Modular Conformal Bootstrap in 2D CFT

• Modular conformal bootstrap analysis is a systematic method that uses modular invariance and positivity constraints to nonperturbatively classify two-dimensional conformal field theories.
• It employs modular linear differential equations and admissibility conditions to quantize parameters like central charge and conformal weights, isolating physically consistent RCFTs.
• Numerical and algebraic techniques, including semi-definite programming, provide rigorous spectrum bounds and extend the approach to higher-genus and flavored bootstrap analyses.

Modular Conformal Bootstrap Analysis

Modular conformal bootstrap is a program to nonperturbatively constrain, classify, and solve two-dimensional conformal field theories (2d CFTs) by systematically combining their modular invariance properties—most notably the invariance of the torus partition function under the modular group PSL(2,ℤ)—with algebraic and positivity requirements associated to operator product expansions and representation theory. This approach applies both to chiral (holomorphic, e.g., rational conformal field theories, RCFTs) and non-chiral (full) theories. The logic of the modular bootstrap is to regard modular data as nontrivial constraints on the local spectra, fusion rules, and sometimes even higher-point or higher-genus correlation data, and to use these to algorithmically characterize the landscape of consistent CFTs.

1. Core Principles of Modular Bootstrap

The modular bootstrap relies on three deeply interrelated ingredients:

1. Modular invariance: Any consistent 2d CFT must have a torus partition function Z(τ, τ̄) that is invariant under modular transformations τ→γτ=(aτ+b)/(cτ+d), γ∈PSL(2,ℤ). Explicitly,

$$Z\left(\frac{a\tau + b}{c\tau + d}, \frac{a\bar\tau+b}{c\bar\tau + d}\right) = Z(\tau, \bar\tau)$$

This holds both for chiral characters (holomorphic/anti-holomorphic) and full theories (Mukhi, 2019).

2. Spectral decomposition in terms of characters: The partition function is typically written

$$Z(\tau, \bar\tau) = \sum_{i,j} M_{ij}\, \chi_i(\tau)\, \overline{\chi_j(\bar\tau)}$$

where the $\chi_i(\tau)$ are characters of modules of the chiral algebra, and $M$ encodes how left- and right-sectors combine. For RCFT these are finite sums; for generic theories, the sum/integral can be continuous (Mukhi, 2019, Kaidi et al., 2021).

3. Algebraic/positivity constraints: The q-expansion of each $\chi_i(\tau)$ must have non-negative integer coefficients (interpreted as state degeneracies), with further constraints (such as unique vacuum normalization), and $M$ must be an integer non-negative matrix commuting with the modular S and T matrices (Mukhi, 2019).

2. Modular Linear Differential Equations and Admissible Characters

A central tool for classifying chiral RCFTs is the analysis of modular linear differential equations (MLDEs):

• MLDE structure: The space of independent characters of an RCFT forms a vector valued modular form (VVMF) of weight zero under PSL(2,ℤ). Each component satisfies a holomorphic MLDE of the form

$$L_n \chi(\tau) = 0,\quad L_n = D_k^n + \sum_{r=0}^{n-1} \varphi_{2(n-r)}(\tau)\,D_k^r$$

where $D_k$ is the modular-covariant derivative, and $\varphi_{2m}(\tau)$ are modular forms of weight $2m$ (Mukhi, 2019, Kaidi et al., 2021, Pan et al., 2024).

• Admissibility conditions: For a solution $$\chi_i(\tau) = q^{-c/24 + h_i} \sum_{m=0}^\infty a_m^{(i)} q^m$$, one demands:
  • (a) Vacuum normalization: $h_0=0$, $a_0^{(0)}=1$.
  • (b) Integrality: $\forall i, m$, $a_m^{(i)}\in\mathbb{Z}$.
  • (c) Non-negativity: $a_m^{(i)}\ge 0$ (Mukhi, 2019).
• Quantization and classification: The parameters (central charge, conformal weights, and MLDE coefficients) are quantized by imposing these conditions, resulting in a discrete (sometimes finite) set of 'admissible' vectors for a given order and Wronskian index of the MLDE (Mukhi, 2019, Kaidi et al., 2021, Govindarajan et al., 31 Mar 2025).

3. Full Modular Bootstrap and Numerical Approaches

Beyond chiral classification, the full modular bootstrap addresses constraints on the spectrum and OPE coefficients of the complete non-chiral theory:

• Linear functional and semi-definite programming methods: One seeks a linear functional acting on the modular crossing equations for torus partition functions, imposing positivity on the non-vacuum terms. The existence of such a functional excludes a hypothesized spectrum beyond a certain gap, leading to rigorous upper bounds on the scaling dimension of the first non-vacuum primary (Collier et al., 2016, Afkhami-Jeddi et al., 2019, Afkhami-Jeddi et al., 2020).
• Gap bounds and extremal spectra: The classic Hellerman bound is $\Delta_1 \leq c/6 + O(1)$ (for the lowest-dimension nonidentity primary); improvements via higher-derivative/semi-definite programming methods have led to asymptotics such as

$\Delta_1 \lesssim \frac{c}{9.1}$

for large $c$ (Afkhami-Jeddi et al., 2019). Extremal functionals can reconstruct entire candidate spectra on the boundary of allowed regions (Afkhami-Jeddi, 2021).

• Spin-dependent and twist-dependent bounds: Modular bootstrap admits spin-dependent improvements; for extremal/near-extremal spin, the bound tightens to $\Delta_1 \lesssim c/12+O(1)$ (Ashrafi, 2019). The twist gap for any unitary CFT satisfies $t_{\rm gap} \le (c-1)/12$ (Benjamin et al., 2019, Collier et al., 2016).

4. Modular Bootstrap of Correlation Functions and Higher Genus

The modular bootstrap framework extends to correlation functions and Riemann surfaces of genus $g>1$:

• Twist-operator approach: Torus modular invariance can be recast as crossing symmetry for a four-point twist-field correlator on the sphere, reducing the modular bootstrap to a standard four-point functional bootstrap (Antunes, 2017).
• Genus-two modular bootstrap: Genus-two partition functions yield new nontrivial constraints—'critical surfaces'—on structure constants involving three distinct primaries, probing beyond data accessible on the torus or sphere (Cho et al., 2017).
• Full consistency via Segal's axioms and analytical bootstrap for Liouville theory: For Liouville CFT, a probabilistic realization of Segal's axioms yields multi-integral formulas over the spectrum, constructed from Virasoro conformal blocks and explicit structure constants (DOZZ formula), manifestly enforcing modular and crossing symmetry at all genera (Guillarmou et al., 2021).

5. Exact Solutions, Quantum Codes, and Code CFTs

Modular bootstrap constraints admit classes of explicit analytic solutions:

• RCFTs and code Narain CFTs: There is a correspondence between quantum stabilizer codes and certain Narain CFTs. The partition function is encoded in a refined enumerator polynomial $W_C(x,y,z)$, with modular invariance reducing to simple algebraic constraints (MacWilliams transform and Y-parity) on $W_C$ (Dymarsky et al., 2020).
• Isospectrality and fake solutions: There exist many isospectral, physically inequivalent code CFTs, as well as 'fake' partition functions that satisfy all modular and positivity constraints but are not associated with any known CFT. This demonstrates that modular bootstrap constraints can have spurious solutions not realized by local quantum field theories (Dymarsky et al., 2020).
• Applications to high-dimensional sphere packing: The modular bootstrap for $U(1)^c \times U(1)^c$ current algebra CFTs is equivalent to the Cohn–Elkies linear programming bound for sphere packing in $2c$ dimensions, yielding best-known upper bounds for high-dimensional packings (Afkhami-Jeddi et al., 2020).

6. Limitations, Outlook, and Future Directions

Despite its successes, modular conformal bootstrap analysis encounters both computational and conceptual challenges:

• Infinitude of admissible solutions: For increasing MLDE order (large $n$) and large Wronskian index $\ell$, the set of admissible characters becomes infinite, whereas actual RCFTs grow sparse. Additional structure—such as extended chiral symmetry, fusion positivity, and higher-point consistency—are often required to single out genuine physical theories (Mukhi, 2019, Kaidi et al., 2021).
• Chiral vs. full constraints: The chiral/holomorphic modular bootstrap (classification of vector-valued modular forms/characters) is often decoupled from the full modular bootstrap (spectrum/OPE for the full theory), but the interplay is partly understood only for low-rank cases (Govindarajan et al., 31 Mar 2025, Kaidi et al., 2021).
• Extension to flavored and quasi-modular bootstrap: Flavored/functional bootstrap, including quasi-modular and Jacobi-form techniques, further refines the constraints for chiral algebras with flavor symmetry, granting a more complete control over representations and null states (Pan et al., 2024).
• Nonunitary and noncompact theories: Deformations away from known unitary solutions provide a route to describe nonunitary CFTs and even extend to varying spacetime dimension (e.g., interpolation from Ising to Yang–Lee models) (Afkhami-Jeddi, 2021).
• Outlook: Progress in computational implementations (e.g., efficient Newton-based algorithms), as well as improved mathematical classification of vector-valued modular forms and modular tensor categories, is driving the field towards a comprehensive, algorithmic map of RCFT and the ‘swampland’ of modular CFTs. The modular bootstrap continues to serve as an essential filter, sharply delimiting the landscape of consistent 2d conformal theories (Mukhi, 2019, Govindarajan et al., 31 Mar 2025).