Analytic foundations of the Thermodynamic Bethe Ansatz solution to the Donaldson-Thomas Riemann-Hilbert problem of the A2 quiver (deformed cubic oscillator)

Bridgeland and Masoero (2023) gave the first solutions, beyond the uncoupled case, of the infinite-dimensional Riemann-Hilbert problems attached to the Donaldson-Thomas theory of the A2 quiver, by identifying the unknowns with the generalised monodromy (Stokes data) of the deformed cubic oscillator, a second-order ODE whose isomonodromic deformations are governed by the first Painleve equation. Two analytic gaps were left open and remain open. First, the constructed meromorphic solution has no intrinsic characterisation or uniqueness theorem, which the authors call the 'crucial missing link' in deriving the Joyce connection from Donaldson-Thomas data. Second, the Thermodynamic Bethe Ansatz (TBA) integral equations that formally solve the same Riemann-Hilbert problem have no analytic theory (existence, uniqueness, analytic dependence on the deformation parameter) in this coupled case. This record poses the combined problem: establish a rigorous analytic solution theory for the coupled A2 TBA, or an equivalent intrinsic selection principle, and prove its identification with the monodromy-derived solution. The uncoupled case is fully understood via Gamma-function solutions and Borel-resummed Voros symbols, while all coupled-case TBA results to date remain formal, perturbative or numerical.

Background

The Donaldson-Thomas side. A BPS structure $(\Gamma,Z,\Omega)$ consists of a free abelian group $\Gamma$ of charges carrying an antisymmetric intersection form, a homomorphism $Z\colon\Gamma\to\mathbb{C}$ called the central charge, and integer BPS (Donaldson-Thomas) invariants $\Omega(\gamma)$, which here count semistable objects in the 3-Calabi-Yau triangulated category (a homological-algebraic category of complexes with a Calabi-Yau-type duality) associated with the $A_2$ quiver (the directed graph with two vertices joined by one arrow). Bridgeland attached to any such BPS structure an infinite-dimensional Riemann-Hilbert problem: for each ray $r\subset\mathbb{C}^*$ that is not an active ray (the active rays are the finitely many $\mathbb{R}_{>0}\,Z(\gamma)$ with $\Omega(\gamma)\neq 0$), one seeks a meromorphic function $\hbar\mapsto X_r(\hbar)$ on the half-plane $H_r$ centred on $r$, with values in the torus of characters $\chi\colon\Gamma\to\mathbb{C}^*$, such that (RH1) across the boundary of each convex sector the functions are related by an explicit wall-crossing automorphism $S(\Delta)$ built from the $\Omega(\gamma)$, (RH2) $\exp(Z(\gamma)/\hbar)\,X_{r,\gamma}(\hbar)\to\xi(\gamma)$ as $\hbar\to 0$ in $H_r$ for a prescribed constant term $\xi$, and (RH3) at most polynomial growth as $\hbar\to 0$. Solving this problem is the analytic step needed to construct the Joyce structure, a complex hyperkahler-type geometry on the space of stability conditions, and hence to recover geometry directly from DT invariants.

The differential-equation side. Bridgeland and Masoero study the deformed cubic oscillator $y''(x)=Q(x,\hbar)\,y(x)$ with $Q=\hbar^{-2}Q_0+\hbar^{-1}Q_1+Q_2$, where $Q_0=x^3+ax+b$, the lower-order terms $Q_1=p/(x-q)+r$ and $Q_2=\frac{3}{4(x-q)^2}+\frac{r}{2p(x-q)}+\frac{r^2}{4p^2}$ are fixed by the condition $p^2=q^3+aq+b$, and $\hbar\in\mathbb{C}^*$ is the deformation parameter. The only finite singularity $x=q$ is apparent (encircling it multiplies solutions by $-1$), so the generalised monodromy consists of the Stokes data at the irregular singularity at $x=\infty$, encoded by multipliers of subdominant (sectorially decaying) solutions in the five Stokes sectors of the leading term $x^3$. Varying $(a,b)$ at fixed monodromy gives isomonodromic (monodromy-preserving) deformations governed by the first Painleve equation (the nonlinear ODE $y''=6y^2+x$). The spectral curve $y^2=x^3+ax+b$ is, away from the discriminant locus, an elliptic curve $X_s$; the central charges are period integrals of this curve over cycles $\gamma\in H_1(X_s,\mathbb{Z})$, and the DT invariants of the $A_2$ quiver supply $\Omega$. Their Theorem 1.6 states that the Stokes data, written in Fock-Goncharov coordinates adapted to the WKB triangulation (coordinates on the moduli of framed local systems introduced by Fock and Goncharov), give a meromorphic solution of the $A_2$ DT Riemann-Hilbert problem, the first such construction beyond the uncoupled case, in which the active charges pairwise do not intersect. The target of the monodromy map is the wild character variety, which coincides with the cluster Poisson variety of the $A_2$ quiver (a space glued from algebraic tori by cluster transformations); this is the sense in which the problem below refers to cluster-theoretic data.

The two analytic gaps. First, uniqueness and characterisation: because the $A_2$ solution is meromorphic, the growth condition (RH3) fails at its poles, and the authors note that this 'loses any hope to prove uniqueness'; they explicitly call the missing characterisation of the solution of Theorem 1.6 the one 'crucial missing link' in deriving the flat (Joyce) connection on the space of stability conditions from purely abstract DT data. Second, the TBA: by the ODE/IM correspondence (ordinary differential equations / integrable models), functional relations among spectral determinants of Schrodinger operators can be rewritten as nonlinear convolution-type integral equations known as Thermodynamic Bethe Ansatz equations. In the present context the relevant system arises as the conformal limit of the integral equations of Gaiotto, Moore and Neitzke, and for the cubic oscillator it was written down by Masoero as a deformed TBA of five coupled equations, with an equivalent Y-system (a system of functional relations among the associated Y-functions), reducing at zero deformation to the classical TBA equation of the three-state Potts model. Remark 1.8 of the paper records that these equations solve the DT Riemann-Hilbert problem 'at least formally'; making this precise is the open analytic problem.

What is known in neighbouring cases. For uncoupled BPS structures Bridgeland proved that the Riemann-Hilbert problem has unique solutions given explicitly by products of Gamma functions; Barbieri proved essential uniqueness of the meromorphic solutions of the uncoupled problem; and Iwaki and Kidwai showed that Borel-resummed Voros symbols (the exponentially resummed WKB period functions of exact WKB analysis) of hypergeometric-type quantum curves solve the BPS Riemann-Hilbert problem. On the TBA side, Masoero proved that for a small linear term the monodromy data of the cubic oscillator do satisfy the deformed TBA, and Ito, Marino and Shu derived, and tested numerically, the TBA equations for the exact WKB periods of one-dimensional polynomial potentials including the PT-symmetric cubic oscillator (a potential invariant under combined parity and time reversal). None of these results addresses existence, uniqueness or analytic continuation for the coupled $A_2$ system, and no characterisation of the $A_2$ Riemann-Hilbert solution is known.

Problem Statement

Establish an analytic theory that singles out, uniquely and canonically, the solution of the Donaldson-Thomas Riemann-Hilbert problem for the $A_2$ quiver constructed by Bridgeland and Masoero from the monodromy of the deformed cubic oscillator, and in particular make the Thermodynamic Bethe Ansatz solution of that problem rigorous. Concretely, prove theorems accomplishing the following for the family of $A_2$ BPS structures arising from the deformed cubic oscillator (equivalently, from meromorphic quadratic differentials on the sphere with a single pole of order seven, with the constant term $\xi$ of (RH2) taken in the image of the abelian holonomy map, as fixed by formula (16) in the source construction):

1. Existence and uniqueness of TBA solutions. Show that the coupled TBA integral equations attached to these BPS structures, in the form of the conformal-limit or deformed TBA for the Stokes data of the deformed cubic oscillator or of the equivalent Y-system, admit solutions with the prescribed asymptotics as $\hbar\to 0$, and establish uniqueness within an explicitly specified natural class of functions on the relevant half-planes of the $\hbar$-plane, allowing at most controlled poles; the class in which uniqueness is claimed must be specified independently of the known monodromy solution. Whichever formulation of the equations is used, the identification required in item 3 anchors it to the same Bridgeland-Masoero solution.

2. Analytic continuation and wall-crossing. Establish the analytic dependence of these solutions on $\hbar$ away from the active rays, and prove that their discontinuities across the active rays are exactly the wall-crossing automorphisms required by condition (RH1) of the DT Riemann-Hilbert problem. This item is not an independent deliverable: it is discharged by items 1 and 3 together, since the monodromy-derived solution of Theorem 1.6 is a meromorphic solution of the $A_2$ Riemann-Hilbert problem and therefore already has the stated analytic dependence and wall-crossing discontinuities.

3. Identification with monodromy. Prove that the TBA solutions coincide with the meromorphic solution constructed from the generalised monodromy of the deformed cubic oscillator (Theorem 1.6 of Bridgeland-Masoero), so that the TBA construction, or the analytic conditions isolated in point 1, provides the intrinsic characterisation of the Riemann-Hilbert solution that is currently missing.

A resolution by an equivalent intrinsic route is equally acceptable: any natural analytic or variational conditions, formulated on the BPS and cluster-theoretic data without reference to the particular differential equation, that provably characterise the Bridgeland-Masoero solution among all meromorphic solutions of the $A_2$ Riemann-Hilbert problem, together with the corresponding uniqueness proof, discharges the objective.

References

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