Modular Invariant Completions

• Modular invariant completions are procedures that replace holomorphic or asymptotic data with globally defined objects having exact modular or bimodular transformation laws.
• They employ techniques such as adding missing representation components, non-holomorphic Eichler integrals, and generalized error functions to achieve full invariance.
• Applications span three-manifold invariants, string theory duality, and modular fusion categories, underscoring their importance in both mathematical physics and algebraic structures.

Modular invariant completions are procedures that replace holomorphic, asymptotic, partial, or otherwise non-modular data by objects with exact modular or bimodular transformation laws. In current usage, the term does not designate a single construction. It includes representation-space completions that add missing components of a modular multiplet, non-holomorphic completions built from Eichler integrals or generalized error functions, global duality-invariant completions of moduli-dependent quantities, and categorical or operator-algebraic completions that turn genus-zero or local data into maximal modular structures (Cheng et al., 2024, Bringmann et al., 2015, Cribiori et al., 2023, Doubek et al., 2016, Benedetti et al., 2024).

1. General forms of completion

A recurrent distinction is between adding components and adding analytic corrections. In the three-manifold literature, the family of $\hat Z$-invariants for a fixed manifold may fail to realize the full Weil module unless supersymmetric defects are included; the completion is then holomorphic and takes place in representation space. By contrast, mock or false theta phenomena require adding non-holomorphic Eichler integrals or generalized error-function terms so that each component transforms modularly (Cheng et al., 2024).

A second recurrent pattern is the passage from local or asymptotic expressions to global modular-invariant formulas. In toroidal orbifolds, naive large-volume or weak-coupling estimates for the species scale are replaced by modular invariant functions involving $-i(T-\bar T)|\eta(T)|^4$, producing globally valid expressions over moduli space rather than asymptotic approximations (Cribiori et al., 2023). In large-$N$ $\mathcal N=4$ SYM, an asymptotic $1/N$ expansion with Eisenstein-series coefficients is completed by exponentially suppressed non-holomorphic modular functions $D_N(s;\tau,\bar\tau)$, yielding a duality-invariant non-perturbative completion (Dorigoni et al., 2022).

A third pattern is universal completion by adjunction or maximal extension. In operad theory, the modular operad of open–closed Riemann surfaces is the modular completion of genus-zero data modulo the Cardy relation (Doubek et al., 2016). In algebraic QFT, modular invariance of the torus partition function is identified with completeness in the sense of Haag duality for multi-interval regions, and failure of modular invariance is measured by an index $\mu$ (Benedetti et al., 2024). A distinct but related use of completion appears in modular fusion categories, where incomplete modular data $(S,T)$ are refined by additional link invariants such as the Borromean tensor $B$ (Kulkarni et al., 2018).

2. False theta functions, mock objects, and three-manifold invariants

For three-manifold invariants $\hat Z_b(M_3;\tau)$, "3d Modularity Revisited" identifies two complementary notions of completion. First, for Seifert manifolds with three singular fibres, the span of all $-i(T-\bar T)|\eta(T)|^4$0-invariants with and without defects is conjecturally, and for Brieskorn spheres proved, to be isomorphic—up to overall powers $-i(T-\bar T)|\eta(T)|^4$1 and finite polynomials—to a Weil representation $-i(T-\bar T)|\eta(T)|^4$2. Defects supply the missing components: for $-i(T-\bar T)|\eta(T)|^4$3, with $-i(T-\bar T)|\eta(T)|^4$4 and $-i(T-\bar T)|\eta(T)|^4$5, varying the defect labels $-i(T-\bar T)|\eta(T)|^4$6 runs through the allowed Weil indices and completes the full vector-valued object (Cheng et al., 2024).

Second, the same work studies analytic completion under orientation reversal. For negative orientation, $-i(T-\bar T)|\eta(T)|^4$7 is governed by false theta functions; for positive orientation, one expects a vector-valued (mixed) mock modular form whose shadows are the unary theta functions on the false side. The paper formulates the False–Mock Conjecture and constructs explicit mock $-i(T-\bar T)|\eta(T)|^4$8-invariants for $-i(T-\bar T)|\eta(T)|^4$9 and more general $N$0 using regularized indefinite theta functions of signature $N$1. The completion takes the form

$N$2

and the key structural point is that the mock side is not fixed by shadow data alone: replacing false theta functions by Zwegers’ canonical mock partners gives the right shadow but the wrong polar part (Cheng et al., 2024).

A higher-rank and higher-depth version of the same theme is developed in "Higher Depth False Modular Forms" (Bringmann et al., 2021). There the basic false theta function is

$N$3

with sign data encoded by $N$4. Its completion is a bimodular theta series

$N$5

built from the generalized error function

$N$6

As $N$7, the completion recovers the original false theta series together with explicit lower-depth corrections. This produces a hierarchy parallel to higher-depth mock modular forms and is applied both to characters of $N$8 for $N$9 and to $\mathcal N=4$0-invariants of $\mathcal N=4$1-manifolds associated with gauge group $\mathcal N=4$2 (Bringmann et al., 2021).

The vertex-algebraic side of the three-manifold story is extended further in the cone-VOA construction of (Cheng et al., 2024). There, regularized indefinite theta functions of the same type used in mock $\mathcal N=4$3-invariants appear as graded traces of suitable twisted cone-VOA modules, so modular completion acquires a VOA interpretation: logarithmic VOAs capture the false-theta side, while cone VOAs capture the mock side.

3. Indefinite theta series and two-variable generating functions

In open Gromov–Witten theory of the elliptic orbifold $\mathcal N=4$4, the relevant coefficients of the potential $\mathcal N=4$5 are governed by indefinite theta series of type $\mathcal N=4$6. The central object in (Bringmann et al., 2015) is the degenerate type-$\mathcal N=4$7 series $\mathcal N=4$8, which admits the closed formula

$\mathcal N=4$9

Replacing $1/N$0 by Zwegers’ completed $1/N$1 yields a non-holomorphic completion $1/N$2. This framework produces explicit modular completions for the coefficients $1/N$3 and $1/N$4: the completed $1/N$5 is modular of weight $1/N$6 on $1/N$7, while $1/N$8 is modular of weight $1/N$9 and is polynomial of degree $D_N(s;\tau,\bar\tau)$0 in $D_N(s;\tau,\bar\tau)$1 over holomorphic functions (Bringmann et al., 2015).

A closely related two-variable completion appears in "On the modular completion of certain generating functions" (Bringmann et al., 2018). There the formal meromorphic generating series

$D_N(s;\tau,\bar\tau)$2

is not well defined on $D_N(s;\tau,\bar\tau)$3, because the poles of $D_N(s;\tau,\bar\tau)$4 occur at CM points and hence on a dense subset. The remedy is to replace each coefficient by

$D_N(s;\tau,\bar\tau)$5

and define

$D_N(s;\tau,\bar\tau)$6

The resulting $D_N(s;\tau,\bar\tau)$7 converges locally uniformly to a smooth function that is modular of weight $D_N(s;\tau,\bar\tau)$8 in $D_N(s;\tau,\bar\tau)$9 for $\mu$0 and of weight $\mu$1 in $\mu$2 for $\mu$3. The same $\mu$4 is also the modular completion of a generating function of weakly holomorphic modular forms $\mu$5 of weight $\mu$6, linking meromorphic weight-$\mu$7 objects, Zagier’s traces of singular moduli, and two-variable non-holomorphic modularity (Bringmann et al., 2018).

These examples fix an important point of terminology. In this part of the literature, a modular invariant completion is not merely a correction term attached to a single $\mu$8-series; it is often a smooth object in more variables whose boundary values encode several different holomorphic or meromorphic generating functions at once (Bringmann et al., 2015, Bringmann et al., 2018).

4. Duality-invariant completions in string theory, supersymmetric gauge theory, and enumerative geometry

For the species scale in heterotic $\mu$9 orbifolds, the starting asymptotic estimate is $(S,T)$0. Imposing modular invariance replaces this by a globally valid expression built from Dedekind $(S,T)$1-functions and non-holomorphic prefactors: $(S,T)$2 For the isotropic six-torus this gives

$(S,T)$3

so additive logarithmic corrections are not optional but required by modular invariance. The same completion is recast as $(S,T)$4, tying the modularly completed cutoff to the gravitino mass and to bounds involving the scalar potential $(S,T)$5 (Cribiori et al., 2023).

In integrated correlators of $(S,T)$6 SYM, the issue is not holomorphicity but large-$(S,T)$7 asymptotics. The exact correlator $(S,T)$8 has a lattice-sum representation that is fully duality invariant at finite $(S,T)$9. Its large-$B$0 expansion, however, is asymptotic and termwise modular only through non-holomorphic Eisenstein series. The completion is supplied by new modular functions

$B$1

which admit a Poincaré-series representation and obey a deformed Laplace equation. They furnish the exponentially suppressed sectors required for a duality-invariant non-perturbative completion of the large-$B$2 expansion and have the interpretation of coincident $B$3-string world-sheet instantons (Dorigoni et al., 2022).

Vafa–Witten theory on Hirzebruch and del Pezzo surfaces supplies a third model. There the holomorphic generating functions $B$4 are not genuine Jacobi forms for $B$5; their completions

$B$6

are obtained from a universal formula expressing rank $B$7 in terms of products of lower-rank $B$8 multiplied by non-holomorphic kernels $B$9, themselves built from boosted generalized error functions. The result is a closed formula for both $\hat Z_b(M_3;\tau)$0 and $\hat Z_b(M_3;\tau)$1 for all $\hat Z_b(M_3;\tau)$2 on $\hat Z_b(M_3;\tau)$3 and $\hat Z_b(M_3;\tau)$4, together with new identities for generalized Appell functions arising from fiber–base duality (Alexandrov, 2020).

Across these three settings, the same structural lesson recurs: modular invariant completion is a replacement of asymptotic or mock data by a globally defined object whose extra terms are tightly constrained by modular symmetry and are not determined by local asymptotics alone (Cribiori et al., 2023, Dorigoni et al., 2022, Alexandrov, 2020).

5. Completion as universal extension and as completeness

In operad theory, modular completion is literal. "Open-closed modular operads, Cardy condition and string field theory" proves that the modular operad of diffeomorphism classes of open–closed Riemann surfaces is obtained from its genus-$\hat Z_b(M_3;\tau)$5 part by a universal completion procedure, with the Cardy relation as the necessary quotient: $\hat Z_b(M_3;\tau)$6 There is an equivalent premodular formulation in which the Cardy relation is already built into the source, so no additional quotient is needed. This identifies the full higher-genus open–closed structure with the modular completion of tree-level data and gives a finitary presentation in terms of generators, relations, and Frobenius-algebraic operations (Doubek et al., 2016).

A different but closely related notion appears in algebraic QFT. "Modular invariance as completeness" argues that for unitary $\hat Z_b(M_3;\tau)$7 conformal QFT, $\hat Z_b(M_3;\tau)$8-invariance of the torus partition function is required by locality, whereas $\hat Z_b(M_3;\tau)$9-invariance is not mandatory. Rather, $-i(T-\bar T)|\eta(T)|^4$00-invariance is equivalent to a completeness property: Haag duality for arbitrary multi-interval regions, absence of nontrivial DHR sectors, and two-interval index $-i(T-\bar T)|\eta(T)|^4$01. In the rational setting this equivalence is encoded by the coupling matrix $-i(T-\bar T)|\eta(T)|^4$02 of the torus partition function,

$-i(T-\bar T)|\eta(T)|^4$03

and failure of modular invariance is measured by the index

$-i(T-\bar T)|\eta(T)|^4$04

The same index appears in a limit of Rényi mutual informations, with

$-i(T-\bar T)|\eta(T)|^4$05

In this language, a modular invariant completion of an incomplete theory is a maximal local extension whose torus partition function is modular invariant and whose net is Haag dual (Benedetti et al., 2024).

These results correct a widespread simplification. Locality alone does not force full modular invariance in $-i(T-\bar T)|\eta(T)|^4$06 CFT; only $-i(T-\bar T)|\eta(T)|^4$07-invariance follows directly. $-i(T-\bar T)|\eta(T)|^4$08-invariance marks the passage from a local but possibly incomplete theory to a maximal one (Benedetti et al., 2024).

6. Algebraic completions of differential, categorical, and EFT data

Invariant theory provides a purely algebraic completion mechanism. For elliptic modular forms, derivatives of a form $-i(T-\bar T)|\eta(T)|^4$09 are packaged into a binary form $-i(T-\bar T)|\eta(T)|^4$10. Any invariant $-i(T-\bar T)|\eta(T)|^4$11 of binary forms of degree $-i(T-\bar T)|\eta(T)|^4$12, of degree $-i(T-\bar T)|\eta(T)|^4$13 and order $-i(T-\bar T)|\eta(T)|^4$14, then yields

$-i(T-\bar T)|\eta(T)|^4$15

This subsumes Rankin–Cohen brackets as a special case and generalizes to higher genus and vector-valued modular forms via Schur functors and concomitants of $-i(T-\bar T)|\eta(T)|^4$16-representations. In this setting, modular completion means replacing raw derivative or tensorial data by invariant-theoretic combinations that are designed to transform as honest modular forms (Cléry et al., 2022).

The same principle is made algorithmic in the modular-invariant SMEFT. With flavor symmetry $-i(T-\bar T)|\eta(T)|^4$17, non-dynamical moduli $-i(T-\bar T)|\eta(T)|^4$18, and the MFV-like assumption that all flavor breaking is encoded in the renormalizable Yukawa sector, higher-dimensional operators are organized as modular singlets

$-i(T-\bar T)|\eta(T)|^4$19

In the holomorphic $-i(T-\bar T)|\eta(T)|^4$20 scenario, all modular forms derive from the weight-$-i(T-\bar T)|\eta(T)|^4$21 triplet $-i(T-\bar T)|\eta(T)|^4$22, and the paper gives two equivalent Hilbert-series bases for the complete operator space. It enumerates all independent operators up to dimension $-i(T-\bar T)|\eta(T)|^4$23, including explicit constructions for all dimension-$-i(T-\bar T)|\eta(T)|^4$24 operators and baryon- and lepton-number conserving dimension-$-i(T-\bar T)|\eta(T)|^4$25 operators; the dimension-$-i(T-\bar T)|\eta(T)|^4$26 total is $-i(T-\bar T)|\eta(T)|^4$27 (Kang et al., 30 Jan 2026). In the non-holomorphic case of polyharmonic Maaß forms, multiplication is not closed. The paper therefore shows that adopting the holomorphic organizing idea naively would lead to an infinite proliferation of modular-invariant structures, and imposes a minimal formal organizing principle to retain a finite and complete basis (Kang et al., 30 Jan 2026).

A distinct categorical use of completion appears in modular fusion categories. There the modular data $-i(T-\bar T)|\eta(T)|^4$28 are incomplete invariants. By adjoining the Borromean tensor $-i(T-\bar T)|\eta(T)|^4$29, defined by coloring the Borromean link with three simple objects, one obtains a stronger topological invariant. For $-i(T-\bar T)|\eta(T)|^4$30 with $-i(T-\bar T)|\eta(T)|^4$31, $-i(T-\bar T)|\eta(T)|^4$32 together with $-i(T-\bar T)|\eta(T)|^4$33 distinguishes the $-i(T-\bar T)|\eta(T)|^4$34 non-equivalent modular categories that are not distinguished by modular data alone (Kulkarni et al., 2018). This is not a completion by non-holomorphic terms, but it is a completion of modular invariant data in the sense of classification.

Taken together, these algebraic constructions show that "modular invariant completion" can mean more than analytic repair of a $-i(T-\bar T)|\eta(T)|^4$35-series. It can also mean the canonical enlargement of differential data to modular invariants, the finite organization of EFT operator spaces under modular flavor symmetries, or the augmentation of incomplete modular data by additional topological tensors (Cléry et al., 2022, Kang et al., 30 Jan 2026, Kulkarni et al., 2018).