# Atick, Witten ## The Hagedorn Transition and the Number of Degrees of Freedom of String Theory \[Links: [Inspire](https://inspirehep.net/literature/261790)\] \[Abstract: Extending recent discussions of the [[0439 Hagedorn transition|Hagedorn transition]] in string theory, we argue that this transition is a first-order phase transition with a very large heat (corresponding to a genus-zero contribution to the free energy that appears above the critical temperature). Formally analyzing the $k$-loop contributions to the free energy at temperatures far above the Hagedorn temperature leads to a number of interesting speculations about the underlying degrees of freedom in string theory.\] # Berends, Giele ## Recursive Calculations for Processes with n Gluons \[Links: [Inspire](https://inspirehep.net/literature/252779)\] \[Abstract: A method is presented in which multigluon processes are calculated [[0058 BCFW|recursively]]. The technique is explicitly developed for processes where only gluons are produced and processes where in addition to the gluons also a quark-antiquark pair with or without a vector boson or $e^+ e^−$ pair are present. The recursion relations are used to derive rigorously amplitudes for certain configurations, where most of the gluons have the same helicities. This proves a number of conjectures in the literature. Also expressions for amplitudes with [[0078 Collinear limit|collinear]] or [[0009 Soft theorems|soft]] gluons are derived.\] ## Comments - this is the original paper of [[0353 Berends-Giele recursion relations|Berends-Giele recursion relations]] # Cardy ## Is There a c-Theorem in Four-Dimensions? \[Links: [Inspire](https://inspirehep.net/literature/270790), [PRB](https://doi.org/10.1016/0370-2693(88)90054-8)\] \[Abstract: The difficulties of extending Zamolodchikov's [[0351 Irreversibility theorems|c-theorem]] to dimensions $d \ne 2$ are discussed. It is shown that, for $d$ even, the one-point function of the trace of the stress tensor on the sphere, $S^d$, when suitably regularized, defines a c-function, which, at least to one loop order, is decreasing along RG trajectories and is stationary at RG fixed points, where it is proportional to the usual conformal anomaly. It is shown that the existence of such a c-function, if it satisfies these properties to all orders, is consistent with the expected behavior of QCD in four dimensions.\] ## Refs - [[0351 Irreversibility theorems]] # Coleman ## Black holes as red herrings: Topological fluctuations and the loss of quantum coherence \[Links: [Inspire](https://inspirehep.net/literature/260855)\] \[Abstract: The topological fluctuations recently suggested by Hawking, by Giddings and Strominger, and by Lavrelashvili, Rubakov, and Tinyakov do not lead to an observable loss of quantum coherence.\] # Giddings, Strominger ## Loss of incoherence and determination of coupling constants in quantum gravity \[Links: [Inspire](https://inspirehep.net/literature/23288)\] \[Abstract: The wave function of an interacting 'family’ of one large 'parent’ and many Planck-sized 'baby' universes is computed in a semiclassical approximation using an adaptation of Hartle-Hawking initial conditions. A recently discovered gravitational instanton which exists for general relativity coupled to axions is employed. The outcome of a single experiment in the parent universe is in general described by a mixed state, even if the initial state is pure. However, a sequence of measurements rapidly collapses the wave function of the family of universes into one of an infinite number of ‘coherent’ states for which quantum incoherence is not observed in the parent universe. This provides a concrete illustration of an unexpected phenomena whose existence has been argued for on quite general grounds by Coleman: quantum incoherence due to information loss to baby universes is not experimentally observable. We further argue that all coupling constants governing dynamics in the parent universe depend on the parameters describing the particular coherent state into which the family wave function collapses. In particular, generically terms that violate any global symmetries will be induced in the effective action for the parent universe. These last results have much broader applicability that our specific model.\] # Moore, Seiberg ## Polynomial Equations for Rational Conformal Field Theories \[Links: [Inspire](https://inspirehep.net/literature/261829)\] \[Abstract: Duality of the conformal blocks of a [[0096 Rational CFT|rational conformal field theory]] defines matrices which may be used to construct representations of all monodromies and modular transformations in the theory. These duality matrices satisfy a finite number of independent polynomial equations, which imply constraints on monodromies allowed in rational conformal field theories. The equations include a key identity needed to prove a recent conjecture of Verlinde that the one-loop modular transformation $S$ diagonalizes the fusion rules. Using this formalism we show that duality of the $g =0$ four-point function and [[0612 Modular invariance|modular invariance]] of all one-loop one-point functions guarantee modular invariance to all orders. The equations for duality matrices should be useful in the classification of conformal field theories.\] ## Comment - this is the original paper of [[0602 Moore-Seiberg construction|Moore-Serberg construction]] # Moriah ## Heegaard splittings of Seifert fibered spaces \[Links: [DOI](https://link.springer.com/article/10.1007/BF01388781)\] \[Abstract: In this paper we give a classification theorem of genus two Heegaard splittings of Seifert fibered manifolds over $S^2$ with three exceptional fibers, except for when two of the exceptional fibers hava the same invariants with opposite orientation.\] # Morris, Thorne, Yurtsever ## Wormholes, Time Machines, and the Weak Energy Condition \[Links: [Inspire](https://inspirehep.net/literature/269623)\] \[Abstract: It is argued that, if the laws of physics permit an advanced civilization to create and maintain a wormhole in space for interstellar travel, then that wormhole can be converted into a [[0570 Time machine|time machine]] with which causality might be violatable. Whether wormholes can be created and maintained entails deep, ill-understood issues about cosmic censorship, quantum gravity, and quantum field theory, including the question of whether field theory enforces an averaged version of the weak energy condition.\] # Jacobson ## Fermions in canonical gravity \[Links: [CQG](https://iopscience.iop.org/article/10.1088/0264-9381/5/10/003)\] \[Abstract: Ashtekar's change of variables in the phase space of general relativity is employed in a canonical formulation of interacting gravitational and spinor fields. The formalism is derived from a complex Palatini-type action with the chiral spin connection as an independent variable. The resulting Hamiltonian constraints are polynomial in the conjugate variables. There is some evidence that the distinction between the Einstein-Cartan and torsion-free theories can be coded entirely in the reality condition specifying the Hermitian part of the torsion.\] # Sonoda ## Sewing conformal field theories II \[Links: [Inspire](https://inspirehep.net/literature/262330)\] \[Abstract: We show that any Riemann surface $M$ with punctures can be constructed by [[0602 Moore-Seiberg construction|sewing]] three-punctured spheres. Correspondingly, any correlation function on $M$ can be obtained by sewing three-point functions on a sphere. There is no unique way of sewing three-punctured spheres to construct $M$, and the resulting correlation functions may depend on the precise way of sewing. We show that this dependence is absent, if we assume that four-point correlation function on a sphere and one-point functions on a torus are determined unambiguously.\] # Tseytlin ## Mobius Infinity Subtraction and Effective Action in $\sigma$ Model Approach to Closed String Theory \[Links: [INSPIRE](https://inspirehep.net/literature/262157)\] \[Abstract: We show how to express the closed string tree amplitudes and effective action in terms of the correlators in the 2D $\sigma$-model. The subtraction of Möbius infinities corresponds to taking $\partial\partial \ln \epsilon$ ($\epsilon$ is a 2D UV cutoff) of the regularized correlators. The effective action is represented as the integral of the “central charge” coefficient thus demonstrating the equivalence between the $S$-matrix and $\sigma$-model approaches to string (tree) equations of motion. The $\partial\partial \ln \epsilon$ prescription provides a consistent way of computing tree tadpoles on non-trivial backgrounds and hence of studying string loop corrections to the equations of motion by equating the full tadpoles to zero.\] # Verlinde ## Fusion rules and modular transformations in 2D conformal field theory \[Links: [DOI](https://doi.org/10.1016/0550-3213(88)90603-7)\] \[Abstract: We study conformal field theories with a finite number of primary fields with respect to some chiral algebra. It is shown that the fusion rules are completely determined by the behavior of the characters under the modular group. We illustrate with some examples that conversely the modular properties of the characters can be derived from the fusion rules. We propose how these results can be used to find restrictions on the values of the central charge and conformal dimensions.\] # Witten ## 2 + 1 dimensional gravity as an exactly soluble system \[Links: [DOI](https://doi.org/10.1016/0550-3213(88)90143-5); [Inspire](https://inspirehep.net/literature/264816)\] \[Abstract: By disentangling the hamiltonian constraint equations, [[0002 3D gravity|2 + 1 dimensional gravity]] (with or without a cosmological constant) is shown to be exactly soluble at the classical and quantum levels. Indeed, it is closely related to Yang-Mills theory with purely the [[0089 Chern-Simons theory|Chern-Simons]] action, which recently has turned out to define a soluble quantum field theory. 2 + 1 dimensional gravity has a straightforward renormalized perturbation expansion, with vanishing beta function. 2 + 1 dimensional quantum gravity may provide a testing ground for understanding the role of classical singularities in quantum mechanics, may be related to the discrete series of [[0032 Virasoro algebra|Virasoro]] representations in 1 + 1 dimensions, and may be a useful tool in studying three-dimensional geometry.\]