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Ward Identities in the \(\mathfrak {sl}_3\) Toda Conformal Field Theory

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Abstract

Toda conformal field theories are natural generalizations of Liouville conformal field theory that enjoy an enhanced level of symmetry. In Toda conformal field theories this higher-spin symmetry can be made explicit, thanks to a path integral formulation of the model based on a Lie algebra structure. The purpose of the present document is to explain how this higher level of symmetry can manifest itself within the rigorous probabilistic framework introduced by R. Rhodes, V. Vargas and the first author in Cerclé (Probabilistic construction of simply-laced Toda conformal field theories, arXiv preprint, arXiv:2102.11219, 2021). One of its features is the existence of holomorphic currents that are introduced via a rigorous derivation of the Miura transformation. More precisely, we prove that the spin-three Ward identities, that encode higher-spin symmetry, hold in the \(\mathfrak {sl}_3\) Toda conformal field theory; as an original input we provide explicit expressions for the descendent fields which were left unidentified in the physics literature. This representation of the descendent fields provides a new systematic method to find the degenerate fields of the \(\mathfrak {sl}_3\) Toda (and Liouville) conformal field theory, which in turn implies that certain four-point correlation functions are solutions of a hypergeometric differential equation of the third order.

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Notes

  1. Unless explicitly stated, holomorphic derivatives will be considered throughout the rest of the document.

  2. Note that the expression for Q differs from the one in Liouville theory because of our convention on \(\gamma \). The standard one can be recovered by scaling \(\gamma \) by a multiplicative factor \(\sqrt{2}\). This scaling is due to the fact that the simple roots are not orthonormal but rather satisfy \(\left\langle e_i,e_i \right\rangle =2\).

  3. The interested reader may find details on the role of this transformation in the construction of two-dimensional CFTs having higher-spin symmetry for instance in [17], where the Miura transformation is used to construct representations of W-algebras.

  4. The weights rather belong to the dual space \({\mathfrak {h}}_3^*\) of \({\mathfrak {h}}_3\), but these two spaces may be identified via Riesz representation theorem.

  5. There is a difference here between the form of Eq. (1.22) and what is commonly written down in the physics literature. Indeed it is standard (see e.g. the discussion below Equation (2.8) in [43]) to scale by a multiplicative factor \(i\sqrt{\frac{48}{22+5c}}\) the expression of \(\varvec{{\mathrm {W}}}\), with c the central charge of the theory—and by doing so the expression of the quantum numbers \(w(\alpha )\) and the descendent operators—in order for the WW OPE (the contraction of two W currents) to be written down in an elegant fashion.

  6. This scalar product differs from the Killing form by a multiplicative factor whose value is not relevant in the present document.

  7. Thanks to Riesz representation theorem we will often identify \({\mathfrak {h}}_3\) with its dual space \({\mathfrak {h}}_3^*\).

  8. That is, we apply Stokes’ formula \(\oint _{\partial B(z_0,r)}f(\xi )g(\xi ) \frac{\sqrt{-1}d{\bar{\xi }}}{2}=\int _{B(z_0,r)}\partial _xf(x)g(x)+\partial _xg(x)f(x) d^2x\) to the term \(\partial _{x}\theta _\delta (z_0-x)\left\langle V_{\gamma e_i,\varepsilon }(x)V_{\alpha _0,\varepsilon }(z_0)\varvec{{\mathrm {V}}}_\varepsilon \right\rangle _\delta \).

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Authors and Affiliations

  1. Laboratoire de Mathématiques d’Orsay, Bâtiment 307, Faculté des Sciences d’Orsay, Université Paris-Saclay, 91405, Orsay, France

    Baptiste Cerclé

  2. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014, Helsinki, Finland

    Yichao Huang

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  1. Baptiste Cerclé
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  2. Yichao Huang
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Correspondence to Baptiste Cerclé.

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Communicated by A.Giuliani.

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The authors are indebted to A. Kupiainen, R. Rhodes and V. Vargas for fruitful discussions on Toda theories. Y. Huang is supported by ERC grant QFPROBA, No.741487..

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Cerclé, B., Huang, Y. Ward Identities in the \(\mathfrak {sl}_3\) Toda Conformal Field Theory. Commun. Math. Phys. 393, 419–475 (2022). https://doi.org/10.1007/s00220-022-04370-5

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