Liouville Sector: Quantum & Geometric Insights

• Liouville Sector is a decomposition of Liouville field theory into chiral halves, exposing free massless field structures via canonical transformation.
• It comprises exact symplectic manifolds with well-preserved presymplectic data, enabling constructions in wrapped Fukaya categories and Floer theory.
• The framework underlies dualities in 2D quantum gravity, holography, and superconformal extensions, linking gauge/CFT correspondences with geometric invariants.

The Liouville sector comprises the chiral decomposition and geometric characterization of Liouville field theory and its extensions, essential in symplectic geometry, two-dimensional quantum gravity, conformal field theory, and gauge/CFT dualities. As a fundamental construct, the Liouville sector captures both the analytic structure of chiral Liouville theory and the presymplectic data of Liouville sectors (with or without corners) on exact symplectic manifolds. These structures underpin the foundations of the wrapped Fukaya category, symplectic cohomology, and holographically dual gravity theories. Liouville sectors admit further generalization to superconformal theories, orbifolds, and connections with gauge theory partition functions.

1. Chiral Factorization and the Liouville Sector in Quantum Field Theory

In two-dimensional Liouville field theory on the cylinder, the classical equation $$\partial_+\partial_- \Phi(T,\sigma) + \pi\mu e^{2b\Phi(T,\sigma)}=0$$ decouples into left- and right-moving components ("holomorphic" and "antiholomorphic" chiralities). A Liouville sector is defined as one of these chiral halves: it contains a free massless field in the far past ("in"), a free massless field in the future ("out"), related via a canonical transformation dependent only on oscillator modes and the zero mode. The full quantum S-matrix factorizes as $$S_{\rm Liouville}=P\,R(p)\,S_p^{\rm (chiral)} \otimes S_p^{\rm (anti)}$$, where $R(p)$ is the zero-mode reflection amplitude, $P$ is parity, and $S_p^{\rm (chiral)}$ is determined by a non-local one-dimensional action on the circle. This action arises from a Legendre transform of the generating function of scattering amplitudes and encodes loop corrections and the full operator structure (Jorjadze et al., 2020). This formalism connects operator, semiclassical, and Feynman diagram approaches, placing the analytic content of the Liouville sector on solid quantum mechanical foundations.

2. Presymplectic Geometry and Liouville Sectors with Corners

Liouville sectors form a distinguished class of exact symplectic manifolds $(M,\omega=d\lambda)$ whose "boundary-at-infinity" preserves sufficient presymplectic data to allow glueing and formation of direct products in a manner compatible with the wrapped Fukaya category. The intrinsic (Liouville $\sigma$-sector) definition characterizes these via the characteristic foliation of the boundary, extended to sectors with corners by analyzing the null foliations of coisotropic intersections among transverse coisotropic hypersurfaces. Locally, near codimension-$k$ corners, $(M,\lambda)$ admits Darboux coordinates $(R_1,\dots,R_k;I_1,\dots,I_k)$, and one has canonical product charts $F\times T^*_{\geq 0}{}^k$. Liouville sectors with corners canonically form a monoid, providing the algebraic structure necessary for Künneth-type constructions in Fukaya categories. The automorphism group is characterized, enabling a definition of bundles of Liouville sectors and answering foundational questions about the optimality and generality of sector definitions (Oh, 2021).

3. Infinity-Categorical Structure and Functoriality

The infinity-category of stabilized Liouville sectors arises via localization of the ordinary category of sectors and strict sectorial embeddings, with stabilization accomplished by iterated product with $T^*[0,1]$. The mapping spaces in the infinity-category correspond to colimits of stable sectorial embeddings, and the wrapped Fukaya category realizes functoriality and coherent actions from the automorphism spaces of stabilized Liouville sectors (Lazarev et al., 2021). A symmetric monoidal structure is established via direct product, with universal properties emerging from factorization homology principles. This formal framework enables the construction of homotopy-coherent functors, Künneth equivalences, and Floer-theoretic invariants in families, extending to Lagrangian cobordism categories and microlocal sheaf invariants.

4. Sectorial Almost-Complex Structures, Hamiltonians, and Maximum Principle

Liouville sectors with corners admit specialized packages of Floer data: $\lambda$-sectorial almost complex structures $J$ and sectorial Hamiltonians $H$. Almost-complex structures $J$ are $\omega$-tame and satisfy $-d\psi \circ J = \lambda$ for exhaustion functions $\psi$ adapted to corner smoothing profiles. Sectorial Hamiltonians $H$ are supported on proper exhaustion profiles and direct the Hamiltonian vector field $X_H$ outward in the $\psi$-direction. This construction enables a uniform maximum-principle framework: for any pseudoholomorphic curve $u$, $\psi\circ u$ satisfies subharmonicity or superharmonicity relations, ruling out interior or boundary maxima and confining all relevant Floer-theoretic curves to bounded sublevels of $\psi$. This uniform approach applies to all wrapped Floer and symplectic cohomology operations, ensuring analytic control without recourse to ad hoc dissipative estimates (Oh, 2021).

5. Liouville Sector in Conformal Field Theory, Gravity, and Holography

The Liouville sector is central in 2D gravity, where the Liouville field $S_L[\phi]$ imparts fluctuating geometry on the worldsheet. Primary operators $V_a(z,\bar z)=e^{2a\phi(z,\bar z)}$ realize a continuous spectrum, with structure constants given by the DOZZ formula and operator product expansions involving both continuous and discrete terms. In the heavy-light regime at large central charge $c$ in 2D CFT, heavy operators source classical geometries (solutions of the Liouville equation), while light operators propagate as primaries. The identity block of the heavy-light correlator reproduces the semi-classical Liouville correlator, unifying gravitational backreaction in AdS/CFT correspondence. Ward identities, accessory parameters via monodromy, and path-integral saddle techniques interconnect Liouville theory with classical gravity on hyperbolic spaces and BTZ black hole quotients (Vos, 2020, Aleshkin et al., 2016).

In q-deformed dilaton gravity the Liouville sector emerges as a holographic dual to the boundary q-Schwarzian quantum mechanics, with exact matching of thermodynamics, quantum spectra, and two-point correlators. The reduction of sinh-dilaton gravity to Liouville gravity via field redefinition links the quantum boundary theory and bulk geometries (Blommaert et al., 2023).

6. Extensions: Super Liouville Sectors and Gauge/CFT Dualities

Liouville sectors generalize to superconformal field theories, notably $\mathcal{N}=1$ super-Liouville theory with distinct Neveu-Schwarz (NS) and Ramond (R) sectors. The super-Virasoro algebra, fermionic modes, and spin-field primaries structure the representation theory. Sectors correspond to instanton partition functions in $U(2)$ gauge theories on ALE spaces: boundary holonomy choices select NS or R sectors, and Whittaker vectors encode the sectorial inner products. S-duality manifests as exchange of NS and R holonomy sectors, ensuring closure of the correspondence (Ito, 2011, Zhao, 2011).

Modular transformations in the NS sector of super-Liouville theory are governed by integral kernels built from supersymmetric hyperbolic gamma functions, with explicit computation achieved via degenerations of parafermionic elliptic gamma integrals. The S-move kernel encodes Moore-Seiberg data and realizes duality kernels for Seiberg-Witten theory and partition functions for mass-deformed 3D theories (Apresyan et al., 2023). AGT-like relations extend to orbifold vortex partition functions and mixed sector correlators, with parameter dictionaries precisely matching gauge and CFT observables.

7. Operator Product Expansion and Correlation Numbers in Gravity

The Liouville sector occupies a central position in the computation of correlation numbers in minimal Liouville gravity. Two independent approaches—direct CFT computations using higher equations of motion (HEM) and discrete, matrix model techniques based on the Douglas string equation—demand careful reconciliation due to the presence of discrete terms in the operator product expansion. The necessity of subtracting contributions conflicting with the matter fusion rules ensures consistency across both methods, with the Liouville sector's non-rational OPE encoding the discrete matrix-model degrees of freedom. This mechanism is conjectured to underlie higher-point functions, further confirming the role of the Liouville sector as the unique irrational CFT governing sums over surfaces in 2D quantum gravity (Aleshkin et al., 2016).