# Bianchi, Sagnotti ## On the systematics of open-string theories \[Links: [Inspire](https://inspirehep.net/literature/296509)\] \[Abstract: We clarify the role of the fusion algebra in determining the interactions and the Chan-Paton symmetry of open-string models. Adapting the internal symmetry to the fusion algebra yields corresponding patterns of symmetry breaking, which we illustrate in a number of examples.\] # Brezin, Kazakov ## Exactly Solvable Field Theories of Closed Strings \[Links: [Inspire](https://inspirehep.net/literature/295835)\] # Douglas, Shenker ## Strings in Less Than One-Dimension \[Links: [Inspire](https://inspirehep.net/literature/282514)\] \[Abstract: Starting from the random triangulation definition of two-dimensional euclidean quantum gravity, we define the continuum limit and compute the partition function for closed surfaces of any genus. We discuss the appropriate way to define continuum string perturbation theory in these systems and show that the coefficients (as well as the critical exponents) are universal. The universality classes are just the multicritical points described by Kazakov. We show how the exact non-perturbative string theory is described by a non-linear ordinary differential equation whose properties we study. The behavior of the simplest theory, $c = 0$ pure gravity, is governed by the Painlevé transcendent of the first kind.\] # Gross, Migdal ## Nonperturbative Two-Dimensional Quantum Gravity \[Links: [Inspire](https://inspirehep.net/literature/282525)\] \[Abstract: We propose a nonperturbative definition of two-dimensional quantum gravity, based on a double-scaling limit of the [[0197 Matrix model|random-matrix model]]. We derive an exact differential equation for the partition function of two-dimensional gravity coupled to conformal matter as a function of the string coupling constant that governs the genus expansion of two-dimensional surfaces, and discuss its properties and consequences. We also construct and discuss the correlation functions of an infinite set of local operators for spherical topology.\] # Halliwell, Hartle ## Integration contours for the no-boundary wavefunction of the universe \[Links: [PRD](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.41.1815)\] \[Abstract: \] ## Summary - in the [[0162 No-boundary wavefunction]] proposal, constrain the contour of integration by 5 simple physical reasons - need complex contour to deal with the convergence criterion - [[0160 Complex diffeomorphism]] - caveat: still freedom in the contour # Lee, Wald ## Local symmetries and constraints \[Links: [Inspire](https://inspirehep.net/literature/300075)\] \[Abstract: The general relationship between local symmetries occurring in a Lagrangian formulation of a field theory and the corresponding constraints present in a phase space formulation are studied. First, a prescription—applicable to an arbitrary Lagrangian field theory—for the construction of phase space from the manifold of field configurations on space‐time is given. Next, a general definition of the notion of local symmetries on the manifold of field configurations is given that encompasses, as special cases, the usual gauge transformations of Yang–Mills theory and general relativity. Local symmetries on phase space are then defined via projection from field configuration space. It is proved that associated to each local symmetry which suitably projects to phase space is a corresponding equivalence class of constraint functions on phase space. Moreover, the constraints thereby obtained are always first class, and the Poisson bracket algebra of the constraint functions is isomorphic to the Lie bracket algebra of the local symmetries on the constraint submanifold of phase space. The differences that occur in the structure of constraints in Yang–Mills theory and general relativity are fully accounted for by the manner in which the local symmetries project to phase space: In Yang–Mills theory all the ‘‘field‐independent’’ local symmetries project to all of phase space, whereas in general relativity the nonspatial diffeomorphisms do not project to all of phase space and the ones that suitably project to the constraint submanifold are ‘‘field dependent.’’ As by‐products of the present work, definitions are given of the symplectic potential current density and the symplectic current density in the context of an arbitrary Lagrangian field theory, and the Noether current density associated with an arbitrary local symmetry. A number of properties of these currents are established and some relationships between them are obtained.\] # Mess ## Lorentz spacetimes of constant curvatures \[Links: [arXiv](https://arxiv.org/abs/0706.1570), [PDF](https://arxiv.org/pdf/0706.1570.pdf)\] \[Abstract: \] ## Refs - claim that all [[0002 3D gravity]] in AdS can be obtained by [[0099 Quotient method in AdS3]] # Verlinde ## Conformal field theory, two-dimensional quantum gravity and quantization of Teichmüller space \[Links: [DOI](https://doi.org/10.1016/0550-3213(90)90510-K)\] \[Abstract: We formulate the geometric quantization of Teichmüller space by using its relation with $SL(2,R)$ [[0089 Chern-Simons theory|Chern-Simons gauge theory]] and show that the physical state conditions arising in this formalism are equivalent to the Virasoro [[0106 Ward identity|Ward identities]] satisfied by the [[0031 Conformal block|conformal blocks]] in CFT. We further show that transition amplitudes between the physical states of this quantum system have a direct correspondence with covariant amplitudes of two-dimensional induced quantum gravity. Possible applications of these results to Virasoro modular geometry and [[0002 3D gravity|(2+1)-dimensional quantum gravity]] are indicated.\]