## The Schrödinger-Virasoro Algebra: Mathematical structure and by Jérémie Unterberger, Claude Roger

By Jérémie Unterberger, Claude Roger

This monograph offers the 1st updated and self-contained presentation of a lately came across mathematical structure—the Schrödinger-Virasoro algebra. simply as Poincaré invariance or conformal (Virasoro) invariance play a key rôle in knowing, respectively, hassle-free debris and two-dimensional equilibrium statistical physics, this algebra of non-relativistic conformal symmetries will be anticipated to use itself evidently to the examine of a few types of non-equilibrium statistical physics, or extra in particular within the context of modern advancements with regards to the non-relativistic AdS/CFT correspondence.

The examine of the constitution of this infinite-dimensional Lie algebra touches upon issues as numerous as statistical physics, vertex algebras, Poisson geometry, integrable structures and supergeometry in addition to illustration thought, the cohomology of infinite-dimensional Lie algebras, and the spectral conception of Schrödinger operators.

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**Example text**

1/-Currents or ab-Theory . . . . . . . . . . 1 Definition of the sv-Fields . . . . . . .. . . . . . . . . 2 Construction of the Generalized Polynomial Fields ˛ ˚j;k . . . . . . . . . . . . . . . . . . . . . . . 4 Correlators of the Polynomial and Generalized Polynomial Fields . . . . . . . . . . . . . . . .. . . . . . . . . 5 Construction of the Massive Fields . . . . . . .. . . . . . . . . 57 58 61 68 72 75 77 81 82 86 89 89 95 106 111 Cohomology, Extensions and Deformations .

1 Ermakov-Lewis Invariants and Schr¨odinger– Virasoro Invariance . . . . . . . . . . . . . . . . . . . 2 Solution of the Associated Classical Problem . . . . . 3 Spectral Decomposition of the Model Operators . . . . . . . . . . . . . . . .. . . . . . . . . 4 Monodromy of Non-resonant Harmonic Oscillators (Elliptic Case) . . . . . . .. . . . . . . . . 5 Monodromy of Harmonic Repulsors (Hyperbolic Type) .

V Preface .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . xxiii Geometric Definitions of sv . . . . . . . . . . . . . .. . . . . . . . . 1 From Newtonian Mechanics to the Schr¨odinger– Virasoro Algebra . . . . . . . . . . . . . . . . .. . . . . . . . . 1 From Galilei to Schr¨odinger: Central Extensions and Projective Automorphisms .