Modular and Thermal Bootstrap

Finite size and finite temperature produce two different bootstrap problems. In two dimensions, changing torus cycles constrains the spectrum in a partition function. Thermal trace cyclicity instead constrains ordered local correlators in any dimension. This chapter develops both and explains why modular covariance and thermal periodicity carry different information.

Enter this chapter

Begin with Torus Partition Functions as Bootstrap Data for the modular route. For local finite-temperature observables, begin with Thermal States and One-Point Data.

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.

Topics

1. Torus Partition Functions as Bootstrap Data — Specifies the state trace, vacuum shift, sectors, and spectral measure.
2. Modular Crossing and Spectral Bounds — Uses modular consistency to constrain gaps and distinguish bounds from asymptotic growth.
3. Spin, Charges, and Extended Modular Sectors — Keeps spin structures, charge insertions, and twisted sectors closed under modular transformations.
4. Higher-Genus and Mapping-Class Constraints — Uses sewing and degeneration to constrain three-point data beyond the torus spectrum.
5. Thermal States and One-Point Data — Identifies new expectation values associated with a thermal ensemble.
6. Thermal OPE and KMS Crossing — Combines the local OPE with Kubo–Martin–Schwinger cyclicity.
7. Thermal Inversion and Sum Rules — Extracts thermal coefficients while retaining growth conditions and low-spin arcs.

How the ideas fit together

Both routes compare two descriptions of the same quantity, but the quantities differ. A torus partition function is a global state trace, with a modular transformation exchanging how its cycles are described. The Kubo–Martin–Schwinger (KMS) condition moves a local operator through a thermal density matrix. Inserting an OPE into that condition relates vacuum OPE coefficients to thermal one-point values. Iliesiu et al. 2018, §§ 2.1–2.2.

Positive state degeneracies in a unitary torus trace do not make all thermal OPE coefficients positive: those coefficients also contain state-dependent expectation values. A finite OPE or character truncation further needs control of the omitted tail.

Review the chapter

Why does knowing the vacuum OPE not determine a finite-temperature two-point function?

Solution

The OPE fixes operator coefficients, but taking its thermal expectation value introduces one-point values in the chosen ensemble. Symmetry can determine their tensor form and temperature scaling without fixing their dimensionless coefficients. KMS consistency constrains this additional state data.

Where to go next

Continue to Thermal Inversion and Sum Rules to extract the additional coefficients, or to Higher-Genus and Mapping-Class Constraints for global constraints beyond a torus trace.

References

• 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.