BPZ-Type Equations in 2D CFT

• BPZ-Type Equations are linear PDEs in 2D CFT that arise from degenerate fields with null descendants, forming the basis of conformal blocks analysis.
• They extend to higher orders and richer chiral algebras, linking classical and quantum integrability through modular and isomonodromic structures.
• These equations connect probabilistic models, SLE processes, and gauge theory via the AGT correspondence, offering applications across statistical and quantum physics.

A Belavin–Polyakov–Zamolodchikov (BPZ) equation is a linear partial differential equation satisfied by correlation functions of two-dimensional conformal field theories (CFT) when one or more fields are degenerate—that is, possess null descendants in their representation modules. These equations are central both to the analytic structure of CFT, where they underpin the theory of conformal blocks, and to the connections between CFT, quantum integrability, and probability. BPZ-type equations generalize to higher order, higher-rank chiral algebras, irregular singularities, and diverse geometric settings spanning classical and quantum integrability.

1. Formal Construction of BPZ-Type Equations

Given a chiral algebra (Virasoro or $\mathcal{W}_N$), a degenerate primary field is characterized by a weight such that its Verma module includes a null vector at some level. For Virasoro with central charge $c=1+6Q^2$, $Q=b+1/b$, the Kac table specifies $(m,n)$-degenerate primaries $V_{(m,n)}$ with conformal weight $\Delta_{(m,n)}$ and a null vector at level $mn$: $$\alpha_{(m,n)} = -\frac12[(m-1)b + (n-1)/b], \quad \Delta_{(m,n)} = \alpha_{(m,n)}(Q-\alpha_{(m,n)})$$ Insertion of such a field into a correlation function induces the decoupling of the null descendant, yielding a differential operator acting on the correlator:

• For $(2,1)$ or $(1,2)$: a second-order BPZ ODE
• For $(3,1)$ or $(1,3)$: a third-order BPZ ODE
• General $(r,s)$: an order $rs$ BPZ PDE

In terms of external insertions at points $\{z_i\}$, these equations take the form (for $(2,1)$ for example) (Jeong et al., 2017Poghosyan et al., 2016): $$\left[\frac{1}{b^2} \partial_{z_0}^2 + \sum_{i\neq0} \left( \frac{1}{z_0-z_i}\partial_{z_i} + \frac{\Delta_i}{(z_0-z_i)^2} \right) \right] \langle V_{(2,1)}(z_0)\cdots\rangle = 0$$

BPZ equations of order $r$ arise for $(r,1)$ or $(1,r)$ degenerate insertions. The corresponding operators, as rigorously constructed for Liouville CFT using a probabilistic framework, are (Zhu, 2020): $$D_r = \sum_{k=1}^r \sum_{n_1+\cdots+n_k = r} (\chi)^{r-k} L_{-n_1}(z)\cdots L_{-n_k}(z)$$ where $\chi$ is determined by the type of degeneracy.

2. Modular, Geometric, and Quantum Integrability Connections

BPZ-type equations are deeply linked to modular and isomonodromic structures:

• Modular Linear Differential Equations (MLDE): Virasoro (or general) conformal blocks can be characterized as solutions to modular-invariant linear ODEs. BPZ loci correspond to special values of MLDE parameters where blocks admit explicit Fuchsian representations, e.g., Gauss hypergeometric at level-2, 3F2 at level-3 (Mahanta et al., 2022).
• Accessory Parameters and Monodromy: For classical limits, such as $W_3$ blocks with semi-degenerate fields, closed BPZ-type ODEs involve accessory parameters (coefficients of simple poles in the "projective connection") which encode the local monodromy data of the conformal block. Their determination reduces to an isomonodromic deformation problem (Belavin et al., 29 Dec 2025).
• Quantum Painlevé and ODE/IM: By confluence of singularities, BPZ equations give rise to quantum analogues of Painlevé equations, with $\beta$-ensemble models and their loop equations serving as the prototype. This connection extends to the ODE/IM (Ordinary Differential Equation / Integrable Model) correspondence, relating the spectral data of Schrödinger operators to CFT quantities (Rumanov, 2014).

3. Higher Order, Irregular, and Extended Chiral Symmetry BPZ Equations

Higher-order BPZ equations emerge for higher-level degeneracies or extended chiral algebras:

• Liouville CFT supports arbitrary order BPZ equations associated with $(r,1)$ or $(1,r)$ degenerate insertions (Zhu, 2020).
• For $\mathcal{W}_3$ and more generally $\mathcal{W}_N$, the construction uses null vectors for semi- and fully-degenerate fields. For example, in boundary Toda theory, level-3 W-algebra null vectors yield third-order Fuchsian BPZ ODEs, with the involvement of higher-spin Ward identities (Cerclé et al., 26 Mar 2025, Belavin et al., 29 Dec 2025).

Irregular BPZ equations arise in the presence of irregular singular operators (with essential singularities rather than regular pole-type behavior). The resulting differential equations contain explicit dependence on irregularity parameters and their derivatives (e.g., terms like $\Lambda\,\partial_\Lambda$), as well as new constant shifts reflecting the change in operator product expansions (Gu et al., 2023).

4. Probabilistic, Statistical, and SLE Connections

BPZ-type equations structure random processes related to conformal or Schramm–Loewner Evolution (SLE):

• Radial BPZ Equations: Emergent from commutation relations for multiple SLEs in multiply-connected domains, these encode the effect of boundary insertions of level-2 degenerate fields (Feng et al., 2024). In SLE partition functions,

$$\mathcal{L}_j Z = \left(\frac{\kappa}{2}\,\partial_{\theta_j}^2 + \sum_{\ell\neq j}\left[\cot\frac{\theta_\ell-\theta_j}{2}\,\partial_{\theta_\ell} - \frac{6-\kappa}{4\kappa}\,\sin^{-2}\frac{\theta_\ell-\theta_j}{2}\right] - \aleph\right)Z = 0$$

• One-Arm and FK–Ising Interfaces: The partition functions for scaling limits conditioned on certain connectivity events are positive solutions to inhomogeneous BPZ radial equations, with boundary and positivity properties tightly characterized (Feng et al., 2024).

5. Gauge Theory, AGT, and qq-Characters

The AGT correspondence translates BPZ equations for conformal blocks into differential (and difference-differential) equations for supersymmetric partition functions in four-dimensional $\mathcal{N}=2$ gauge theories:

• Surface Defects and qq-Characters: Tuning Coulomb data to create surface defects produces instanton sums obeying second- or third-order differential equations, mirroring the CFT null vector decoupling (Nekrasov, 2017, Jeong et al., 2017).
• Q-Operator and Nekrasov–Shatashvili Limit: The VEV of the Baxter Q-operator satisfies a mixed difference-differential equation, generalizing the TQ relation of quantum integrable systems. The Nekrasov–Shatashvili limit yields direct agreement with the classical spectral problem, establishing the quantum-classical correspondence (Poghosyan et al., 2016).

6. Structural and Solution Aspects

BPZ-type equations may be viewed as essential linear constraints (null vector decoupling) with the following functional significance:

• Provide a finite-dimensional local system for the space of conformal blocks associated with a given degenerate insertion.
• Allow the construction and characterization of conformal blocks as explicit Dotsenko–Fateev integrals or integrals over Lefschetz thimbles, with algebraic or analytic verification of the BPZ system (Gu et al., 2023).
• In the semiclassical limit, such equations reduce to spectral equations of the Schrödinger or Heun type, with classical accessory parameters playing the dual role of monodromy data and isomonodromic deformations.

7. Examples and Special Cases

Context	BPZ-Type Equation	Order	Structure/Features
Virasoro, $(2,1)$ field	Hypergeometric BPZ ODE	2	Fuchsian, conformal blocks
Virasoro, $(3,1)$ field	3rd-order BPZ ODE	3	3F2, Fibonacci fusion, Heun limit
$W_3$ block	3rd-order BPZ ODE	3	Accessory parameter, heavy-light
$\mathcal{W}_N$ Toda	Like $W_3$, higher order	$N$	W-algebra null vector structure
Irregular block	BPZ with $\Lambda\,\partial_\Lambda$	$r$	Irregularity parameter
Radial SLE partition	Radial BPZ PDE	2	Boundary data, positivity, fusion
$\mathcal{N}=2$ gauge	BPZ $\Leftrightarrow$ Q-operator eqn	2-3	Mixed diff/difference, qq-chars

In summary, BPZ-type equations form the analytic backbone of conformal field theory and its various generalizations. They produce a rich tapestry of connections across representation theory, special functions, integrability, quantum algebras, and probability, with a galaxy of explicit forms and applications documented in both rigorous and mathematical physics literature (Belavin et al., 29 Dec 2025Zhu, 2020Gu et al., 2023Cerclé et al., 26 Mar 2025Mahanta et al., 2022Nekrasov, 2017Jeong et al., 2017Poghosyan et al., 20161408.38472411.16051).