Knizhnik Zamolodchikov Bernard Equations Research Articles

Open-string integrals with multiple unintegrated punctures at genus one

We study integrals appearing in intermediate steps of one-loop open-string amplitudes, with multiple unintegrated punctures on the A-cycle of a torus. We construct a vector of such integrals which closes after taking a total differential with respect to the N unintegrated punctures and the modular parameter τ. These integrals are found to satisfy the elliptic Knizhnik-Zamolodchikov-Bernard (KZB) equations, and can be written as a power series in α′ — the string length squared- in terms of elliptic multiple polylogarithms (eMPLs). In the N-puncture case, the KZB equation reveals a representation of B1,N, the braid group of N strands on a torus, acting on its solutions. We write the simplest of these braid group elements — the braiding one puncture around another — and obtain generating functions of analytic continuations of eMPLs. The KZB equations in the so-called universal case is written in terms of the genus-one Drinfeld-Kohno algebra mathfrak{t} 1,N ⋊ mathfrak{d} , a graded algebra. Our construction determines matrix representations of various dimensions for several generators of this algebra which respect its grading up to commuting terms.

Folded and contracted solutions of coupled classical dynamical Yang–Baxter and reflection equations

In this paper we give a concrete recipe how to construct triples of algebra-valued meromorphic functions on a complex vector space a satisfying three coupled classical dynamical Yang–Baxter equations and an associated classical dynamical reflection equation. Such triples provide the local factors of a consistent system of first order differential operators on a, generalising asymptotic boundary Knizhnik–Zamolodchikov–Bernard (KZB) equations.The recipe involves folding and contracting a-invariant and θ-twisted symmetric classical dynamical r-matrices along an involutive automorphism θ. In case of the universal enveloping algebra of a simple Lie algebra g we determine the subclass of Schiffmann’s classical dynamical r-matrices which are a-invariant and θ-twisted.The paper starts with a section highlighting the connections between asymptotic (boundary) KZB equations, representation theory of semisimple Lie groups, and integrable quantum field theories.

Two dialects for KZB equations: generating one-loop open-string integrals

Two different constructions generating the low-energy expansion of genus-one configuration-space integrals appearing in one-loop open-string amplitudes have been put forward in refs. [1–3]. We are going to show that both approaches can be traced back to an elliptic system of Knizhnik-Zamolodchikov-Bernard(KZB) type on the twice-punctured torus.We derive an explicit all-multiplicity representation of the elliptic KZB system for a vector of iterated integrals with an extra marked point and explore compatibility conditions for the two sets of algebra generators appearing in the two differential equations.

Planck constant as spectral parameter in integrable systems and KZB equations

We construct special rational ${\rm gl}_N$ Knizhnik-Zamolodchikov-Bernard (KZB) equations with $\tilde N$ punctures by deformation of the corresponding quantum ${\rm gl}_N$ rational $R$-matrix. They have two parameters. The limit of the first one brings the model to the ordinary rational KZ equation. Another one is $\tau$. At the level of classical mechanics the deformation parameter $\tau$ allows to extend the previously obtained modified Gaudin models to the modified Schlesinger systems. Next, we notice that the identities underlying generic (elliptic) KZB equations follow from some additional relations for the properly normalized $R$-matrices. The relations are noncommutative analogues of identities for (scalar) elliptic functions. The simplest one is the unitarity condition. The quadratic (in $R$ matrices) relations are generated by noncommutative Fay identities. In particular, one can derive the quantum Yang-Baxter equations from the Fay identities. The cubic relations provide identities for the KZB equations as well as quadratic relations for the classical $r$-matrices which can be halves of the classical Yang-Baxter equation. At last we discuss the $R$-matrix valued linear problems which provide ${\rm gl}_{\tilde N}$ Calogero-Moser (CM) models and Painleve equations via the above mentioned identities. The role of the spectral parameter plays the Planck constant of the quantum $R$-matrix. When the quantum ${\rm gl}_N$ $R$-matrix is scalar ($N=1$) the linear problem reproduces the Krichever's ansatz for the Lax matrices with spectral parameter for the ${\rm gl}_{\tilde N}$ CM models. The linear problems for the quantum CM models generalize the KZ equations in the same way as the Lax pairs with spectral parameter generalize those without it.

Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles

We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB) equations related to the WZW-theory corresponding to the adjoint $G$-bundles of different topological types over complex curves $\Sigma_{g,n}$ of genus $g$ with $n$ marked points. The bundles are defined by their characteristic classes - elements of $H^2(\Sigma_{g,n},\mathcal{Z}(G))$, where $\mathcal{Z}(G)$ is a center of the simple complex Lie group $G$. The KZB equations are the horizontality condition for the projectively flat connection (the KZB connection) defined on the bundle of conformal blocks over the moduli space of curves. The space of conformal blocks has been known to be decomposed into a few sectors corresponding to the characteristic classes of the underlying bundles. The KZB connection preserves these sectors. In this paper we construct the connection explicitly for elliptic curves with marked points and prove its flatness.

Q-Deformed KZB Heat Equation: Completeness, Modular Properties and SL(3, [formula omitted

We study the properties of one-dimensional hypergeometric integral solutions of the q-difference (“quantum”) analogue of the Knizhnik–Zamolodchikov–Bernard equations on tori. We show that they also obey a difference KZB heat equation in the modular parameter, give formulae for modular transformations, and prove a completeness result, by showing that the associated Fourier transform is invertible. These results are based on SL(3, Z ) transformation properties parallel to those of elliptic gamma functions.

Wakimoto Modules and Knizhnik-Zamolodchikov-Bernard Equations

This article is a short review on the construction of solutions of the KnizhnikZamolodchikov (KZ) equation and the Knizhnik-Zamolodchikov-Bernard (KZB) equation in integral form with the help of the Wakimoto realization of affine Lie algebras. Correlation functions of the (chiral) Wess-Zumino-Witten models on a Riemann sphere satisfy the well-known Knizhnik-Zamolodchikov (KZ) equations. 1), 2) There are many works on integral representation of solutions of the KZ equations. The most general solutions were found in Ref. 3) from the viewpoint of the theory of hypergeometric type integrals. There is another approach which makes use of free field theories on Riemann surfaces and the Wakimoto realization of affine Lie algebras; e.g., Refs. 4) 8). We review this result in §3. Bernard 9) found a system of equations for the genus one case, which is now called the Knizhnik-Zamolodchikov-Bernard (KZB) equations. Felder and Varchenko found an integral representation of solutions of this system in Ref. 10) by the same method as in Ref. 3). In our paper 11) we applied the free field theory to the KZB equations. We will review this result in §4. The key lemma is the screening current Ward identity used in Refs. 7) and 8) (with necessary modification). For the sl(2) case Bernard and Felder found the same result in Ref. 6) by using Wakimoto modules.

On the Quantization of Isomonodromic Deformations on the Torus

The quantization of isomonodromic deformation of a meromorphic connection on the torus is shown to lead directly to the Knizhnik–Zamolodchikov–Bernard equations in the same way as the problem on the sphere leads to the system of Knizhnik–Zamolodchikov equations. The Poisson bracket required for a Hamiltonian formulation of isomonodromic deformations is naturally induced by the Poisson structure of Chern–Simons theory in a holomorphic gauge fixing. This turns out to be the origin of the appearance of twisted quantities on the torus.