False Theta Function Duality

• False theta function duality is a framework defining theta-like q-series modified by sign factors or truncations that break classical modularity.
• The duality includes mechanisms such as modular completions and inversion duality, linking partial theta functions with Appell–Lerch sums via formal q→q⁻¹ transformations.
• This structural principle underpins applications in quantum modular forms, 3-manifold invariants, combinatorial identities, and representation theory.

False theta function duality refers to a family of structural correspondences attached to theta-like $q$-series whose sign insertions, half-lattice truncations, or quadrant restrictions destroy the exact modular self-duality of ordinary theta functions. In the recent literature, the expression is used in several related senses rather than as the name of a single canonical theorem: modular completion can restore a Fourier self-duality-like transformation law; the formal inversion $q\mapsto q^{-1}$ relates partial theta identities to Appell–Lerch sums; Hecke-type double sums exhibit opposite behavior in the false-theta and Appell–Lerch regimes according to the sign of the relevant quadratic-form discriminant; and radial limits of false theta series encode quantum modular forms and $3$-manifold invariants (Bringmann et al., 2019, Mortenson, 2012, Mortenson, 2022, Murakami, 2022).

1. Basic objects and the main senses of duality

A false theta function is theta-like but non-modular because the summand is altered by a sign factor or by a one-sided restriction. A representative example is

$$\psi(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} \left(n+\tfrac12\right)(-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}, \qquad q=e^{2\pi i\tau},\ \zeta=e^{2\pi i z},$$

to be compared with the ordinary Jacobi theta function

$$\vartheta(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} (-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}.$$

The extra factor $\left(n+\tfrac12\right)$, or more generally a sign insertion, destroys the Fourier self-duality that underlies ordinary modularity (Bringmann et al., 2019).

The literature also distinguishes false theta functions from partial theta functions. A partial theta function is “half of a theta series,” typically unilateral, whereas a false theta function is a bilateral theta-type series with the “wrong signs.” One recurring observation is that a false theta function can sometimes be expressed as a sum of two partial theta functions, so the two notions are closely related but not identical (Mortenson, 2012).

Sense of duality	Mechanism	Representative source
Fourier/modular completion	Error-function smoothing with auxiliary $w$ restores Jacobi-type transformation	(Bringmann et al., 2019)
Partial theta/Appell–Lerch	Formal $q\mapsto q^{-1}$ and bilateral extension	(Mortenson, 2012)
Discriminant-sign duality	$b-ac<0$ gives theta $\times$ false theta; $q\mapsto q^{-1}$0 gives Appell–Lerch $q\mapsto q^{-1}$1 theta	(Mortenson, 2022)
Radial-limit duality	False theta $q\mapsto q^{-1}$2-series recover WRT invariants at roots of unity	(Murakami, 2022)
Higher-rank/depth duality	Rank-two sign sums become iterated holomorphic Eichler-type integrals	(Bringmann et al., 2021)

These meanings are compatible rather than interchangeable. Some papers explicitly present a duality principle, while others speak of “duality-like” behavior, “mirror” identities, or a structural contrast between false-theta and mock/Appell–Lerch regimes (Mortenson, 2012, Bringmann et al., 2021).

2. Modular completion and restored Fourier self-duality

A central modern formulation of false theta duality is the modular-completion framework. In the positive-definite lattice setting, with bilinear form $q\mapsto q^{-1}$3, quadratic form $q\mapsto q^{-1}$4, characteristic vector $q\mapsto q^{-1}$5, coset $q\mapsto q^{-1}$6, and $q\mapsto q^{-1}$7 satisfying $q\mapsto q^{-1}$8, the false theta series is

$q\mapsto q^{-1}$9

Its completion is obtained by replacing the discontinuous sign-type factor with an error-function smoothing depending on $3$0: $3$1 The original false theta is recovered as a boundary value when $3$2 with $3$3 in the admissible range (Bringmann et al., 2019).

The conceptual point is that the completion restores a self-duality-like Fourier property. The crucial lemma proves that the smoothed kernel satisfies

$3$4

which is the precise duality-like mechanism underlying the modular $3$5-transformation (Bringmann et al., 2019). Ordinary theta functions are modular because the Gaussian is stable under Fourier transform; false theta functions are not modular because the sign factor is not; the completion replaces the sign by a smoothed object that is Fourier self-dual up to scalar factors.

The same framework yields an explicit obstruction-to-modularity formula. For

$3$6

the modular defect is expressed by an Eichler-type integral involving $3$7. This makes the non-modularity computable rather than merely qualitative, and it is the analytic source of subsequent quantum-modular and Rademacher-type applications (Bringmann et al., 2019).

3. Inversion duality, Appell–Lerch sums, and discriminant-sign dichotomies

A second major meaning of false theta duality arises from formal inversion and bilateral extension. The guiding principle is that if an Eulerian $3$8-series identity valid for $3$9 is formally transformed by $$\psi(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} \left(n+\tfrac12\right)(-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}, \qquad q=e^{2\pi i\tau},\ \zeta=e^{2\pi i z},$$0, then an Appell–Lerch expression on one side often turns into a partial theta function on the other, and conversely. In this sense Appell–Lerch sums and partial theta functions are treated as dual manifestations of the same $$\psi(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} \left(n+\tfrac12\right)(-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}, \qquad q=e^{2\pi i\tau},\ \zeta=e^{2\pi i z},$$1-series phenomenon, with the Appell–Lerch sum functioning as a kind of analytic completion or bilateral extension of a partial theta function (Mortenson, 2012).

This duality is not the same as modular Fourier duality. It is instead an identity-theoretic and analytic mirror principle. The paper develops it for classical mock theta functions, using Bailey pairs and conjugate Bailey pairs, and emphasizes that mixed identities often contain false-theta-type correction terms. The resulting picture places false theta functions inside a broader partial-theta/Appell–Lerch duality rather than outside it (Mortenson, 2012).

A related but distinct dichotomy appears for Hecke-type double sums

$$\psi(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} \left(n+\tfrac12\right)(-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}, \qquad q=e^{2\pi i\tau},\ \zeta=e^{2\pi i z},$$2

When $$\psi(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} \left(n+\tfrac12\right)(-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}, \qquad q=e^{2\pi i\tau},\ \zeta=e^{2\pi i z},$$3, there is a general decomposition into finite sums of products of classical theta functions and false theta functions. This is the negative-discriminant regime. The same Hecke-type template behaves differently when $$\psi(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} \left(n+\tfrac12\right)(-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}, \qquad q=e^{2\pi i\tau},\ \zeta=e^{2\pi i z},$$4: in that regime one obtains Appell–Lerch sums plus theta functions. The sign of the quadratic-form discriminant therefore governs whether the non-theta part is false-theta-like or mock/Appell–Lerch-like (Mortenson, 2022).

This contrast reappears in admissible $$\psi(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} \left(n+\tfrac12\right)(-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}, \qquad q=e^{2\pi i\tau},\ \zeta=e^{2\pi i z},$$5 string functions. For positive fractional level and the $$\psi(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} \left(n+\tfrac12\right)(-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}, \qquad q=e^{2\pi i\tau},\ \zeta=e^{2\pi i z},$$6-level case, the Hecke double sums lead to Appell–Lerch sums and mock theta functions; for negative admissible level, the same framework yields explicit false theta combinations. The paper explicitly emphasizes the structural contrast between the positive- and negative-discriminant regimes (Borozenets et al., 2024).

Hecke–Rogers double-sum identities supply another duality-like correspondence. In the “plus-sign-flipped” setting, combinations of shifted two-dimensional Hecke–Rogers sums collapse to one-dimensional false theta series, so that a double-lattice object is transformed into a sign-weighted unary theta-type object by symmetry, shift, and sign cancellation (Mortenson, 2020).

4. Higher-rank duality and iterated Eichler-type integrals

For higher-rank false theta functions, duality is formulated through depth and iterated Eichler integrals. A generic rank-two false theta function is

$$\psi(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} \left(n+\tfrac12\right)(-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}, \qquad q=e^{2\pi i\tau},\ \zeta=e^{2\pi i z},$$7

The fundamental lemma converts the product of sign functions into a double integral with square-root kernels and an explicit $$\psi(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} \left(n+\tfrac12\right)(-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}, \qquad q=e^{2\pi i\tau},\ \zeta=e^{2\pi i z},$$8-correction term. As a consequence, $$\psi(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} \left(n+\tfrac12\right)(-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}, \qquad q=e^{2\pi i\tau},\ \zeta=e^{2\pi i z},$$9 is represented as a sum of a modular theta contribution and an iterated holomorphic Eichler-type integral (Bringmann et al., 2021).

This realizes rank-two false theta functions as depth-two analogues of the rank-one situation. The associated completion $$\vartheta(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} (-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}.$$0 depends on two variables $$\vartheta(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} (-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}.$$1 and transforms under simultaneous modular action on both variables. Differentiation in $$\vartheta(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} (-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}.$$2 lowers the depth, producing lower-rank pieces in the same way that higher-depth mock modular forms reduce to lower-depth objects under suitable differential operators. The paper uses this structure for generic parafermion characters of type $$\vartheta(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} (-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}.$$3 and $$\vartheta(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} (-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}.$$4, for superconformal Schur indices, and for $$\vartheta(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} (-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}.$$5-invariants of certain plumbing $$\vartheta(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} (-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}.$$6-graphs (Bringmann et al., 2021).

False-indefinite theta functions extend the same idea to quadratic forms of signature $$\vartheta(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} (-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}.$$7. In that setting the holomorphic object is an Eichler-type integral associated to a vector-valued Maass form, and the modular defect is a controlled Mordell-type integral rather than an arbitrary error term. This provides the analytic control needed for precision asymptotics and circle-method arguments, while preserving the same dual organization: a sign-weighted Lorentzian-lattice $$\vartheta(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} (-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}.$$8-series on one side and a modular or Maass-theoretic completion on the other (Bringmann et al., 2024).

A recurrent theme is that false theta duality at higher rank is not an involutive self-reciprocity in the classical theta sense. It is a correspondence between a holomorphic sign-sum presentation and a modular-completion or Eichler-integral presentation, with explicit boundary terms and depth reduction (Bringmann et al., 2021, Bringmann et al., 2024).

5. Quantum modularity, radial limits, and $$\vartheta(z;\tau)\coloneqq i \sum_{n\in \mathbb Z} (-1)^n q^{\frac12\left(n+\frac12\right)^2}\zeta^{n+\frac12}.$$9-manifold invariants

A major domain of false theta function duality is the radial-limit correspondence with quantum modular and topological invariants. In the unary family

$\left(n+\tfrac12\right)$0

the completion $\left(n+\tfrac12\right)$1 satisfies an Eichler-integral relation, and the modular transformation law shows that the obstruction to modularity extends to a real-analytic function on $\left(n+\tfrac12\right)$2. The conclusion is that $\left(n+\tfrac12\right)$3 is a vector-valued quantum modular form (Bringmann et al., 2019).

For plumbed $\left(n+\tfrac12\right)$4-manifolds beyond the weakly negative definite regime, indefinite false theta functions become candidates for homological blocks. In the case of certain plumbing indefinite $\left(n+\tfrac12\right)$5-graphs, the candidate block $\left(n+\tfrac12\right)$6 is an explicit sign-weighted quadrant sum attached to an indefinite quadratic form, and its radial limits recover normalized Witten–Reshetikhin–Turaev invariants: $\left(n+\tfrac12\right)$7 The paper stresses that ordinary Zwegers-type indefinite theta functions are not the correct objects in this setting because their radial limits vanish; the indefinite false theta correction is essential (Murakami, 2022).

The Poincaré homology sphere is the decisive worked example. For a suitable $\left(n+\tfrac12\right)$8-vertex $\left(n+\tfrac12\right)$9-graph, the construction yields

$w$0

and the paper proves that this coincides with the original GPPV homological block (Murakami, 2022).

A related phenomenon appears in Hikami’s observations on Habiro’s unified WRT invariant. For the Poincaré sphere,

$w$1

so the value at a root of unity is one-half of the radial limit from inside the unit disc. The explanation is a decomposition into two false-theta-type contributions: one has a convergent radial limit and reproduces the root-of-unity value, while the other generally diverges. In the special case $w$2 the two contributions coincide, producing the factor $w$3 (Matsusaka, 2022).

These results give a precise meaning to “radial-limit duality”: the false theta series is not merely analogous to the topological invariant but encodes it through analytically controlled root-of-unity limits (Murakami, 2022, Matsusaka, 2022).

6. Representation theory, combinatorics, arithmetic, and conceptual scope

In logarithmic and nonrational CFT, regularized partial and false theta functions occur directly as characters. For the singlet vertex algebra $w$4, atypical module characters are expressed by differences of partial theta functions, hence by false-theta-type objects after antisymmetrization. Regularization by $w$5 yields modular-like $w$6- and $w$7-transformations with an explicit integral kernel and a theta correction, and these formulas support a Verlinde-type fusion rule (Creutzig et al., 2013). This is a duality-like structure in which the false theta character is related to its inverse-modulus transform not by a finite matrix alone but by an integral transform plus a correction term.

False theta duality also appears as representation duality between single-variable false theta series and fermionic multisums. Infinite families of identities rewrite false theta functions as constrained multi-$w$8-hypergeometric sums motivated by characters of vertex operator superalgebras. A particularly sharp phenomenon is parity splitting: an odd number of summation variables yields a false theta identity, whereas the even-variable companion yields a modular product identity (Jennings-Shaffer et al., 2020).

Partition theory supplies further examples of modular/false-modular complementarity. Companions to Capparelli’s identities express refined partition generating functions as a Jacobi theta core plus false theta corrections $w$9 and $q\mapsto q^{-1}$0; the two companions exchange these correction terms in a complementary way (Bringmann et al., 2014). Ramanujan’s false theta identities also admit bijective interpretations in which a sign-reversing involution cancels all weighted overpartition data except a fixed-point family, so the false theta coefficients arise as a residual combinatorial set rather than as a modular transform (Burson, 2018).

Arithmetic manifestations persist even when no direct modularity is available. For reciprocals of false theta functions, congruence families for the coefficients $q\mapsto q^{-1}$1 are proved modulo $q\mapsto q^{-1}$2 and $q\mapsto q^{-1}$3, and the authors state that the generalized theorem suggests a family-level phenomenon for primes $q\mapsto q^{-1}$4, hinting at a kind of “arithmetic duality” in the reciprocal false theta setting (Jin et al., 3 Aug 2025).

Several common confusions are ruled out by the literature. First, false theta duality is not the classical functional equation of Riemann’s theta function: the algebraic functional equation for the theta multiplier and determinant bundles concerns classical theta only and does not address false theta functions (Candelori, 2015). Second, false theta duality is not a single universal involution. Depending on context, it may mean Fourier-restored modularity after completion, inversion duality with Appell–Lerch sums, discriminant-sign bifurcation of Hecke-type sums, or radial-limit correspondence with quantum invariants (Bringmann et al., 2019, Mortenson, 2012, Mortenson, 2022, Murakami, 2022). Third, indefinite theta and indefinite false theta are not interchangeable: for the plumbing $q\mapsto q^{-1}$5-graphs considered in the homological-block setting, the naive indefinite theta object has vanishing radial limits, whereas the indefinite false theta function gives the nontrivial invariant (Murakami, 2022).

Taken together, these developments show that false theta function duality is best understood as a structural principle organizing non-modular theta-like $q\mapsto q^{-1}$6-series. The principle links sign-truncated lattice sums to modular completions, Eichler integrals, Appell–Lerch mirrors, fermionic multisums, quantum modular defects, and topological radial limits, while preserving the defining feature of the subject: the false theta function remains holomorphic, but its modular failure is explicit, computable, and often itself the source of the duality.