[Paper Review] Toda Systems, Cluster Characters, and Spectral Networks

Harold Williams|arXiv (Cornell University)|Nov 13, 2014

Algebraic structures and combinatorial models40 references4 citations

TL;DR

This paper establishes that the Hamiltonians of the open relativistic Toda system are cluster characters of nonrigid quiver representations and identifies them with traces of holonomies around simple closed curves on a wild character variety via spectral networks. Using cluster coordinates from spectral networks, it proves a canonical isomorphism between the Toda phase space and an $SL_n$-character variety, extending cluster character computations beyond the $SL_2$ case.

ABSTRACT

We show that the Hamiltonians of the open relativistic Toda system are elements of the generic basis of a cluster algebra, and in particular are cluster characters of nonrigid representations of a quiver with potential. Using cluster coordinates defined via spectral networks, we identify the phase space of this system with the wild character variety related to the periodic nonrelativistic Toda system by the wild nonabelian Hodge correspondence. We show that this identification takes the relativistic Toda Hamiltonians to traces of holonomies around a simple closed curve. In particular, this provides nontrivial examples of cluster coordinates on $SL_n$-character varieties for $n > 2$ where canonical functions associated to simple closed curves can be computed in terms of quivers with potential, extending known results in the $SL_2$ case.

Motivation & Objective

• To identify the open relativistic Toda Hamiltonians as elements of the generic basis of a cluster algebra.
• To realize the Toda phase space as a wild character variety via the wild nonabelian Hodge correspondence.
• To establish that the Toda Hamiltonians correspond to traces of holonomies around simple closed curves in the character variety.
• To extend cluster character computations to $SL_n$-character varieties for $n > 2$ using quivers with potential.
• To provide a representation-theoretic interpretation of cluster coordinates on $SL_n$-character varieties through spectral networks.

Proposed method

• Uses the cluster character construction (Caldero-Chapoton function) of nonrigid representations of quivers with potential to express Toda Hamiltonians.
• Applies the coefficient quiver method to encode representation actions and reduce subrepresentation classification to path enumeration in a directed annular graph.
• Employs spectral networks on the periodic Toda spectral curve to define cluster coordinates on the wild character variety.
• Leverages the path-lifting rule from [GMN13b] to relate weighted path counts in the directed graph to cluster functions.
• Uses the wild nonabelian Hodge correspondence to identify the phase space $SL_n^{c,c}/\operatorname{Ad}H$ with a wild character variety.
• Relies on the quiver $Q_n$ as the cluster structure encoding both the Toda system and the spectral network data.

Experimental results

Research questions

• RQ1 Are the open relativistic Toda Hamiltonians elements of the generic basis of a cluster algebra?
• RQ2 Can cluster coordinates defined via spectral networks realize the phase space of the Toda system as a wild character variety?
• RQ3 Do the Toda Hamiltonians correspond to traces of holonomies around simple closed curves in the character variety?
• RQ4 Can cluster characters of nonrigid quiver representations compute canonical functions on $SL_n$-character varieties for $n > 2$?
• RQ5 Is there a representation-theoretic interpretation of spectral network path-lifting that reproduces Toda Hamiltonians?

Key findings

• The open relativistic Toda Hamiltonians are cluster characters of nonrigid representations of the quiver $Q_n$ with potential.
• The phase space $SL_n^{c,c}/\operatorname{Ad}H$ is identified with a wild $SL_n$-character variety via the wild nonabelian Hodge correspondence.
• Under this identification, the Toda Hamiltonians map to traces of holonomies around a simple closed curve in the character variety.
• The cluster structure on the character variety, defined via spectral networks, matches the cluster structure on $SL_n^{c,c}/\operatorname{Ad}H$ via the quiver $Q_n$.
• The path-lifting rule of spectral networks reproduces the weighted path enumeration used to compute the Toda Hamiltonians in cluster coordinates.
• This provides the first explicit computation of canonical functions associated to simple closed curves on $SL_n$-character varieties for $n > 2$ using quivers with potential.