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arXiv:1705.05865

Genus Two Modular Bootstrap

Minjae Cho, Scott Collier, Xi Yin

Published May 16, 2017• 4 citations
arXivHTMLPDF

Abstract

We study the Virasoro conformal block decomposition of the genus two partition function of a two-dimensional CFT by expanding around a Z3-invariant Riemann surface that is a three-fold cover of the Riemann sphere branched at four points, and explore constraints from genus two modular invariance and unitarity. In particular, we find 'critical surfaces' that constrain the structure constants of a CFT beyond what is accessible via the crossing equation on the sphere.

DOI
10.48550/arxiv.1705.05865

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We study the Virasoro conformal block decomposition of the genus two partition function of a two-dimensional CFT by expanding around a Z3-invariant Riemann surface that is a three-fold cover of the Riemann sphere branched at four points, and explore constraints from genus two modular invariance and unitarity. In particular, we find 'critical surfaces' that constrain the structure constants of a CFT beyond what is accessible via the crossing equation on the sphere.
This thesis studies how self-consistency constraints on conformal field theory like observables can be used to extract information about CFT or geometric data. The observables considered are associated with genus-0, genus-1, and genus-2 geometries: the four-punctured sphere, the torus S1 x Sd-1, and the genus-2 manifold (S1 x Sd-1)♯(S1 x Sd-1). Although these observables differ in geometric and analytic complexity, they are united by a common principle: the same observable admits multiple equivalent decompositions, and the requirement that these descriptions agree imposes nontrivial constraints on the underlying spectrum and operator product expansion. The first part develops a hyperbolic bootstrap for compact hyperbolic spin surfaces. Using an analogy between harmonic analysis on hyperbolic manifolds and the conformal bootstrap, we derive rigorous bounds on Laplacian and Dirac spectral gaps by combining representation-theoretic methods with semidefinite programming. The second part studies thermal partition functions with angular twists. We show that, even in the absence of full modular invariance in dimensions greater than two, the high-temperature behavior of these observables is controlled by thermal effective field theory, leading to universal relations among twisted partition functions. The third part studies the genus-2 partition function in conformal field theory. By equating different decompositions of this observable, we derive a genus-2 crossing equation and use it to relate thermal one-point data to heavy operator product coefficients.
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.
A bstract In this paper, we attempt to explore the landscape of two-dimensional conformal field theories (2d CFTs) by efficiently searching for numerical solutions to the modular bootstrap equation using machine-learning-style optimization. The torus partition function of a 2d CFT is fixed by the spectrum of its primary operators and its chiral algebra, which we take to be the Virasoro algebra with c > 1. We translate the requirement that this partition function is modular invariant into a loss function, which we then minimize to identify possible primary spectra. Our approach involves two technical innovations that facilitate finding reliable candidate CFTs. The first is a strategy to estimate the uncertainty associated with truncating the spectrum to the lowest dimension operators. The second is the use of a new singular-value-based optimizer (Sven) that is more effective than gradient descent at navigating the hierarchical structure of the loss landscape. We numerically construct candidate truncated CFT partition functions with central charges between 1 and 87\frac{8}{7}78​ , a range devoid of known examples, and argue that these candidates likely come from a continuous space of modular bootstrap solutions. We also provide evidence for a more stringent constraint on the spectral gap near c = 1 than the existing bound of Δgap≤c6+13{\Delta}_{\text {gap}}\le \frac{c}{6}+\frac{1}{3}Δgap​≤6c​+31​ .
A genus-2 crossing equation in d≥2d\geq 2d≥2
We explore a "genus-2" crossing equation obeyed by CFTs in general dimensions d≥2d\geq 2d≥2. This crossing equation relates two different decompositions of the "genus-2 partition function" -- namely the partition function on the connected sum M2=(S1×Sd−1)♯(S1×Sd−1)M_2=(S^1\times S^{d-1})\sharp (S^1\times S^{d-1})M2​=(S1×Sd−1)♯(S1×Sd−1). The "sunrise" channel decomposition expresses M2M_2M2​ as a pair of three-punctured spheres glued together with cylinders, while the "dumbbell" channel decomposition expresses M2M_2M2​ as a gluing of two one-point functions on S1×Sd−1S^1\times S^{d-1}S1×Sd−1. We introduce coordinates to describe each channel, and write down Casimir equations obeyed by the corresponding blocks. We also explain why equality between the two channels guarantees mapping class group invariance of the genus-2 partition function in 3d CFTs. As an application of the genus-2 crossing equation, we derive a novel relation between asymptotics of "heavy-heavy-heavier" OPE coefficients and squares of thermal one-point coefficients in 3d CFTs. Along the way, we demonstrate how expectation values of conformal generators can help locate saddle points in large quantum number limits.