R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

We define new invariants of 3d Calabi-Yau categories endowed with a stability structure. Intuitively, they count the number of semistable objects with fixed class in the K-theory of the category ('number of BPS states with given charge' in physics language). Formally, our motivic DT-invariants are elements of quantum tori over a version of the Grothendieck ring of varieties over the ground field. Via the quasi-classical limit 'as the motive of affine line approaches to 1' we obtain numerical DT-invariants which are closely related to those introduced by Behrend. We study some properties of both motivic and numerical DT-invariants including the wall-crossing formulas and integrality. We discuss the relationship with the mathematical works (in the non-triangulated case) of Joyce, Bridgeland and Toledano-Laredo, as well as with works of physicists on Seiberg-Witten model (string junctions), classification of N=2 supersymmetric theories (Cecotti-Vafa) and structure of the moduli space of vector multiplets. Relating the theory of 3d Calabi-Yau categories with distinguished set of generators (called cluster collection) with the theory of quivers with potential we found the connection with cluster transformations and cluster varieties (both classical and quantum).

Stability structures, motivic Donaldson-Thomas invariants and cluster transformations

We consider BPS states in a large class of d=4, N=2 field theories, obtained by reducing six-dimensional (2,0) superconformal field theories on Riemann surfaces, with defect operators inserted at points of the Riemann surface. Further dimensional reduction on S^1 yields sigma models, whose target spaces are moduli spaces of Higgs bundles on Riemann surfaces with ramification. In the case where the Higgs bundles have rank 2, we construct canonical Darboux coordinate systems on their moduli spaces. These coordinate systems are related to one another by Poisson transformations associated to BPS states, and have well-controlled asymptotic behavior, obtained from the WKB approximation. The existence of these coordinates implies the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum. This construction provides a concrete realization of a general physical explanation of the wall-crossing formula which was proposed in 0807.4723. It also yields a new method for computing the spectrum using the combinatorics of triangulations of the Riemann surface.

Wall-crossing, Hitchin Systems, and the WKB Approximation

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmuller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. It is related to the motivic dilogarithm.

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Moduli spaces of local systems and higher Teichmüller theory

We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of coefficients, describe this complex explicitly in terms of "tagged triangulations" of the surface, and determine its homotopy type and its growth rate.

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Cluster algebras and triangulated surfaces. Part I: Cluster complexes

Un ensemble amasse est une paire (X, A) d'espaces positifs (i.e. de varietes munies d'un atlas positif) munis de l'action d'un groupe discret. L'espace A est relie au spectre d'une algebre amassee [12]. Les deux espaces sont lies par un morphisme p: A → X. L'espace A est muni d'une 2-forme fermee, eventuellement degeneree, et l'espace X est muni d'une structure de Poisson. L'application p est compatible avec ces structures. Le dilogarithme avec ses avatars motiviques et quantiques joue un role fondamental dans la structure d'un ensemble amasse. Nous definissons une deformation non-commutative de l'espace X. Nous montrons que, dans le cas ou le parametre de la deformation q est une racine de l'unite, l'algebre deformee a un centre qui contient l'algebre des fonctions sur l'espace X originel. Notre exemple principal est celui de l'espace des modules associe dans [6] a une surface topologique S munie d'un nombre fini de points distingues sur le bord et a un groupe algebrique semi-simple G. C'est un avatar algebro-geometrique de la theorie de Teichmuller d'ordre superieur sur la surface S a valeurs dans G. Nous evoquons l'existence d'une dualite entre les espaces A et X. Une des manifestations de cette dualite est une conjecture de dualite affirmant que les points tropicaux d'un espace parametrent une base dans l'espace d'une certaine classe de fonctions sur l'espace Langlands-dual. Nous demontrons cette conjecture dans un certain nombre d'exemples.

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Cluster ensembles, quantization and the dilogarithm

Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the A-space. We develope general properties of cluster ensembles, including its group of symmetries - the cluster modular group, and a relation with the motivic dilogarithm. We define a q-deformation of the X-space. Formulate general duality conjectures regarding canonical bases in the cluster ensemble context. We support them by constructing the canonical pairing in the finite type case. 
Interesting examples of cluster ensembles are provided the higher Teichmuller theory, that is by the pair of moduli spaces corresponding to a split reductive group G and a surface S defined in math.AG/0311149. 
We suggest that cluster ensembles provide a natural framework for higher quantum Teichmuller theory.

We construct, using the quantum dilogarithm, a series of *-representations of quantized cluster varieties. This includes a construction of infinite dimensional unitary projective representations of their discrete symmetry groups - the cluster modular groups. The examples of the latter include the classical mapping class groups of punctured surfaces. 
One of applications is quantization of higher Teichmuller spaces. 
The constructed unitary representations can be viewed as analogs of the Weil representation. In both cases representations are given by integral operators. Their kernels in our case are the quantum dilogarithms. 
We introduce the symplectic/quantum double of cluster varieties and related them to the representations.

The quantum dilogarithm and representations of quantum cluster varieties

We explicitly describe a noncommutative deformation of the *-algebra of functions on the Teichmuller space of Riemann surfaces with holes that is equivariant with respect to the action of the mapping class group.

V. V. Fock

Papers

Moduli spaces of local systems and higher Teichmüller theory

Cluster ensembles, quantization and the dilogarithm

Cluster ensembles, quantization and the dilogarithm

The quantum dilogarithm and representations of quantum cluster varieties

Quantum Teichmuller space