2020/11/30 by Kim, Hyun Kyu
#13F60 #14D20 #14D23 #14M35 #57K31 #58D27 #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Mathematical Physics (math-ph) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2011.14765
In this paper we partially settle Fock-Goncharov's duality conjecture for cluster varieties associated to their moduli spaces of \rm G-local systems on a punctured surface \frakS with boundary data, when \rm G is a group of type A2, namely \rm SL3 and \rm PGL3. Based on Kuperberg's \rm SL3-webs, we introduce the notion of \rm SL3-laminations on \frakS defined as certain \rm SL3-webs with integer weights. We introduce coordinate systems for \rm SL3-laminations, and show that \rm SL3-laminations satisfying a congruence property are geometric realizations of the tropical integer points of the cluster \mathscrA-moduli space \mathscrA_\rm SL3,\frakS. Per each such \rm SL3-lamination, we construct a regular function on the cluster \mathscrX-moduli space \mathscrX_\rm PGL3,\frakS. We show that these functions form a basis of the ring of all regular functions. For a proof, we develop \rm SL3 quantum and classical trace maps for any triangulated bordered surface with marked points, and state-sum formulas for them. We construct quantum versions of the basic regular functions on \mathscrX_\rm PGL3,\frakS. The bases constructed in this paper are built from non-elliptic webs, hence could be viewed as higher `bangles' bases, and the corresponding `bracelets' versions can also be considered as direct analogs of Fock-Goncharov's and Allegretti-Kim's bases for the \rm SL2-\rm PGL2 case.