[Paper Review] CFT exercises for the needs of AGT

А. Миронов, Сергей Андреевич Миронов+2 more|arXiv (Cornell University)|Aug 14, 2009

Black Holes and Theoretical PhysicsPhysics and Astronomy21 references69 citations

TL;DR

This paper provides explicit, computable formulas for $W_3$-algebra triple correlators and the Shapovalov matrix at levels one and two, derived via free field realizations and verified using the Wick theorem. These results are essential for constructing $W_N$-symmetric conformal blocks in the AGT correspondence, particularly for $N=3$, enabling systematic checks and generalizations of the AGT relation between conformal blocks and Nekrasov partition functions.

ABSTRACT

An explicit check of the AGT relation between the W_N-symmetry controlled conformal blocks and U(N) Nekrasov functions requires knowledge of the Shapovalov matrix and various triple correlators for W-algebra descendants. We collect simplest expressions of this type for N=3 and for the two lowest descendant levels, together with the detailed derivations, which can be now computerized and used in more general studies of conformal blocks and AGT relations at higher levels.

Motivation & Objective

• To provide explicit, computable expressions for $W_3$-algebra triple correlators and the Shapovalov matrix at low levels, which are essential for conformal block computations in the AGT correspondence.
• To bridge the gap in CFT literature by collecting and deriving formulas for $W$-algebra descendants and structure constants that are otherwise missing or implicit.
• To enable computerized computation of conformal blocks and verification of the AGT relation at higher levels by supplying foundational formulas.
• To serve as a pedagogical and technical reference for researchers working on $W_N$-symmetric CFTs and their relation to $N=2$ supersymmetric gauge theories.

Proposed method

• Derive triple correlators and Shapovalov matrix elements using the free field realization of $c=1$ conformal field theory and the Wick theorem.
• Apply recursive relations between correlators with different numbers of descendants, derived from conformal symmetry and $W_3$ algebra structure.
• Use the normal-ordered exponential vertex operators and their correlation functions to compute structure constants and selection rules.
• Verify key results via explicit computation in the free field model, particularly for $W_3$ descendants at levels one and two.
• Employ the inverse Shapovalov form and triple vertices to reconstruct 4-point conformal blocks in the AGT framework.
• Implement special state conditions (e.g., $W$-primary states) to simplify and constrain the correlator expressions.

Experimental results

Research questions

• RQ1 What are the explicit expressions for $W_3$-algebra triple correlators involving descendants at levels one and two?
• RQ2 How can the Shapovalov matrix elements for $W_3$-descendants be computed and used in conformal block constructions?
• RQ3 What are the recursive relations between triple correlators with different numbers of $W$-descendants?
• RQ4 How do the structure constants and conformal dimensions transform under $W$-algebra symmetry in the context of the AGT correspondence?
• RQ5 Can the free field model be used to systematically derive and verify $W_3$-correlators relevant for AGT?

Key findings

• The paper derives and verifies explicit expressions for $W_3$-algebra triple correlators involving $W_{-1}$ and $L_{-1}$ descendants at levels one and two, which are essential for AGT block computations.
• The Shapovalov matrix elements for $W_3$-descendants at levels one and two are computed and validated using the free field model, providing a computational basis for higher-level studies.
• The recursive relations between correlators with different numbers of descendants are derived and confirmed via Wick contractions in the $c=1$ free field theory.
• The results are shown to satisfy the matching condition between conformal blocks and Nekrasov functions in the AGT correspondence, particularly for $N=3$.
• The paper provides a complete set of frame-boxed formulas for triple vertices and Shapovalov inverses that are directly usable in numerical and symbolic computations of conformal blocks.
• The derivation confirms that the free field model correctly reproduces the $W_3$-algebra structure, including the role of $W$-currents and their normal-ordering, validating the approach for higher-level generalizations.