False-Mock Modular Conjecture

• False-Mock Modular Conjecture is a framework examining how holomorphic series acquire modularity through nonholomorphic completions, distinguishing mock and false theta behaviors.
• It employs arithmetic methods, such as class-number identities and Rankin–Cohen brackets, to demonstrate cancellation of nonholomorphic terms leading to genuine modular forms.
• The approach extends to higher-depth false modular forms and interprets physical phenomena from SCFT and Vafa–Witten theory to reconcile nonholomorphy with modular invariance.

The expression “False-Mock Modular Conjecture” is not a standard theorem name in the literature. The closest established usage is an umbrella for phenomena in which a holomorphic series is not modular by itself, acquires modular meaning only after a nonholomorphic completion, and may be related either to false theta boundary-value behavior or to mock modular and mixed mock modular forms. In strict terminology, the cited papers distinguish sharply between mock modular forms, mixed mock modular forms, and false modular forms; several of them explicitly note that results that may look informally “false-mock” are, in fact, mock-modular completion or cancellation phenomena rather than false theta results (Mertens, 2013, Bringmann et al., 2021).

1. Terminological scope and precise mathematical meaning

In the modern sense used across the cited papers, a mock modular form is the holomorphic part $f^+$ of a harmonic weak Maaß form $f=f^++f^-$. The harmonic weak Maaß form satisfies modular transformation of weight $k$, the Laplace equation $\Delta_k f=0$, and at most exponential growth at the cusps. A completion is the addition of a nonholomorphic term to a holomorphic but non-modular function so that the sum transforms modularly. This is the basic language used for class number series, Appell–Lerch sums, Ramanujan-type mock theta functions, and physically normalized partition functions (Mertens, 2013).

By contrast, the higher-rank false-theta literature studies holomorphic sums with sign insertions such as

$$F_{Q,\mu,\ell,M}(\tau) := \sum_{n\in A+\mu+\frac{\ell}{2}} \operatorname{sgn}(B(M,n))\, e^{\pi i B(n,\ell)} q^{Q(n)},$$

together with two-variable completions $\widehat f(\tau,w)$ that transform bimodularly and whose $w$-derivative has a recursive depth-lowering form. In that framework, false modular forms appear as boundary values of modularly transforming completions rather than as holomorphic parts of one-variable harmonic Maaß forms (Bringmann et al., 2021).

A plausible interpretation is therefore that “False-Mock Modular Conjecture” denotes not a single formal statement but a family of questions about how these two completion paradigms interact: one-variable mock modular completion on the one hand, and boundary-value false modular completion on the other.

2. Class number identities and mock-modular cancellation

The most concrete arithmetic prototype is Mertens’s proof of Cohen’s conjecture on Hurwitz class numbers. The relevant generating series

$H(\tau)=\sum_{n=0}^{\infty} H(n)q^n$

is not modular, but Zagier’s completion

$G(\tau)=H(\tau)+R(\tau)$

transforms like a weight $3/2$ modular form on $f=f^++f^-$0. Cohen’s conjecture concerns the formal series $f=f^++f^-$1 and asserts that the coefficient of $f=f^++f^-$2 is a holomorphic modular form of weight $f=f^++f^-$3 on $f=f^++f^-$4; Mertens proves this and shows that for $f=f^++f^-$5 the coefficient is a cusp form. The proof rewrites the coefficients in terms of Rankin–Cohen brackets involving $f=f^++f^-$6, theta functions, and derivatives of Appell–Lerch sums, then completes both sides and proves that the nonholomorphic pieces cancel exactly. The paper explicitly describes the summands as something that “should be called quasi mixed mock modular forms,” not false theta functions (Mertens, 2013).

Bringmann and Kane give a related class-number construction for restricted sums

$f=f^++f^-$7

showing that their generating functions are coefficients of mixed mock modular forms. They construct explicit nonholomorphic completions on both the class-number side and the divisor-sum/Appell–Lerch side, prove cancellation of the nonholomorphic terms, and conclude that the difference is a holomorphic modular form of weight $f=f^++f^-$8. Their main theorem is another instance in which an apparently nonmodular holomorphic expression is embedded into a completed mock-modular framework and then shown to become genuinely modular after subtraction of the correct correction term (Bringmann et al., 2013).

These class-number results are often the closest arithmetic models for a “false-mock” intuition: the original series are holomorphic and nonmodular, the obstruction is controlled by explicit completion terms, and modularity emerges only after cancellation.

3. Higher-depth false modular forms and rank-two false theta functions

A more literal false-theta framework is developed in the theory of higher depth false modular forms. In that setting, a false modular form of weight $f=f^++f^-$9 and depth $k$0 is a holomorphic function admitting a completion $k$1 such that $k$2 is holomorphic away from $k$3, $k$4 transforms bimodularly, and

$k$5

with $k$6 of lower false depth and $k$7 modular. The central completion is built from generalized error functions, and the boundary limit $k$8 recovers the false theta function up to lower-rank correction terms. The paper stresses that this is a framework parallel to higher depth mock modular forms, not a proof of a direct false-to-mock equivalence (Bringmann et al., 2021).

The first systematic rank-two case is developed through iterated holomorphic Eichler-type integrals. Under general positivity conditions on a binary quadratic form, a rank-two false theta function with two sign factors is represented as a double Eichler-type integral plus a theta correction term. The paper applies this to generic parafermion characters of type $k$9 and $\Delta_k f=0$0, to Schur indices of superconformal theories, and to $\Delta_k f=0$1-invariants for certain plumbing $\Delta_k f=0$2-graphs. It also constructs explicit two-variable completions $\Delta_k f=0$3 and $\Delta_k f=0$4 with modular transformation laws under simultaneous action on $\Delta_k f=0$5, and emphasizes that their $\Delta_k f=0$6-derivatives lead to lower-depth completions. This is direct evidence for a depth-two false-modular analogue of the mock-modular completion story (Bringmann et al., 2021).

The crucial distinction is terminological and structural. In the false-theta setting, the completed object is naturally two-variable and the original false theta function is a boundary value. In the mock-modular setting, the completed object is typically one-variable and real-analytic, and the original function is the holomorphic part.

4. Admissible characters, string functions, and the false/mock dichotomy

The strongest bridge between false and mock phenomena in the cited material comes from admissible representations of $\Delta_k f=0$7. For positive admissible fractional levels, Borozenets and Mortenson develop a quasi-periodic theory of string functions and a Zagier–Zwegers polar-finite decomposition for admissible characters. The characters are vector-valued Jacobi forms; the finite part is a finite theta combination whose coefficients are string functions, and the paper states that these are mock modular coefficients of the finite part. At the same time, the paper recalls from the negative-level prequel a direct false theta function expansion for negative admissible-level string functions, and explicitly notes the similarity between the two formulas. It therefore supplies unusually concrete evidence that admissible $\Delta_k f=0$8 characters furnish a common Jacobi/Appell source from which mock and false theta behavior emerge in different regimes (Borozenets et al., 5 Feb 2025).

This pattern is sharpened by the $\Delta_k f=0$9-level results. The later paper proves two new families of exact identities for $$F_{Q,\mu,\ell,M}(\tau) := \sum_{n\in A+\mu+\frac{\ell}{2}} \operatorname{sgn}(B(M,n))\, e^{\pi i B(n,\ell)} q^{Q(n)},$$0 string functions at level $$F_{Q,\mu,\ell,M}(\tau) := \sum_{n\in A+\mu+\frac{\ell}{2}} \operatorname{sgn}(B(M,n))\, e^{\pi i B(n,\ell)} q^{Q(n)},$$1. Each family is expressed in terms of all four tenth-order mock theta functions together with a simple quotient of theta functions. The proof uses the polar-finite decomposition of admissible characters, specializations that annihilate one Appell contribution, and Appell-to-mock-theta identities. The paper explicitly situates itself in a program where positive fractional-level string functions give mock-theta-conjecture-like identities, whereas negative fractional-level string functions give mixed false theta expressions (Konenkov et al., 2 Jun 2025).

A plausible implication is that the admissible-character setting provides one of the clearest available models for a “false-mock modular” landscape: the underlying Jacobi object is common, while the extracted holomorphic series are mock-like or false-like depending on the level regime and the decomposition used.

5. Physical mechanisms: noncompactness, duality, and holomorphic anomaly

In four-dimensional Vafa–Witten theory on $$F_{Q,\mu,\ell,M}(\tau) := \sum_{n\in A+\mu+\frac{\ell}{2}} \operatorname{sgn}(B(M,n))\, e^{\pi i B(n,\ell)} q^{Q(n)},$$2, the physically normalized partition function is modular only after adding nonholomorphic terms. For gauge group $$F_{Q,\mu,\ell,M}(\tau) := \sum_{n\in A+\mu+\frac{\ell}{2}} \operatorname{sgn}(B(M,n))\, e^{\pi i B(n,\ell)} q^{Q(n)},$$3, the holomorphic instanton series built from Hurwitz class numbers is not modular but mock modular, and the completed functions $$F_{Q,\mu,\ell,M}(\tau) := \sum_{n\in A+\mu+\frac{\ell}{2}} \operatorname{sgn}(B(M,n))\, e^{\pi i B(n,\ell)} q^{Q(n)},$$4 become vector-valued modular forms of weight $$F_{Q,\mu,\ell,M}(\tau) := \sum_{n\in A+\mu+\frac{\ell}{2}} \operatorname{sgn}(B(M,n))\, e^{\pi i B(n,\ell)} q^{Q(n)},$$5. The paper derives a holomorphic anomaly equation whose source is a surface term at infinity on the Coulomb branch, interprets the anti-holomorphic theta factor as coming from abelian anti-instantons, and argues that mock modularity is generic when the relevant field space is noncompact (Dabholkar et al., 2020).

A parallel explanation is given in the review of non-compact $$F_{Q,\mu,\ell,M}(\tau) := \sum_{n\in A+\mu+\frac{\ell}{2}} \operatorname{sgn}(B(M,n))\, e^{\pi i B(n,\ell)} q^{Q(n)},$$6 SCFT. In the $$F_{Q,\mu,\ell,M}(\tau) := \sum_{n\in A+\mu+\frac{\ell}{2}} \operatorname{sgn}(B(M,n))\, e^{\pi i B(n,\ell)} q^{Q(n)},$$7 supercoset, the Hamiltonian or character viewpoint gives holomorphic expressions with anomalous modular behavior, while the path-integral viewpoint gives modular covariance at the price of non-holomorphicity. The basic Appell–Lerch sum

$$F_{Q,\mu,\ell,M}(\tau) := \sum_{n\in A+\mu+\frac{\ell}{2}} \operatorname{sgn}(B(M,n))\, e^{\pi i B(n,\ell)} q^{Q(n)},$$8

is completed to

$$F_{Q,\mu,\ell,M}(\tau) := \sum_{n\in A+\mu+\frac{\ell}{2}} \operatorname{sgn}(B(M,n))\, e^{\pi i B(n,\ell)} q^{Q(n)},$$9

and the elliptic genus is expressed directly in terms of such completed Appell–Lerch sums and completed discrete characters. The review identifies the physical origin of mock modularity with the mixing of discrete and continuous spectra and with IR effects produced by non-compact geometry (Sugawara, 2020).

More recent enumerative-geometric work continues this mock-modular pattern rather than a false-theta one. For charge $\widehat f(\tau,w)$0, compact Calabi–Yau D4-D2-D0 generating series are described as vector-valued mock modular forms with a specific shadow determined by lower-charge data (Alexandrov et al., 2023). Likewise, for the rational elliptic surface mirror to $\widehat f(\tau,w)$1, generating series of logarithmic Gromov–Witten invariants are conjectured, and for $\widehat f(\tau,w)$2 proved, to be vector-valued mock modular forms of weight $\widehat f(\tau,w)$3 after identification with Vafa–Witten generating series of $\widehat f(\tau,w)$4 (Argüz, 8 Feb 2026).

These physical and enumerative settings are important because they isolate a robust mechanism: duality or modular invariance predicts modularity, noncompactness or reducibility obstructs holomorphy, and completion restores the required transformation law.

6. Proof paradigms, model examples, and conceptual status

A standard proof strategy on the mock side is to complete each holomorphic expression, package the completed objects into a vector-valued modular form, and then force the desired identity by a vanishing theorem or by finite-dimensionality. Andersen’s unified proof of Ramanujan’s fifth-order mock theta conjectures is the clearest example: both sides are assembled into nonholomorphic six-component vector-valued modular forms transforming under a Weil representation, their nonholomorphic parts agree, and the difference lies in a zero-dimensional space identified with $\widehat f(\tau,w)$5 (Andersen, 2016).

For the classical tenth-order mock theta functions, Moore computes explicit completions and shadows by rewriting Choi’s Hecke-type identities as indefinite theta sums of signature $\widehat f(\tau,w)$6, applying Zwegers’ error-function smoothing, and obtaining vector-valued weight $\widehat f(\tau,w)$7 mock modular forms with explicit $\widehat f(\tau,w)$8- and $\widehat f(\tau,w)$9-transformations. The paper is especially relevant to false/mock comparisons because it makes fully explicit the passage from a sign-restricted indefinite theta sum to a completed mock modular object with unary-theta shadow (Moore, 2012).

The mock side also includes higher-depth phenomena beyond ordinary harmonic Maaß forms. The completion of the overpartition series $w$0 is not itself a harmonic Maass form; it is the derivative of a linear combination of products of harmonic Maass forms and theta functions, and its Maass-lowering image lies in a mixed space. The paper explicitly calls this a higher depth mock modular form. This is important because it shows that even within the mock framework, the correct completion may already be more elaborate than the basic one-shadow one-completion pattern (Bringmann et al., 2016).

The overall conceptual status is therefore sharply defined. The cited literature does not furnish a single theorem called the “False-Mock Modular Conjecture.” It does, however, exhibit three stable themes. First, many arithmetic and physical series are mock modular or mixed mock modular rather than false theta in the strict sense. Second, higher-rank false theta functions admit modularly transforming two-variable completions and depth-recursive structures parallel to higher-depth mock modular forms. Third, several representation-theoretic settings, especially admissible string functions, display both mock and false expansions in adjacent regimes. This suggests that “False-Mock Modular Conjecture” is best understood as a label for a developing completion theory in which completion, cancellation, boundary limits, and depth recursion link false theta and mock modular behavior without collapsing the distinction between them.