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Modular Bootstrap and Discrete Symmetries in 2D CFTs

DARIA UNGOLO

Published Dec 4, 2025• 0 citations
OpenAlex

Abstract

Two-dimensional Conformal Field Theories (CFTs) allow us to describe quantum field theories at critical points and in general systems without a characteristic scale. It is possible to implement such theories on topologically non-trivial surfaces. If we formulate a two-dimensional CFT on a toroidal surface, a new symmetry emerges known as modular invariance. This symmetry corresponds to the set of transformations that act without changing the toroidal surface, consequently, the partition function of the theory must remain invariant under these transformations. This invariance is used as a constraint by the modular bootstrap method to obtain information about the spectrum without using the Lagrangian and operator content of the theory. By imposing that the theory is consistent with modular symmetry, it is possible to determine the limits of the spectrum beyond which modular invariance can no longer be satisfied, thus constraining the space of admissible two-dimensional conformal field theories. To study CFTs that present discrete global symmetries, it is essential to introduce Topological Defect Lines (TDLs). These are one-dimensional, topologically invariant operators that implement the action of the symmetry group on the Hilbert space. The presence of TDLs modifies the topology of the system and introduces new partition functions associated with different symmetry sectors. Under modular transformations, the TDLs mix among themselves, and consequently, the corresponding partition functions also mix, thus promoting the modular constraint to a vector-valued condition. This formalism makes it possible to implement discrete symmetries also considering the possible presence of anomalies. The objective of this thesis is therefore to understand how discrete symmetries and anomalies affect the spectrum of a CFT. Based on the modular bootstrap framework developed by Lin and Shao for the Z2 symmetry, the analysis was first analytically extended to the Abelian case Z3. This was useful in extending the discussion to a more complex case, namely the non-Abelian symmetry S3. This extension constitutes the original theoretical contribution of the thesis, providing the explicit formulation of modular bootstrap equations for CFTs with non-Abelian discrete symmetry. The analytical modular constraints were reformulated numerically, bridging the modular bootstrap framework with semidefinite programming in SDPB. This analysis paves the way for a numerical study based on convex optimization techniques to explore the constraints arising from modular bootstrap equations.

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Two-dimensional Conformal Field Theories (CFTs) allow us to describe quantum field theories at critical points and in general systems without a characteristic scale. It is possible to implement such theories on topologically non-trivial surfaces. If we formulate a two-dimensional CFT on a toroidal surface, a new symmetry emerges known as modular invariance. This symmetry corresponds to the set of transformations that act without changing the toroidal surface, consequently, the partition function of the theory must remain invariant under these transformations. This invariance is used as a constraint by the modular bootstrap method to obtain information about the spectrum without using the Lagrangian and operator content of the theory. By imposing that the theory is consistent with modular symmetry, it is possible to determine the limits of the spectrum beyond which modular invariance can no longer be satisfied, thus constraining the space of admissible two-dimensional conformal field theories. To study CFTs that present discrete global symmetries, it is essential to introduce Topological Defect Lines (TDLs). These are one-dimensional, topologically invariant operators that implement the action of the symmetry group on the Hilbert space. The presence of TDLs modifies the topology of the system and introduces new partition functions associated with different symmetry sectors. Under modular transformations, the TDLs mix among themselves, and consequently, the corresponding partition functions also mix, thus promoting the modular constraint to a vector-valued condition. This formalism makes it possible to implement discrete symmetries also considering the possible presence of anomalies. The objective of this thesis is therefore to understand how discrete symmetries and anomalies affect the spectrum of a CFT. Based on the modular bootstrap framework developed by Lin and Shao for the Z2 symmetry, the analysis was first analytically extended to the Abelian case Z3. This was useful in extending the discussion to a more complex case, namely the non-Abelian symmetry S3. This extension constitutes the original theoretical contribution of the thesis, providing the explicit formulation of modular bootstrap equations for CFTs with non-Abelian discrete symmetry. The analytical modular constraints were reformulated numerically, bridging the modular bootstrap framework with semidefinite programming in SDPB. This analysis paves the way for a numerical study based on convex optimization techniques to explore the constraints arising from modular bootstrap equations.
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We study constraints on the space of d=2d=2d=2 fermionic CFTs as a function of non-perturbative anomalies exhibited under a fermionic discrete symmetry group GfG^fGf, focusing our attention also on cases where GfG^fGf is non-abelian or presents a non-trivial twist of the Z2f\mathbb{Z}^f_2Z2f​ subgroup. For the cases we selected, among our results we find that modular bootstrap consistency bounds predict the presence of relevant/marginal operators only for some groups and anomalies. From this point of view, the appearance in the analysis of several kinks around irrelevant operators with Δ>2Δ>2Δ>2 means that for fermionic systems with increasingly larger symmetry groups modular bootstrap is able to give less constraining bounds than its bosonic counterpart. Within our analysis we show how the anomaly constraints on fermionic CFTs can be effectively recovered from the structure of the abelian subgroups of GfG^fGf. Finally, we extend the previous surgery description of bordism invariants that describe 3d3d3d abelian spin-TQFTs, in order to include the case of theories with Spin-Z2l+1f\mathrm{Spin}\text{-}\mathbb{Z}^f_{2^{l+1}}Spin-Z2l+1f​ structures.
We develop a neural network bootstrap framework for reconstructing partition functions of two-dimensional conformal field theories (CFTs) based on modular invariance and the Cardy condition, which are recast as crossing equations for four-point correlators. For torus partition functions, we use the twist-field representation in the symmetric-orbifold description to map modular S-invariance to four-point crossing and focus on the diagonal kinematics of four insertions on a line. For annulus partition functions, we formulate open/closed channel duality as crossing symmetry for mixed four-point functions of defect-changing operators in interface CFT. In both cases, the reconstruction problem is formulated in the anchored-bootstrap form, where the crossing constraints are supplemented by minimal spectral input (a gap) and anchor data. We solve this under-determined problem by using lightweight feed-forward neural networks to parametrise the correlators and their corresponding partition functions. A key ingredient of this approach is the spectral bias of the neural networks in the lazy training regime, which selects specific crossing-symmetric configurations. This reformulation unifies standard modular and annulus constraints in two dimensions with the anchored neural approach for CFT correlators, providing a new way to reconstruct full partition functions from sparse data with remarkable accuracy.
We develop a functional-analytic framework for modular constraints in two-dimensional conformal field theory based on the reflection symmetry [Formula: see text]. This symmetry isolates a natural reflection-odd subspace of [Formula: see text] in which all deviations from modular self-duality live. For any modular-invariant partition function, the vacuum-heavy decomposition determines a uniquely defined reflection-odd modular defect [Formula: see text] whose completed Mellin transform satisfies an odd functional equation. Pairing D with compactly supported reflection-odd test functions yields a structured hierarchy of linear functionals. Under mild assumptions on the light spectrum, these pairings obey positivity conditions that lead to smeared Cardy inequalities comparing the heavy spectrum to the Cardy density through positive Laplace kernels. We also show that the vanishing of all such pairings is equivalent to averaged Cardy saturation in [Formula: see text], giving a complete reflection-odd characterization of modular self-duality. This provides a simple and flexible functional basis for modular bootstrap analyses, replacing ad hoc constructions with a unified reflection-odd hierarchy suited for both analytic arguments and numerical implementation.
We consider unitary, modular invariant, two-dimensional CFTs which are invariant under the parity transformation P P . Combining P P with modular inversion S S leads to a continuous family of fixed points of the SP S P transformation. A particular subset of this locus of fixed points exists along the line of positive left- and right-moving temperatures satisfying \beta_L \beta_R = 4\pi^2 β L β R = 4 π 2 . We use this fixed locus to prove a conjecture of Hartman, Keller, and Stoica that the free energy of a large- c c CFT _2 2 with a suitably sparse low-lying spectrum matches that of AdS _3 3 gravity at all temperatures and all angular potentials. We also use the fixed locus to generalize the modular bootstrap equations, obtaining novel constraints on the operator spectrum and providing a new proof of the statement that the twist gap is smaller than (c-1)/12 ( c − 1 ) / 12 when c>1 c > 1 . At large c c we show that the operator dimension of the first excited primary lies in a region in the (h,\overline{h}) ( h , h ¯ ) -plane that is significantly smaller than h+\overline{h}<c/6 h + h ¯ > c / 6 . Our results for the free energy and constraints on the operator spectrum extend to theories without parity symmetry through the construction of an auxiliary parity-invariant partition function.