# Affleck, Ludwig ## Universal noninteger ‘‘ground-state degeneracy’’ in critical quantum systems \[Links: [Inspire](https://inspirehep.net/literature/29628), [PRL](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.67.161)\] \[Abstract: One-dimensional critical quantum systems have a universal, intensive ‘‘ground-state degeneracy’’, $g$, which depends on the universality class of the boundary conditions, and is in general noninteger. This is calculated, using the conjectured boundary conditions corresponding to a multichannel Kondo impurity and shown to agree with Bethe-ansatz results. $g$ is argued to decrease under renormalization from a less stable to a more stable critical point and plays a role in boundary critical phenomena quite analogous to that played by $c$, the [[0306 Weyl anomaly|conformal anomaly]], in the bulk case.\] ## Refs - [[0351 Irreversibility theorems]] # Bianchi, Pradisi, Sagnotti ## Planar duality in the discrete series \[Links: [Inspire](https://inspirehep.net/literature/318242)\] \[Abstract: The models of the discrete series of Belavin, Polyakov and Zamolodchikov may be generalized by allowing for suitable open-string sectors. The resulting open-string disk amplitudes have planar duality and proper factorization in the presence of a global internal Chan-Paton symmetry.\] # Bianchi, Sagnotti ## Twist symmetry and open string Wilson lines \[Links: [Inspire](https://inspirehep.net/literature/302836)\] \[Abstract: The Möbius amplitude plays an important role in open-string theories, since it determines which sectors of a given model consist of unoriented open strings. It also fixes the Chan-Paton representations of all their states, according to the behavior under the interchange of the ends of open strings (“twist”). In this paper we discuss the role played by conventional Wilson lines in Chan-Paton symmetry breaking, and we show that the presence of an extended symmetry algebra allows, in general, a number of choices for the behavior of massive states under twist. This freedom may be ascribed to additional discrete Wilson lines, and yields consistent modifications of the group assignments, that are illustrated in a number of examples.\] # Cardy ## Operator content and modular properties of higher-dimensional conformal field theories \[Links: [Inspire](https://inspirehep.net/literature/30207), [DOI](https://doi.org/10.1016/0550-3213(91)90024-R)\] \[Abstract: Possible generalizations of the principle of modular invariance to conformal field theories in dimensions $d >2$ are considered. It is shown that the partition function in a geometry with the topology of $S^{d −1}\times S^1$ encodes the operator content of the theory, and that this result implies that the asymptotic behavior of the density of scaling dimensions is determined by the universal coefficient of the Casimir term. These relations are verified for the case of a free massless scalar field, and it shon that, if the field is conformally coupled, the above partition functions enjoy modular properties which generalize those of the case $d =2$. A naive extension to the continuum limit of the three-dimensional Ising model fails, however.\] # Cardy, Lewellen ## Bulk and boundary operators in conformal field theory \[Links: [Inspire](https://inspirehep.net/literature/29279)\] \[Abstract: In [[0548 Boundary CFT|conformal field theory on a manifold with a boundary]], there is a short-distance expansion expressing local bulk operators in terms of boundary operators at an adjacent boundary. We show how the coefficients of such an expansion are given solely by data appearing in the bulk theory on the sphere and torus. In particular, the coefficients of the identity operator, which fix the one-point functions, are determined by the elements of the matrix $S$ which implements [[0612 Modular invariance|modular transformations]] on the torus. The other coefficients are related, in addition, to the elements of the matrices implementing duality transformations on the [[0031 Conformal block|conformal blocks]] of the four-point functions on the sphere. Some examples are given.\] # Deutsch ## Quantum statistical mechanics in a closed system \[Links: [DOI](https://doi.org/10.1103/PhysRevA.43.2046)\] \[Abstract: A closed quantum-mechanical system with a large number of degrees of freedom does not necessarily give time averages in agreement with the microcanonical distribution. For systems where the different degrees of freedom are uncoupled, situations are discussed that show a violation of the usual statistical-mechanical rules. By adding a finite but very small perturbation in the form of a [[0197 Matrix model|random matrix]], it is shown that the results of quantum statistical mechanics are recovered. Expectation values in energy eigenstates for this perturbed system are also discussed, and deviations from the microcanonical result are shown to become exponentially small in the number of degrees of freedom.\] ## Summary - OG for [[0040 Eigenstate thermalisation hypothesis|ETH]] - studies [[0541 Thermalisation|thermalisation]] for closed quantum systems # Ginsparg (Lectures) \[Links: [arXiv](https://arxiv.org/abs/hep-th/9108028), [PDF](https://arxiv.org/pdf/hep-th/9108028.pdf)\] \[Abstract: These lectures consisted of an elementary introduction to conformal field theory, with some applications to statistical mechanical systems, and fewer to string theory. Contents: 1. Conformal theories in $d$ dimensions 2. Conformal theories in 2 dimensions 3. The central charge and the Virasoro algebra 4. Kac determinant and unitarity 5. Identification of $m = 3$ with the critical Ising model 6. Free bosons and fermions 7. Free fermions on a torus 8. Free bosons on a torus 9. Affine Kac-Moody algebras and coset constructions 10. Advanced applications\] # Halliwell, Hartle ## Wave functions constructed from an invariant sum over histories satisfy constraints \[Links: [PRD](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.43.1170)\] \[Abstract: \] ## Comments - shows that [[0426 Cauchy slice holography]] may be generalised to more general theories ## Summary - [[0345 Wheeler-DeWitt (WdW) equation]] is satisfied for wavefunctions obtained by sum over histories (in the [[0162 No-boundary wavefunction]] proposal) as long as the path integral is defined to have an invariant action, invariant measure, and an invariant class of paths summed over where the invariance is generated by the constraints (e.g. diffeomorphism generated by the Hamiltonian and momentum constraints) # Preskill, Schwarz, Shapere, Trivedi, Wilczek ## Limitations on the statistical description of black holes \[Links: [DOI]([https://doi.org/10.1142/S0217732391002773](https://doi.org/10.1142/S0217732391002773))\] \[Abstract: We argue that the description of a block hole as a statistical (thermal) object must break down as the extreme (zero-temperature) limit is a approached, due to uncontrollable thermodynamic fluctuatations. For the recently discovered charged black holes, the analysis is significantly different, but again indicates that a statistical decription of the extreme hole is inappropriate. These holes invite a more normal elementary particle interpretation than is possible for Reissner-Nordström hole.\] # Reshetikhin, Turaev ## Invariants of 3-manifolds via link polynomials and quantum groups \[Links: [DOI](https://doi.org/10.1007/BF01239527)\] # Witten (Jan) ## Two-dimensional gravity and intersection theory on moduli space \[Links: [Inspire](https://inspirehep.net/literature/307956)\] # Witten (Oct) ## On quantum gauge theories in two dimensions \[Links: [DOI](https://doi.org/10.1007/BF02100009)\] \[Abstract: Two dimensional quantum Yang-Mills theory is studied from three points of view: (i) by standard physical methods; (ii) by relating it to the large $k$ limit of three dimensional [[0089 Chern-Simons theory|Chern-Simons theory]] and [[0003 2D CFT|two dimensional conformal field theory]]; (iii) by relating its weak coupling limit to the theory of Reidemeister-Ray-Singer torsion. The results obtained from the three points of view agree and give formulas for the volumes of the moduli spaces of representations of fundamental groups of two dimensional surfaces.\]