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Topological Elliptic Genera I -- The mathematical foundation

Ying-Hsuan Lin, Mayuko Yamashita

TL;DR

This work develops Topological Elliptic Genera, a framework of genuinely equivariant TMF refinements that lift classical elliptic genera to spectral invariants twisted by $G$-representations. Central to the approach is the construction of $U(1)$-, $Sp(1)$-, and $O(n)$-topological elliptic genera, organized into a coherent trio that interrelates via external and internal structures and the level-rank dualities in TMF. The paper provides foundational definitions, a detailed character formula for TMF-valued genera, and powerful applications, notably new divisibility constraints for Euler numbers of tangential $SU(k)$ and $Sp(k)$-manifolds, along with torsion-detection phenomena in $MSp$. A key bridge to physics is established through level-rank duality isomorphisms in TMF, aligning topological refinements with known dualities in conformal field theory. The results open a path to further exploration of these genera and their physical interpretations in Part II and subsequent installments.

Abstract

We construct {\it Topological Elliptic Genera}, homotopy-theoretic refinements of the elliptic genera for $SU$-manifolds and variants including the Witten-Landweber-Ochanine genus. The codomains are genuinely $G$-equivariant Topological Modular Forms developed by Gepner-Meier, twisted by $G$-representations. As the first installment of a series of articles on Topological Elliptic Genera, this issue lays the mathematical foundation and discusses immediate applications. Most notably, we deduce an interesting divisibility result for the Euler numbers of $Sp$-manifolds.

Topological Elliptic Genera I -- The mathematical foundation

TL;DR

This work develops Topological Elliptic Genera, a framework of genuinely equivariant TMF refinements that lift classical elliptic genera to spectral invariants twisted by -representations. Central to the approach is the construction of -, -, and -topological elliptic genera, organized into a coherent trio that interrelates via external and internal structures and the level-rank dualities in TMF. The paper provides foundational definitions, a detailed character formula for TMF-valued genera, and powerful applications, notably new divisibility constraints for Euler numbers of tangential and -manifolds, along with torsion-detection phenomena in . A key bridge to physics is established through level-rank duality isomorphisms in TMF, aligning topological refinements with known dualities in conformal field theory. The results open a path to further exploration of these genera and their physical interpretations in Part II and subsequent installments.

Abstract

We construct {\it Topological Elliptic Genera}, homotopy-theoretic refinements of the elliptic genera for -manifolds and variants including the Witten-Landweber-Ochanine genus. The codomains are genuinely -equivariant Topological Modular Forms developed by Gepner-Meier, twisted by -representations. As the first installment of a series of articles on Topological Elliptic Genera, this issue lays the mathematical foundation and discusses immediate applications. Most notably, we deduce an interesting divisibility result for the Euler numbers of -manifolds.

Paper Structure

This paper contains 53 sections, 55 theorems, 372 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations and conventions
  3. Preliminaries
  4. Generalities on genuinely equivariant spectra
  5. Equivariant $\mathrm{TMF}$ and their twists
  6. The construction of Gepner-Meier GepnerMeier
  7. Specialization to elliptic curves over $\mathbb{C}$
  8. Genuinely equivariant refinement of the sigma orientation
  9. A remark on $\mathbf{R}\mathrm{O}(G)$-grading versus $G$-equivariant twists
  10. On tangential and normal Thom spectra
  11. The definitions of topological elliptic genera
  12. The $U(1)$-topological elliptic genus $\mathrm{Jac}_{U(1)_k} \colon MTSU(k) \to \mathrm{TJF}_k$
  13. The general construction
  14. Functoriality
  15. Examples: The trio of $U$-$Sp$ and $O$-topological elliptic genera
  16. ...and 38 more sections

Key Result

Theorem 1.17

For any closed strictSee Definition def_strict_str. tangential $Sp(k)$-manifold $M_{4k}$ of real dimension $4k$, its Euler number satisfies

Figures (2)

  • Figure 1: The cell diagram of $\mathrm{TJF}_k$
  • Figure 2: The cell diagram of $\mathrm{TEJF}_{2k}$

Theorems & Definitions (163)

  • Theorem 1.17: Theorem \ref{['thm_divisibility_constraints_concrete']} (1)
  • Remark 2.10
  • Remark 2.23
  • Example 2.25: $G=U(1)$: Topological Jacobi Forms
  • Example 2.27: $G=Sp(1)$: Topological Even Jacobi Forms
  • Definition 2.45: Looijenga's line bundle $\mathcal{A}(\xi)$ GukovKrushkalMekerPei
  • Definition 2.51: multi-variable Jacobi Forms and $G$-equivariant Modular Forms
  • Lemma 2.55: ando2010circleequivariant and GukovKrushkalMekerPei
  • Remark 2.59: The relation between Euler class $\chi(V) \in \mathrm{TMF}[V]^G$ and $\Phi_V$
  • Example 2.62: $G=U(n)$
  • ...and 153 more