---
title: Modular Invariant Completions
url: https://www.emergentmind.com/topics/modular-invariant-completions
type: topic
---

# Modular Invariant Completions

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 [2403.14920], [1506.07833], [2306.08673], [1611.08759], [2408.04011].

## 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 [2403.14920].

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 [2306.08673]. 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 [2210.14038].

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 [1611.08759]. 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 \(\mu\) [2408.04011]. 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 \(B\) [1806.03158].

## 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 \(\hat Z\)-invariants with and without defects is conjecturally, and for Brieskorn spheres proved, to be isomorphic—up to overall powers \(q^\Delta\) and finite polynomials—to a Weil representation \(\Theta^{m+K}\). Defects supply the missing components: for \(M_3=\Sigma(p_1,p_2,p_3)\), with \(m=p_1p_2p_3\) and \(r_{\boldsymbol\nu}=m-\sum_i(1+\nu_i)\bar p_i\), varying the defect labels \(\boldsymbol\nu\) runs through the allowed Weil indices and completes the full vector-valued object [2403.14920].

Second, the same work studies **analytic completion** under orientation reversal. For negative orientation, \(\hat Z\) 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 \(\hat Z\)-invariants for \(-\Sigma(2,3,6r+1)\) and more general \(-\Sigma(s,t,str+1)\) using regularized indefinite theta functions of signature \((1,1)\). The completion takes the form
\[
\widehat f_r(\tau)=f_r(\tau)-\vartheta_r^*(\tau)-\sum_i g_{i,r}(\tau)\,\vartheta_i^*(\tau),
\]
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 [2403.14920].

A higher-rank and higher-depth version of the same theme is developed in "Higher Depth False Modular Forms" [2109.00394]. There the basic false theta function is
\[
F_{Q,\mu,\ell,M}(\tau)=\sum_{n\in A+\mu+\frac{\ell}{2}} \operatorname{sgn}(B(M,n))\,e^{\pi i B(n,\ell)}\,q^{Q(n)},
\]
with sign data encoded by \(M=(m_1,\dots,m_r)\). Its completion is a bimodular theta series
\[
\widehat\Phi_{Q,\mu,\ell,M}(z,\zeta;\tau,w)
\]
built from the generalized error function
\[
E_{Q,M}(x)=\det(A)^{1/2}\int_{\mathbb R^n}\operatorname{sgn}(B(M,y))\,e^{-2\pi Q(y-x)}\,d^ny.
\]
As \(w\to\tau+i\infty\), 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 \(W^0(p)_{A_n}\) for \(1\le n\le 3\) and to \(\hat Z\)-invariants of \(3\)-manifolds associated with gauge group \(\mathrm{SU}(3)\) [2109.00394].

The vertex-algebraic side of the three-manifold story is extended further in the cone-VOA construction of [2403.14920]. There, regularized indefinite theta functions of the same type used in mock \(\hat Z\)-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 \(\mathbb P^1_{2,3,6}\), the relevant coefficients of the potential \(W_q(x,y,z)\) are governed by indefinite theta series of type \((1,2)\). The central object in [1506.07833] is the degenerate type-\((1,2)\) series \(F(z_1,z_2,z_3;\tau)\), which admits the closed formula
\[
F(z_1,z_2,z_3;\tau)
=
\vartheta(z_1;\tau)\,\mu(z_1,z_2;\tau)\,\mu(z_1,z_3;\tau)
-
\frac{\eta^3(\tau)\,\vartheta(z_2+z_3;\tau)}{\vartheta(z_2;\tau)\,\vartheta(z_3;\tau)}
\,\mu(z_1,z_2+z_3;\tau).
\]
Replacing \(\mu\) by Zwegers’ completed \(\widehat\mu\) yields a non-holomorphic completion \(\widehat F\). This framework produces explicit modular completions for the coefficients \(C_{yz^2}\) and \(C_{yz^4}\): the completed \(\mathcal C_{yz^2}\) is modular of weight \(2\) on \(SL_2(\mathbb Z)\), while \(\mathcal C_{yz^4}\) is modular of weight \(5/2\) and is polynomial of degree \(2\) in \(R(0;\tau)\) over holomorphic functions [1506.07833].

A closely related two-variable completion appears in "On the modular completion of certain generating functions" [1804.07589]. There the formal meromorphic generating series
\[
A(\tau,z)=\sum_{d>0} F_d(z)e(d\tau)
\]
is not well defined on \(\mathbb H\times\mathbb H\), because the poles of \(F_d(z)\) occur at CM points and hence on a dense subset. The remedy is to replace each coefficient by
\[
F_d^*(z;v)=F_d(z)+\widetilde F_d(z;v),
\]
and define
\[
A^*(\tau,z)=\sum_{d\in\mathbb Z}\bigl(F_d(z)+\widetilde F_d(z;v)\bigr)e(d\tau).
\]
The resulting \(A^*(\tau,z)\) converges locally uniformly to a smooth function that is modular of weight \(2\) in \(z\) for \(\Gamma\) and of weight \(3/2\) in \(\tau\) for \(\Gamma_0(4)\). The same \(A^*(\tau,z)\) is also the modular completion of a generating function of weakly holomorphic modular forms \(g_D(\tau)\) of weight \(3/2\), linking meromorphic weight-\(2\) objects, Zagier’s traces of singular moduli, and two-variable non-holomorphic modularity [1804.07589].

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 \(q\)-series; it is often a **smooth object in more variables** whose boundary values encode several different holomorphic or meromorphic generating functions at once [1506.07833], [1804.07589].

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

For the species scale in heterotic \(T^6/(\mathbb Z_3\times\mathbb Z_3)\) orbifolds, the starting asymptotic estimate is \(\mathcal S_{sp}\simeq \mathcal V_k\,g_s^{-2}\). Imposing modular invariance replaces this by a globally valid expression built from Dedekind \(\eta\)-functions and non-holomorphic prefactors:
\[
\mathcal S_{sp}(T_i,\bar T_i)\simeq -\sum_{i=1}^{3}\log\left[-i(T_i-\bar T_i)\,|\eta(T_i)|^4\right].
\]
For the isotropic six-torus this gives
\[
\mathcal S_{sp}\simeq -\log\Big[(-i(T-\bar T))^3|\eta(T)|^{12}\Big]
\simeq \mathcal V_6^{1/3}-3\log\big(\mathcal V_6^{1/3}\big),
\]
so additive logarithmic corrections are not optional but required by modular invariance. The same completion is recast as \(\mathcal S_{sp}\simeq -\log m_{3/2}^2\), tying the modularly completed cutoff to the gravitino mass and to bounds involving the scalar potential \(V\) [2306.08673].

In integrated correlators of \(\mathcal N=4\) SYM, the issue is not holomorphicity but large-\(N\) asymptotics. The exact correlator \(\mathcal C_{G_N}(\tau,\bar\tau)\) has a lattice-sum representation that is fully duality invariant at finite \(N\). Its large-\(N\) expansion, however, is asymptotic and termwise modular only through non-holomorphic Eisenstein series. The completion is supplied by new modular functions
\[
D_N(s;\tau,\bar\tau):=\sum_{(m,n)\ne(0,0)}e^{-4\sqrt{N Y_{mn}(\tau,\bar\tau)}}\,Y_{mn}(\tau,\bar\tau)^{-s},
\]
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-\(N\) expansion and have the interpretation of coincident \((p,q)\)-string world-sheet instantons [2210.14038].

Vafa–Witten theory on Hirzebruch and del Pezzo surfaces supplies a third model. There the holomorphic generating functions \(h_{N,\mu}(\tau,z)\) are not genuine Jacobi forms for \(N\ge2\); their completions
\[
\widehat h_{N,\mu}(\tau,z)
\]
are obtained from a universal formula expressing rank \(N\) in terms of products of lower-rank \(h_{N_i,\mu_i}\) multiplied by non-holomorphic kernels \(R_n(\{\gamma_i\};\tau_2,\beta)\), themselves built from boosted generalized error functions. The result is a closed formula for both \(h_{N,\mu}\) and \(\widehat h_{N,\mu}\) for all \(N\) on \(\mathbb F_m\) and \(\mathbb B_m\), together with new identities for generalized Appell functions arising from fiber–base duality [2005.03680].

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 [2306.08673], [2210.14038], [2005.03680].

## 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-\(0\) part by a universal completion procedure, with the Cardy relation as the necessary quotient:
\[
\mathcal{QOC}\cong \mathrm{Mod}(\mathcal{OC})/\text{Cardy}.
\]
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 [1611.08759].

A different but closely related notion appears in algebraic QFT. "Modular invariance as completeness" argues that for unitary \(2d\) conformal QFT, \(T\)-invariance of the torus partition function is required by locality, whereas \(S\)-invariance is not mandatory. Rather, \(S\)-invariance is equivalent to a completeness property: Haag duality for arbitrary multi-interval regions, absence of nontrivial DHR sectors, and two-interval index \(\mu=1\). In the rational setting this equivalence is encoded by the coupling matrix \(M\) of the torus partition function,
\[
S\,M=M\,S,\qquad T\,M=M\,T,
\]
and failure of modular invariance is measured by the index
\[
\mu_{\mathcal T}=\left(\frac{\sum_i d_i^2}{\sum_{ij} d_i M_{ij} d_j}\right)^2.
\]
The same index appears in a limit of Rényi mutual informations, with
\[
U_2(1)=-\frac12\log \mu_{\mathcal T}.
\]
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 [2408.04011].

These results correct a widespread simplification. Locality alone does not force full modular invariance in \(2d\) CFT; only \(T\)-invariance follows directly. \(S\)-invariance marks the passage from a local but possibly incomplete theory to a maximal one [2408.04011].

## 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 \(f\in M_k(\Gamma)\) are packaged into a binary form \(F(\tau)\in \mathrm{Sym}^r(V)\). Any invariant \(I\) of binary forms of degree \(r\), of degree \(d\) and order \(n\), then yields
\[
\mathcal V(I,f)(\tau)\in M_{dk+2n}(\Gamma).
\]
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 \(GL(g)\)-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 [2211.05611].

The same principle is made algorithmic in the modular-invariant SMEFT. With flavor symmetry \(A_4^{(q)}\times A_4^{(e)}\), non-dynamical moduli \(\tau_q,\tau_e\), and the MFV-like assumption that all flavor breaking is encoded in the renormalizable Yukawa sector, higher-dimensional operators are organized as modular singlets
\[
\bigl[Y_{\mathbf r}^{(k_Y)},\,Y_{\mathbf r'}^{(k_Y')*},\,\mathcal O\bigr]_{\mathbf1}.
\]
In the holomorphic \(A_4\) scenario, all modular forms derive from the weight-\(2\) triplet \(Y^{(2)}_{\mathbf3}\), and the paper gives two equivalent Hilbert-series bases for the complete operator space. It enumerates all independent operators up to dimension \(7\), including explicit constructions for all dimension-\(5\) operators and baryon- and lepton-number conserving dimension-\(6\) operators; the dimension-\(6\) total is \(2961\) [2601.23060]. 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 [2601.23060].

A distinct categorical use of completion appears in modular fusion categories. There the modular data \((S,T)\) are incomplete invariants. By adjoining the Borromean tensor \(B\), defined by coloring the Borromean link with three simple objects, one obtains a stronger topological invariant. For \(\mathcal Z(\mathrm{Vec}_G^\omega)\) with \(G=\mathbb Z/q\mathbb Z\rtimes \mathbb Z/p\mathbb Z\), \(B\) together with \(T\) distinguishes the \(p\) non-equivalent modular categories that are not distinguished by modular data alone [1806.03158]. 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 \(q\)-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 [2211.05611], [2601.23060], [1806.03158].

Source: https://www.emergentmind.com/topics/modular-invariant-completions