Hamiltonian Gravity and Noncommutative Geometry

A version of foliated spacetime is constructed in which the spatial geometry is described as a time dependent noncommutative geometry. The ADM version of the gravitational action is expressed in terms of these variables. It is shown that the vector constraint is obtained without the need for an extraneous shift vector in the action.Comment: 22 pages, AMS-LaTeX. Some improvements - mainly to sections 8 and 9. Typographical errors to equations in appendix correcte

M[any] Vacua of IIB

Description of the spectrum of fluctuations around a commutative vacuum solution, as well as around a solution with degenerate commutator in IIB matrix model is given in terms of supersymmetric Yang-Mills (YM) model. We construct explicitly the map from Hermitian matrices to YM fields and study the dependence of the spectrum and respective YM model on the symmetries of the solution. The gauge algebra of the YM model is shown to contain local reparameterisation algebra as well as Virasoro one.Comment: 17 pages, Virasoro algebra explicitely given, LaTeX style change, minor text change

A constructive commutative quantum Lovasz Local Lemma, and beyond

The recently proven Quantum Lovasz Local Lemma generalises the well-known Lovasz Local Lemma. It states that, if a collection of subspace constraints are "weakly dependent", there necessarily exists a state satisfying all constraints. It implies e.g. that certain instances of the kQSAT quantum satisfiability problem are necessarily satisfiable, or that many-body systems with "not too many" interactions are always frustration-free. However, the QLLL only asserts existence; it says nothing about how to find the state. Inspired by Moser's breakthrough classical results, we present a constructive version of the QLLL in the setting of commuting constraints, proving that a simple quantum algorithm converges efficiently to the required state. In fact, we provide two different proofs, one using a novel quantum coupling argument, the other a more explicit combinatorial analysis. Both proofs are independent of the QLLL. So these results also provide independent, constructive proofs of the commutative QLLL itself, but strengthen it significantly by giving an efficient algorithm for finding the state whose existence is asserted by the QLLL. We give an application of the constructive commutative QLLL to convergence of CP maps. We also extend these results to the non-commutative setting. However, our proof of the general constructive QLLL relies on a conjecture which we are only able to prove in special cases.Comment: 43 pages, 2 conjectures, no figures; unresolved gap in the proof; see arXiv:1311.6474 or arXiv:1310.7766 for correct proofs of the symmetric cas

Open problems, questions, and challenges in finite-dimensional integrable systems

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference “Finite-dimensional Integrable Systems, FDIS 2017” held at CRM, Barcelona in July 2017.Postprint (updated version

Algebraic Geometry Approach to the Bethe Equation for Hofstadter Type Models

We study the diagonalization problem of certain Hofstadter-type models through the algebraic Bethe ansatz equation by the algebraic geometry method. When the spectral variables lie on a rational curve, we obtain the complete and explicit solutions for models with the rational magnetic flux, and discuss the Bethe equation of their thermodynamic flux limit. The algebraic geometry properties of the Bethe equation on high genus algebraic curves are investigated in cooperationComment: 28 pages, Latex ; Some improvement of presentations, Revised version with minor changes for journal publicatio