Hecke transformations between topological sectors of WZW conformal blocks: intertwiners of the sector-wise elliptic KZB connections

For the Wess–Zumino–Witten (WZW) theory on elliptic curves, the space of conformal blocks decomposes into sectors labeled by the characteristic classes of the underlying adjoint $G$-bundles, and flat Knizhnik–Zamolodchikov–Bernard (KZB) connections have been constructed sector by sector. What is missing is the inter-sector structure of this theory: the explicit Hecke transformation operators that intertwine conformal blocks of different characteristic classes, together with their intertwining property. The problem asks for this construction, completing the program announced (but not executed) in the 2012 source paper. Two further aspects of the sector-wise theory were likewise not carried out in that paper — the behaviour over the full moduli space of bundles of the dynamical twist ambiguities $\delta r(\mathbf u)$ of the sector $r$-matrices, and the explicit trigonometric and rational degenerations of the sector-adapted connections — and they are recorded below as background rather than as parts of the requested construction.

Background

The Wess–Zumino–Witten (WZW) model is a two-dimensional conformal field theory whose chiral symmetry algebra is the affine Kac–Moody algebra (the central extension of the algebra of loops taking values in the Lie algebra of a simple complex Lie group $G$). Its chiral correlation functions are the conformal blocks: multilinear functionals on tensor products of integrable highest-weight representations of the affine algebra that satisfy the Ward identities (the symmetry constraints of the theory). On a curve $\Sigma_{g,n}$ of genus $g$ with $n$ marked points, the conformal blocks form a vector bundle over the moduli space of complex structures, and the Knizhnik–Zamolodchikov–Bernard (KZB) equations — the genus-zero version is due to Knizhnik and Zamolodchikov, the arbitrary-genus version to Bernard — are the horizontality condition for a projectively flat connection (the KZB connection) on this bundle. For genus one, Felder and Wieczerkowski constructed this connection in terms of classical dynamical elliptic $r$-matrices: antisymmetric tensors $r(\mathbf u,z)$ depending on a spectral parameter $z$ on the elliptic curve and on dynamical variables $\mathbf u$ in a Cartan subalgebra, satisfying the classical dynamical Yang–Baxter equation (the consistency condition guaranteeing flatness).

When the structure group is not simply connected, the WZW theory on a curve is really a theory of holomorphic $G$-bundles, and the topology of these bundles is captured by characteristic classes: elements of $H^2(\Sigma_{g,n},\mathcal Z(G))$, where $\mathcal Z(G)$ is the center of $G$. The spaces of conformal blocks and the integrable affine modules decompose into sectors labeled by these classes. Levin, Olshanetsky, Smirnov and Zotov classified the dynamical elliptic $r$-matrices arising as sections of bundles of arbitrary topological type (the dynamical variables turn out to be coordinates on the moduli space of the underlying bundles) and, in the source paper of this problem, constructed explicitly the flat KZB connection in every sector of the elliptic-curve theory, proving its flatness. In the trivial sector this reduces to the Felder–Wieczerkowski construction; the other sectors are genuinely new families of KZB equations.

Classically, changing the topological type of a bundle is achieved by a Hecke modification: a singular gauge transformation at a point of the curve that changes the characteristic class. The same authors showed, at the classical level, that such modifications implement the symplectic Hecke correspondence mapping Calogero–Moser type integrable systems (integrable many-body systems with pair interactions) to integrable tops (Euler-top type systems of rigid-body rotations) and Painlevé-type equations (nonlinear ordinary differential equations with movable branch points). The quantum counterpart — operators intertwining conformal blocks belonging to different sectors — was explicitly deferred to a sequel paper, which has never appeared.

Two further issues attach to this construction without being posed as open questions by the source paper. First, Lemma 4.1 of the source paper shows that any dynamical $r$-matrix satisfying its defining conditions 1–5 differs from the canonical one by a twist $\delta r(\mathbf u)$, a Cartan-valued term generated locally by a gauge function $f(\mathbf u)$ (the proof of this generation is local, via the Poincaré lemma); how these twists behave globally over the full moduli space of bundles — whether they are globally trivializable, what obstructs trivialization, and how they affect the monodromy and the global definition of the KZB connection — was not addressed there and is not raised there as a question. Second, the trigonometric and rational degeneration limits of the sector-adapted elliptic connections were remarked in the source paper to be obtainable via procedures described in the literature it cites, but were never derived. A further issue noted in the source paper — the bundle interpretation and $r$-matrices for Dynkin-diagram automorphisms that do not come from the center (existing for $A_n$, $D_n$ and $E_6$), where the paper states that 'so far the underlying vector bundle structure is unclear' — was subsequently addressed at the classical level by Levin, Olshanetsky and Zotov, who constructed the corresponding moduli spaces, twisted dynamical $r$-matrices and KZB-type connections for bundles defined by arbitrary finite-order automorphisms.

Problem Statement

The objective is to complete the inter-sector structure of the sector-wise KZB theory for WZW conformal blocks on elliptic curves, given the sector-wise flat connections constructed in the source paper. Concretely, for a simple complex Lie group $G$ with non-trivial center $\mathcal Z(G)$ and elliptic curves $\Sigma_{1,n}$ with $n$ marked points, where the conformal blocks decompose into sectors labeled by the characteristic classes in $H^2(\Sigma_{1,n},\mathcal Z(G))$, the problem asks for the following construction.

Construct the Hecke transformation operators: explicit operators that intertwine the conformal blocks — equivalently, the sector-wise KZB connections — of sectors with different characteristic classes, and establish their intertwining property. This is precisely the content deferred to the never-published sequel of the source paper, which announced that 'In the subsequent paper we will describe the transformation operators that intertwine the different sectors (the Hecke transformations).'

This construction concerns the same objects as the source paper — the moduli space of adjoint $G$-bundles of non-trivial topological type over elliptic curves and its sector-wise dynamical $r$-matrices — and completes the picture that the sector-wise local construction began, by connecting the sectors that the constructed KZB connection preserves. The problem is solved when this construction is carried out at the level of generality of the source paper (arbitrary characteristic classes, arbitrary simple $G$ with non-trivial center), with a proof of the intertwining property. The global behaviour of the dynamical twists $\delta r(\mathbf u)$ over the moduli space of bundles and the trigonometric and rational degenerations of the sector-adapted connections are related questions that the source paper did not carry out; they are described in Background as context and are not part of the requested construction.

References

1. A. M. Levin, M. A. Olshanetsky, A. V. Smirnov, A. V. Zotov, 'Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles', Symmetry, Integrability and Geometry: Methods and Applications 8 (2012) 095, DOI 10.3842/SIGMA.2012.095, arXiv:1207.4386, https://arxiv.org/abs/1207.4386 (source paper of this problem).
2. A. M. Levin, M. A. Olshanetsky, A. V. Smirnov, A. V. Zotov, 'Characteristic Classes and Hitchin Systems. General Construction', Communications in Mathematical Physics 316 (2012) 1–44, DOI 10.1007/s00220-012-1585-x, arXiv:1006.0702, https://arxiv.org/abs/1006.0702.
3. A. M. Levin, M. A. Olshanetsky, A. V. Smirnov, A. V. Zotov, 'Characteristic Classes of SL(N)-Bundles and Quantum Dynamical Elliptic R-Matrices', Journal of Physics A: Mathematical and Theoretical 46 (2013) 035201, DOI 10.1088/1751-8113/46/3/035201, arXiv:1208.5750, https://arxiv.org/abs/1208.5750.
4. A. M. Levin, M. A. Olshanetsky, A. V. Zotov, 'Classification of Isomonodromy Problems on Elliptic Curves', Russian Mathematical Surveys 69 (2014), no. 1, 35–118, DOI 10.1070/RM2014v069n01ABEH004878, arXiv:1311.4498, https://arxiv.org/abs/1311.4498.
5. A. Levin, M. Olshanetsky, A. Zotov, 'Geometry of Higgs bundles over elliptic curves related to automorphisms of simple Lie algebras, Calogero-Moser systems and KZB equations', Theoretical and Mathematical Physics 188 (2016), no. 2, 1121–1154, DOI 10.1134/S0040577916080018, arXiv:1507.04265, https://arxiv.org/abs/1507.04265.
6. A. Levin, M. Olshanetsky, A. Zotov, 'Planck Constant as Spectral Parameter in Integrable Systems and KZB Equations', Journal of High Energy Physics 10 (2014) 109, DOI 10.1007/JHEP10(2014)109, arXiv:1408.6246, https://arxiv.org/abs/1408.6246.
7. G. Felder, C. Wieczerkowski, 'Conformal blocks on elliptic curves and the Knizhnik-Zamolodchikov-Bernard equations', Communications in Mathematical Physics 176 (1996) 133–161, DOI 10.1007/BF02099366, arXiv:hep-th/9411004, https://arxiv.org/abs/hep-th/9411004.
8. P. Etingof, O. Schiffmann, 'Twisted traces of intertwiners for Kac-Moody algebras and classical dynamical r-matrices corresponding to generalized Belavin-Drinfeld triples', Mathematical Research Letters 6 (1999), no. 6, 593–612, DOI 10.4310/mrl.1999.v6.n6.a1, arXiv:math/9908115, https://arxiv.org/abs/math/9908115.
9. V. Toledano-Laredo, Y. Yang, 'The elliptic Casimir connection of a simple Lie algebra', arXiv:1805.12261, https://arxiv.org/abs/1805.12261 (preprint, 56 pages).