Categorified canonical bases and framed BPS states

Abstract

We consider a cluster variety associated to a triangulated surface without punctures. The algebra of regular functions on this cluster variety possesses a canonical vector space basis parametrized by certain measured laminations on the surface. To each lamination, we associate a graded vector space, and we prove that the graded dimension of this vector space gives the expansion in cluster coordinates of the corresponding basis element. We discuss the relation to framed BPS states in N = 2 field theories of class S.

A note on discrete dynamical systems in theories of class S

“…These theories admit supersymmetric line defects which are associated to the spectral problem of computing framed BPS states [25]. In many cases such a problem can be approached by localization on quiver moduli spaces [9,[12][13][14][15], or on the moduli spaces of semiclassical configurations [5,32], by using cluster algebras [3,10,11,41], or via BPS graphs or spectral networks [21,22,27,34].…”

“…They in general violate the Dirac quantization condition with an arbitrary Wilson line of SU(3). As a result the set of allowed Wilson lines in this theory is reduced 3 and corresponds precisely to the representations of SU(3) which are invariant under the center Z 3 . On the other hand the set of dyonic line defects of the pure SU(3) theory constructed in [11] is still present.…”

“…Remark. We know that the correct operation to generate defects in (SU(3)/Z 3 ) n theories is the half-monodromy of SU (3). But what happens if we insist on using the 1/6 monodromy (4.2)?…”

Quantization of canonical bases and the quantum symplectic double

“…One can form the set A (Z t ) of points of this cluster variety valued in the semifield Z t of tropical integers, and there is a canonical map I A : A (Z t ) → O(X ) from this set into the algebra of regular functions on X [7,8]. Building on the ideas of [7], it has been shown in many cases that the image of this map is a canonical basis for the algebra of regular functions (see [4], Theorem 1.1).…”

Quantum line defects and refined BPS spectra

“…A very interesting prescription for the categorification of certain canonical basis associated to cluster varieties was proposed in [3]. Line defects can be described in theories of class S[A 1 ] in terms of laminations over the curve C. The construction of [3] takes certain laminations and writes the associated IR line operator as a sum of cluster variables, whose coefficients are graded dimensions of the singular cohomology of the quiver grassmannian associated with framed quivers. It would be very interesting to understand the relation between our construction and [3].…”