Comparison of spaces associated to DGLA via higher holonomy
Paul Bressler Alexander Gorokhovsky Ryszard Nest Boris Tsygan




P.Bressler]Paul Bressler
Departamento de Matemáticas, Universidad de Los Andes, Bogotá, Colombia paul.bressler@gmail.com

A.Gorokhovsky]Alexander Gorokhovsky
Department of Mathematics, UCB 395,
University of Colorado, Boulder, CO 80309-0395, USA
Alexander.Gorokhovsky@colorado.edu

R.Nest]Ryszard Nest
Department of Mathematics,
Copenhagen University, Universitetsparken 5, 2100 Copenhagen, Denmark
 rnest@math.ku.dk

B.Tsygan]Boris Tsygan
Department of
Mathematics, Northwestern University, Evanston, IL 60208-2730, USA
b-tsygan@northwestern.edu

R. Nest was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).



For a differential graded Lie algebra 𝔤 whose components vanish in degrees below -1 we construct an explicit equivalence between the nerve of the Deligne 2-groupoid and the simplicial set of 𝔤-valued differential forms introduced by V. Hinich. The construction uses the theory of non-abelian multiplicative integration.


Comparison of spaces associated to DGLA via higher holonomy
    [
    December 30, 2023
===========================================================





§ INTRODUCTION

The principal result of the present note is a construction of a weak equivalence of two spaces (simplicial sets) naturally associated with a nilpotent differential grade Lie algebra subject to certain restrictions.

To a nilpotent DGLA 𝔤 which satisfies the additional condition

    𝔤^i = 0 for i<-1

P. Deligne <cit.> and, independently, E. Getzler <cit.> associated a (strict) 2-groupoid which we denote ^2(𝔤) and refer to as the Deligne 2-groupoid. In an earlier paper by the authors (<cit.>) it was shown that the simplicial nerve 𝔑^2(𝔤) of the 2-groupoid ^2(𝔤), 𝔤 a nilpotent DGLA satisfying (<ref>) is weakly equivalent to another simplicial set, denoted Σ(𝔤), introduced by V. Hinich <cit.> by constructing natural maps from 𝔑^2(𝔤) and Σ(𝔤) to a third simplicial set and showing that they are equivalences.

In this note we outline a construction of the map

    𝕀(𝔤) Σ(𝔤) →𝔑^2(𝔤)

omitting technical details. The main tool in our construction is the theory of non-abelian multiplicative integration on surfaces.

In <cit.>, Yekutieli constructed a theory of multiplicative integration on surfaces based on Riemann products(cf. also Aref'eva's work <cit.>; a sketch of a construction, also based on Riemann products, is due to Kontsevich, cf. <cit.>) and does not apply in the context at hand. In <cit.> Kapranov develops a theory of higher holonomy based on Chen's iterated integrals. In the present work we take the approach based on the fundamental theorem of calculus.

The paper is organized as follows. In Section <ref> we recall some basic notions of differential and integral calculus. In Section <ref> we review the theory of connections and holonomy. In Section <ref> we define the two-dimensional holonomy of connection-curvature pairs and discuss some of its properties. In Section <ref> we recall the definitions of the simplicial sets 𝔑^2(𝔤) and Σ(𝔤), construct the map (<ref>) and state its properties in Theorem <ref> and Theorem <ref>.



 §.§ Notation

Throughout the paper


  * k is a field of characteristic zero


  * for a variety X over k: 𝔸^n_X := X×𝔸^n_k; for a commutative k-algebra R: 𝔸^n_R := 𝔸^n_SpecR


  * for a variety _X  Y → X over X and a 𝒪_X-module ℰ we denote by ℰ_Y the 𝒪_Y-module _X^*ℰ


  * for an X-variety Y → X over X we denote by 𝚍_/X𝒪_Y →Ω^1_Y/X the relative differential


  * for an nilpotent 𝒪_X-Lie algebra 𝔤 we denote by exp(𝔤) the algebraic group over X defined as exp(𝔤)(Y) = 𝔤_Y(Y) equipped with the group structure given by the Baker-Campbell-Hausdorff series


  * for a group G and g∈ G we denote (g) the automorphism h ↦ ghg^-1




§ CALCULUS



 §.§ Logarithmic derivatives

Suppose that 𝔤 is a nilpotent Lie algebra. For D∈(𝔤), a = exp(x)∈exp(𝔤), x ∈𝔤. Define Dlog(a)∈𝔤 by

    Dlog(a) := (Da)· a^-1= (exp((x)) - 1/(x))(Dx).

For D∈(𝔤), a, b ∈exp(𝔤), x∈𝔤,

    Dlog(ab)     =     Dlog(a) + _a(Dlog(b)) 
    
    Dlog(a^-1)     =     -_a^-1(Dlog(a)) 
    _xlog(a)     =     x - _a(x)


For D_1,D_2∈(𝔤), a∈exp(𝔤)

    D_1(D_2log(a)) - D_2(D_1log(a)) = 
     [D_1,D_2]log(a) + [D_1log(a), D_2log(a)]




 §.§ Initial value problems

Let _X 𝔸^1_X → X denote the projection so that _X *𝒪_𝔸^1_X = 𝒪_X[s].

Suppose that ℰ is a vector bundle on X. For


  * A∈_𝒪_𝔸^1_X(ℰ_𝔸^1_X) nilpotent

  * a section σ X →𝔸^1_X of the projection _X

  * a section e ∈Γ(X;ℰ)

there exists a unique section u ∈Γ(𝔸^1_X; ℰ_𝔸^1_X) which satisfies

    ∂ u∂ s + A(u) = 0 
    σ^*(u) = e .




 §.§ Integration

Let x = s⊗ 1 and y = 1⊗ s denote the coordinate functions on 𝔸^1_X×_X𝔸^1_X. Let Δ denote the subvariety defined by the equation x=y.

For α∈Ω^1_𝔸^1_X/X we denote by

    ∫_x^y α∈Γ(𝔸^1_X×_X𝔸^1_X;𝒪_𝔸^1_X×_X𝔸^1_X)

the solution of the initial value problem

    ∂ u∂ y = _2^*α
    
    u|_Δ = 0

in the unknown u = u(x,y) ∈Γ(𝔸^1_X×_X𝔸^1_X;𝒪_𝔸^1_X×_X𝔸^1_X).



§ DIFFERENTIAL GEOMETRY




 §.§ Connections and curvature

Suppose that 𝔤 is a nilpotent 𝒪_X-Lie algebra. For A∈Γ(𝔸^k_X; Ω^1_𝔸^k_X/X⊗𝔤) 
we consider connection ∇ := 𝚍_/X + A. Its curvature is defined by F = F(∇) := 𝚍_/XA + 12 [A,A] ∈Γ(𝔸^k_X; Ω^2_𝔸^k_X/X⊗𝔤).

Suppose now that ℰ is an 𝒪_X-module and ρ𝔤→_𝒪_X(ℰ) is a representation of 𝔤. The connection ∇ gives rise to the connection ∇^ℰ on ℰ_𝔸^k_X.

The sheaf Ω^1_𝔸^k_X/X of relative differentials is equipped with the canonical relative connection. Hence, the connection ∇ induces a connection on the 𝒪_𝔸^k_X-module Ω^1_𝔸^k_X/X⊗ℰ which will be denoted ∇^ℰ as well.

An example of the above situation is given by the adjoint representation 𝔤→(𝔤) giving rise to the connection ∇^𝔤.



 §.§ Logarithmic covariant derivative


For u = exp(x)∈exp(𝔤)(𝔸^k_X), let

    d log (u):= (exp((x)) - 1/(x))(𝚍x)

and

    ∇log(u) := d log u + A .





For ξ_1, ξ_2 ∈𝒯_𝔸^k_X/X such that [ξ_1, ξ_2] = 0 the identity

    ∇^𝔤_ξ_1∇_ξ_2log u-
    ∇^𝔤_ξ_2∇_ξ_1log u
    = 
    
    [∇_ξ_1log u, ∇_ξ_2log u]+
    ι_ξ_1ι_ξ_2F

holds.




 §.§ Holonomy

Suppose that 𝔤 is a nilpotent 𝒪_X-Lie algebra. For A∈Γ(𝔸^1_X; Ω^1_𝔸^1_X/X⊗𝔤), ∇ := 𝚍_/X + A we denote by

    (x,y) ↦_x^y ∇∈exp(𝔤)(𝔸^1_X×_X𝔸^1_X)

the solution of the initial value problem

    _2^*∇_∂ /∂ ylog(u) = 0 
    
    u|_Δ = 1 ,

where Δ is the subvariety defined by the equation x=y.


Suppose that ℰ is an 𝒪_X-module and ρ𝔤→_𝒪_X(ℰ) is a representation of 𝔤 which is nilpotent in the sense that there exists N such that (ρ)^N = 0. The representation ρ induces the map of groups exp(ρ) exp(𝔤) →(ℰ). Since the two compositions 𝔸^1_X×_X𝔸^1_X 𝔸^1_X  X, where _i are the canonical projections, are equal it follows that _𝒪_𝔸^1_X ×_X 𝔸^1_X(_1^*ℰ_𝔸^1_X, _2^*ℰ_𝔸^1_X) = _𝒪_𝔸^1_X ×_X 𝔸^1_X(ℰ_𝔸^1_X ×_X 𝔸^1_X) and there is a canonical map

    exp(𝔤)(𝔸^1_X×_X𝔸^1_X) →(_1^*ℰ_𝔸^1_X, _2^*ℰ_𝔸^1_X) .


We denote by _x^y ∇^ℰ the image of _x^y ∇ under (<ref>).


 The following useful identities hold.

    (_x^y ∇^ℰ)∂ /∂ y(_x^y ∇^ℰ)^-1 = ∇^ℰ_∂ /∂ y


    (_x^y ∇) ·( _y^z ∇) = _x^z ∇






 §.§ Holonomy and curvature

Suppose that 𝔤 is a nilpotent 𝒪_X-Lie algebra, A∈Γ(𝔸^2_X; Ω^1_𝔸^2_X/X⊗𝔤), ∇ := 𝚍_/X + A, F = F(∇) := 𝚍_/XA + 12 [A,A] ∈Γ(𝔸^2_X; Ω^2_𝔸^2_X/X⊗𝔤).

The second projection 𝔸^2_X →𝔸^1_X identifies 𝔸^2_X with 𝔸^1_𝔸^1_X. Let A∈Γ(𝔸^2_X; Ω^1_𝔸^2_X/𝔸^1_X⊗𝔤) denote the image of A,  ∇ := d_/𝔸^1_X + A. Hence, the holonomy

    ((x_1,y_2),(y_1,y_2)) ↦_x_1^y_1∇∈exp(𝔤)(𝔸^2_X×_𝔸^1_X𝔸^2_X)

is defined and will be denoted, by abuse of notation, by _x_1^y_1∇.




    ∇_∂ /∂ y_2log(_x_1^y_1∇) = ∫_t=x_1^t=y_1(_y_1^t ∇^𝔤)^-1(ι_∂ /∂ y_2F)



We will show that both sides of (<ref>) are solutions of the initial value problem

    ∇^𝔤_∂ /∂ y_1Φ =  ι_∂ /∂ y_1ι_∂ /∂ y_2F
    Φ|_x_1=y_1 = 0 .

It is clear that both sides of (<ref>) satisfy the initial condition. By (<ref>),

    ∇^𝔤_∂ /∂ y_1∇_∂ /∂ y_2log(_x_1^y_1∇)-
    ∇^𝔤_∂ /∂ y_2∇_∂ /∂ y_1log(_x_1^y_1∇)
    =
    
    [∇_∂ /∂ y_1log(_x_1^y_1∇), ∇_∂ /∂ y_2log(_x_1^y_1∇)]+
    ι_∂ /∂ y_1ι_∂ /∂ y_2F

 By definition, ∇_∂ /∂ y_1log(_x_1^y_1∇) = 0. Hence,

    ∇^𝔤_∂ /∂ y_1∇_∂ /∂ y_2log(_x_1^y_1∇)=ι_∂ /∂ y_1ι_∂ /∂ y_2F

which is to say, the left-hand side satisfies the differential equation. Differentiating the right-hand side we obtain

    ∇^𝔤_∂ /∂ y_1∫_x_1^y_1(_y_1^y ∇^𝔤)^-1(ι_∂ /∂ y_2F) 
    
    = (_x_1^y_1∇^𝔤)∂ /∂ y_1(_x_1^y_1∇^𝔤)^-1∫_x_1^y_1(_y_1^y ∇^𝔤)^-1(ι_∂ /∂ y_2F) 
    
    = (_x_1^y_1∇^𝔤)∂ /∂ y_1∫_x_1^y_1(_x_1^y ∇^𝔤)^-1(ι_∂ /∂ y_2F) = (_x_1^y_1∇^𝔤)ι_∂ /∂ y_1(_x_1^y_1∇^𝔤)^-1(ι_∂ /∂ y_2F) 
    
    = ι_∂ /∂ y_1ι_∂ /∂ y_2F

Thus, the right-hand side satisfies the differential equation as well.


For a nilpotent 𝒪_X-Lie algebra 𝔤, A∈Γ(𝔸^k_X; Ω^1_𝔸^k_X/X⊗𝔤), ∇ := 𝚍_/X + A, a pair of points (x_1,…,x_k), (y_1,…, y_k) ∈𝔸^k_X(X) and an index 1⩽ j⩽ k such that x_i = y_i for i≠ j let

    _x_j^y_j∇ := _x_j^y_j∇^j ,

where Y = 𝔸^k-1_X and 𝔸^k_X → Y is the projection along the j^th coordinate, ∇^j = 𝚍_/Y + A^j, A^j ∈Γ(𝔸^1_Y; Ω^1_𝔸^k_Y/Y⊗𝔤) is the image of A.




 §.§ Special broken lines


A special broken line (of length n) γ in 𝔸^k_X is a sequence of points (p^(i))_i=0^n, p^(i) = (p^(i)_1,…, p^(i)_k) such that for any 0 ⩽ i < n-1 there exists a 1⩽ j_i⩽ k such that p^(i)_l = p^(i+1)_l for all l≠ j_i.

A special broken line is closed if p^(0) = p^(n).


For a nilpotent 𝒪_𝔸^k_X-Lie algebra 𝔤, A∈Γ(𝔸^k_X; Ω^1_𝔸^k_X/X⊗𝔤), ∇ := 𝚍_/X + A and a special broken line γ = (p^(i))_i=0^n in 𝔸^k_X, we define the holonomy of ∇ along γ as the product

    _γ∇ := (_p^(n-1)_j_n-1^p^(n)_j_n-1∇) ·(_p^(n-1)_j_n-2^p^(n-1)_j_n-2∇)·⋯·(_p^(0)_j_0^p^(1)_j_0∇)


For a special broken line γ = (p^(i))_i=0^n we denote by -γ the special broken line (q^(i))_i=0^n with q^(i) = p^(n-i).



    _-γ∇ = (_γ∇)^-1



For a closed special broken line γ = (p^(i))_i=0^n we denote by τγ the closed special broken line (q^(i))_i=0^n with q^(i) = p^(i+1 n+1).



    _τγ∇ = (_p^(0)_j_0^p^(1)_j_0∇)(_γ∇)

where j_0 is as in Definition <ref>.






§ TWO-DIMENSIONAL HOLONOMY




 §.§ Connection-curvature pairs

Data of a connection-curvature pair on 𝔸^k_X consists, by definition, of


  * a nilpotent crossed module[A crossed module is the same thing as a dgla concentrated in degrees 0 and -1. A nilpotent crossed module is one which corresponds to a nilpotent dgla.] in 𝒪_X-Lie algebras 𝔥𝔤(𝔥)


  * A∈Γ(𝔸^k_X; Ω^1_𝔸^k_X/X⊗𝔤)


  * β∈Γ(𝔸^k_X; Ω^2_𝔸^k_X/X⊗𝔥)

Let ∇ := 𝚍_/X + A, F = F(∇) := 𝚍_/XA + 12 [A,A].



These data are required to satisfy


  * ρ(F) = (β),

  * the Bianchi identity ∇β = 0.




 §.§ Two-dimensional holonomy

For a connection-curvature pair (∇, β) on 𝔸^2_X we denote by

    _x_2^y_2_x_1^y_1 (∇,β) ∈exp(𝔥)(𝔸^2_X×_X𝔸^2_X)

the solution of the initial value problem

    ∇_∂ /∂ y_2log(u) = ∫_x_1^y_1(_y_1^y ∇^𝔥)^-1(ι_∂ /∂ y_2β)
    
    u|_Δ_1 = 1

where Δ_1 is given by the equation x_1 = y_1.


For a connection-curvature pair (∇, β) on 𝔸^k_X and


  * points (x_1,…,x_k), (y_1,…, y_k) ∈𝔸^k_X(X);


  * indices 1⩽ j_1, j_2 ⩽ k such that x_i = y_i for i≠ j_1 and i≠ j_2

let

    _x_j_2^y_j_2_x_j_1^y_j_1 (∇,β) := _x_j_2^y_j_2_x_j_1^y_j_1 (∇^(j_1,j_2),β^(j_1,j_2))

where Y = 𝔸^k-2_X and 𝔸^k_X → Y is the projection along the j_1^th and j_2^th coordinates, ∇^(j_1,j_2) = 𝚍_/Y + A^(j_1,j_2), A^(j_1,j_2)∈Γ(𝔸^2_Y; Ω^1_𝔸^k_Y/Y⊗𝔤) is the image of A, β^(j_1,j_2)∈Γ(𝔸^2_Y; Ω^1_𝔸^k_Y/Y⊗𝔥) is the image of β.




 §.§ Rectangles

A rectangle R in 𝔸^2_X is a sequence of points (p^(i))_i=0^3, p^(i) = (p^(i)_1,p^(i)_2) ∈𝔸^2_X(X), such that

    p^(i)_1 = p^(i+1)_1 ⇔ p^(i+1)_2 = p^(i+2)_2

for all i  4. Rectangles are in one-to-one correspondence with pairs of points in 𝔸^2_X(X). The pair ((x_1,x_2), (y_1,y_2)) determines the rectangle ((x_1,x_2),(y_1,x_2)(y_1,y_2)(x_1,y_2)).

For a rectangle R = ((x_1,x_2),(y_1,x_2)(y_1,y_2)(x_1,y_2)) the closed special broken line of length four

    ∂ R := ((x_1,x_2),(y_1,x_2)(y_1,y_2)(x_1,y_2),(x_1,x_2)) ,

will be called the boundary of R.

For a rectangle R = ((x_1,x_2),(y_1,x_2)(y_1,y_2)(x_1,y_2)) in 𝔸^2_X let

    _R(∇,β) := _x_2^y_2_x_1^y_1 (∇,β) .




 §.§ Chains of rectangles

A chain of rectangles S = (γ,x,y) in 𝔸^k+1_X is given by the following data:


  * A special broken line γ = (p^(i))_i=0^n, p^(i) = (p^(i)_1,…, p^(i)_k)
in 𝔸^k_𝔸^1_X.

  * Points x,y ∈𝔸^1_X(X). 

The chain of rectangles S consists of rectangles  

    R_i:= ( (x, p^(i-1)_j_i-1), (y, p^(i-1)_j_i-1), (y, p^(i)_j_i-1), (x, p^(i)_j_i-1)),

1⩽ i ⩽ n, in the notations of Definition <ref>.
The
boundary of S, denoted ∂ S is the closed special broken line

    ∂ S := ((p^(0),x),(p^(0),y),(p^(1),y),…,(p^(n),y),(p^(n),x),…,(p^(0),x))

in 𝔸^k+1_X.

For 1⩽ i ⩽ n let γ_⩽ i = (p^(l))_l=0^i, S_⩽ i = (γ_⩽ i,x,y). Note that γ_⩽ n = γ and S_⩽ n = S. 
Define

    _S_⩽ 1(∇,β) := _R_1 (∇,β)

and for 2⩽ i ⩽ n let

    _S_⩽ i(∇,β) := (_γ_⩽ i-1∇)(_R_i (∇,β))·_S_⩽ i-1(∇,β)




 §.§ Green's theorem for chains of rectangles



Suppose that S is a chain of rectangles in 𝔸^k+1_X and (∇, β) is a connection-curvature pair as in <ref>. Then,

    _∂ S∇ = exp(δ)(_S(∇,β)) .





 §.§ Parallelepipeds

A parallelepiped is a pair of points in 𝔸^3_X.

In what follows we use that first coordinate projection to identify 𝔸^3_X with 𝔸^2_𝔸^1_X. A parallelepiped Q = ((x_1,x_2,x_3),(y_1,y_2,y_3))∈𝔸^3_X×_X𝔸^3_X(X) gives rise to the rectangle R = ((x_2,x_3),(y_2,x_3)(y_2,y_3)(x_2,y_3)) in 𝔸^2_𝔸^1_X and the chain of rectangles S = (∂ R, x_1,y_1).


Suppose that Q is a parallelepiped in 𝔸^3_X and (∇, β) is a connection-curvature pair as in <ref>.

    _S(∇,β) = (._x_1^y_1∇|_(x_2,x_3))(._R(∇,β)|_y_1)·(._R(∇,β)|_x_1)^-1





 §.§ Parametrization of simplexes

Recall that the n-simplex Δ^n is defined as the hyperplane in 𝔸^n+1 given by the equation t_0+… +t_n = 1.

We parameterize Δ^1, Δ^2 and Δ^3, respectively, by


  Δ^1: t_0 = 1-x, t_1 = x;


  Δ^2: t_0 = 1-x_1, t_1 = x_1(1-x_2), t_2 = x_1x_2;


  Δ^3: t_0 = 1-x_1,t_1 = x_1(1-x_2), t_2 = x_1x_2(1-x_3), t_3 = x_1x_2x_3 .

Let Φ^i denote the parameterization of Δ^i, i=1,2,3.


    _Δ^1∇ := _0^1Φ^1*∇



    _Δ^2(∇,β) := _0^1_0^1(Φ^2*∇,Φ^2*β)


With these notations, for a connection-curvature pair (∇,β) on 𝔸^3_X  Gauss-Ostrogradsky Theorem <ref> says

    (_Δ^1 f^*∇)(_Δ^2∂_0^*(∇,β) ) = 
    (_Δ^2∂_2^*(∇,β))^-1·(_Δ^2∂_1^*(∇,β))·(_Δ^2∂_3^*(∇,β)),

where f = ∂_3∘∂_2 = ∂_2∘∂_2 is the map Δ^1 →Δ^3 induced by 0 ↦ 0, 1 ↦ 1.



§ FROM Σ TO 𝔑^2




Throughout this section we assume that 𝔤 is a nilpotent DGLA which satisfies 𝔤^i = 0 for i ⩽ -2. Recall that in <cit.> we showed that the simplicial nerve 𝔑^2(𝔤) of the 2-groupoid ^2(𝔤) is weakly equivalent to another simplicial set Σ(𝔤).
In this section we apply the theory of multiplicative integration outlined above to construct the map

    𝕀(𝔤) Σ(𝔤) →𝔑^2(𝔤)

omitting technical details. We will rely heavily on results and notations from <cit.>.

In what follows we denote by Ω_n, n = 0, 1, 2, … the commutative differential graded algebra over ℚ with generators t_0,…,t_n of degree zero and dt_0, …, dt_n of degree one subject to the relations t_0 + ⋯ + t_n = 1 and dt_0 + ⋯ + dt_n = 0. The differential d Ω_n →Ω_n[1] is defined by t_i ↦ dt_i and dt_i ↦ 0. The assignment [n] ↦Ω_n extends in a natural way to a simplicial commutative differential graded algebra.

Recall that the set of Maurer-Cartan elements of 𝔤, denoted (𝔤) is defined as

    (𝔤) := {γ∈𝔤^1 |δγ + 1/2[γ,γ] = 0} .




 §.§ The simplicial set Σ(𝔤)

For a nilpotent L_∞-algebra 𝔤 and a non-negative integer n let

    Σ_n(𝔤) =(𝔤⊗Ω_n) .

Equipped with structure maps induced by those of Ω_∙ the assignment n ↦Σ_n(𝔤) defines a simplicial set denoted Σ(𝔤).

The simplicial set Σ(𝔤) was introduced by V. Hinich in <cit.> for DGLA
and used by E. Getzler in <cit.> (where it is denoted _∙(𝔤)) for
general nilpotent L_∞-algebras.



 §.§ The Deligne 2-groupoid

We denote by ^2(𝔤) the Deligne 2-groupoid as
defined by P. Deligne <cit.> and independently by E. Getzler, <cit.>. Below we review the construction of Deligne 2-groupoid of a nilpotent DGLA following <cit.> and references therein.

Since 𝔤 is a nilpotent DGLA it follows that 𝔤^0 is a nilpotent Lie algebra. The unipotent group exp𝔤^0 acts on the space 𝔤^1 by affine transformations. The action of exp X, X∈𝔤^0, on γ∈𝔤^1 is given by the formula

    (exp X) ·γ= γ- ∑_i=0^∞(
    X)^i/(i+1)!(δ X + [γ, X]) .

The action of gauge transformations (<ref>) preserves the subset of Maurer-Cartan elements (𝔤)⊂𝔤^1.

We denote by ^1(𝔤) the Deligne groupoid defined as the groupoid associated with the action of the group exp𝔤^0 by gauge
transformations on the set (𝔤).









We denote by ^2(𝔤) the Deligne 2-groupoid whose objects and the 1-morphisms of ^2(𝔤) are those of ^1(𝔤). The 2-morphisms of ^2(𝔥)are defined as follows.

For γ∈(𝔤) let [·, ·]_γ denote the Lie bracket on 𝔤^-1 defined by

    [a, b]_γ=[a, δ b+[γ,  b]].

Equipped with this bracket, 𝔤^-1 becomes a nilpotent Lie
algebra. We denote by exp_γ𝔤^-1 the
corresponding unipotent group, and by

    exp_γ𝔤^-1→exp_γ𝔤^-1

the corresponding exponential map. If γ_1, γ_2 are two Maurer-Cartan
elements, then the group exp_γ_2𝔤^-1 acts on
_^1(𝔤)(γ_1, γ_2). For exp_γ_2 t
∈exp_γ_2𝔤^-1 and _^1(𝔤)(γ_1, γ_2) the action is given by

    (exp_γ_2 t) · (exp X) =
    exp(δ t+[γ_2,t]) exp X ∈exp𝔤^0 .

By definition, _^2(𝔤)(γ_1, γ_2) is the groupoid associated with the above action.



 §.§ The simplicial nerve of the Deligne 2-groupoid

We will need the following explicit description of the simplicial nerve of ^2(𝔤). See <cit.> for details.


The simplicial nerve 𝔑^2(𝔤) admits the following explicit
description:


  * 𝔑_0^2(𝔤) = (𝔤)


  * For n≥ 1 there is a canonical bijection between 𝔑_n^2(𝔤) and the set of data of the form

    ((μ_i)_0≤ i≤ n, (g_ij)_0≤ i < j≤ n, (c_ijk)_0≤ i < j<k≤ n) ,

where


  * (μ_i) is an (n+1)-tuple of objects of Maurer-Cartan elements of 𝔤,

  * (g_ij) is a collection of 1-morphisms (gauge transformations) g_ijμ_j→μ_i

  * (c_ijk) is a collection of 2-morphisms c_ijk:g_ijg_jk→ g_ik which satisfies

    c_ijlc_jkl=c_iklc_ijk

(in the set of 2-morphisms g_ijg_jkg_kl→ g_il).




For a morphism f [m] → [n] in Δ the induced structure map
f^*𝔑_n^2(𝔤) →𝔑_m^2(𝔤) is given (under the above bijection) by f^*((μ_i),(g_ij),(c_ijk)) = ((ν_i),(h_ij),(d_ijk)),
where ν_i = μ_f(i), h_ij = g_f(i),f(j), d_ijk = c_f(i),f(j),f(k).




 §.§ From Σ to 𝔑^2

Recall that, for n = 0,1,2,…, Σ_n(𝔤) = (Ω_n⊗𝔤) ⊂(Ω_n⊗𝔤)^1 = Ω_n^0⊗𝔤^1 ⊕Ω_n^1⊗𝔤^0 ⊕Ω_n^2⊗𝔤^-1. Thus, every element μ∈Σ_n(𝔤) is a triple: μ = (μ^0,1,μ^1,0,μ^2,-1).

For n = 0,1,2,… the map

    𝕀_n(𝔤) Σ_n(𝔤) →𝔑_n^2(𝔤)

is defined as follows:


  n=0 The map 𝕀_0(𝔤) Σ_0(𝔤) = (𝔤) →(𝔤) = 𝔑_0^2(𝔤) is defined to be the identity.


  n=1 For μ = (μ^0,1,μ^1,0) ∈Σ_1(𝔤),

    𝕀_1(𝔤)(μ) = ((∂_i^*μ^0,1)_i=0,, _Δ^1(d+μ^1,0)).



  n≥ 2 For μ = (μ^0,1,μ^1,0,μ^2,-1) ∈Σ_n(𝔤),

    𝕀_n(𝔤)(μ) = ((μ^0,1_i)_0⩽ i ⩽ n, (_Δ^1(d+μ^1,0_ij)_0⩽ i<j ⩽ n, (_Δ^2(d+μ^1,0_ijk,μ^2,-1_ijk))_0⩽ i<j<k ⩽ n),

where


  * μ^0,1_i = f^*μ^0,1, where f Δ^0 →Δ^n is the map induced by [0] → [n]  0 ↦ i;


  * μ^1,0_ij = f^*μ^1,0, where f Δ^1 →Δ^n is the map induced by [1] → [n]  0 ↦ i, 1 ↦ j;


  * μ^2,-1_ijk = f^*μ^2,-1, where f Δ^2 →Δ^n is the map induced by [1] → [n]  0 ↦ i, 1 ↦ j, 2 ↦ k.




Suppose that 𝔤 is a nilpotent DGLA which satisfies 𝔤^i = 0 for i ⩽ -2. The collection of maps 𝕀_n(𝔤), n = 0,1,2,…, defines a morphism of simplicial sets

    𝕀(𝔤) Σ(𝔤) →𝔑^2(𝔤)

natural in 𝔤 which satisfies


  * 𝕀_0(𝔤) Σ_0(𝔤) →𝔑_0^2(𝔤) is the identity map after the identification Σ_0(𝔤) = (𝔤) = 𝔑_0^2(𝔤).


  * The restriction of 𝕀 to the subcategory of abelian algebras (a.k.a. complexes) coincides with the integration map

    ∫Σ(𝔞) → K(𝔞[1]) = 𝔑^2(𝔞),

𝔞 abelian. Note that the integration map is a morphism of simplicial groups.


  * If 𝔞↪𝔤 is a central subalgebra the diagram

    Σ(𝔤) @>𝕀>> 𝔑^2(𝔤) 
    
    @VVV @VVV 
    Σ(𝔤/𝔞)_0 @>𝕀>> 𝔑^2(𝔤/𝔞)_0

is a morphism of principal fibrations relative to the morphism of groups (<ref>), where Σ(𝔤/𝔞)_0 (respectively, 𝔑^2(𝔤/𝔞)_0) denotes the image of the map Σ(𝔤) →Σ(𝔤/𝔞) (respectively, 𝔑^2(𝔤) →𝔑^2(𝔤/𝔞)).



Theorem <ref> and induction on the nilpotency length imply the following statement.


The map 𝕀(𝔤) is an equivalence.


plain

