---
title: Holomorphic Modular Bootstrap in 2D CFTs
url: https://www.emergentmind.com/topics/holomorphic-modular-bootstrap
type: topic
---

# Holomorphic Modular Bootstrap in 2D CFTs

Searching arXiv for recent and foundational papers on holomorphic modular bootstrap.
Holomorphic modular bootstrap is a collection of methods for constraining or classifying two-dimensional chiral conformal field theories by combining holomorphy with modular covariance of torus observables. In the RCFT setting, the basic objects are holomorphic characters assembled into a vector-valued modular form, and one seeks “admissible” $q$-series with non-negative integer coefficients that transform under $SL(2,\mathbb Z)$ as characters of some RCFT [2503.23761]. In chiral theories one may also bootstrap the ordinary torus partition function by truncating modular crossing to a closed polynomial system [1903.06272], refine the analysis by flavor fugacities and quasi-modularity [2409.01095], extend modular constraints to generalized Gibbs ensembles built from higher-spin holomorphic currents [2603.28244], or impose genus-two Siegel modular invariance to constrain spectra and OPE data [1705.05862]. The subject therefore encompasses several related but technically distinct programs unified by the use of holomorphic modular structure as a bootstrap principle.

## 1. Core formulation in terms of characters and modular covariance

In a Rational Conformal Field Theory of central charge $c$ and $n$ primaries, one has $n$ linearly independent torus characters
\[
\chi_i(\tau)=q^{h_i-c/24}(1+\cdots),\qquad q=e^{2\pi i\tau},
\]
which transform among themselves under $SL(2,\mathbb Z)$ [2503.23761]. The holomorphic modular bootstrap consists in finding all sets of $n$ $q$-series with non-negative integer coefficients, termed the “admissible” characters, which transform under $SL(2,\mathbb Z)$ as the characters of some RCFT [2503.23761].

A standard formulation packages the characters into a vector
\[
\chi(\tau)=\bigl(\chi_1(\tau),\ldots,\chi_n(\tau)\bigr)^T,
\]
viewed as a weight-zero vector-valued modular form with multiplier $\rho$, so that
\[
\chi(\tau+1)=\rho(T)\chi(\tau),\qquad \chi(-1/\tau)=\rho(S)\chi(\tau)
\]
[2503.23761]. In unitary RCFTs $\rho(T)$ is diagonal with entries $e^{2\pi i(h_i-c/24)}$, while admissibility in the Bantay–Gannon sense requires $\rho(S)^2=I$ and $(\rho(ST^{-1}))^3=I$ for weight $k=0$ [2503.23761].

An older and closely related formulation begins from a generic modular linear differential equation (MLDE) of order $n$,
\[
\Bigl[D^n + \sum_{k=0}^{n-1} \varphi_{2(n-k)}(\tau)\,D^k\Bigr]\chi(\tau)=0,
\]
with $D$ the Serre covariant derivative and $\varphi_{2m}(\tau)$ modular forms of weight $2m$ [1910.02973]. Solving the MLDE near $q=0$ yields
\[
\chi_i(\tau)=q^{\alpha_i}\sum_{k=0}^{\infty} a_k^{(i)}q^k,\qquad \alpha_i=h_i-\frac{c}{24},
\]
and the bootstrap imposes integrality, non-negativity, normalization $a^{(0)}_0=1$, and linear independence [1910.02973]. This MLDE language remains central in later work on classification, exact $S$-matrices, and admissibility criteria [2107.13557; 2602.14665; 2604.11277].

A key integer datum is the Wronskian index $\ell$. In the MLDE formulation, the Wronskian determinant of the character vector controls the pole structure of the differential equation and organizes the classification problem [1910.02973]. For $d\le 5$, one exact set of constraints is
\[
d=2:\ \ell \text{ even};\qquad d=3:\ \ell\in 3\mathbb Z;\qquad d=4:\ \ell \text{ even};\qquad d=5:\ \text{no restriction}
\]
[2107.13557]. This discrete structure is one reason the holomorphic bootstrap often reduces an apparently continuous search to a finite or rigid problem.

## 2. MLDEs, Wronskian organization, and low-rank classification

The MLDE approach was developed as a classification program for rational conformal field theories by postulating a modular differential equation of fixed order and Wronskian index and then demanding that its solutions be admissible characters [1910.02973]. The status report literature emphasizes that this organizes known results for small numbers of characters and clarifies where finite classification is possible [1910.02973].

A later refinement uses representation theory of ${\rm PSL}(2,\mathbb Z_n)$ to classify allowed central charges and conformal weights for theories with any number of characters $d$, thereby avoiding bottlenecks in earlier approaches [2107.13557]. The basic input is that integrality of $q$-expansions with rational exponents implies that the vector-valued modular form is invariant under a principal congruence subgroup $\Gamma(n)$, so that $\rho$ factors through $SL(2,\mathbb Z_n)$ [2107.13557]. One then uses the character tables of irreducible representations of $SL(2,\mathbb Z_n)$ to determine the possible exponents $\alpha_i$ modulo $1$ [2107.13557]. This collapses the search for $(c,h_i)$ to a finite list of exponent tuples mod $1$ [2107.13557].

For $d\le 5$, the same work states that all monic MDEs are rigid, in the sense that once the exponents $\alpha_i$ are fixed, the coefficients $\phi_k$ are uniquely determined; the only non-monic rigid case up to $d=5$ is $d=2,\ell=2$ [2107.13557]. It then tabulates physically sensible RCFT characters for $d=2,3,4,5$, including the familiar two-character $\ell=0$ solutions such as the Lee–Yang model and the Deligne–Cvitanović series, infinite three-character families such as $\mathrm{Spin}(n)_1$, and specific four- and five-character examples [2107.13557].

The 2026 update extends this program to MLDEs with up to six characters and Wronskian index $<6$ in one-accessory-parameter cases with $c_{\rm eff}\le 24$ [2604.11277]. It gives explicit one-parameter MLDE families for $(n,\ell)=(4,2),(4,4),(5,2),(6,0)$ and defines an admissible solution by non-negative integer $q$-coefficients in all components together with $a_{0,0}=1$ [2604.11277]. It also introduces the distinction between admissible and “tenable”: an admissible solution is called tenable if its Verlinde fusion coefficients are non-negative integers [2604.11277]. This suggests a sharper separation between modular admissibility and full RCFT consistency.

## 3. Vector-valued modular forms and the generation of new admissible solutions

A complementary formulation replaces direct MLDE scanning by the theory of vector-valued modular forms (VVMFs) [2503.23761]. In this approach, the space $M_k(\rho)$ of weakly holomorphic VVMFs of weight $k$ and multiplier $\rho$ is the basic object, and one uses a theorem of Gannon that for an admissible representation $\rho$ of rank $n$, the graded space $M_*(\rho)=\oplus_k M_k(\rho)$ is a free module of rank $n$ over the ring $\mathbb C[J]$ [2503.23761].

This has a direct bootstrap consequence. Once one known admissible character vector $\chi^{(0)}(\tau)$ is given for a multiplier $\rho$, one may generate new solutions with the same multiplier by applying invariant differential operators $V_r$ built from the Serre derivative and Eisenstein series, deriving new quasi-character vectors, and then taking linear combinations with polynomials in the Klein $J$-invariant [2503.23761]. The explicit template described is
\[
\chi^{\rm new}(\tau)=(J(\tau)+b)\,\chi^{(0)}(\tau)-\sum_i n_i\,\chi^{(i)}(\tau),
\]
with integer parameters chosen so that the leading pole cancels as desired and the resulting $q$-expansion has non-negative integer coefficients [2503.23761].

The method is illustrated in the two-character case, where it reproduces all known admissible solutions with Wronskian indices $6$ and $8$ [2503.23761]. It is also worked out for examples with up to six characters, including Ising, three-state Potts, $F_4$ level $2$, and tricritical Ising [2503.23761]. The emphasis is that packaging all characters into a single VVMF trades the explicit construction of high-order MLDEs for the structure of $M_k(\rho)$ as a free module over $\mathbb C[J]$ [2503.23761].

This VVMF approach is not a rejection of MLDEs but a reorganization of the same modular data. A plausible implication is that it is especially effective when the number of characters becomes large, since the paper explicitly states that the “straight ahead” MLDE approach becomes hard to implement in that regime [2503.23761].

## 4. Quasi-characters, sign structure, and construction of admissible families

Quasi-characters occupy a central place in the two-character holomorphic bootstrap. They are solutions of simple MLDEs with integral Fourier coefficients after overall normalization, but unlike admissible characters they may contain negative coefficients [2507.07170]. For rank $2$ and $\ell=0$, the relevant equation is the MMS equation
\[
(D_\tau^2+\mu E_4(\tau))\chi=0,
\]
with
\[
\mu=-\frac{c(c+4)}{576},\qquad \alpha_0=-\frac{c}{24},\qquad \alpha_1=\alpha_0+h,\qquad h=\frac{c+4}{24}
\]
[2507.07170]. Integral solutions arise in seven sub-series with
\[
c=24M+j,\qquad j\in\{1,2,14/5,4,26/5,6,7\},\qquad M\in\mathbb Z
\]
[2507.07170].

An important result is that any admissible two-character VVMF with $\ell\equiv 0\!\!\pmod 6$ can be written as a finite linear combination of quasi-characters from one of these seven series [2507.07170]. More precisely,
\[
\chi_i=\sum_{M=M_{\min}}^{M_{\max}} N_M\,\chi_i^{[24M+j]}
\]
produces a new VVMF with Wronskian index $\ell=6(M_{\max}-M_{\min})$ [2507.07170]. In this sense quasi-characters form an explicit basis for the space of admissible two-character solutions of arbitrarily large Wronskian index [2507.07170].

The 2025 analysis proves conjectures about the sign pattern of quasi-character coefficients [2507.07170]. In each $\ell=0$ series exactly one of the two characters has an alternating-sign pattern up to order
\[
n_*=2|M|\simeq \frac{|c-j|}{12}\simeq \frac{c}{12},
\]
after which its coefficients stabilize to a definite sign, while the other character is of definite sign for all $n$ [2507.07170]. The paper gives both an asymptotic proof, using the Frobenius recursion for $n\ll c$ and a Rademacher formula for $n\gg c$, and an inductive proof based on sign properties of the recursion kernel [2507.07170].

This sign theorem has a direct bootstrap use: because only finitely many negative coefficients occur before stabilization, the admissibility problem for linear combinations of quasi-characters reduces to a finite set of linear Diophantine constraints [2507.07170]. The paper illustrates this with explicit $\ell=6$ and $\ell=12$ constructions [2507.07170]. This is one of the clearest examples where the holomorphic bootstrap converts an infinite positivity requirement into a finite combinatorial one.

## 5. Fast modular crossing and the chiral or holomorphic truncation scheme

A different branch of the subject begins not from RCFT characters and MLDEs, but from the modular crossing equation for torus partition functions [1903.06272]. For a two-dimensional CFT with torus partition function $Z(\tau,\bar\tau)$, modularity under $\tau\to -1/\tau$ yields a crossing equation, and in the spinless case this can be written as
\[
\sum_{\Delta\ge 0}\rho_\Delta\,[G_\Delta(\beta)-G_\Delta(1/\beta)]=0
\]
[1903.06272]. The key observation is that by acting with a finite set of linear functionals and truncating to the first $N=P/2+1$ primaries, one obtains a finite closed system of polynomial equations [1903.06272]. Whenever the corresponding extremal functional is positive above the first nonzero root, the solution gives a rigorous upper bound on the exact gap [1903.06272].

For the spinless bootstrap, the functionals are
\[
\mathcal F_k[f](\beta)\equiv
\frac1{2(2k-1)!}\bigl[(1+\beta)^2\partial_\beta\bigr]^{2k-1}
\frac{1+\beta}{2\sqrt\beta}\,f(\beta)\Big|_{\beta=1},\qquad k=1,\dots,P,
\]
and acting on the reduced blocks yields Laguerre polynomials
\[
f_k(\Delta)=L_{2k-1}\!\bigl[4\pi(\Delta-\tfrac{c-1}{12})\bigr]
\]
up to a special shifted vacuum formula [1903.06272]. The resulting truncation equations are exactly polynomial:
\[
\sum_{\mu=0}^{P/2} a_\mu\,L_{2k-1}\!\bigl[4\pi(\Delta_\mu-\tfrac{c-1}{12})\bigr]=0,\qquad k=1,\dots,P
\]
[1903.06272]. A fast Newton-type algorithm then solves these systems efficiently by exploiting empirical self-similarity of the truncated spectrum as $P$ increases [1903.06272].

The holomorphic version is obtained, in the words of the summary, by “simply omitting the anti-holomorphic sector from the usual spin-zero modular bootstrap” [1903.06272]. One replaces the torus partition function by a single chiral sum over holomorphic characters, retains the first $P/2+1$ chiral primaries, and solves the same finite system
\[
\sum_{\mu=0}^{P/2} a_\mu\,f_k(h_\mu)=0,\qquad k=1,\dots,P
\]
[1903.06272]. In the holomorphic case, positivity is demanded only for $h\ge h_1$ [1903.06272]. The summary also notes two simplifications: the vacuum character often has no level-1 null, so $G_0(\beta)=\beta^{-c/24}$, and for rational holomorphic VOAs one may incorporate extended-algebra characters to produce even stronger finite crossing systems [1903.06272].

A striking example is the chiral theory at $c=12$, where numerics converge rapidly to
\[
Z(\tau)=\sum_{h\ge 0}\rho_h\,q^{h-1}=j(\tau)-744,\qquad q=e^{2\pi i\tau},
\]
with the bound on the gap converging to $\Delta_1=2$ to better than $10^{-30}$ at $P=2000$ and the first degeneracies matching exactly the expansion of $j(\tau)-744$ [1903.06272]. This illustrates that in some meromorphic cases the truncated polynomial bootstrap effectively reconstructs the exact chiral partition function.

## 6. Flavored, quasi-modular, and generalized-Gibbs extensions

Holomorphic modular bootstrap has also been refined beyond unflavored characters. One such refinement is the “holomorphic quasi-modular bootstrap,” which introduces flavor fugacities $b_i$ conjugate to Cartan charges and derives flavored modular differential equations whose coefficients are quasi-Jacobi forms [2409.01095]. For a highest-weight module $M_a$ of a Kac–Moody algebra at level $k$, the flavored character is
\[
\chi_a(\tau,\{b_i\})=\mathrm{tr}_{M_a}\,q^{L_0-\frac{c}{24}}\prod_i b_i^{h^i_0},
\]
and modular transformations act as
\[
S:(\tau,b_i)\to(-1/\tau,b_i/\tau),\qquad T:(\tau,b_i)\to(\tau+1,b_i)
\]
up to known Gaussian phases and cocycles [2409.01095].

Using Zhu’s recursion, Pan–Zeng obtain partial differential equations in $(\tau,b_i)$ with coefficients built from $E_2,E_4,E_6$ and twisted Eisenstein series $E_k[\pm1;b^\alpha]$ [2409.01095]. A central role is played by a special null state $|\mathcal N_T\rangle$ of the vacuum module,
\[
|\mathcal N_T\rangle=L_{-2}^N|0\rangle + C_2\text{--terms},\qquad L_n|\mathcal N_T\rangle=0\ (n>0),
\]
together with stronger constraints proposed in affine theories:
\[
J_0^a|\mathcal N_T\rangle=0,\qquad L_2|\mathcal N_T\rangle=0,\qquad h^i_{n\ge 2}|\mathcal N_T\rangle=0
\]
[2409.01095]. These conditions are stated to fix simultaneously the level $k$, central charge $c$, and the precise descendant form of $|\mathcal N_T\rangle$ [2409.01095]. Solving the flavored MLDEs by a Frobenius ansatz recovers spectra in both untwisted and twisted sectors, with twisted sectors produced by half-integral spectral-flow translations $b_i\to b_i+n_i\tau$ [2409.01095].

A separate extension concerns generalized Gibbs ensembles built from zero modes of higher-spin holomorphic currents [2603.28244]. In a chiral CFT one considers
\[
Z(\tau,\{\alpha_i\})=\Big\langle \exp\!\Big(\sum_i \alpha_i I_0^{(i)}\Big)\Big\rangle_\tau,
\]
where $I^{(i)}(z)$ is a holomorphic current of spin $w_i$ and $I_0^{(i)}$ its zero mode [2603.28244]. The modular $S$ transformation acts as
\[
\tau\to -\frac1\tau,\qquad \alpha_i\to \alpha_i/\tau^{w_i},
\]
and the paper proves that the asymptotic small-$\alpha$ expansion of the transformed partition function is completely determined by OPE data of the currents [2603.28244].

The technical core is a Zhu-recursion analysis of integrated torus $n$-point functions leading to a universal recursion
\[
f_n=\Bigl(\frac{3n}{2}-2\Bigr)\frac1{n-1}\,\delta_W f_{n-1}-\frac12\,\delta_W^2 f_{n-2}+\mathcal O(\tau^2),
\]
and ultimately to composite operators satisfying
\[
\langle[W^{n+1}]\rangle_\tau=\delta_W\,\langle[W^n]\rangle_\tau
\]
[2603.28244]. The decisive algebraic point is that only the second-order pole in the OPE enters:
\[
W(z)W(w)\sim \frac{(WW)_2(w)}{(z-w)^2}+\cdots
\]
[2603.28244]. For a basis of currents $J^i$, the modular transformation is therefore controlled entirely by the second-order-pole structure constants $c^{ij}{}_k$ [2603.28244]. The paper interprets the resulting family of functional equations as a “holomorphic modular bootstrap” for chiral algebras deformed by higher-spin currents [2603.28244]. This suggests a broadened notion of the subject in which the bootstrap targets not only spectra of primary weights but also higher-spin OPE structure constants.

## 7. Higher-genus and non-RCFT variants

The holomorphic modular bootstrap also has a genus-two form for purely chiral, or meromorphic, two-dimensional CFTs [1705.05862]. At genus two, the partition function $Z_2(\Omega)$ is a Siegel modular object,
\[
Z_2(\Omega)=W_2(\Omega)/F_2(\Omega)^{c/2},
\]
where $W_2(\Omega)$ is a holomorphic Siegel modular form of degree two and weight $k=c/2$ [1705.05862]. Because the graded ring of even Siegel modular forms is finitely generated by $E_4(\Omega),E_6(\Omega),\chi_{10}(\Omega),\chi_{12}(\Omega)$, the space of candidate genus-two partition functions at fixed central charge is finite-dimensional [1705.05862].

In Schottky coordinates $(p_1,p_2,x)$, the genus-two partition function has an expansion
\[
Z_2(p_1,p_2,x)=\sum_{h_1,h_2=0}^\infty C_{h_1,h_2}(x)\,p_1^{h_1}p_2^{h_2},
\]
or equivalently a conformal-block decomposition in which coefficients are sums of squared three-point couplings [1705.05862]. Matching the finite-dimensional Siegel-modular ansatz to the conformal-block expansion yields algebraic formulas for spectral multiplicities and OPE sums in terms of finitely many “light data” [1705.05862]. Unitarity then imposes polynomial inequalities $N_h(L)\ge 0$ and $C_{h_1h_2h_3}(L)\ge 0$, from which one derives upper and lower bounds on multiplicities, light OPE sums, averaged OPE coefficients, and the maximal spectral gap [1705.05862]. This genus-two bootstrap differs from character classification, but it is holomorphic and modular in exactly the sense central to the broader subject.

Another neighboring use of holomorphic modular bootstrap techniques appears in the study of D4–D2–D0 BPS indices on compact Calabi–Yau threefolds [2204.02207]. There the generating functions $h_{r,\mu}(\tau)$ transform as vector-valued modular forms for $r=1$ and are reconstructed from polar data using an explicit overcomplete basis of vector-valued modular forms [2204.02207]. For $r>1$ they become mock modular, and the bootstrap requires both polar parts and shadows [2204.02207]. Although this is not an RCFT classification problem, it demonstrates that the same modular-reconstruction logic extends to enumerative invariants.

A final development is the extraction of exact modular $S$-matrices intrinsic to the MLDE setup [2602.14665]. By rewriting an MLDE as a Fuchsian equation in the variable $z=J(\tau)/1728$ or $w=1728/J(\tau)$, one identifies the $T$- and $S$-matrices with local monodromies around regular singular points and computes $S$ from a numerical connection matrix:
\[
S=C^{-1}M_1C
\]
[2602.14665]. Exact algebraic entries are then reconstructed using the fact that normalized $S$-matrix entries lie in a cyclotomic field $\mathbb Q(\zeta_N)$ [2602.14665]. The 2026 update explicitly incorporates this result into the classification program [2604.11277]. A plausible implication is that the holomorphic modular bootstrap has moved from character-level classification toward a more internally complete determination of full modular data.

Holomorphic modular bootstrap is therefore best understood not as a single algorithm but as a family of modularly constrained inverse problems for chiral two-dimensional theories. Its main formulations—MLDE classification, VVMF generation, quasi-character assembly, fast chiral modular crossing, flavored and GGE deformations, and genus-two Siegel bootstrap—share the same structural principle: finite or rigid modular data, when combined with positivity, integrality, or factorization, can severely constrain or in favorable cases determine spectra, partition functions, and modular representation data [1903.06272; 2503.23761; 2409.01095; 2603.28244; 1705.05862].

Source: https://www.emergentmind.com/topics/holomorphic-modular-bootstrap