Holomorphic Modular Bootstrap in 2D CFTs

• Holomorphic modular bootstrap is a framework for classifying two-dimensional chiral CFTs by using modular covariance and holomorphy to generate admissible q-series and RCFT characters.
• It employs modular linear differential equations and vector-valued modular forms to convert continuous spectral constraints into finite, combinatorial conditions such as positivity and integrality.
• The approach extends to flavored, quasi-modular, and higher-genus settings, enabling precise determination of spectral gaps, OPE data, and modular S-matrices in varied chiral theories.

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 (Govindarajan et al., 31 Mar 2025). In chiral theories one may also bootstrap the ordinary torus partition function by truncating modular crossing to a closed polynomial system (Afkhami-Jeddi et al., 2019), refine the analysis by flavor fugacities and quasi-modularity (Pan et al., 2024), extend modular constraints to generalized Gibbs ensembles built from higher-spin holomorphic currents (Ashok et al., 30 Mar 2026), or impose genus-two Siegel modular invariance to constrain spectra and OPE data (Keller et al., 2017). 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 $SL(2,\mathbb Z)$ (Govindarajan et al., 31 Mar 2025). 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 (Govindarajan et al., 31 Mar 2025).

A standard formulation packages the characters into a vector

$SL(2,\mathbb Z)$0

viewed as a weight-zero vector-valued modular form with multiplier $SL(2,\mathbb Z)$1, so that

$SL(2,\mathbb Z)$2

(Govindarajan et al., 31 Mar 2025). In unitary RCFTs $SL(2,\mathbb Z)$3 is diagonal with entries $SL(2,\mathbb Z)$4, while admissibility in the Bantay–Gannon sense requires $SL(2,\mathbb Z)$5 and $SL(2,\mathbb Z)$6 for weight $SL(2,\mathbb Z)$7 (Govindarajan et al., 31 Mar 2025).

An older and closely related formulation begins from a generic modular linear differential equation (MLDE) of order $SL(2,\mathbb Z)$8,

$SL(2,\mathbb Z)$9

with $c$0 the Serre covariant derivative and $c$1 modular forms of weight $c$2 (Mukhi, 2019). Solving the MLDE near $c$3 yields

$c$4

and the bootstrap imposes integrality, non-negativity, normalization $c$5, and linear independence (Mukhi, 2019). This MLDE language remains central in later work on classification, exact $c$6-matrices, and admissibility criteria (Kaidi et al., 2021, Govindarajan et al., 16 Feb 2026, Govindarajan et al., 13 Apr 2026).

A key integer datum is the Wronskian index $c$7. In the MLDE formulation, the Wronskian determinant of the character vector controls the pole structure of the differential equation and organizes the classification problem (Mukhi, 2019). For $c$8, one exact set of constraints is

$c$9

(Kaidi et al., 2021). 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 (Mukhi, 2019). The status report literature emphasizes that this organizes known results for small numbers of characters and clarifies where finite classification is possible (Mukhi, 2019).

A later refinement uses representation theory of $n$0 to classify allowed central charges and conformal weights for theories with any number of characters $n$1, thereby avoiding bottlenecks in earlier approaches (Kaidi et al., 2021). The basic input is that integrality of $n$2-expansions with rational exponents implies that the vector-valued modular form is invariant under a principal congruence subgroup $n$3, so that $n$4 factors through $n$5 (Kaidi et al., 2021). One then uses the character tables of irreducible representations of $n$6 to determine the possible exponents $n$7 modulo $n$8 (Kaidi et al., 2021). This collapses the search for $n$9 to a finite list of exponent tuples mod $n$0 (Kaidi et al., 2021).

For $n$1, the same work states that all monic MDEs are rigid, in the sense that once the exponents $n$2 are fixed, the coefficients $n$3 are uniquely determined; the only non-monic rigid case up to $n$4 is $n$5 (Kaidi et al., 2021). It then tabulates physically sensible RCFT characters for $n$6, including the familiar two-character $n$7 solutions such as the Lee–Yang model and the Deligne–Cvitanović series, infinite three-character families such as $n$8, and specific four- and five-character examples (Kaidi et al., 2021).

The 2026 update extends this program to MLDEs with up to six characters and Wronskian index $n$9 in one-accessory-parameter cases with $$\chi_i(\tau)=q^{h_i-c/24}(1+\cdots),\qquad q=e^{2\pi i\tau},$$0 (Govindarajan et al., 13 Apr 2026). It gives explicit one-parameter MLDE families for $$\chi_i(\tau)=q^{h_i-c/24}(1+\cdots),\qquad q=e^{2\pi i\tau},$$1 and defines an admissible solution by non-negative integer $$\chi_i(\tau)=q^{h_i-c/24}(1+\cdots),\qquad q=e^{2\pi i\tau},$$2-coefficients in all components together with $$\chi_i(\tau)=q^{h_i-c/24}(1+\cdots),\qquad q=e^{2\pi i\tau},$$3 (Govindarajan et al., 13 Apr 2026). It also introduces the distinction between admissible and “tenable”: an admissible solution is called tenable if its Verlinde fusion coefficients are non-negative integers (Govindarajan et al., 13 Apr 2026). 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) (Govindarajan et al., 31 Mar 2025). In this approach, the space $$\chi_i(\tau)=q^{h_i-c/24}(1+\cdots),\qquad q=e^{2\pi i\tau},$$4 of weakly holomorphic VVMFs of weight $$\chi_i(\tau)=q^{h_i-c/24}(1+\cdots),\qquad q=e^{2\pi i\tau},$$5 and multiplier $$\chi_i(\tau)=q^{h_i-c/24}(1+\cdots),\qquad q=e^{2\pi i\tau},$$6 is the basic object, and one uses a theorem of Gannon that for an admissible representation $$\chi_i(\tau)=q^{h_i-c/24}(1+\cdots),\qquad q=e^{2\pi i\tau},$$7 of rank $$\chi_i(\tau)=q^{h_i-c/24}(1+\cdots),\qquad q=e^{2\pi i\tau},$$8, the graded space $$\chi_i(\tau)=q^{h_i-c/24}(1+\cdots),\qquad q=e^{2\pi i\tau},$$9 is a free module of rank $SL(2,\mathbb Z)$0 over the ring $SL(2,\mathbb Z)$1 (Govindarajan et al., 31 Mar 2025).

This has a direct bootstrap consequence. Once one known admissible character vector $SL(2,\mathbb Z)$2 is given for a multiplier $SL(2,\mathbb Z)$3, one may generate new solutions with the same multiplier by applying invariant differential operators $SL(2,\mathbb Z)$4 built from the Serre derivative and Eisenstein series, deriving new quasi-character vectors, and then taking linear combinations with polynomials in the Klein $SL(2,\mathbb Z)$5-invariant (Govindarajan et al., 31 Mar 2025). The explicit template described is

$SL(2,\mathbb Z)$6

with integer parameters chosen so that the leading pole cancels as desired and the resulting $SL(2,\mathbb Z)$7-expansion has non-negative integer coefficients (Govindarajan et al., 31 Mar 2025).

The method is illustrated in the two-character case, where it reproduces all known admissible solutions with Wronskian indices $SL(2,\mathbb Z)$8 and $SL(2,\mathbb Z)$9 (Govindarajan et al., 31 Mar 2025). It is also worked out for examples with up to six characters, including Ising, three-state Potts, $n$0 level $n$1, and tricritical Ising (Govindarajan et al., 31 Mar 2025). The emphasis is that packaging all characters into a single VVMF trades the explicit construction of high-order MLDEs for the structure of $n$2 as a free module over $n$3 (Govindarajan et al., 31 Mar 2025).

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 (Govindarajan et al., 31 Mar 2025).

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 (Das et al., 9 Jul 2025). For rank $n$4 and $n$5, the relevant equation is the MMS equation

$n$6

with

$n$7

(Das et al., 9 Jul 2025). Integral solutions arise in seven sub-series with

$n$8

(Das et al., 9 Jul 2025).

An important result is that any admissible two-character VVMF with $n$9 can be written as a finite linear combination of quasi-characters from one of these seven series (Das et al., 9 Jul 2025). More precisely,

$q$0

produces a new VVMF with Wronskian index $q$1 (Das et al., 9 Jul 2025). In this sense quasi-characters form an explicit basis for the space of admissible two-character solutions of arbitrarily large Wronskian index (Das et al., 9 Jul 2025).

The 2025 analysis proves conjectures about the sign pattern of quasi-character coefficients (Das et al., 9 Jul 2025). In each $q$2 series exactly one of the two characters has an alternating-sign pattern up to order

$q$3

after which its coefficients stabilize to a definite sign, while the other character is of definite sign for all $q$4 (Das et al., 9 Jul 2025). The paper gives both an asymptotic proof, using the Frobenius recursion for $q$5 and a Rademacher formula for $q$6, and an inductive proof based on sign properties of the recursion kernel (Das et al., 9 Jul 2025).

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 (Das et al., 9 Jul 2025). The paper illustrates this with explicit $q$7 and $q$8 constructions (Das et al., 9 Jul 2025). 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 (Afkhami-Jeddi et al., 2019). For a two-dimensional CFT with torus partition function $q$9, modularity under $SL(2,\mathbb Z)$0 yields a crossing equation, and in the spinless case this can be written as

$SL(2,\mathbb Z)$1

(Afkhami-Jeddi et al., 2019). The key observation is that by acting with a finite set of linear functionals and truncating to the first $SL(2,\mathbb Z)$2 primaries, one obtains a finite closed system of polynomial equations (Afkhami-Jeddi et al., 2019). Whenever the corresponding extremal functional is positive above the first nonzero root, the solution gives a rigorous upper bound on the exact gap (Afkhami-Jeddi et al., 2019).

For the spinless bootstrap, the functionals are

$SL(2,\mathbb Z)$3

and acting on the reduced blocks yields Laguerre polynomials

$SL(2,\mathbb Z)$4

up to a special shifted vacuum formula (Afkhami-Jeddi et al., 2019). The resulting truncation equations are exactly polynomial: $SL(2,\mathbb Z)$5 (Afkhami-Jeddi et al., 2019). A fast Newton-type algorithm then solves these systems efficiently by exploiting empirical self-similarity of the truncated spectrum as $SL(2,\mathbb Z)$6 increases (Afkhami-Jeddi et al., 2019).

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” (Afkhami-Jeddi et al., 2019). One replaces the torus partition function by a single chiral sum over holomorphic characters, retains the first $SL(2,\mathbb Z)$7 chiral primaries, and solves the same finite system

$SL(2,\mathbb Z)$8

(Afkhami-Jeddi et al., 2019). In the holomorphic case, positivity is demanded only for $SL(2,\mathbb Z)$9 (Afkhami-Jeddi et al., 2019). The summary also notes two simplifications: the vacuum character often has no level-1 null, so $SL(2,\mathbb Z)$00, and for rational holomorphic VOAs one may incorporate extended-algebra characters to produce even stronger finite crossing systems (Afkhami-Jeddi et al., 2019).

A striking example is the chiral theory at $SL(2,\mathbb Z)$01, where numerics converge rapidly to

$SL(2,\mathbb Z)$02

with the bound on the gap converging to $SL(2,\mathbb Z)$03 to better than $SL(2,\mathbb Z)$04 at $SL(2,\mathbb Z)$05 and the first degeneracies matching exactly the expansion of $SL(2,\mathbb Z)$06 (Afkhami-Jeddi et al., 2019). 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 $SL(2,\mathbb Z)$07 conjugate to Cartan charges and derives flavored modular differential equations whose coefficients are quasi-Jacobi forms (Pan et al., 2024). For a highest-weight module $SL(2,\mathbb Z)$08 of a Kac–Moody algebra at level $SL(2,\mathbb Z)$09, the flavored character is

$SL(2,\mathbb Z)$10

and modular transformations act as

$SL(2,\mathbb Z)$11

up to known Gaussian phases and cocycles (Pan et al., 2024).

Using Zhu’s recursion, Pan–Zeng obtain partial differential equations in $SL(2,\mathbb Z)$12 with coefficients built from $SL(2,\mathbb Z)$13 and twisted Eisenstein series $SL(2,\mathbb Z)$14 (Pan et al., 2024). A central role is played by a special null state $SL(2,\mathbb Z)$15 of the vacuum module,

$SL(2,\mathbb Z)$16

together with stronger constraints proposed in affine theories: $SL(2,\mathbb Z)$17 (Pan et al., 2024). These conditions are stated to fix simultaneously the level $SL(2,\mathbb Z)$18, central charge $SL(2,\mathbb Z)$19, and the precise descendant form of $SL(2,\mathbb Z)$20 (Pan et al., 2024). 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 $SL(2,\mathbb Z)$21 (Pan et al., 2024).

A separate extension concerns generalized Gibbs ensembles built from zero modes of higher-spin holomorphic currents (Ashok et al., 30 Mar 2026). In a chiral CFT one considers

$SL(2,\mathbb Z)$22

where $SL(2,\mathbb Z)$23 is a holomorphic current of spin $SL(2,\mathbb Z)$24 and $SL(2,\mathbb Z)$25 its zero mode (Ashok et al., 30 Mar 2026). The modular $SL(2,\mathbb Z)$26 transformation acts as

$SL(2,\mathbb Z)$27

and the paper proves that the asymptotic small-$SL(2,\mathbb Z)$28 expansion of the transformed partition function is completely determined by OPE data of the currents (Ashok et al., 30 Mar 2026).

The technical core is a Zhu-recursion analysis of integrated torus $SL(2,\mathbb Z)$29-point functions leading to a universal recursion

$SL(2,\mathbb Z)$30

and ultimately to composite operators satisfying

$SL(2,\mathbb Z)$31

(Ashok et al., 30 Mar 2026). The decisive algebraic point is that only the second-order pole in the OPE enters: $SL(2,\mathbb Z)$32 (Ashok et al., 30 Mar 2026). For a basis of currents $SL(2,\mathbb Z)$33, the modular transformation is therefore controlled entirely by the second-order-pole structure constants $SL(2,\mathbb Z)$34 (Ashok et al., 30 Mar 2026). The paper interprets the resulting family of functional equations as a “holomorphic modular bootstrap” for chiral algebras deformed by higher-spin currents (Ashok et al., 30 Mar 2026). 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 (Keller et al., 2017). At genus two, the partition function $SL(2,\mathbb Z)$35 is a Siegel modular object,

$SL(2,\mathbb Z)$36

where $SL(2,\mathbb Z)$37 is a holomorphic Siegel modular form of degree two and weight $SL(2,\mathbb Z)$38 (Keller et al., 2017). Because the graded ring of even Siegel modular forms is finitely generated by $SL(2,\mathbb Z)$39, the space of candidate genus-two partition functions at fixed central charge is finite-dimensional (Keller et al., 2017).

In Schottky coordinates $SL(2,\mathbb Z)$40, the genus-two partition function has an expansion

$SL(2,\mathbb Z)$41

or equivalently a conformal-block decomposition in which coefficients are sums of squared three-point couplings (Keller et al., 2017). 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” (Keller et al., 2017). Unitarity then imposes polynomial inequalities $SL(2,\mathbb Z)$42 and $SL(2,\mathbb Z)$43, from which one derives upper and lower bounds on multiplicities, light OPE sums, averaged OPE coefficients, and the maximal spectral gap (Keller et al., 2017). 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 (Alexandrov et al., 2022). There the generating functions $SL(2,\mathbb Z)$44 transform as vector-valued modular forms for $SL(2,\mathbb Z)$45 and are reconstructed from polar data using an explicit overcomplete basis of vector-valued modular forms (Alexandrov et al., 2022). For $SL(2,\mathbb Z)$46 they become mock modular, and the bootstrap requires both polar parts and shadows (Alexandrov et al., 2022). 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 $SL(2,\mathbb Z)$47-matrices intrinsic to the MLDE setup (Govindarajan et al., 16 Feb 2026). By rewriting an MLDE as a Fuchsian equation in the variable $SL(2,\mathbb Z)$48 or $SL(2,\mathbb Z)$49, one identifies the $SL(2,\mathbb Z)$50- and $SL(2,\mathbb Z)$51-matrices with local monodromies around regular singular points and computes $SL(2,\mathbb Z)$52 from a numerical connection matrix: $SL(2,\mathbb Z)$53 (Govindarajan et al., 16 Feb 2026). Exact algebraic entries are then reconstructed using the fact that normalized $SL(2,\mathbb Z)$54-matrix entries lie in a cyclotomic field $SL(2,\mathbb Z)$55 (Govindarajan et al., 16 Feb 2026). The 2026 update explicitly incorporates this result into the classification program (Govindarajan et al., 13 Apr 2026). 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 (Afkhami-Jeddi et al., 2019, Govindarajan et al., 31 Mar 2025, Pan et al., 2024, Ashok et al., 30 Mar 2026, Keller et al., 2017).