[논문 리뷰] The two-variable elliptic genus in odd dimensions

Yong Wang|arXiv (Cornell University)|2026. 01. 09.

Geometry and complex manifolds인용 수 0

한 줄 요약

두 변수 elliptic genus를 odd-dimensional spin manifolds에 대해 Toeplitz 연산자의 지수 및 holomorphic SL(2,Z)-Jacobi 형태로 정의하고, Gamma 부분군을 사용하여 관련 genus를 구축하며, anomaly cancellation 공식과 divisibility 결과를 도출한다.

ABSTRACT

A kind of two-variable elliptic genus for almost-complex manifolds was introduced by Ping Li and its various properties were established by him. In this paper, we define a two-variable elliptic genus for odd dimensional spin manifolds which is the index for some Toeplitz operator and a holomorphic $SL(2,Z)$-Jacobi form. We also define some two-variable elliptic genera for almost-complex manifolds and odd dimensional spin manifolds which are holomorphic $Γ_0(2)$, $Γ^0(2)$, $Γ_θ$-Jacobi forms. By these Jacobi forms, we can get some $SL(2,{\bf Z})$ and $Γ^0(2)$ modular forms. By these $SL(2,{\bf Z})$ and $Γ^0(2)$ modular forms, we get some interesting anomaly cancellation formulas for almost complex manifolds and odd spin manifolds. As corollaries, we get some divisibility results of the holomorphic Euler characteristic number and the index of Toeplitz operators. In addition, we also define some another two-variable elliptic genera for even (rep. odd ) dimensional manifolds which are meromorphic $Γ_0(2)$, $Γ^0(2)$, $Γ_θ$-Jacobi forms.

연구 동기 및 목표

• Motivate and define a two-variable elliptic genus for odd-dimensional spin manifolds.
• Construct the genus as the index of a Toeplitz operator and as a holomorphic SL(2,Z)-Jacobi form.
• Extend the framework to genera associated with almost-complex and spin manifolds under modular subgroups.
• Derive anomaly cancellation formulas and corollaries on divisibility of holomorphic Euler characteristics and Toeplitz indices.

제안 방법

• Define E(M,W,τ,z) and the Toeplitz-operator index Ell(M,W,g,τ,z).
• Introduce virtual bundles Q(E) and derive ch(Q(E),g^{Q(E)},d,τ) and modular properties.
• Prove Ell(M,W,g,τ,z) is a weak Jacobi form under specified vanishing conditions (c1(W)=p1(M)=p1(W), H^3(M,R)=0, simple connectedness).
• Obtain anomaly cancellation formulas via relations between SL(2,Z) modular forms and Gamma0(2), Gamma^0(2), Gamma_theta Jacobi forms.
• Derive divisibility results for holomorphic Euler characteristics and Toeplitz indices as corollaries.
• Extend the construction to even dimensional cases yielding meromorphic Jacobi forms for Gamma0(2), Gamma^0(2), Gamma_theta.

실험 결과

연구 질문

• RQ1 Can a two-variable elliptic genus be defined for odd-dimensional spin manifolds so that it arises as a Toeplitz operator index and as a holomorphic Jacobi form?
• RQ2 What modular and Jacobi form structures do these odd-dimensional genera possess under SL(2,Z) and its subgroups?
• RQ3 Do anomaly cancellation formulas emerge from the modular properties of the constructed genera?
• RQ4 What divisibility properties of holomorphic Euler characteristics and Toeplitz indices can be deduced from these modular relations?
• RQ5 How can the odd-dimensional construction be related to two-variable elliptic genera for almost-complex and even/odd spin manifolds using Gamma subgroups?

주요 결과

• The two-variable elliptic genus Ell(M,W,g,τ,z) is defined as the index of a Toeplitz operator twisted by E(M,W,τ,z) and Q(E), and is expressible via characteristic forms and theta functions.
• Under c1(W)=0, p1(M)=p1(W), H^3(M,R)=0, and simple connectedness, Ell(M,W,g,τ,z) is a weak Jacobi form of weight (d+1−l) and index l/2.
• The construction yields modular forms for SL(2,Z) and for subgroups Γ0(2), Γ^0(2), Γ_theta, enabling anomaly cancellation formulas for odd spin and almost-complex manifolds.
• Corollaries include divisibility results for the holomorphic Euler characteristic and the index of Toeplitz operators, as well as explicit relations among coefficients a_n(M,W,g,τ) in the q-expansion.
• Theorem 3.7 provides specific divisibility relations: for certain values of d+1−l (e.g., 4, 6, 8, 10) the coefficients a_0^1 and a_1^1 are multiples of 240, 504, 480, 264 respectively, with corresponding structural constraints.