Topological wave functions and heat equations

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 $⟨\Psi |$ and a particular vacuum state $|\Omega ⟩$. 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 $G^{\prime }$ in three dimensions, and on its relations to hypermultiplet couplings, nonabelian Donaldson-Thomas theory and black hole degeneracies.

50 pages; v2: small typos fixed, references added; v3: cosmetic changes, published version; v4: typos fixed, small clarification added

References in corpus (21)

• Black Hole Attractors and the Topological String
• Quantum Foam and Topological Strings
• Hartle-Hawking Wave-Function for Flux Compactifications
• Charge Orbits of Symmetric Special Geometries and Attractors
• S-duality and Topological Strings
• BPS black holes, quantum attractor flows and automorphic forms
• Attractors and the Holomorphic Anomaly
• Minimal representations, spherical vectors, and exceptional theta series I
• BPS Black Hole Degeneracies and Minimal Automorphic Representations
• On the partition sum of the NS five-brane
• Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups
• Off-shell N=2 tensor supermultiplets
• Towards Integrability of Topological Strings I: Three-forms on Calabi-Yau manifolds
• Unitary Realizations of U-duality Groups as Conformal and Quasiconformal Groups and Extremal Black Holes of Supergravity Theories
• Gauge theory, topological strings, and S-duality
• $R^4$ couplings, the fundamental membrane and exceptional theta correspondences
• A Unified Approach to the Minimal Unitary Realizations of Noncompact Groups and Supergroups
• Minimal Unitary Realizations of Exceptional U-duality Groups and Their Subgroups as Quasiconformal Groups
• The Automorphic Membrane
• Minimal representations: spherical vectors and automorphic functionals
• Topological Strings and (Almost) Modular Forms

Cited by in corpus (38)

• Membranes at Quantum Criticality
• Split States, Entropy Enigmas, Holes and Halos
• Direct Integration of the Topological String
• A holomorphic and background independent partition function for matrix models and topological strings
• Quantum Attractor Flows
• Non-supersymmetric Black Holes and Topological Strings
• Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory
• Modular anomaly equation, heat kernel and S-duality in N=2 theories
• Fivebrane instantons, topological wave functions and hypermultiplet moduli spaces
• Horava-Lifshitz f(R) Gravity
• Topological String Theory on Compact Calabi-Yau: Modularity and Boundary Conditions
• Linear perturbations of Hyperkahler metrics
• Quasi-conformal actions, quaternionic discrete series and twistors: SU(2,1) and G_2(2)
• Quantum Entanglement of Baby Universes
• 5D Black Holes and Non-linear Sigma Models
• Lectures on Spectrum Generating Symmetries and U-duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace
• Higher genus Gromov-Witten invariants of the Grassmannian, and the Pfaffian Calabi-Yau threefolds
• Lectures on Mirror Symmetry and Topological String Theory
• Modular and duality properties of surface operators in N=2* gauge theories
• Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges
• Hitchin functionals are related to measures of entanglement
• Jordan C*-Algebras and Supergravity
• Geometric quantization and the metric dependence of the self-dual field theory
• Automorphic Instanton Partition Functions on Calabi-Yau Threefolds
• S-duality and the prepotential in N=2* theories (I): the ADE algebras
• The automorphic NS5-brane
• Extremal Black Holes as Qudits
• Chiral observables and S-duality in N = 2* U(N) gauge theories
• Topological Strings on Elliptic Fibrations
• Non-holomorphic deformations of special geometry and their applications
• Quantization and holomorphic anomaly
• On genus one fibered Calabi-Yau threefolds with 5-sections
• Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes
• Jordan Algebras and Extremal Black Holes
• U-Duality and the Leech Lattice
• The Attractor Conjecture for Calabi-Yau variations of Hodge structures
• Topics in Open Topological Strings
• Geometric Transition as a Change of Polarization