C3k2 Block in Conformal Field Theory

• C3k2 Block is a multifaceted construct in conformal field theory defined through representation-theoretic vector bundles and analytic formulations.
• It employs advanced differential operators, including a fourth-order operator, to encode tensor structures in multipoint conformal blocks across various dimensions.
• Its algebraic geometric interpretation as divisors in moduli spaces bridges symplectic and special linear cases, with implications for statistical lattice models.

The C3k2 block is a technical construct arising in the paper of multipoint conformal blocks, particularly in the analysis of higher-dimensional conformal field theories (CFTs) and in the algebraic geometry of moduli spaces associated with Lie algebraic data. Its origin traces to both representation-theoretic formulations (as in conformal blocks vector bundles on moduli spaces) and to analytic constructions (as explicit joint eigenfunctions of integrable models in CFT via chain-like OPE diagrams and advanced embedding space formalism). The term "C3k2 block" is therefore context-dependent, describing either a divisor or bundle in type C conformal block theory, a multipoint CFT vertex in integrable systems, or a component in the tensor-structure decomposition of correlation functions in low-dimensional CFTs.

1. Definition and Representation-Theoretic Context

The C3k2 block in algebraic geometry refers to a conformal blocks vector bundle

$V(\mathfrak{sp}_{2\ell}, \vec{\lambda}, 1)$

on the moduli space $\bar{M}_{0,n}$ associated with the symplectic Lie algebra $\mathfrak{sp}_{2\ell}$ at level one. Here, $\vec{\lambda}$ is an $n$-tuple of dominant weights (with the parity condition that $|\vec{\lambda}|$ is even) (Hobson, 2016). In parallel, for type A, the construction uses bundles $V(\mathfrak{sl}_2, \vec{\lambda}, \ell)$ where the weight sequence is interpreted in terms of integral multiples of the fundamental weight, restricted by the level.

A C3k2 block thus represents a section of these vector bundles, supported by explicit combinatorial and representation-theoretic data.

In the context of multipoint conformal blocks in CFT, the C3k2 block refers to a single-vertex system ("comb channel vertex") involving three fields in a chain-like OPE diagram for d = 3 or d = 4, and is characterized by a unique conformal cross-ratio and a fourth-order differential operator encoding the tensor structures at the vertex (Buric et al., 2021).

2. Differential Operators and Integrable Model Structure

Recent advancements identify multipoint conformal blocks as joint eigenfunctions of a commutative algebra of differential operators derived from Gaudin integrable systems. Specifically, the vertex operator (that labels the three-point tensor structure) for the C3k2 block is realized as a fourth-order differential operator acting on the single cross-ratio $\mathcal{X}$ intrinsic to the vertex (Buric et al., 2021). Explicitly, after the change of variable $$\mathcal{X} = \wp(\omega_3)^2 / [\wp(\omega_3)^2 - \wp(z)^2]$$, the operator is

$$L_{ext}(z, \partial_z) = \partial_z^4 + \sum_{p=0}^2 g_p(z)\partial_z^p$$

with $g_p(z)$ determined by the dimensions and spins of the fields. This operator coincides (up to scaling and shifts) with the Hamiltonian of the crystallographic (lemniscatic) elliptic Calogero-Moser-Sutherland (CMS) system, originally classified by Etingof et al.

3. Embedding Space Formalism and Tensor Structures

The C3k2 block construction in analytic CFT approaches relies on an embedding space formalism capable of encoding both symmetric traceless and mixed-symmetry tensor fields. Polarization vectors in auxiliary $(d+2)$-dimensions encode the field indices, and gauge invariance plus transversality constraints are imposed for irreducibility (Buric et al., 2021, Fortin et al., 2022). Conformally invariant three-point functions are expressed as

$$\langle \varphi_1\varphi_2\varphi_3 \rangle = \Omega(X_i, Z_i) t(\mathcal{X})$$

where $t(\mathcal{X})$ encodes the tensor structure for the single cross-ratio at the vertex.

This formalism is essential for systematically constructing the corresponding differential operators (Casimirs for links, vertex operators for junctions) and for expressing the blocks as power series in the cross ratios, with all tensorial data built in.

4. Explicit Power Series Construction and Hypergeometric Representations

In three dimensions, conformal blocks (including C3k2-type components) are constructed by reducing a four-point function, via the embedding space operator product expansion, to sums over three-point functions and acting with explicit differential projection operators. After conformal substitution, these components are realized as explicit power series in two conformal cross ratios (commonly denoted $x_3, x_4$ or $u, v$) (Fortin et al., 2022). The coefficients—parametrized by Pochhammer symbols and combinatorial factors—encode the Lorentz/tensor structures admissible for the exchanged representations.

More recently, compact representations for three-dimensional conformal blocks have been achieved by dressing the block with fractional derivative operators,

$$\hat{G}_{\Delta, \ell}(z, \bar{z}) = \mathcal{T}^{1/2} G_{\Delta, \ell}$$

where $\mathcal{T}^\delta = T_z^\delta T_{\bar{z}}^\delta$ is realized via modified Riemann–Liouville half-derivatives. This leads to a factorized form,

$$\hat{G}_{\Delta, \ell}^{3d}(z, \bar{z}) = [z^{(\Delta-1)/2} {}_4F_3(\ldots; z)] \times [\bar{z}^{(\Delta-1)/2} {}_4F_3(\ldots; \bar{z})]$$

with all descendant summations encoded in hypergeometric ${}_4F_3$ functions (Song, 2023).

5. Algebraic Geometry: Divisor Cones and Rank-One Classification

The algebraic geometry of C3k2 blocks, specifically for type C at level one, yields divisors

$c_1(V_{\mathfrak{sp}_{2\ell}, \vec{\lambda}, 1})$

which coincide with $c_1(V_{\mathfrak{sl}_2, \vec{\lambda}, \ell})$ if and only if the vector bundle has rank one or zero. The underlying degree computations for four-pointed curves ensure this property. For instance, when $\vec{\lambda} = (a, b, c, d)$ and $a+b+c+d = 2(\ell + s)$, explicit formulas yield

$$\deg(V_{\mathfrak{sl}_2, \vec{\lambda}, \ell}) = (\mathrm{rank\;} V) \cdot s$$

and

$$\deg(V_{\mathfrak{sp}_{2\ell}, \vec{\lambda}, 1}) = \max(0, (\ell+1-a)(\ell+2s-a)/2)$$

in the rank one case (Hobson, 2016).

Crucially, the cone generated by these (rank one) divisors is shown to be polyhedral—finitely generated—mirroring known results for types A and D at level one.

6. Connections to Probabilistic Models and Combinatorial Algebra

The analytic and algebraic structures generalize to probabilistic contexts in two dimensions, such as the representation-theoretic construction of degenerate conformal blocks for the $W_3$ algebra at $c=2$ (Lafay et al., 19 Feb 2024). Here, an explicit basis is given via Specht polynomials associated to rectangular Young tableaux of three columns, reflecting the $\mathfrak{sl}_3$ content. The space of such blocks is shown to be an irreducible module for the Kuperberg diagram algebra, and a conjectural correspondence is established to scaling limits in the triple dimer model.

In practical terms, explicit formulas for connection probabilities in lattice models (e.g., those involving dimer covers reduced to webs) are matched to ratios of Specht polynomial functions, confirming the algebraic-geometric block constructions.

7. Implications and Perspectives

The identification of C3k2 blocks as both analytic objects in integrable models of CFT and divisors in moduli space theories yields multiple consequences:

• Establishes exact correspondences between divisors for symplectic (type C) and special linear (type A) Lie algebras in rank one scenarios, enabling transfer of geometric and birational results.
• Connects conformal block theory in higher dimensions with integrable systems (Gaudin, elliptic CMS), providing systematic tools for analytic bootstrap studies and explicit computation of higher-point functions.
• Embedding space formalisms and compact (hypergeometric/fractional calculus-based) representations facilitate efficient analytic and numerical work in 3D and 2D CFTs.
• The combinatorial algebra approach using Specht modules and $\mathfrak{sl}_3$ webs bridges conformal field theory with scaling limits in probabilistic lattice models.

This synthesis demonstrates the centrality of the C3k2 block in unifying geometric, analytic, and algebraic methods across conformal field theory and algebraic geometry, and in connecting representation-theoretic constructions to concrete models in statistical mechanics.