomaly for a conformal theory on a curved space background.\] ## Comment - this paper implicitly assumes parity; a version that does not assume so is given by [[2011#Giombi, Prakash, Yin]] # Susskind, Thorlacius, Uglum ## The Stretched Horizon and Black Hole Complementarity \[Links: [arXiv](https://arxiv.org/abs/hep-th/9306069), [PDF](https://arxiv.org/pdf/hep-th/9306069)\] \[Abstract: Three postulates asserting the validity of conventional quantum theory, semi-classical general relativity and the statistical basis for thermodynamics are introduced as a foundation for the study of black hole evolution. We explain how these postulates may be implemented in a ''stretched horizon'' or membrane description of the black hole, appropriate to a distant observer. The technical analysis is illustrated in the simplified context of 1+1 dimensional dilaton gravity. Our postulates imply that the dissipative properties of the stretched horizon arise from a course graining of microphysical degrees of freedom that the horizon must possess. A principle of [[0347 Black hole complementarity|black hole complementarity]] is advocated. The overall viewpoint is similar to that pioneered by 't Hooft but the detailed implementation is different.\] # Verlinde, Verlinde ## QCD at High Energies and Two-Dimensional Field Theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/9302104), [PDF](https://arxiv.org/pdf/hep-th/9302104.pdf)\] \[Abstract: Previous studies of high-energy scattering in QCD have shown a remarkable correspondence with two-dimensional field theory. In this paper we formulate a simple effective model in which this two-dimensional nature of the interactions is manifest. Starting from the (3+1)-dimensional Yang-Mills action, we implement the high energy limit $s\! >\! > \! t$ via a scaling argument and we derive from this a simplified effective theory. This effective theory is still (3+1)-dimensional, but we show that its interactions can to leading order be summarized in terms of a two-dimensional sigma-model defined on the transverse plane. Finally, we verify that our formulation is consistent with known perturbative results. This is a revised and extended version of hep-th 9302104. In particular, we have added a section that clarifies the connection with Lipatov's gluon emission vertex.\] ## Comments - gives a [[0117 Shockwave|shockwave]] effective action # Wald ## Black hole entropy is Noether charge \[Links: [arXiv](https://arxiv.org/abs/gr-qc/9307038), [PDF](https://arxiv.org/pdf/gr-qc/9307038.pdf)\] \[Abstract: We consider a general, classical theory of gravity in $n$ dimensions, arising from a diffeomorphism invariant Lagrangian. In any such theory, to each vector field, $\xi^a$, on spacetime one can associate a local symmetry and, hence, a Noether current $(n−1)$-form, $\mathbf{j}$, and (for solutions to the field equations) a Noether charge $(n−2)$-form, $\mathbf{Q}$. Assuming only that the theory admits stationary black hole solutions with a bifurcate Killing horizon, and that the canonical mass and angular momentum of solutions are well defined at infinity, we show that the first law of [[0127 Black hole thermodynamics|black hole mechanics]] always holds for perturbations to nearby stationary black hole solutions. The quantity playing the role of black hole entropy in this formula is simply $2\pi$ times the integral over $\Sigma$ of the Noether charge $(n−2)$-form associated with the horizon Killing field, normalized so as to have unit surface gravity. Furthermore, we show that this [[0004 Black hole entropy|black hole entropy]] always is given by a local geometrical expression on the horizon of the black hole. We thereby obtain a natural candidate for the entropy of a dynamical black hole in a general theory of gravity. Our results show that the validity of the "[[0005 Black hole second law|second law]]" of black hole mechanics in dynamical evolution from an initially stationary black hole to a final stationary state is equivalent to the positivity of a total Noether flux, and thus may be intimately related to the [[0116 Positive energy theorem|positive energy]] properties of the theory. The relationship between the derivation of our formula for black hole entropy and the derivation via "Euclidean methods" also is explained.\] ## Summary - this is the origin of the [[0559 Wald entropy|Wald entropy]] - assumes the theory admits stationary black holes with a bifurcate Killing horizon, and the canonical mass and angular momentum are well-defined at infinity => 1st law holds for perturbations to *nearby stationary* black hole solutions - entropy is the Noether charge associated with the horizon Killing field - explains the relation to Euclidean methods ## Important points - need to show that the entropy is like $S=\int_{\Sigma} F$ where $F$ is *locally* constructed out of the metric and other dynamical fields ## Refs - earlier calculation of [[0341 Lovelock gravity|Lovelock gravity]] entropy [[1993#Jacobson, Myers]] - used Hamiltonian methods - that is before this paper: i.e. not using [[0019 Covariant phase space|CPS]] - [[1993#Jacobson, Kang, Myers]]: [[0018 JKM ambiguity|JKM ambiguity]] - [[1994#Iyer, Wald]]: dynamic metric