The non-compact elliptic genus: mock or modular

Abstract

We analyze various perspectives on the elliptic genus of non-compact supersymmetric coset conformal field theories with central charge larger than three. We calculate the holomorphic part of the elliptic genus via a free field description of the model, and show that it agrees with algebraic expectations. The holomorphic part of the elliptic genus is directly related to an Appell-Lerch sum and behaves anomalously under modular transformation properties. We analyze the origin of the anomaly by calculating the elliptic genus through a path integral in a coset conformal field theory. The path integral codes both the holomorphic part of the elliptic genus, and a non-holomorphic remainder that finds its origin in the continuous spectrum of the non-compact model. The remainder term can be shown to agree with a function that mathematicians introduced to parameterize the difference between mock theta functions and Jacobi forms. The holomorphic part of the elliptic genus thus has a path integral completion which renders it non-holomorphic and modular.

13 pages

References in corpus (2)

• Mock Theta Functions
• N=2 Superconformal Algebra and the Entropy of Calabi-Yau Manifolds

Cited by in corpus (70)

• Quantum Black Holes, Wall Crossing, and Mock Modular Forms
• 2d Index and Surface operators
• The Omega deformed B-model for rigid N=2 theories
• 3d Modularity
• A Twisted Non-compact Elliptic Genus
• Resurgence in complex Chern-Simons theory
• D3-instantons, Mock Theta Series and Twistors
• Non-holomorphic Modular Forms and SL(2,R)/U(1) Superconformal Field Theory
• String theory at the tip of the cigar
• Multiple D3-instantons and mock modular forms I
• The toroidal block and the genus expansion
• Elliptic genera of ALE and ALF manifolds from gauged linear sigma models
• M String, Monopole String and Modular Forms
• A holomorphic anomaly in the elliptic genus
• From 3d duality to 2d duality
• Localization and real Jacobi forms
• S-duality and refined BPS indices
• Lessons on Black Holes from the Elliptic Genus
• A 2d (0,2) appetizer
• BPS invariants of N=4 gauge theory on a surface
• Vafa-Witten theory and iterated integrals of modular forms
• Single-centered black hole microstate degeneracies from instantons in supergravity
• Moonshine in Fivebrane Spacetimes
• Mirror Symmetry and Bosonization in 2d and 3d
• Wall-crossing made smooth
• Quantum Dynamics of Supergravity on R^3 x S^1
• Duality and Mock Modularity
• Black holes and higher depth mock modular forms
• dS${}_2$ Supergravity
• Elliptic genera and real Jacobi forms
• A Puncture in the Euclidean Black Hole
• Quantum hypermultiplet moduli spaces in N=2 string vacua: a review
• A 5d/2d/4d correspondence
• Elliptic Genera of Non-compact Gepner Models and Mirror Symmetry
• Comments on Non-holomorphic Modular Forms and Non-compact Superconformal Field Theories
• K3 String Theory, Lattices and Moonshine
• The Black Hole Interior and a Curious Sum Rule
• APS $\eta$-invariant, path integrals, and mock modularity
• Superstrings in Thermal Anti-de Sitter Space
• Gauge Threshold Corrections for N = 2 Heterotic Local Models with Flux, and Mock Modular Forms
• Emergent Strings in Type IIB Calabi--Yau Compactifications
• Higher Poles and Crossing Phenomena from Twisted Genera
• String Theory in Polar Coordinates and the Vanishing of the One-Loop Rindler Entropy
• Umbral Moonshine and K3 Surfaces
• Counting Strings, Wound and Bound
• ADE Double Scaled Little String Theories, Mock Modular Forms and Umbral Moonshine
• Pure and Twisted Holography
• An Elliptic Triptych
• Stringy Information and Black Holes
• Compact Formulas for the Completed Mock Modular Forms
• Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes
• Three-Charge Black Holes and Quarter BPS States in Little String Theory
• Squashed toric sigma models and mock modular forms
• Donaldson-Witten theory, surface operators and mock modular forms
• Mock modularity from black hole scattering states
• Squashed Toric Manifolds and Higher Depth Mock Modular Forms
• On Mathieu moonshine and Gromov-Witten invariants
• Twisted Massive Non-Compact Models
• Three Avatars of Mock Modularity
• Topological correlators of $SU(2)$, $\mathcal{N}=2^{\ast }$ SYM on four-manifolds
• Mock modularity at work, or black holes in a forest
• `Analytic Continuation' of N=2 Minimal Model
• Duality in N=4 Liouville Theory and Moonshine Phenomena
• Umbral Moonshine and String Duality
• Modular Transformations of Ramanujan's Tenth Order Mock Theta Functions
• Quantum Black Hole Entropy and Localization in Supergravity
• Non-Compact SCFT and Mock Modular Forms
• TASI Lectures on Moonshine
• Infinite Type D-modules and Higher Depth Mock Modular Forms I
• Topological Mathieu Moonshine