Higher-Genus and Mapping-Class Constraints

Higher genus adds information that no torus trace contains: sewing a surface through pairs of pants weights propagation channels by products of three-point coefficients. Equality under a mapping-class move therefore constrains both the spectrum and OPE data. The price is a larger moduli space, channel-dependent positivity, and nontrivial control of conformal blocks and sewing tails.

Required background. Spin, charges, and extended modular sectors supplies vector-valued modular actions, and chiral blocks, sewing, and modularity supplies the genus-one sewing language. Helpful background. Smooth manifolds and tensors reviews the geometric structures used to describe moduli and homology cycles.

Literature cutoff. The account of higher-genus bootstrap methods and limitations below reflects sources available through 2026-08-09. It states durable sewing and factorization principles, not a claim that present block algorithms solve the general higher-genus problem.

Sewing a genus-two surface

A closed oriented surface of genus $g\ge2$ can be cut into $2g-2$ pairs of pants joined by $3g-3$ tubes. Choose local coordinates at every circular boundary and a complete basis of states on each tube. Propagation through a tube with plumbing parameter $q_a$ contributes

$$q_a^{L_0-c_L/24} \bar q_a^{\bar L_0-c_R/24}, \qquad \lvert q_a\rvert<1.$$

The inverse two-point Gram matrix must be inserted when the state basis is not orthonormal. The local coordinates, Gram normalization, and Weyl-anomaly factor are part of the sewing prescription; changing them while keeping the coefficients fixed changes the amplitude.

For genus two, a convenient pants decomposition has two three-holed spheres joined by three tubes. In a diagonal, reflection-positive theory and an orthonormal primary basis, its primary-level structure is

$$Z_2(\mathbf q,\bar{\mathbf q}) =\sum_{i,j,k} \lvert C_{ijk}\rvert^2 \mathcal F_{ijk}(\mathbf q) \bar{\mathcal F}_{ijk}(\bar{\mathbf q}),$$

where $\mathbf q=(q_1,q_2,q_3)$ and each block includes all descendants in the three channels. More generally the coefficient is

$$C_{ijk}\, \bar C_{\bar i\bar j\bar k}$$

contracted with the left-right pairing and inverse Gram matrices; it need not be an absolute square. The special positive form is valuable for functionals, but it is a property of the pairing and reflection frame, not of every higher-genus expansion. Genus-two Virasoro blocks and a positive threefold-pillow frame are constructed in Cho, Collier, and Yin 2019, §§ 2–3.

The leading primary term in a narrow-tube limit behaves schematically as

$$\mathcal F_{ijk}(\mathbf q) \sim \prod_{a=1}^{3} q_a^{h_{i_a}-c_L/24} \left(1+O(q_a)\right),$$

with $(i_1,i_2,i_3)=(i,j,k)$. The precise universal prefactor depends on the chosen sewing coordinates and conformal frame. A numerical comparison is meaningful only after that prefactor and all three coordinate conventions agree.

Period matrices and mapping-class action

Choose a symplectic homology basis $(A_I,B_I)$, $I=1,\ldots,g$. The period matrix obeys

$$\Omega=\Omega^{\mathsf T}, \qquad \operatorname{Im}\Omega>0.$$

A change of symplectic basis

$$\Gamma= \begin{pmatrix}A&B\\C&D\end{pmatrix} \in Sp(2g,\mathbb Z)$$

acts by

$$\Omega\longmapsto (A\Omega+B)(C\Omega+D)^{-1}.$$

At genus one this reduces to the familiar modular action on $\tau$. At higher genus, however, the full object is a surface with a marking, local sewing data, and conformal blocks. The mapping-class group acts projectively on chiral blocks, while a consistent nonchiral amplitude combines left and right sectors with the appropriate anomaly factor. For $g>1$, the symplectic action records the effect on homology but does not replace the full mapping-class representation on blocks; for $g\ge4$, period matrices also lie on the Jacobian locus rather than filling the entire Siegel upper half-space.

Let $P$ and $P'$ be two pants decompositions related by a mapping-class element $\gamma$. The physical consistency equation is

$$\sum_{\boldsymbol i} \mathcal C^{(P)}_{\boldsymbol i} \mathcal F^{(P)}_{\boldsymbol i}(m) = \mathcal A_\gamma(m) \sum_{\boldsymbol j} \mathcal C^{(P')}_{\boldsymbol j} \mathcal F^{(P')}_{\boldsymbol j}(\gamma\cdot m),$$

where $m$ denotes moduli and sewing coordinates, $\mathcal C$ contains the contracted three-point data, and $\mathcal A_\gamma$ is the known multiplier or conformal-anomaly factor in the selected trivialization. Treating $\mathcal A_\gamma$ as one without first canceling the anomaly is an incorrect equation.

A crossing-symmetric point fixed by part of the mapping-class group permits derivative functionals, much as $\tau=i$ does at genus one. The key difference is that the unknown nonnegative quantities can be $\lvert C_{ijk}\rvert^2$ rather than only degeneracies. On a one-complex-dimensional $\mathbb Z_3$-symmetric locus, such functionals constrain three-point data, but that locus does not cover the full three-complex-dimensional genus-two moduli space Cho, Collier, and Yin 2019, §§ 3–4.

Degeneration limits are nonnegotiable checks

Every proposed higher-genus amplitude must reproduce lower-complexity surfaces at the boundary of moduli space.

Separating degeneration

Pinching a cycle that separates a genus-two surface produces two tori connected by a long tube. If $t$ is the plumbing parameter, then after extracting the universal sewing factor,

$$Z_2 \longrightarrow Z_1(\tau_1)Z_1(\tau_2) +\sum_{\mathcal O\ne\mathbf1} t^{h_{\mathcal O}} \bar t^{\bar h_{\mathcal O}} \langle\mathcal O\rangle_{\tau_1} \langle\mathcal O^\dagger\rangle_{\tau_2} +\cdots.$$

The leading vacuum term checks normalization. The subleading terms check torus one-point functions and the inverse two-point metric. A theory with vanishing torus one-point functions in a symmetry sector must not acquire them through a mismatched sewing basis.

Nonseparating degeneration

Pinching a nonseparating cycle lowers the genus by one and leaves two punctures. Schematically,

$$Z_g \longrightarrow \sum_i t^{h_i-c_L/24} \bar t^{\bar h_i-c_R/24} \left\langle \mathcal O_i(p)\mathcal O_i^\dagger(q) \right\rangle_{g-1}.$$

This checks the spectrum, conjugation, and two-point normalization. The $c/24$ shifts shown here belong to the long-cylinder propagation convention; if a different universal anomaly prefactor is extracted, the displayed powers must be adjusted consistently.

Pants-channel factorization

Taking all three genus-two tubes long isolates products of three-point coefficients. In the positive diagonal frame, the first nonvacuum term identifies the lightest triple with nonzero $C_{ijk}$. The torus spectrum alone cannot decide whether that coefficient vanishes, which is exactly why genus two adds information.

Mapping-class closure and truncation control

The following table records the checks required before a truncated sewing equation can support a bound. It separates exact geometric identities from estimates on discarded states.

Channel or object	Data propagated	Required transformation or limit	Positivity domain	Omitted-tail condition
Three-tube genus-two channel	three primaries, descendants, and two three-point vertices	specified mapping-class move between pants decompositions	$\lvert C_{ijk}\rvert^2\ge0$ only in a reflection-positive diagonal frame	group terms by total sewing energy and bound the absolute coefficient sum
Separating node	operator exchanged between two tori	$t\to0$ gives two torus amplitudes after the universal prefactor	vacuum term positive in a unitary normalized trace	first omitted operator contributes at its declared $\Delta$; include torus one-point bounds
Nonseparating node	conjugate pair inserted on genus $g-1$	$t\to0$ gives a two-point function on the lower-genus surface	positivity depends on conjugate pairing and frame	retain inverse Gram norms and bound descendant as well as primary tails
Period-matrix description	marked $\Omega$ and chiral block vector	$(A\Omega+B)(C\Omega+D)^{-1}$ plus projective block action	no component-wise positivity in a generic complex frame	transformed point must remain inside a domain where both expansions converge
Total-energy truncation $n\le N$	absolute grouped coefficient $A_n$	apply the mapping-class equation before comparing truncations	use $A_n\ge0$ only after taking absolute values	if $A_n\le Ce^{\kappa n}$ and $re^\kappa<1$, then $\lvert R_N\rvert\le C(re^\kappa)^{N+1}/(1-re^\kappa)$

In the last row, $r$ is an upper bound for the magnitude of every effective sewing monomial at the comparison point, after the universal vacuum powers are removed. The estimate is conservative because it discards cancellations. If no bound on $A_n$ is available, the truncation error is unknown; comparing two cutoffs is then an empirical convergence test, not a certified remainder.

Mapping-class closure also requires more than checking one generator on one locus. Record the chosen generators, their projective relations, the moduli region covered by the block expansion, and all sector mixing. An equality on the $\mathbb Z_3$-symmetric genus-two locus is a genuine constraint but not the full mapping-class system.

What higher genus has established

Genus-two modular bootstrap has demonstrated that mapping-class consistency can constrain products of three-point coefficients beyond sphere crossing and torus modular invariance Cho, Collier, and Yin 2019, §§ 1 and 3. In holomorphic CFTs, Siegel modular forms make parts of the genus-two problem finite-dimensional and permit analytic constraints Keller, Mathys, and Zadeh 2018, §§ 2–4. In special Narain and code-CFT families, higher-genus theta-polynomial structure can distinguish theories sharing the same genus-one partition function Henriksson, Kakkar, and McPeak 2023, §§ 3–5.

These results do not imply that arbitrary nonrational CFTs have a numerically tractable, globally convergent genus-$g$ block basis. Claims about optimal current bounds, software performance, or general uniqueness need a dated computation with its block and truncation inputs.

Scope beyond this page

This page develops physical CFT sewing equations, degeneration limits, and the checks needed to use them. Rigorous vertex-operator-algebra constructions, conformal-block bundles, and mapping-class theorems continue in Mathematical QFT. Handlebody saddles and holographic interpretations continue in Holography and Quantum Gravity. Curved-background and cosmological applications continue in QFT in Curved Spacetime. None of those interpretations is required for the sewing constraint itself.

Common failure modes

Suppressing local-coordinate data. A conformal block is tied to local coordinates and a normalization of states. Comparing coefficients across conventions without the transition factor creates artificial disagreement.

Assuming every coefficient is a square. Absolute-square positivity holds in selected reflection-positive pairings. A generic non-diagonal CFT or mapping-class frame has a more general Hermitian contraction.

Checking only the leading degeneration. Vacuum factorization can pass while subleading conjugation, Gram-matrix, or torus one-point factors are wrong. Test at least the first nonvacuum exchange.

Treating a symmetric locus as the full moduli space. A one-dimensional slice can give rigorous inequalities on that slice, but it does not impose all genus-two mapping-class equations.

Exercises

Explain why a genus-two pants decomposition contains two three-point vertices and three propagation tubes.

Solution

For a closed genus-$g$ surface, a pants decomposition has $2g-2$ pairs of pants and $3g-3$ internal circles. At $g=2$, these numbers are two and three. Each pair of pants carries one three-point amplitude, and gluing the three pairs of matching boundaries produces three propagators. Hence the primary coefficient is a product of two three-point coefficients contracted along three state labels.

Assume grouped absolute sewing coefficients satisfy $A_n\le5e^{0.2n}$ and all effective sewing monomials obey magnitude at most $r=e^{-1}$. Bound the contribution with $n>N$.

Solution

Since $re^{0.2}=e^{-0.8}<1$,

$$\lvert R_N\rvert \le \sum_{n=N+1}^{\infty}5e^{-0.8n} =\frac{5e^{-0.8(N+1)}}{1-e^{-0.8}}.$$

The estimate applies only after the universal vacuum powers have been removed and only in the convergence region used to derive the coefficient bound.

Continue to thermal data

Thermal States and One-Point Data begins the independent finite-temperature route in general dimension. It uses a density matrix and spacetime symmetries rather than a higher-genus mapping-class transformation.

References

• Cho, Minjae, Scott Collier, and Xi Yin. “Genus Two Modular Bootstrap.” Journal of High Energy Physics 2019, no. 4 (2019): 022. doi:10.1007/JHEP04(2019)022. Open preprint.
• Henriksson, Johan, Ashish Kakkar, and Brian McPeak. “Narain CFTs and Quantum Codes at Higher Genus.” Journal of High Energy Physics 2023, no. 4 (2023): 011. doi:10.1007/JHEP04(2023)011. Open preprint.
• Keller, Christoph A., Grégoire Mathys, and Ida G. Zadeh. “Bootstrapping Chiral CFTs at Genus Two.” Advances in Theoretical and Mathematical Physics 22, no. 6 (2018): 1447–1487. doi:10.4310/ATMP.2018.v22.n6.a3. Open preprint.