Stability data, irregular connections and tropical curves

Abstract

Abstract. We study a class of meromorphic connections ∇(Z) on P 1 , parametrised by the central charge Z of a stability condition, with values in a Lie algebra of formal vector fields on a torus. Their definition is motivated by the work of Gaiotto, Moore and Neitzke on wall-crossing and three-dimensional field theories. Our main results concern two limits of the families ∇(Z) as we rescale the central charge Z → RZ. In the R → 0 "conformal limit" we recover a version of the connections introduced by Bridgeland and Toledano Laredo (and so the Joyce holomorphic generating functions for enumerative invariants), although with a different construction yielding new explicit formulae. In the R → ∞ "large complex structure" limit the connections ∇(Z) make contact with the Gross-Pandharipande-Siebert approach to wall-crossing based on tropical geometry. Their flat sections display tropical behaviour, and also encode certain tropical/relative Gromov-Witten invariants.

A Riemann–Hilbert problem for uncoupled BPS structures

“…The same class of Riemann-Hilbert problems was considered by Filippini, Garcia-Fernandez and Stoppa in [7], motivated by the physics work [10]. Their solution takes values in the automorphism group of an algebraic torus.…”

“…The functionX has the form of a classical solution to a Riemann-Hilbert problem. Such an integral expression is the basis solution for an analogous Riemann-Hilbert problem considered in [7], section 4.3. We can look atX as a piecewise function in 0 < | Im(θ)| < 2π, Im(w) = 0, satisfying the symmetryX 4) and with discontinuities prescribed by S. This can be shown via a direct integral contour argument.…”

“…In this section we briefly describe the relation between their approach and ours. The problem in [7] is stated for "positive BPS structures". It is viewed as the "conformal limit" of a Riemann-Hilbert problem stated in [10], which has a different asymptotic behaviour at infinity.…”

“…. , n. The reader can see [7,Section 4] for the explicit formulae and the proof in the "positive" case and compare with [3, Section 4] for the notation in C[Γ] [[s]]. Note also that, due to a different sign conventions, the central charge in [7] is −Z.…”

Variations of BPS Structure and a Large Rank Limit

“…the solution to the infinite dimensional, birational RH problem of (Γ, Z, Ω) at ξ ∈ T is the leading order term in the → 0, N → ∞ limit of a sum of simple oscillators, in the nonempty open sector of H ℓ where ℜ(Z(γ i )/t) > 0 for all i. Π is the finite dimensional analogue of the torus character projecting along the β j component as in Theorem 1. In terms of matrix entries we have (12) where for a matrix A we write A (kl) = A kl + A lk . We will see that in fact there is an explicit formula…”