Topological wave functions and heat equations

TL;DR

The paper strengthens the wave-function perspective on holomorphic anomaly equations and casts them as heat-type constraints on a holomorphic section, achieving a purely holomorphic reformulation. In Hermitian symmetric tube domains, it shows the general solution can be written as a matrix element in the Schrödinger-Weil representation, tying the anomaly to representation-theoretic identities and the metaplectic correction. It further argues for a one-parameter generalization of the topological amplitude linked to the minimal representation of a larger duality group G', with potential connections to hypermultiplet dynamics, nonabelian Donaldson-Thomas theory, and black hole degeneracies. The work lays a representation-theoretic and geometric quantization framework for understanding the topology-quantum structure of topological strings and their nonperturbative completions, suggesting deep automorphic and geometric quantization structure behind OSV-like conjectures.

Abstract

It is generally known that the holomorphic anomaly equations in topological string theory reflect the quantum mechanical nature of the topological string partition function. We present two new results which make this assertion more precise: (i) we give a new, purely holomorphic version of the holomorphic anomaly equations, clarifying their relation to the heat equation satisfied by the Jacobi theta series; (ii) in cases where the moduli space is a Hermitian symmetric tube domain $G/K$, we show that the general solution of the anomaly equations is a matrix element $\IP{Ψ| g | Ω}$ of the Schrödinger-Weil representation of a Heisenberg extension of $G$, between an arbitrary state $\braΨ$ and a particular vacuum state $\ketΩ$. Based on these results, we speculate on the existence of a one-parameter generalization of the usual topological amplitude, which in symmetric cases transforms in the smallest unitary representation of the duality group $G'$ in three dimensions, and on its relations to hypermultiplet couplings, nonabelian Donaldson-Thomas theory and black hole degeneracies.

Figures (4)

• Figure 1: Euclidean Jordan algebras of degree 3 and their corresponding $G$, $K$, $V$ and $n_v$. The groups $G$, $K$ are given only up to a finite cover or finite quotient. $Herm_3$ denotes $3 \times 3$ Hermitian matrices; $\Gamma_{n-1,1}$ is a Jordan algebra of degree 2 defined by a quadratic form of Lorentzian signature Gunaydin:1983bi.
• Figure 2: A list of the generators of the Lie algebra $\widetilde{{\mathfrak g}}$, with two gradings, by $D$ (horizontal) and by $\Delta$ (vertical).
• Figure 3: Isometry groups $G'$ of the moduli spaces occurring upon dimensional reduction from $4$ to $3$ dimensions, with their maximal compact subgroups $K'$, and the corresponding $G$, $K$ from $4$ dimensions. All groups are given only up to finite covering or finite quotient.
• Figure 4: A list of the generators of the Lie algebra ${\mathfrak g}'$, with two gradings, by $D$ (horizontal) and by $\Delta$ (vertical). This extends Figure \ref{['fj-picture']}, which was the analogous picture for $\widetilde{{\mathfrak g}} \subset {\mathfrak g}'$.