Minimal Model Holography

We review the duality relating 2d W_N minimal model CFTs, in a large N 't Hooft like limit, to higher spin gravitational theories on AdS_3.Comment: 54 pages, 1 figure; Contribution to J. Phys. A special volume on "Higher Spin Theories and AdS/CFT" edited by M. R. Gaberdiel and M. Vasiliev. v2. minor change

Torus Knot and Minimal Model

We reveal an intimate connection between the quantum knot invariant for torus knot T(s,t) and the character of the minimal model M(s,t), where s and t are relatively prime integers. We show that Kashaev's invariant, i.e., the N-colored Jones polynomial at the N-th root of unity, coincides with the Eichler integral of the character.Comment: 10 page

Even spin minimal model holography

The even spin W^e_\infty algebra that is generated by the stress energy tensor together with one Virasoro primary field for every even spin s \geq 4 is analysed systematically by studying the constraints coming from the Jacobi identities. It is found that the algebra is characterised, in addition to the central charge, by one free parameter that can be identified with the self-coupling constant of the spin 4 field. We show that W^e_\infty can be thought of as the quantisation of the asymptotic symmetry algebra of the even higher spin theory on AdS_3. On the other hand, W^e_\infty is also quantum equivalent to the so(N) coset algebras, and thus our result establishes an important aspect of the even spin minimal model holography conjecture. The quantum equivalence holds actually at finite central charge, and hence opens the way towards understanding the duality beyond the leading 't Hooft limit.Comment: 32 pages, v2: reference added, minor changes in tex

Minimal Model for Sand Dunes

We propose a minimal model for aeolian sand dunes. It combines an analytical description of the turbulent wind velocity field above the dune with a continuum saltation model that allows for saturation transients in the sand flux. The model provides a qualitative understanding of important features of real dunes, such as their longitudinal shape and aspect ratio, the formation of a slip face, the breaking of scale invariance, and the existence of a minimum dune size.Comment: 4 pages, 4 figures, replaced with publishd versio

Triality in Minimal Model Holography

The non-linear W_{\infty}[\mu] symmetry algebra underlies the duality between the W_N minimal model CFTs and the hs[\mu] higher spin theory on AdS_3. It is shown how the structure of this symmetry algebra at the quantum level, i.e. for finite central charge, can be determined completely. The resulting algebra exhibits an exact equivalence (a`triality') between three (generically) distinct values of the parameter \mu. This explains, among other things, the agreement of symmetries between the W_N minimal models and the bulk higher spin theory. We also study the consequences of this triality for some of the simplest W_{\infty}[\mu] representations, thereby clarifying the analytic continuation between the`light states' of the minimal models and conical defect solutions in the bulk. These considerations also lead us to propose that one of the two scalar fields in the bulk actually has a non-perturbative origin.Comment: 29 pages; v2. Typos correcte

The W_N minimal model classification

We first rigourously establish, for any N, that the toroidal modular invariant partition functions for the (not necessarily unitary) W_N(p,q) minimal models biject onto a well-defined subset of those of the SU(N)xSU(N) Wess-Zumino-Witten theories at level (p-N,q-N). This permits considerable simplifications to the proof of the Cappelli-Itzykson-Zuber classification of Virasoro minimal models. More important, we obtain from this the complete classification of all modular invariants for the W_3(p,q) minimal models. All should be realised by rational conformal field theories. Previously, only those for the unitary models, i.e. W_3(p,p+1), were classified. For all N our correspondence yields for free an extensive list of W_N(p,q) modular invariants. The W_3 modular invariants, like the Virasoro minimal models, all factorise into SU(3) modular invariants, but this fails in general for larger N. We also classify the SU(3)xSU(3) modular invariants, and find there a new infinite series of exceptionals.Comment: 25 page

The Minimal Model Program Revisited

We give a light introduction to some recent developments in Mori theory, and to our recent direct proof of the finite generation of the canonical ring.Comment: to appear in Contributions to Algebraic Geometry, EMS Series of Congress Report

Minimal model theory for log surfaces

We discuss the log minimal model theory for log surfaces. We show that the log minimal model program, the finite generation of log canonical rings, and the log abundance theorem for log surfaces hold true under assumptions weaker than the usual framework of the log minimal model theory.Comment: 34 pages, v2: Section 8 is new, v3: minor revisio