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 fN_{\mathrm{f}} \in \mathbb{N} write

    (1)Conf {1,,N f}( 2)( 2) n\FatDiagonal \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,,z N f):Conf {1,,n}( 2) N f. (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)𝒜 N f pbSpan(𝒟 N f pb)/4T \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 nn 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)ω KZΩ(Conf {1,,N f}(),𝒜 N f pb) \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)ω KZi<j{1,,n}d dRlog(z iz j)t ij, \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 iith and the jjth strand.

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

ϕdϕ+ω KVϕ \nabla \phi \;\coloneqq\; d \phi + \omega_{KV} \wedge \phi

for any smooth function

ϕ:Conf {1,,N f}()𝒜 N f pbMod \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

KZϕ=0 \nabla_{KZ} \phi \;=\; 0

is called the Knizhnik-Zamolodchikov equation.

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

(V 1,,V N f)(𝔤Rep /) N f, ( 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:𝒜 N f pfEnd 𝔤(V 1V N f) 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)ω KZi<j{1,,n}d dRlog(z iz j)w V(t ij)Ω(Conf {1,,N f}(),End(V 1V N f)). \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 ω ZK\omega_{ZK} (Def. ) is flat:

dω ZK+ω ZKω ZK=0. 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)

Last revised on January 19, 2020 at 14:29:22. See the history of this page for a list of all contributions to it.