Algorithms in algebraic number theory

In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers.Comment: 34 page

Standard model plethystics

We study the vacuum geometry prescribed by the gauge invariant operators of the minimal supersymmetric standard model via the plethystic program. This is achieved by using several tricks to perform the highly computationally challenging Molien-Weyl integral, from which we extract the Hilbert series, encoding the invariants of the geometry at all degrees. The fully refined Hilbert series is presented as the explicit sum of 1422 rational functions. We found a good choice of weights to unrefine the Hilbert series into a rational function of a single variable, from which we can read off the dimension and the degree of the vacuum moduli space of the minimal supersymmetric standard model gauge invariants. All data in Mathematica format are also presented

EPFL Lectures on Conformal Field Theory in D>= 3 Dimensions

This is a writeup of lectures given at the EPFL Lausanne in the fall of 2012. The topics covered: physical foundations of conformal symmetry, conformal kinematics, radial quantization and the OPE, and a very basic introduction to conformal bootstrap.Comment: 68 pages; v2 - misprints correcte

State-Dependent Bulk-Boundary Maps and Black Hole Complementarity

We provide a simple and explicit construction of local bulk operators that describe the interior of a black hole in the AdS/CFT correspondence. The existence of these operators is predicated on the assumption that the mapping of CFT operators to local bulk operators depends on the state of the CFT. We show that our construction leads to an exactly local effective field theory in the bulk. Barring the fact that their charge and energy can be measured at infinity, we show that the commutator of local operators inside and outside the black hole vanishes exactly, when evaluated within correlation functions of the CFT. Our construction leads to a natural resolution of the strong subadditivity paradox of Mathur and Almheiri et al. Furthermore, we show how, using these operators, it is possible to reconcile small corrections to effective field theory correlators with the unitarity of black hole evaporation. We address and resolve all other arguments, advanced in arxiv:1304.6483 and arxiv:1307.4706, in favour of structure at the black hole horizon. We extend our construction to states that are near equilibrium, and thereby also address the "frozen vacuum" objections of arxiv:1308.3697. Finally, we explore an intriguing link between our construction of interior operators and Tomita-Takesaki theory.Comment: (v1) 92 pages; mathematica file included with source. (v2) 97 pages; added discussion of non-Abelian symmetries; fixed typo

NEEXP is Contained in MIP*

We study multiprover interactive proof systems. The power of classical multiprover interactive proof systems, in which the provers do not share entanglement, was characterized in a famous work by Babai, Fortnow, and Lund (Computational Complexity 1991), whose main result was the equality MIP = NEXP. The power of quantum multiprover interactive proof systems, in which the provers are allowed to share entanglement, has proven to be much more difficult to characterize. The best known lower-bound on MIP* is NEXP ⊆ MIP*, due to Ito and Vidick (FOCS 2012). As for upper bounds, MIP* could be as large as RE, the class of recursively enumerable languages. The main result of this work is the inclusion of NEEXP = NTIME[2^(2poly(n))] ⊆ MIP*. This is an exponential improvement over the prior lower bound and shows that proof systems with entangled provers are at least exponentially more powerful than classical provers. In our protocol the verifier delegates a classical, exponentially large MIP protocol for NEEXP to two entangled provers: the provers obtain their exponentially large questions by measuring their shared state, and use a classical PCP to certify the correctness of their exponentially-long answers. For the soundness of our protocol, it is crucial that each player should not only sample its own question correctly but also avoid performing measurements that would reveal the other player's sampled question. We ensure this by commanding the players to perform a complementary measurement, relying on the Heisenberg uncertainty principle to prevent the forbidden measurements from being performed