nLab Knizhnik-Zamolodchikov-Kontsevich construction -- definition

For the Definition of the Knizhnik-Zamolodchikov connection we need the following notation:

1. configuration spaces of points

   For $N_{\mathrm{f}} \in \mathbb{N}$ write

   (1)$$\underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{R}^2) \;\coloneqq\; (\mathbb{R}^2)^n \backslash FatDiagonal$$

   for the ordered configuration space of n points in the plane, regarded as a smooth manifold.

   Identifying the plane with the complex plane $\mathbb{C}$, we have canonical holomorphic coordinate functions

   (2)$$(z_1, \cdots, z_{N_{\mathrm{f}}}) \;\colon\; \underset{{}^{\{1,\cdots,n\}}}{Conf}(\mathbb{R}^2) \longrightarrow \mathbb{C}^{N_{\mathrm{f}}} \,.$$

2. horizontal chord diagrams

   (3)$$\mathcal{A}^{{}^{pb}}_{N_{\mathrm{f}}} \;\coloneqq\; Span\big(\mathcal{D}^{{}^{pb}}_{N_{\mathrm{f}}}\big)/4T$$

   for the quotient vector space of the linear span of horizontal chord diagrams on $n$ strands by the 4T relations (infinitesimal braid relations), regarded as an associative algebra under concatenation of strands (here).

Definition

(Knizhnik-Zamolodchikov form)

The universal Knizhnik-Zamolodchikov form is the horizontal chord diagram-algebra valued differential form (3) on the configuration space of points (1)

(4)$$\omega_{KZ} \;\in\; \Omega \big( \underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{C}) \,, \mathcal{A}^{{}^{pb}}_{N_{\mathrm{f}}} \big)$$

given in the canonical coordinates (2) by:

(5)$$\omega_{KZ} \;\coloneqq\; \underset{ i \lt j \in \{1, \cdots, n\} }{\sum} d_{dR} log\big( z_i - z_j \big) \otimes t_{i j} \,,$$

where

is the horizontal chord diagram with exactly one chord, which stretches between the $i$th and the $j$th strand.

Regarded as a connection form for a connection on a vector bundle, this defines the universal Knizhnik-Zamolodchikov connection $\nabla_{KZ}$, with covariant derivative

$$\nabla \phi \;\coloneqq\; d \phi + \omega_{KV} \wedge \phi$$

for any smooth function

$$\phi \;\colon\; \underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{C}) \longrightarrow \mathcal{A}^{{}^{pb}}_{N_{\mathrm{f}}} Mod$$

with values in modules over the algebra of horizontal chord diagrams modulo 4T relations.

The condition of covariant constancy

$$\nabla_{KZ} \phi \;=\; 0$$

is called the Knizhnik-Zamolodchikov equation.

Finally, given a metric Lie algebra $\mathfrak{g}$ and a tuple of Lie algebra representations

$$( V_1, \cdots, V_{N_{\mathrm{f}}} ) \;\in\; (\mathfrak{g} Rep_{/\sim})^{N_{\mathrm{f}}} \,,$$

the corresponding endomorphism-valued Lie algebra weight system

$$w_{V} \;\colon\; \mathcal{A}^{{}^{pf}}_{N_{\mathrm{f}}} \longrightarrow End_{\mathfrak{g}}\big( V_1 \otimes \cdots V_{N_{\mathrm{f}}} \big)$$

turns the universal Knizhnik-Zamolodchikov form (4) into a endomorphism ring-valued differential form

(6)$$\omega_{KZ} \;\coloneqq\; \underset{ i \lt j \in \{1, \cdots, n\} }{\sum} d_{dR} log\big( z_i - z_j \big) \otimes w_V(t_{i j}) \;\in\; \Omega \big( \underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{C}) \,, End\big(V_1 \otimes \cdot V_{N_{\mathrm{f}}} \big) \big) \,.$$

The universal formulation (4) is highlighted for instance in Bat-Natan 95, Section 4.2, Lescop 00, p. 7. Most authors state the version after evaluation in a Lie algebra weight system, e.g. Kohno 14, Section 5.

Proposition

(Knizhnik-Zamolodchikov connection is flat)

The Knizhnik-Zamolodchikov connection $\omega_{ZK}$ (Def. ) is flat:

$$d \omega_{ZK} + \omega_{ZK} \wedge \omega_{ZK} \;=\; 0 \,.$$

Proposition

(Kontsevich integral for braids)

The Dyson formula for the holonomy of the Knizhnik-Zamolodchikov connection (Def. ) is called the Kontsevich integral on braids.

(e.g. Lescop 00, side-remark 1.14)