Non-Holomorphic Modular Forms

• Non-holomorphic modular forms are real-analytic functions on the upper half-plane that transform with a specific weight under a modular group, generalizing classical holomorphic forms.
• They include structures like almost holomorphic, harmonic Maass, and polyharmonic forms, each with detailed Fourier expansions and differential properties.
• These forms have significant applications in number theory and mathematical physics, underpinning connections between modular invariance, string theory amplitudes, and genetic periods.

Non-holomorphic modular forms are real-analytic functions on the upper half-plane (or higher-rank analogues) that transform with a specific weight under a modular group, generalizing the classical holomorphic modular forms by relaxing the holomorphy condition. They encapsulate a wide spectrum of analytic, arithmetic, and geometric phenomena not visible to their holomorphic counterparts. Modern developments include harmonic and polyharmonic Maaß forms, equivariant iterated Eisenstein integrals, real-analytic Siegel modular forms, and non-holomorphic completions of mock modular forms and Jacobi forms. Their structure underpins key links between modular forms and periods, mixed motives, mathematical physics, string theory amplitudes, and modular flavor symmetry.

1. Core Definitions and Principal Classes

A non-holomorphic modular form of (complex) weight $k$ for a discrete subgroup $\Gamma\subset SL_2(\mathbb{R})$ (possibly with multiplier system) is a real-analytic function $F:\mathbb{H}\to\mathbb{C}$ that satisfies

$$F(\gamma\tau) = j(\gamma,\tau)^k \nu(\gamma)\, F(\tau)$$

for all $\gamma\in\Gamma$, with $j(\gamma,\tau)=c\tau+d$ and $\nu$ a character or multiplier. The distinction arises from relaxing the holomorphy in $\tau$. Prototypical examples include:

• Almost Holomorphic Modular Forms: Polynomials in $y^{-1}$ with holomorphic coefficients, e.g.,

$$F(\tau,\bar\tau) = \sum_{s=0}^d F_s(\tau)(2iy)^{-s}$$

where $F_s$ are holomorphic, with modular transformations as above. The Eisenstein series $\widehat E_2(\tau)=E_2(\tau) - \frac{3}{\pi y}$ is a classical example (Zemel, 2013).

• Harmonic Maass Forms: Functions annihilated by the (weight $k$) hyperbolic Laplacian,

$\Delta_k F = 0$

allowing moderate (exponential-type) growth at cusps. Their Fourier expansions naturally split into a holomorphic "mock" part and a non-holomorphic part involving incomplete $\Gamma$-functions (Ahlgren et al., 2013, Bringmann, 2017).

• Polyharmonic (Higher Depth) Maaß Forms: Solutions to $(\Delta_k)^r F=0$ for integers $r\geq 1$, with important representations in modular flavor symmetry models (Qu et al., 2024).
• Non-holomorphic Siegel Modular Forms: As found in higher-genus theta series and their completions, which involve explicit non-holomorphic corrections and transform under $Sp_{2n}(\mathbb{Z})$ with prescribed weight (Christensen, 2024, Roehrig, 2021).
• Non-holomorphic Completions of Mock Modular and Jacobi Forms: Completions that restore modularity by attaching controlled non-holomorphic "shadow" terms (Eguchi et al., 2010, Bringmann, 2017, Bringmann et al., 2018).

2. Structural Features and Algebraic Frameworks

Non-holomorphic modular forms exhibit rich algebraic and analytic structures, frequently including:

• Fourier Expansions:

$$F(\tau) = \sum_{n} a^+(n) q^n + \sum_{n} a^-(n) \Gamma(w,2\pi n y) q^{-n}$$

with holomorphic and non-holomorphic (incomplete $\Gamma$-function) parts; "plus" and "minus" parts are explicitly computable in various models (Ahlgren et al., 2013, Bringmann, 2017).

• Differential and Laplace Equations:

Non-holomorphic modular forms are frequently characterized by (generalized) Laplacian conditions—homogeneous ($\Delta_k F=0$ for harmonic) or inhomogeneous (as in the algebra of equivariant Eisenstein integrals) (Brown, 2017).

• Algebraic Closure and Filtrations:

The algebras of non-holomorphic modular forms are bigraded by weight $(r,s)$ with ring structures paralleling the holomorphic case. Filtrations exist by "length" (number of iterated integrals), motivic weight (for period structure), and M-filtration inherited from Lie-theoretic constructions (Brown, 2017, Brown, 2017, Dorigoni et al., 2024).

• Single-valued Periods and Zeta-values:

Fourier or Laurent coefficients often lie in the $\mathbb{Q}$-algebra generated by single-valued multiple zeta values, connecting modular forms to mixed Tate motives and periods (Brown, 2017, Dorigoni et al., 2024).

3. Model Constructions and Prototypical Examples

Selected constructions illustrate the diversity of non-holomorphic modular forms:

• Weak Harmonic Maass Forms of Weight $5/2$ on $SL_2(\mathbb{Z})$: Ahlgren–Andersen construct canonical bases so that the non-holomorphic part of the first basis element encodes partition numbers $p(n)$ (Ahlgren et al., 2013). The structure is governed by Maass–Poincaré series, $s$-derivatives thereof, explicit Kloosterman–Bessel sums for Fourier coefficients, and differential $\xi$ operators relating weights $5/2$ and $-1/2$.
• Equivariant Iterated Eisenstein Integrals: Brown's real-analytic modular forms via iterated integrals of holomorphic Eisenstein series, corrected by explicit "single-valued" twists to enforce modular equivariance. The resulting algebra, $\mathcal{M\!I}^E$, is closed under bigraded ring operations and encompasses modular graph functions relevant for string perturbation theory. Fourier expansions involve $q, \bar{q}, \log|q|$ with coefficients in single-valued multiple zeta values. Differential and Laplace actions are explicitly computable (Brown, 2017, Dorigoni et al., 2024, Brown, 2017).
• Non-holomorphic Siegel Modular Forms and Geometric Theta Lifts: Genus-$2$ Siegel modular forms built as completions of linking-number generating series for geodesics in arithmetic hyperbolic $3$-folds. The non-holomorphic correction is constructed via Kudla–Millson theta-integrals over boundary components, restoring modular transformation properties and delivering polynomial bounds for linking numbers (Christensen, 2024).
• Non-holomorphic Jacobi Forms and Taylor Coefficients: Taylor expansion coefficients of non-holomorphic (e.g., completed) Jacobi forms exhibit modular or quasimodular transformation properties upon suitable completion (by $v$ or $E_2$-dependent terms), relevant to combinatorics (e.g., rank moments) and enumerative geometry (Joyce invariants) (Bringmann, 2017).
• Non-holomorphic Completions in Conformal Field Theory: In string theory, partition functions on non-compact backgrounds (e.g., $SL(2,\mathbb{R})/U(1)$) manifest essential non-holomorphic dependence: modular completion of discrete characters using $\bar{\tau}$-dependent shadows is necessary for full modular invariance (Eguchi et al., 2010).

4. Connections to Periods, Motives, and Arithmetic

Non-holomorphic modular forms serve as a bridge between modular/automorphic theory and deep structures in arithmetic geometry:

• Single-valued Periods: The algebraic structure of many non-holomorphic modular forms is controlled by single-valued multiple zeta values, which are periods of mixed Tate motives (Brown, 2017, Dorigoni et al., 2024, Brown, 2017).
• Real-analytic Eisenstein Series and Iterated Integrals: These functions encode periods, (quasi-)periods, and L-values for cusp forms and modular graphs. In vector-valued frames, the coefficients are motivic and are interpretable in terms of the de Rham and Betti realizations of elliptic motives (Brown, 2017, Brown, 2017).
• Non-holomorphic Drinfeld Modular Forms: In positive characteristic, nearly-holomorphic analogues admit an algebraic theory paralleling that of classical modular forms, connecting special values at CM points with periods of CM Drinfeld modules and transcendence results (Gezmiş et al., 3 Mar 2025).

5. Representation Theory, Differential Operators, and Modular Symmetries

• Vector-valued and Symmetric Power Representations: Comprehensive structure theorems show that spaces of (almost-)holomorphic, quasimodular, and vector-valued non-holomorphic modular forms are unified via symmetric-power representations (the "Shimura isomorphism") (Zemel, 2013).
• Differential Lifts and Maaß Operators: Operators such as the Maass lowering $\xi_k$ and raising $D$ map between non-holomorphic modular forms of complementary weights, closely relating holomorphic, harmonic, and polyharmonic forms. In modular flavor symmetry, such operators are instrumental in constructing Yukawa structures from polyharmonic Maaß forms transforming in finite modular group multiplets (Qu et al., 2024).
• Decomposition into Multiplets: For level-$N$ modular forms, spaces of (poly)harmonic Maaß forms decompose into irreducible multiplets under the finite modular group $\Gamma_N'$ or $\Gamma_N$, with explicit classification available for low levels and weights (Qu et al., 2024).

6. Applications in Mathematical Physics, String Theory, and Number Theory

Non-holomorphic modular forms are essential in diverse applications:

• String Amplitudes and Modular Graph Functions: The structure of modular graph functions arising in the low-energy genus-one expansion of closed string amplitudes is captured precisely by the algebra of non-holomorphic modular forms generated by equivariant Eisenstein integrals and zeta-generators (Dorigoni et al., 2024, Brown, 2017, Brown, 2017).
• Modular Completions in Quantum Field Theory: The analytic properties of partition functions and indices in various 2d conformal field theories (notably non-compact targets) fundamentally rely on non-holomorphic modular completions to achieve correct modular transformation behavior and account for holomorphic anomaly (Eguchi et al., 2010).
• Algebraic and Diophantine Applications: Via the theory of regularized inner products, non-holomorphic modular forms provide exact formulae (e.g., for partition functions) and encode deeper relations between modular forms and arithmetic, including transcendence and algebraicity of special values at CM points (Ahlgren et al., 2013, Gezmiş et al., 3 Mar 2025).

7. Outlook and Structural Analogies

• The schematic parallelism between genus-zero single-valued multiple polylogarithms and genus-one single-valued iterated Eisenstein integrals, especially the role of zeta-generators and Hopf algebra structures, provides a template for further development across motivic, arithmetic, and string-theoretic contexts (Dorigoni et al., 2024).
• Open directions include the classification of period structures arising in higher-depth modular completions, the explicit implementation of these forms in configuration-space integrals in string theory, and systematically extending these frameworks to real and $p$-adic (e.g., Drinfeld) modular settings (Brown, 2017, Gezmiş et al., 3 Mar 2025).

A plausible implication is that non-holomorphic modular forms, with their intrinsic period data and motivic structures, provide the universal analytic and arithmetic language for understanding modular phenomena in mathematics and physics beyond the reach of holomorphic theory alone.