[Paper Review] Integrality of Framing and Geometric Origin of 2-functions

Schwarz, Albert, Vologodsky, Vadim+1 more|arXiv (Cornell University)|Feb 23, 2017

Black Holes and Theoretical Physics19 references3 citations

TL;DR

This paper establishes the integrality of framing operations for 2-functions with algebraic coefficients and identifies their geometric origin in the q-expansion of truncated normal functions on degenerating families of Calabi-Yau threefolds. Using p-adic Hodge theory and Frobenius structures, it proves that these functions are invariant under framing and arise naturally from algebraic cycles, extending the integrality of Gromov-Witten invariants to number fields.

ABSTRACT

We say that a formal power series $\sum a_n z^n$ with rational coefficients is a 2-function if the numerator of the fraction $a_{n/p}-p^2 a_n$ is divisible by $p^2$ for every prime number $p$. One can prove that 2-functions with rational coefficients appear as building block of BPS generating functions in topological string theory. Using the Frobenius map we define 2-functions with coefficients in algebraic number fields. We establish two results pertaining to these functions. First, we show that the class of 2-functions is closed under the so-called framing operation (related to compositional inverse of power series). Second, we show that 2-functions arise naturally in geometry as $q$-expansion of the truncated normal function associated with an algebraic cycle extending a degenerating family of Calabi-Yau 3-folds.

Motivation & Objective

• To extend the notion of 2-functions to coefficients in algebraic number fields using the Frobenius map.
• To prove that the framing operation—related to compositional inverse of power series—preserves integrality in this generalized setting.
• To establish a geometric realization of such 2-functions as q-expansions of truncated normal functions associated with algebraic cycles on degenerating Calabi-Yau threefolds.
• To demonstrate that the integrality of invariants in Gromov-Witten theory persists in the context of algebraic number fields via p-adic methods.
• To clarify the categorical and arithmetic meaning of irrational invariants in mirror symmetry by linking them to cohomological invariants with algebraic coefficients.

Proposed method

• Define 2-functions with coefficients in algebraic number fields using the Frobenius endomorphism on p-adic cohomology.
• Use the p-adic B-model and logarithmic de Rham cohomology over Q to analyze period integrals and their monodromy.
• Construct a Frobenius-equivariant connection matrix Φp and derive compatibility conditions with the cup product pairing via I = (0 0 0 1; 0 0 1 0; 0 -1 0 0; -1 0 0 0).
• Apply Fontaine-Lafaille theory to relate extension classes in the limiting mixed Hodge structure to p-adic invariants.
• Derive differential equations for the connection matrix and use them to prove integrality of the infinitesimal invariant via δ²n₀ = Frobₚ(D) − D.
• Verify integrality of the mirror map, Griffiths-Yukawa coupling, and cohomology basis through p-adic valuation arguments and Frobenius compatibility.

Experimental results

Research questions

• RQ1 Can the concept of 2-functions be generalized to include coefficients in algebraic number fields while preserving integrality under framing?
• RQ2 How does the framing operation—defined as a compositional inverse in the context of power series—act on 2-functions with algebraic coefficients?
• RQ3 What is the geometric origin of 2-functions with algebraic coefficients in the context of Calabi-Yau threefolds and their degenerations?
• RQ4 Can the integrality of Gromov-Witten invariants be extended beyond rational coefficients to algebraic number fields using p-adic Hodge theory?
• RQ5 What is the role of the truncated normal function and its q-expansion in realizing 2-functions as geometric invariants?

Key findings

• The class of 2-functions with algebraic coefficients is closed under the framing operation, as shown in Theorem 8.
• The 2-function property is preserved under the compositional inverse operation, which corresponds to framing in the context of generating functions.
• 2-functions arise geometrically as the q-expansion of the truncated normal function associated with an algebraic cycle extending a degenerating family of Calabi-Yau threefolds, as established in Theorem 22.
• The integrality of the infinitesimal invariant is proven via the identity Frobₚ(D) − D = −δ²n₀, where δ is the logarithmic derivative and n₀ is a p-adic analytic function.
• The cohomology basis, mirror map, and Griffiths-Yukawa coupling are all shown to be integral in the p-adic sense, with denominators not divisible by p.
• The compatibility of the Frobenius structure with the cup product pairing leads to the conclusion that the diagonal entries of the connection matrix satisfy δmₛₛ = 0 and m₀₀ = m₁₁ = m₂₂ = m₃₃ = 1, with m₁₀ = 0 and m₃₂ = −m₁₀ = 0.