[Paper Review] Generalized Donaldson-Thomas invariants on the local projective plane

Yukinobu Toda|arXiv (Cornell University)|May 14, 2014

Advanced Algebra and Geometry17 references3 citations

TL;DR

This paper establishes that the generating series of generalized Donaldson-Thomas invariants on the local projective plane with arbitrary positive rank are expressible as finite linear combinations of modular forms and theta series for indefinite lattices. The series converge absolutely to holomorphic functions on the upper half-plane, providing a modular structure consistent with S-duality conjectures in string theory.

ABSTRACT

We show that the generating series of generalized Donaldson-Thomas invariants on the local projective plane with any positive rank is described in terms of modular forms and theta type series for indefinite lattices. In particular it absolutely converges to give a holomorphic function on the upper half plane.

Motivation & Objective

• To understand the structure of generalized Donaldson-Thomas invariants on the local projective plane, a non-compact Calabi-Yau 3-fold, for arbitrary positive rank.
• To resolve the complexity of explicit formulas for non-coprime rank and degree pairs by identifying a unified modular structure.
• To demonstrate that the generating series of these invariants converge absolutely and exhibit modular-like properties.
• To provide a framework linking algebraic geometry invariants to modular forms and theta functions on indefinite lattices.

Proposed method

• The paper constructs generalized theta series from lattice data involving a non-degenerate lattice, rational vectors, and bilinear forms, ensuring convergence via conditions on signs and bilinear pairings.
• It introduces a new class of theta series (5) that generalize classical theta series and mock theta series, allowing for indefinite quadratic forms.
• The proof uses a combinatorial decomposition of the moduli space via torus localization and graph-theoretic data (G, ψ), reducing the invariant computation to sums over subgraphs and lattice points.
• The generating series is expressed as a finite linear combination of these generalized theta series, scaled by powers of the Dedekind eta function and inverse theta series.
• Key identities are derived using intersection theory on the blow-up of P², particularly involving canonical divisor and curve classes.
• The convergence of the series is established via analytic estimates on the growth of coefficients, relying on the structure of the bilinear forms and sign functions.

Experimental results

Research questions

• RQ1 Can the generating series of generalized DT invariants on the local P² be expressed in terms of modular forms and theta series for indefinite lattices?
• RQ2 Does the generating series converge absolutely to a holomorphic function on the upper half-plane for any positive rank?
• RQ3 Is there a unified modular structure underlying DT invariants even when rank and degree are not coprime, resolving the complexity of explicit formulas?
• RQ4 Can the S-duality conjecture be supported by showing that the invariants transform in a modular way under the action of SL(2,Z)?
• RQ5 What is the precise algebraic and analytic structure of the generating series beyond the rank one and two cases?

Key findings

• The generating series of generalized DT invariants for any positive rank r and integer l is shown to be a finite linear combination of generalized theta series and modular forms.
• The series converges absolutely to a holomorphic function on the upper half-plane, as established by convergence estimates on the theta series (5).
• For any r ≥ 1 and l ∈ ℤ, the series DT(r,l) is expressed as q^{r/8} η(q)^{-3r} times a rational combination of generalized theta series in q^{1/N} for some N ≥ 1.
• The construction includes classical theta series (positive definite case) and mock theta series (b=1, k=0) as special cases, unifying known results.
• The result provides a structural explanation for the modular-like behavior observed in low-rank cases, such as rank two and three, without requiring explicit closed forms.
• The method avoids direct computation of Euler characteristics of moduli spaces, instead using lattice-theoretic and graph-theoretic decomposition to derive the modular structure.