Modular Crossing and Spectral Bounds

Modular invariance becomes a bootstrap equation once the vacuum is isolated and every remaining character contribution is treated as unknown spectral data. Positivity can then exclude a proposed gap at finite energy. Cardy growth is a different result: it follows from a high-temperature inverse transform and needs additional hypotheses about the vacuum, the measure, and the averaging scale.

Required background. Torus Partition Functions as Bootstrap Data fixes the trace and modular conventions; crossing and positivity supplies the separating-functional logic. Helpful background. Linear functionals and positivity develops numerical certificates, while Tauberian theorems and asymptotic spectral data explains why an averaged asymptotic statement is generally stronger than a pointwise one.

From S invariance to a positive sum rule

Set the spatial radius to one and restrict first to a rectangular torus,

$$\tau=\frac{i\beta}{2\pi}, \qquad Z(\beta)=\operatorname{Tr} e^{-\beta(L_0+\bar L_0-c_{\mathrm{tot}}/24)}, \qquad c_{\mathrm{tot}}=c_L+c_R.$$

The $S$ transformation gives

$$Z(\beta)=Z(\beta'), \qquad \beta'=\frac{4\pi^2}{\beta}.$$

For a state or character contribution of shifted energy

$$E=\Delta-\frac{c_{\mathrm{tot}}}{24},$$

introduce the crossing vector

$$F_E(\beta)=e^{-\beta E}-e^{-\beta' E}.$$

After packaging all vacuum descendants into $F_{\mathrm{vac}}$ and choosing a basis for nonvacuum primaries, modular crossing has the schematic form

$$F_{\mathrm{vac}}(\beta) +\sum_{\Delta,J}d_{\Delta,J}F_{\Delta,J}(\beta)=0, \qquad d_{\Delta,J}\ge0.$$

The last inequality uses a unitary Hilbert-space trace and a basis in which multiplicities are nonnegative. A continuous spectrum replaces the sum by an integral against a positive measure. If characters mix signs, if a sector is twined, or if the theory is nonunitary, this scalar positivity argument must be modified rather than assumed.

The self-dual point is $\beta=2\pi$. Every undifferentiated $F_E$ vanishes there, but odd derivatives under $\beta\leftrightarrow4\pi^2/\beta$ do not. For example,

$$\left.\beta\partial_\beta F_E(\beta)\right|_{\beta=2\pi} =-4\pi E e^{-2\pi E}.$$

Applying this derivative to crossing gives the exact moment identity

$$\sum_{\text{all states}}d_E\,E e^{-2\pi E}=0,$$

with the vacuum included. Since $E_{\mathrm{vac}}=-c_{\mathrm{tot}}/24$, positive-energy states must compensate its negative contribution. Useful gap bounds combine several odd derivatives so that the resulting function has a controlled sign on an entire proposed spectral region.

The exclusion-functional theorem

Let $\mathcal A(\Delta_*)$ be a proposed spectrum—for example, all nonvacuum scalar primaries satisfying $\Delta\ge\Delta_*$, with other spins subject to declared unitarity bounds. Suppose a linear functional $\alpha$ on the crossing functions satisfies

$$\alpha(F_{\mathrm{vac}})>0, \qquad \alpha(F_{\Delta,J})\ge0 \quad\text{for every }(\Delta,J)\in\mathcal A(\Delta_*).$$

Acting on crossing would make a sum of nonnegative terms equal zero while leaving one strictly positive term. Therefore no spectrum contained in $\mathcal A(\Delta_*)$ exists: the assumed gap is excluded.

This is a finite spectral conclusion even when $\alpha$ uses only finitely many derivatives. Its validity does not come from the derivative order; it comes from proving the sign condition on the full allowed continuum of $(\Delta,J)$. Polynomial or semidefinite representations can make that proof rigorous. Sampling the sign at finitely many dimensions is only a diagnostic.

Hellerman’s low-order construction gives a useful benchmark. Under its stated assumptions—unitarity, modular invariance, a discrete spectrum, $c_L,c_R>1$, and no extended chiral algebra—the lightest nontrivial primary obeys

$$0<\Delta_1<\frac{c_L+c_R}{12}+0.473695\ldots.$$

This is a finite-$c$ bound, not a large-$c$ or Cardy estimate Hellerman 2011, §§ 2–3. Changing the chiral algebra changes the vacuum character and hence the crossing vectors; the numerical constant cannot simply be transplanted.

Systematic derivative functionals improve or refine such bounds, but every reported result must state the derivative basis, spin resolution, assumed spectral gaps, character convention, arithmetic precision, and how positivity was certified Friedan and Keller 2013, §§ 2–4.

Degeneracy bounds and truncated problems

A degeneracy can be bounded by normalizing a functional on a selected contribution. If a state at $(\Delta_0,J_0)$ has multiplicity $d_0$, crossing implies

$$d_0\alpha(F_{\Delta_0,J_0}) =-\alpha(F_{\mathrm{vac}}) -\sum_{(\Delta,J)\ne(\Delta_0,J_0)} d_{\Delta,J}\alpha(F_{\Delta,J}).$$

If $\alpha(F_{\Delta_0,J_0})=1$ and the remaining functional values are nonnegative, then

$$d_0\le-\alpha(F_{\mathrm{vac}}),$$

provided the right side is nonnegative. The direction changes if the normalization or signs change. This elementary step is why every functional convention should be written before quoting a number.

Two truncations are often present:

1. a derivative truncation, restricting $\alpha$ to finitely many derivatives at the self-dual point; and
2. a spectral or character truncation, replacing an infinite sum, measure, or descendant series by a finite representation.

The first can still yield a theorem if global positivity is certified. The second needs an omitted-tail bound. For a positive state density binned into unit energy intervals, suppose

$$d_n\le A e^{\kappa n} \quad\text{for }n>N, \qquad \beta>\kappa.$$

Then the discarded canonical tail obeys

$$\sum_{n>N}d_ne^{-\beta n} \le \frac{A e^{-(\beta-\kappa)(N+1)}} {1-e^{-(\beta-\kappa)}}.$$

Without a proved or explicitly estimated $A,\kappa$, a stable-looking truncated residual is not an error bound.

Cardy growth requires a second argument

Now take $\beta\to0^+$. Modular invariance maps this limit to $\beta'=4\pi^2/\beta\to\infty$. If the vacuum is unique and dominates the low-temperature trace,

$$Z(\beta') =e^{\beta'c_{\mathrm{tot}}/24} \left[1+o(1)\right].$$

It follows that

$$\log Z(\beta) =\frac{\pi^2c_{\mathrm{tot}}}{6\beta} +o(\beta^{-1}).$$

A formal inverse Laplace transform with shifted energy $E=\Delta-c_{\mathrm{tot}}/24$ has saddle

$$\beta_*=\pi\sqrt{\frac{c_{\mathrm{tot}}}{6E}},$$

and leading entropy

$$S(E)\sim2\pi\sqrt{\frac{c_{\mathrm{tot}}E}{6}}.$$

This is the spin-summed leading Cardy exponent in the present convention. A left-right refined formula requires a two-variable transform and control over the angular potential. Cardy’s original modular argument identifies the universal high-energy growth Cardy 1986, pp. 186–204.

The saddle is not a license to claim a pointwise degeneracy for every individual $E$. A robust statement specifies the spectral object being estimated—cumulative counting function, windowed density, or distribution—and the window width. For a positive modular spectral density, Tauberian methods turn canonical asymptotics into upper and lower microcanonical bounds with controlled errors Mukhametzhanov and Zhiboedov 2019, §§ 2–5. Narrower windows or fixed-spin claims require stronger hypotheses.

The required checks are distinct:

Claim	Energy regime	Essential hypotheses	What can invalidate it
Functional gap exclusion	finite	exact crossing, correct vacuum block, nonnegative multiplicities or measure, global functional sign	missed sector, wrong character basis, uncertified sign
Degeneracy bound	finite	the above plus a normalized target contribution	target mixing, multiplicity convention, omitted tail
Canonical Cardy free energy	$\beta\to0^+$	$S$ covariance, unique or declared vacuum sector, low-temperature dominance	extra zero-energy states, light-state accumulation, multiplier mismatch
Microcanonical Cardy growth	$E\to\infty$	canonical asymptotics plus positivity and an inverse-transform or Tauberian theorem	signed measure, window too narrow, nonuniform angular-potential limit

Light states, spin, and multipliers

Vacuum dominance means more than writing the vacuum term first. An accumulating continuum near $E_{\mathrm{vac}}$, an extensive ground-state degeneracy, or a sector with a lower effective vacuum energy can compete with it. In a flavored trace, the dominant polar term may depend on the chemical-potential contour. In a fermionic theory, $S$ can exchange spin structures, so the high-temperature behavior of one component is controlled by the low-temperature spectrum of another.

At nonzero $\operatorname{Re}\tau$, the trace resolves spin. Positivity then applies to the Hilbert-space multiplicities, while phases from $e^{2\pi i(\operatorname{Re}\tau)J}$ can make the evaluated function complex. A spin-resolved functional must preserve the full $S,T$ transformation law. If $c_L\ne c_R$, the multiplier must also be retained; imposing the nonanomalous scalar equation would solve a different problem.

Common failure modes

Using a finite grid as a positivity proof. A functional can change sign between sampled dimensions or at an untested spin. Certify the continuum condition analytically or with a rigorous polynomial representation.

Mixing descendants and primaries. A state-level density and a Virasoro-primary density have different vacuum factors and different effective central-charge shifts. State which one the crossing vectors represent.

Calling Cardy a finite bound. The leading exponent controls an asymptotic density or averaged count under inverse-transform hypotheses. It does not bound the first excitation at finite energy.

Ignoring the tail of a truncation. Increasing a cutoff until a plot looks stable is useful evidence, but it is not a quantified remainder. Record the tail estimate and vary derivative and spectral cutoffs independently.

Exercises

Derive the first-derivative moment equation from $Z(\beta)=Z(4\pi^2/\beta)$.

Solution

Write $Z(\beta)=\sum_Ed_Ee^{-\beta E}$. Differentiating the crossing difference and evaluating at $\beta=2\pi$ gives

$$0=\sum_Ed_E \left.\beta\partial_\beta F_E\right|_{2\pi} =-4\pi\sum_Ed_EE e^{-2\pi E}.$$

Dividing by the nonzero common factor yields the stated identity.

Assume $Z(\beta)\sim e^{A/\beta}$ as $\beta\to0^+$ with $A>0$. Find the leading inverse-Laplace exponent at large $E$.

Solution

The exponent in the inverse transform is $\Phi(\beta)=\beta E+A/\beta$. Its stationary point is $\beta_*=\sqrt{A/E}$, and

$$\Phi(\beta_*)=2\sqrt{AE}.$$

For $A=\pi^2c_{\mathrm{tot}}/6$, this becomes $2\pi\sqrt{c_{\mathrm{tot}}E/6}$. Establishing the prefactor or a windowed error requires more than this saddle.

Continue the calculation

Spin, Charges, and Extended Modular Sectors replaces the scalar equation by a modularly closed sector vector.

A reproducible calculation should compare derivative orders and spectral cutoffs, require an omitted-character tail estimate, and keep the thermal KMS fixture separate.

References

• Cardy, John L. “Operator Content of Two-Dimensional Conformally Invariant Theories.” Nuclear Physics B 270, no. 2 (1986): 186–204. doi:10.1016/0550-3213(86)90552-3.
• Friedan, Daniel, and Christoph A. Keller. “Constraints on 2d CFT Partition Functions.” Journal of High Energy Physics 2013, no. 10 (2013): 180. doi:10.1007/JHEP10(2013)180. Open preprint.
• Hellerman, Simeon. “A Universal Inequality for CFT and Quantum Gravity.” Journal of High Energy Physics 2011, no. 8 (2011): 130. doi:10.1007/JHEP08(2011)130. Open preprint.
• Mukhametzhanov, Baur, and Alexander Zhiboedov. “Modular Invariance, Tauberian Theorems, and Microcanonical Entropy.” Journal of High Energy Physics 2019, no. 10 (2019): 261. doi:10.1007/JHEP10(2019)261. Open preprint.