Modular and Thermal Bootstrap

Two different consistency principles organize finite-size and finite-temperature CFT data. In two dimensions, changing the cycles used to describe a torus constrains a global trace over the Hilbert space. In any dimension, cyclicity of a thermal trace constrains ordered local correlators through the KMS condition. Both become bootstrap equations, but they act on different observables and require different hypotheses.

Helpful background. The chapter uses chiral blocks, sewing, and modularity to decompose torus amplitudes, OPE convergence and associativity to control local thermal expansions, and nonrational modular spectral densities when a discrete character sum must be replaced by an integral.

Enter this chapter

The chapter has a deliberate fork.

Question	Observable	Consistency operation	Unknown data	Typical limitation
Two-dimensional torus bootstrap	$Z(\tau,\bar\tau)$, possibly with sector or charge labels	$S$ and $T$ changes of the torus cycles	dimensions, spins, degeneracies, sector measures	anomaly multipliers, continuum spectra, or incomplete sector closure
Higher-genus bootstrap	$Z_g(\Omega,\bar\Omega)$ or sewn conformal blocks	mapping-class transformations and degeneration limits	spectrum and products of three-point coefficients	block technology, moduli coverage, and truncation tails
Local thermal bootstrap	ordered correlators on $S^1_\beta\times\mathbb R^{d-1}$	KMS cyclicity plus the local OPE	products of vacuum OPE coefficients and thermal one-point coefficients	no general coefficient positivity and a finite OPE domain
Thermal inversion	an analytically continued thermal two-point function	a discontinuity integral at fixed spin	thermal one-point data in one OPE	polynomial-growth assumptions, arc terms, and low-spin ambiguities

Torus modular covariance is therefore not another name for KMS periodicity. The first compares equivalent presentations of a compact two-dimensional surface and uses the full state trace. The second moves an operator through a density matrix and depends on operator ordering. A two-dimensional torus partition function can of course be interpreted thermally after choosing one cycle as Euclidean time, but that special identification does not turn a local KMS equation in general dimension into modular invariance. The standard torus construction and the finite-temperature OPE make this distinction explicit Di Francesco, Mathieu, and Sénéchal 1997, § 10.1, pp. 336–340 Iliesiu et al. 2018, § 2, pp. 5–13.

The modular route

Start with Torus Partition Functions as Bootstrap Data. It fixes the trace, Casimir shifts, character basis, continuum measure, and the distinction between invariance and covariance under $S$ and $T$.

Then use Modular Crossing and Spectral Bounds to isolate the vacuum and construct positive sum rules. That page sharply separates two conclusions:

• a finite-dimensional exclusion functional can prove a finite spectral bound once its sign conditions hold on the entire allowed spectrum; and
• Cardy growth is an asymptotic inverse-transform statement that additionally needs vacuum dominance, positivity or controlled signed measures, and Tauberian or saddle-point hypotheses.

Refinements belong on Spin, Charges, and Extended Modular Sectors. Spin structures, symmetry twists, charge chemical potentials, defects, and extended characters generally form vectors under the modular group. Testing one component is not enough: the declared set must close under both generators, including anomaly phases.

Higher-Genus and Mapping-Class Constraints adds data that the genus-one trace cannot see. Sewing a genus-two surface exposes products of three-point coefficients, while separating and nonseparating degenerations recover lower-genus factorization limits. The physical sewing equations are developed here; rigorous conformal-block bundles and general mapping-class theorems, handlebody interpretations, and curved-background applications are taken up in the corresponding later volumes.

The thermal route

Thermal States and One-Point Data begins with a normalized density matrix and fixes the geometry, ensemble, stress-tensor normalization, and selection rules. On the flat thermal cylinder, a spin-$J$ primary that contains a rotational singlet can acquire a one-point function proportional to $\beta^{-\Delta}$; its dimensionless coefficient is new state data.

Thermal OPE and KMS Crossing inserts those one-point functions into a convergent local OPE. For an identical bosonic scalar, cyclicity of the trace gives reflection about $\tau=\beta/2$. The resulting crossing equation is linear in thermal coefficients, but they need not be positive. A generalized-free image sum supplies a benchmark whose periodicity and omitted-image tail can be checked directly.

Finally, Thermal Inversion and Sum Rules explains two complementary extractions. Derivatives at the KMS-symmetric point give linear sum rules. Analytic continuation and a discontinuity integral can recover thermal coefficients as poles in dimension, but only above the spin threshold implied by large-boost growth; at lower spin, arc or subtraction terms must be retained.

A disciplined workflow

For a modular problem, record the left and right central charges, vacuum multiplicity, allowed spins, spectral measure, modular multiplier, sector vector, and convergence domain before applying a functional. For a thermal problem, record the spacetime dimension, ensemble, normalization of the density matrix and operators, Euclidean ordering, statistics phase, OPE domain, analytic continuation, and tail estimate before truncating or inverting.

A strong conclusion is the one that survives those declarations. Exact modular covariance does not by itself validate a numerical truncation. A small KMS residual does not by itself prove that an omitted OPE tail is small. A high-temperature free energy does not by itself give pointwise degeneracies. An inversion integral without its arc analysis does not determine low-spin data.

Review the chapter

By the end of the seven leaves, you should be able to:

1. construct a torus trace and verify the Ising character transformations under $S$ and $T$;
2. formulate a positive modular sum rule and explain exactly what a finite functional proves;
3. build a sector vector that closes under modular transformations;
4. identify the three-point data and factorization checks in a genus-two sewing expansion;
5. parameterize thermal one-point functions in a fixed ensemble;
6. derive scalar KMS crossing and bound a generalized-free image tail; and
7. state the analytic, growth, spin, and arc hypotheses of a thermal inversion.

References

• Di Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Graduate Texts in Contemporary Physics. New York: Springer, 1997. doi:10.1007/978-1-4612-2256-9.
• Iliesiu, Luca, Murat Koloğlu, Raghu Mahajan, Eric Perlmutter, and David Simmons-Duffin. “The Conformal Bootstrap at Finite Temperature.” Journal of High Energy Physics 2018, no. 10 (2018): 070. doi:10.1007/JHEP10(2018)070. Open preprint.