Dilogarithm identities for conformal field theories and cluster algebras: Simply laced case

Abstract

Abstract. The dilogarithm identities for the central charges of conformal field theories of simply laced type were conjectured by Bazhanov, Kirillov, and Reshetikhin. Their functional generalizations were conjectured by Gliozzi and Tateo. They have been partly proved by various authors. We prove these identities in full generality for any pair of Dynkin diagrams of simply laced type based on the cluster algebra formulation of the Y-systems.

Reduction cohomology of Riemann surfaces

“…The condition (2.5) for mutation (4.26) to be involutive gives us a cohomological condition (vanishing square of the n-th coboundary operator) on modular forms. Cluster algebras introduced in [FZ1] have numerous applications in various areas of mathematics [FG1, FG2, FG3, FG4, GSV1, GSV2, GSV3, FST, KS, N1, N2, DFK, GLS, HL, Ke1,Ke2,Na,Sch,N1,N2,Sch]. In this subsection we recall [Zu] definition of a vertex cluster algebra and show its cohomological nature.…”

Schottky vertex operator cluster algebras

“…The theory of cluster algebras is connected to many different areas of mathematics, e.g., the representation theory of finite dimensional algebras, Lie theory, Poisson geometry and Teichmüller theory [30,31,32]. Among these topics are dilogarithm identities for conformal field theories [50,51], quantum algebras [33,34], quivers [36,37,52]. Cluster algebras have numerous applications [22,23,24,25,30,31,18,38,50,51,11].…”

“…Among these topics are dilogarithm identities for conformal field theories [50,51], quantum algebras [33,34], quivers [36,37,52]. Cluster algebras have numerous applications [22,23,24,25,30,31,18,38,50,51,11]. Cluster algebras appear in several applications in Conformal Field Theory [11,50,51].…”

“…Cluster algebras have numerous applications [22,23,24,25,30,31,18,38,50,51,11]. Cluster algebras appear in several applications in Conformal Field Theory [11,50,51]. It is natural to find an analogue of a cluster algebra structure in the language of vertex operator algebras.…”

Relation between $f$-vectors and $d$-vectors in cluster algebras of finite type or rank 2

“…These cluster algebras are introduced by [10] and are classified completely by [11]. Cluster algebras of finite type have connections with Dynkin diagrams or (real) root systems in Lie algebras, and they are applied to the logarithm identities, T, Y -systems and so on [15][16][17]20]. On the other hand, cluster algebras of rank 2 are cluster algebras which have two cluster variables in every cluster.…”