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This course provides a thorough introduction to the bosonic string based on the Polyakov path integral and conformal field theory. We introduce central ideas of string theory, the tools of conformal field theory, the Polyakov path integral, and the covariant quantization of the string. We discuss string interactions and cover the tree-level and one loop amplitudes. More advanced topics such as T-duality and D-branes will be taught as part of the course. The course is geared for M.Sc. and Ph.D. students enrolled in Collaborative Ph.D. Program in Theoretical Physics. Required previous course work: Quantum Field Theory (AM516 or equivalent). The course evaluation will be based on regular problem sets that will be handed in during the term. The primary text is the book: 'String theory. Vol. 1: An introduction to the bosonic string. J. Polchinski (Santa Barbara, KITP) . 1998. 402pp. Cambridge, UK: Univ. Pr. (1998) 402 p.' All interested students should contact Alex Buchel at [email protected] as soon as possible.

Because the gravitational Hamiltonian is a pure boundary term on-shell, asymptotic gravitational fields store information in a manner not possible in local field theories. Two properties follow from this purely gravitational behavior. The first, 'Boundary Unitarity,' holds under AdS-like boundary conditions. This is the statement that the algebra of boundary observables is independent of time; i.e., that the algebra of boundary observables at any one time t_1 in fact coincides with the algebra of boundary observables at any other time t_2. As a result, any information available at the boundary at time t_1 remains available at any other time t_2. The second, 'Perturbative Holography,' holds under either AdS-like or asymptotically flat boundary conditions. In the AdS context, it is the statement that the algebra of boundary observables at any time t includes all perturbative observables anywhere in the spacetime. In the asymptotically flat context, Perturbative Holography is that statement that the algebra of observables on I^+ within any neighborhood of i^0 contains all perturbative observables. Perturbative Holography holds about any classical solution with a regular past infinity; i.e., spacetimes which collapse to form classical black holes are explicitly allowed. We derive the above properties and discuss their implications for information in black hole evaporation.

In this talk we introduce loop quantum gravity and we apply the theory to the black hole singularity problem. The Schwarzschild black hole solution inside the event horizon coincides with the Kantowski-Sachs space-time and we can study a simple spherically symmetric mini-superspace model. We show the classical black hole singularity is controlled by the quantum theory and the space-time can be dynamically extended beyond the classical singularity. We consider a semiclassical analysis of the black hole in LQG and we focus our attention on the space-time structure. The semiclassical solution is regular everywhere and similar to the Reissner-Nordstrom metric. The LQG black hole metric interpolates between two asymptotically flat regions, the 'r=infinity' region and the 'r=0' region. The metric is self-dual in the sense it is invariant under the symmetry (r ->1/r) which relates small and large distances. We study the thermodynamics of the semiclassical solution.

This course provides a thorough introduction to the bosonic string based on the Polyakov path integral and conformal field theory. We introduce central ideas of string theory, the tools of conformal field theory, the Polyakov path integral, and the covariant quantization of the string. We discuss string interactions and cover the tree-level and one loop amplitudes. More advanced topics such as T-duality and D-branes will be taught as part of the course. The course is geared for M.Sc. and Ph.D. students enrolled in Collaborative Ph.D. Program in Theoretical Physics. Required previous course work: Quantum Field Theory (AM516 or equivalent). The course evaluation will be based on regular problem sets that will be handed in during the term. The primary text is the book: 'String theory. Vol. 1: An introduction to the bosonic string. J. Polchinski (Santa Barbara, KITP) . 1998. 402pp. Cambridge, UK: Univ. Pr. (1998) 402 p.' All interested students should contact Alex Buchel at [email protected] as soon as possible.

I present a viewpoint on black hole thermodynamics according to which the entropy: derives from horizon 'degrees of freedom''; is finite because the deep structure of spacetime is discrete; is ``objective'' thanks to the distinguished coarse graining provided by the horizon; and obeys the second law of thermodynamics precisely because the effective dynamics of the exterior region is not unitary.

We introduce a two-body quantum Hamiltonian model of spin-1/2 on a 2D spatial lattice with exact topological degeneracy in all coupling regimes. There exists a gapped phase in which the low-energy sector reproduces an effective color code model. High energy excitations fall into three families of anyonic fermions that turn out to be strongly interacting. The model exhibits a Z_2xZ_2 gauge group symmetry and string-net integrals of motion, which are related to the existence of topological charges that are invisible to moving high-energy fermions.

In the 60’s, the analytic S-matrix program was developed in an attempt to describe the strong interactions – at the time, this was a theory of massive particles like pions. The S-matrix is an object that encodes the information of the probability of producing a certain set of final particles from a given set of initial particles. Eventually, the S-matrix program was replaced by Quantum Field Theory and in particular by Quantum Chromo Dynamics as the description of the strong interactions. In recent years there has been a resurrection of the S-matrix paradigm. The current view is that S-matrix techniques are most natural and powerful in theories of massless particles! Moreover, from this new perspective, the simplest quantum field theory to consider is now believed to be the maximally supersymmetric gravity theory. If the expectation is correct then N=8 supergravity will turn out to be a finite theory of gravity in perturbation theory.

The essential ingredients of a quantum theory are usually a Hilbert space of states and an algebra of operators encoding observables. The mathematical operations available with these structures translate fairly well into physical operations (preparation, measurement etc.) in a non-relativistic world. This correspondence weakens in quantum field theory, where the direct operational meaning of the observable algebra structure (encoded usually through commutators) is lost. The situation becomes even worse when we want to give a more dynamical role to spacetime as for example in attempts to formulate a quantum theory of gravity. I argue that a revision of the structures that we think of as fundamental in a quantum theory is in order. I go on to outline a proposal in this direction, based on the so called 'general boundary formulation', emphasizing the operational meaning of the ingredients. If time permits I will also comment on the relation to the framework of algebraic quantum field theory.

Inflationary scenarios with detectable primordial tensor perturbations typically require symmetries that can protect the potential over a super-Planckian field excursion. An old and natural idea is for the inflaton to be an axion protected by a shift symmetry. However, this has appeared difficult to realize in string theory because axion periodicities are sub-Planckian in known examples. I will explain how in compactifications containing wrapped fivebranes, the effective axion range is increased by monodromy: a single axion period can be traversed many times. The resulting potential is approximately linear and can source technically natural large-field inflation. As a result of the all-orders axionic shift symmetry, the potential receives negligible corrections from moduli stabilization.

In the past couple of years many new developments have been made in the techniques used for computing one-loop gauge theory amplitudes. These developments have mainly involved exploiting generalized unitarity techniques to construct the coefficients of the basis integral functions which make up a one-loop amplitude. I will outline these new developments along with their application to both QCD and N=8 supergravity amplitudes.

In topological quantum computation, a quantum algorithm is performed by braiding and fusion of certain quasi-particles called anyons. Therein, the performed quantum circuit is encoded in the topology of the braid. Thus, small inaccuracies in the world-lines of the braided anyons do not adversely affect the computation. For this reason, topological quantum computation has often been regarded as error-resilient per se, with no need for quantum error-correction. However, newer work [1], [2] shows that even topological computation is plagued with (small) errors. As a consequence, it requires error-correction, too, and in the scaling limit causes a poly-logarithmic overhead similar to systems without topological error-correction. I will discuss Nussinov and Ortiz' recent result [2] that the toric code is not fault-tolerant in a Hamiltonian setting, and outline its potential implications for topological quantum computation in general. [1] Nayak, C., Simon, S. H., Stern, A. et al. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083-1159 (2008). [2] Z. Nussinov and G. Ortiz, arXiv:0709.2717 (condmat)