Bruce K Driver
email: bdriver@ucsd.eduf.gabriel@warwick.ac.ukbhall@nd.edutkemp@math.ucsd.edu

Franck Gabriel

Brian C Hall

Todd Kemp

The Makeenko-Migdal equation for Yang-Mills theory on compact surfaces

proofs given by the first, third, and fourth authors for the plane case extend essentially without change to compact surfaces.

Introduction

The Euclidean Yang-Mills field theory on a surface Σ describes a random connection on a principal bundle over Σ for a compact Lie group K, known as the structure group. Work of A. Sengupta [Sen1,Sen2,Sen3,Sen4] gave a formula for the expectation value of any gauge-invariant function defined in terms of parallel transport along the edges of a graph G in Σ. (Related work was done by D. Fine [Fine1,Fine2] and E. Witten [Witt1,Witt2].) This theory was then further developed [Lévy1] and generalized [Lévy2] in the work of T. Lévy. Sengupta's formula (generalizing Driver's formula [Dr,Theorem 6.4] in the plane case) is given in terms of the heat kernel on the group K. (See Section 2.) One noteworthy feature of the formula is its invariance under area-preserving diffeomorphisms of Σ.

The typical objects of study in the theory are the Wilson loop functionals, given by E{trace(hol(L))}, (1) where E denotes the expectation value with respect to the Yang-Mills measure, hol(L) denotes the holonomy of the connection around a loop L traced out t 4 t 1 t 2 t 3 v Fig. 1. A typical loop L for the Makeenko-Migdal equation in a graph G, and the trace is taken in some fixed representation of K. The diffeomorphism-invariance of the theory is reflected in Sengupta's formula: the expectation (1) is given as a function (determined by the topology of the graph and of Σ) of all the areas of the faces of G.

A key identity for calculating Wilson loops is the Makeenko-Migdal equation [MM, Equation 3] for Yang-Mills theory. For the plane case, the Makeenko-Migdal equation takes the form (3) below, as worked out by V. A. Kazakov and I. K. Kostov in [KK, Equation 24] (see also [K,Equation 9] and [GG, Equation 6.4]. We take K = U (N ) and we use the bi-invariant metric on U (N ) whose value on the Lie algebra u(N ) = T e (U (N )) is a scaled version of the Hilbert-Schmidt inner product:

X, Y = N trace(X * Y ).

(2)

We then express the Wilson loop functionals using the normalized trace, tr(A) := 1 N trace(A).

We now consider a loop L in the plane with simple crossings, and we let v be one such crossing. We let t 1 , t 2 , t 3 , and t 4 denote the areas of the faces adjacent to v, as in Figure 1. We also let L 1 denote the portion of the loop from the beginning to the first return to v and let L 2 denote the loop from the first return to the end, as in Figure 2. The planar Makeenko-Migdal equation then gives a formula for the alternating sum of the derivatives of the Wilson loop functional with respect to these areas:

∂ ∂t 1 - ∂ ∂t 2 + ∂ ∂t 3 - ∂ ∂t 4 E{tr(hol(L))} = E{tr(hol(L 1 ))tr(hol(L 2 ))}. (3)
We follow the convention that if any of the adjacent faces is the unbounded face, the corresponding derivative on the left-hand side of (3) is omitted. Note also that the faces F 1 , F 2 , F 3 , and F 4 are not necessarily distinct, so that the same derivative may occur more than once on the left-hand side of (3). The first rigorous proof of (3) was given by Lévy in [Lévy3,Proposition 6.24]. A second proof was given by A. Dahlqvist in [Dahl,Proposition 7.2]. Both of these proofs proceed by computing the individual time derivatives on the lefthand side of (3). These formulas involve calculations along a sequence of faces proceeding from a face adjacent to v to the unbounded face. After taking the alternating sum of derivatives, both Lévy and Dahlqvist obtain a cancellation that allows the result to simplify to the right-hand side of (3). In [DHK2], three of the authors of the present paper gave three new proofs of (3). All of these proofs were "local" in nature, meaning that the calculations involve only faces and edges adjacent to the crossing v.

The goal of the present paper is to demonstrate that two of the proofs of (3) in [DHK2] can be applied almost without change to the case of an arbitrary compact surface Σ, possibly with boundary. In particular, due to the local nature of the proofs in [DHK2], we do not require the presence of an unbounded face.

Let us say that a graph G in Σ is admissible if G contains the entire boundary of Σ and each component of the complement of G is homeomorphic to a disk. (Actually, according to Proposition 1.3.10 of [Lévy2], if each component of the complement is a disk, the graph necessarily contains the entire boundary of Σ.)

Theorem 1 (Makeenko-Migdal Equation for Surfaces). Let Σ be a compact surface, possibly with boundary. Let K = U (N ) and let E denote expectation value with respect to the normalized Yang-Mills measure over Σ, possibly with constraints on the holonomies around the boundary components. Suppose that L is a closed curve that can be traced out on an admissible graph G in Σ. Suppose v is a simple crossing of L in the interior of Σ and let L 1 and L 2 denote the two pieces of the curve cut at v. Then we have

∂ ∂t 1 - ∂ ∂t 2 + ∂ ∂t 3 - ∂ ∂t 4 E{tr(hol(L))} = E{tr(hol(L 1 ))tr(hol(L 2 ))}.
We will actually prove an abstract Makeenko-Migdal equation (generalizing Proposition 6.22 in [Lévy3]) that applies to an arbitrary structure group K and that implies Theorem 1 as a special case. As in the plane case, the abstract Makeenko-Migdal equation allows one to compute alternating sums of derivatives of other sorts of functions; see Section 2.5 of [DHK2] for examples.

For any fixed N, the Makeenko-Migdal equation in (3) or in Theorem 1 is not especially helpful in computing Wilson loop functionals. After all, even though the loops L 1 and L 2 are simpler than L, the right-hand side of (3) involves the expectation of a product of traces rather than a product of expectations. Thus, the right-hand side cannot be considered as a recursively known quantity. In the plane case, however, it is known that the Yang-Mills theory for U (N ) has a large-N limit, and that in this limit, all traces become deterministic. (This deterministic limit is known as the master field and was investigated by various authors, including I. M. Singer [Sing], R. Gopakumar and D. Gross [GG], and M. Anshelevich and Sengupta [AS]. A detailed proof of the existence and deterministic nature of the limit was provided by Lévy in [Lévy3,Section 5].) Thus, in the large-N limit in the plane case, there is no difference between the expectation of a product and a product of expectations and (3) becomes

∂ ∂t 1 - ∂ ∂t 2 + ∂ ∂t 3 - ∂ ∂t 4 τ (hol(L)) = τ (hol(L 1 ))τ (hol(L 2 )), (4)
where τ (•) is the limiting value of E{tr(•)}.

In the plane case, Lévy also establishes the following "unbounded face condition." If t denotes the area of any face F that adjoins the unbounded face, we have

∂ ∂t τ (hol(L)) = - 1 2 τ (hol(L)
), (F adjoins the unbounded face).

(5) (See Axiom Φ 4 in Section 0.5 of [Lévy3] and compare Theorem 2 in [DHK2].) Lévy then shows that the large-N limit of U (N ) Yang-Mills theory on the plane is completely determined by the large-N Makeenko-Migdal equation ( 4) and the unbounded face condition ( 5), together with some continuity and invariance properties [Lévy3, Section 0.5].

It is currently not rigorously known whether Yang-Mills theory on a compact surface Σ admits a large-N limit. (However, see for example [DK], where the large-N limit of Yang-Mills theory on the 2-sphere is explored non-rigorously.) If the limit does exist and is deterministic (as in the plane case), it is reasonable to expect that the limiting theory would satisfy (4) (this is assumed in [DK]). One would have to justify interchanging the derivatives with the large-N limit in Theorem 1. On the other hand, since Σ does not have an unbounded face, the unbounded face condition in (5) does not even make sense. Thus, even if (4) holds for the large-N limit of Yang-Mills theory on Σ, this relation may not allow for a complete characterization of the limit. Nevertheless, if the large-N limit on Σ exists and satisfies (4), this relation should contain a lot of information about the limiting theory.

The authors thank Ambar Sengupta for many useful discussions of Yang-Mills theory on surfaces.

Yang-Mills theory on surfaces

The Yang-Mills measure for a graph G in a surface Σ has been described by Sengupta, first for closed surfaces in [Sen1] (see also [Sen2]) and then for surfaces with boundary [Sen3], possibly incorporating constraints on the holonomy around the boundary. Related work was done by Fine [Fine1,Fine2] and Witten [Witt1,Witt2]. Sengupta's results were further developed and generalized by Lévy in [Lévy1] and [Lévy2].

We consider a compact, connected surface Σ, possibly with boundary. We do not require that Σ be orientable. We then consider a connected compact group K, called the structure group, equipped with a fixed bi-invariant Riemannian metric. (If K is not simply connected, the Yang-Mills measure as described below may incorporate contributions from inequivalent principal K-bundles over Σ.) We also consider the heat kernel ρ t on K at the identity, that is, the unique function such that

∂ρ t ∂t = 1 2 ∆ρ t
and such that for any continuous function f on K,

lim t→0 K f (x)ρ t (x) dx = f (id),
where id is the identity element of K and dx is the normalized Haar measure.

2.1. The unconstrained Yang-Mills measure on a graph. We begin by precisely defining the appropriate notion of a graph in Σ. By an edge we will mean a continuous map γ : [0, 1] → Σ, assumed to be injective except possibly that γ(0) = γ(1). We identify two edges if they differ by an orientation-preserving reparametrization. Two edges that differ by an orientation-reversing reparametrization are said to be inverses of each other. A graph is then a finite collection of edges (and their inverses) that meet only at their endpoints. Given a graph G, we choose arbitrarily one element out of each pair consisting of an edge and its inverse. We then refer to the chosen edges as the positively oriented edges.

We call a graph G in Σ admissible if G contains the entire boundary of Σ and each face F of G (i.e., each component of the complement of G in Σ) is homeomorphic to an open disk. Thus, the boundary of F can be represented by a single loop in G. To each positively oriented edge e in G we associate an edge variable x ∈ K, and then correspondingly associate x -1 to the inverse of e. We then form a measure on K n , where n is the number of edges, as follows. For each face F of G, we consider the "holonomy" h, which is just the product of edge variables (and their inverses) along the boundary of F . We then consider first an un-normalized measure μ on K n , given by

dμ(x) = i ρ |Fi| (h i ) dx,
where dx is the product of the normalized Haar measures in the edge variables. Note: since the Haar measure on K is symmetric (i.e. invariant under x → x -1 ), the measure μ is independent of the choice of which edges in G are positively oriented.

We consider also the normalized measure

dµ(x) = 1 Z dμ(x),
where

Z = K n i ρ |Fi| (h i ) dx
is the partition function of the graph. This formula for µ is Sengupta's formula [Sen1, Theorem 5.3], which he derives from a rigorous version of the usual pathintegral formula. (As with μ, µ is independent of which edges are chosen to be positively oriented.) We use the notation E for the expectation value with respect to the normalized Yang-Mills measure:

E{f } := K n f (x) dµ(x).
It is known that the partition function Z depends only on the area and diffeomorphism class of Σ and not on the choice of graph; see Proposition 5.2 in [Sen1]. (For the independence of Z from the graph, it is essential that we use normalized Haar measures in the definition of the un-normalized measure μ.) If, for example, Σ = S 2 , then Z is given by

Z S 2 = ρ A (id),
where A is the area of the sphere and id is the identity element of K. In particular, for a fixed diffeomorphism class of surface and fixed topological type of the embedded graph, Z depends only on the sum of the areas t i of the faces of G.

Although the formula for the Yang-Mills measure on a surface is similar to the formula [Dr,Theorem 6.4] in the plane case, the two measures behave differently. In the plane case, the holonomies h i around the bounded faces of a graph are independent heat-kernel distributed random variables [Lévy3,Proposition 4.4]. For a general compact surface Σ, the h i 's are neither independent nor heat kernel distributed. For the case of a simple closed curve in S 2 , for example, we may represent the curve by a graph with a single edge, with edge variable x. The holonomies associated to the two faces of the graph are then h 1 = x and h 2 = x -1 , so that the Yang-Mills measure for this graph is

dµ(x) = 1 ρ s+t (id) ρ s (x)ρ t (x -1 ) dx,
where s and t are the areas of the two faces. (This formula may be interpreted as saying that the holonomy around the loop is distributed as a Brownian bridge at time s, where the bridge returns to the identity at time s + t.) Even for this simple example, there is no easy way to compute the expected trace of the holonomy around the loop.

Although the Yang-Mills measure on a surface is more difficult to compute with than the measure on the plane, we will show that two of the proofs of the Makeenko-Migdal equation given in [DHK2] go through essentially without change. To illustrate this point, consider the graph in Figure 3, which we regard

t 1 t 2 t 3 t 4 t 5 e 1 e 2 e 3
e 4

e 5 e 6 Fig. 3. A graph embedded in S 2 with five faces as being embedded in S 2 . If x i is the edge variable associated to the edge e i , and t j is the area of F j , the un-normalized Yang-Mills measure takes the form

dμ(x) = ρ t1 (x -1 2 x 1 )ρ t2 (x -1 3 x 6 x 2 )ρ t3 (x -1 4 x 3 )ρ t4 (x -1 1 x -1 5 x 4 )ρ t5 (x -1 6 x 5 ) dx. ( 6
)
(Note that the boundary of, say, F 1 is e 1 e -1 2 , but since parallel transport is order-reversing, the holonomy around F 1 is represented as x -1 2 x 1 .) If the graph were embedded in the plane instead of the sphere, we would simply omit the factor of ρ t5 (x -1 6 x 5 ), since in that case, F 5 would be the unbounded face, which does not contribute to Driver's formula. We see, then, that replacing the plane by some other surface does not change the "local" structure of the un-normalized Yang-Mills measure. If, for example, we wish to establish the Makeenko-Migdal equation for the central vertex in Figure 3, the first two proofs in [DHK2] apply without change, since the "local Makeenko-Migdal equation" in Theorem 6 there can be applied to the integration over the variables x 1 , . . . , x 4 . (In particular, since our proofs in the plane case make no reference to the unbounded face, the absence of an unbounded face on Σ does not cause a difficulty.) Once the Makeenko-Migdal equation for the un-normalized measure is established, it is then a simple matter to establish it for the normalized measure as well.

2.2. The constrained Yang-Mills measure on a graph. It is possible to modify the construction in the preceding subsection by constraining the holonomy around one or more of the boundary components to lie in a fixed conjugacy class. If the boundary component in question consists of a sequence e 1 , . . . , e k of edges with edge variables x 1 , . . . , x k , the holonomy around the component will be x k x k-1 • • • x 1 , since holonomy is order reversing. (Note that this boundary component will usually not be the boundary of one of the faces of G.) To constrain

x k x k-1 • • • x 1 to lie in C, we insert a δ-function δ(x k x k-1 • • • x 1 c -1
) and then integrate over c ∈ C. Thus, integration with respect to the un-normalized constrained measure μ takes the form

f (x) dμ(x) = K n C1 • • • C N f (x) i ρ |Fi| (h i ) × j δ(x j kj x j kj -1 • • • x j 1 c -1 j ) dx dvol(c 1 ) • • • dvol(c N ), (7)
where C 1 , . . . , C N are the conjugacy classes to which various boundary holonomies are constrained and where dvol is the normalized, Ad-invariant volume measure on the given conjugacy class. (See Theorem 4 in [Sen3] and compare Section 1.5 of [Lévy1].) In ( 7), we may interpret δ(•) as the small-ε limit of ρ ε (•). Alternatively, we may think of δ(

x k x k-1 • • • x 1 c -1
) as a rule telling us that instead of integrating over x k , we simply evaluate

x k to (x k-1 • • • x 1 c -1 ) -1 .
We may then construct a normalized measure by dividing by a normalization constant, which we refer to as the constrained partition function. Similarly to the unconstrained case, the constrained partition function depends only on the area, the topological type of the surface, and the constraints, but not on the choice of graph. (See the formula for N T (c) in Theorem 4 of [Sen3] and compare Proposition 4.3.5 in [Lévy2] in a more general setting.)

In Figure 4, for example, if the holonomy around the boundary of a disk is constrained to lie in C, the expected trace of the holonomy around the inner loop would be computed as

1 Z C K 3 tr(x -1 )ρ s (x -1 )ρ t (y -1 zyx)δ(zc -1 ) dx dy dz dvol(c) = 1 Z C K 2 tr(x -1 )ρ s (x -1 )ρ t (y -1 cyx) dx dy dvol(c) = 1 Z C K tr(x -1 )ρ s (x -1 )ρ t (cx) dx dvol(c),
where in the last expression, we have used the Ad-invariance of dvol(c) to eliminate the y variable. By contrast, if we left the boundary holonomy unconstrained, we would integrate tr(x -1 )ρ s (x -1 )ρ t (y -1 zyx) over K 3 , in which case the result would simplify to K tr(x -1 )ρ s (x -1 ) dx (with no normalization factor necessary).

The Makeenko-Migdal equation for surfaces

Throughout this section, we assume G is an admissible graph in Σ, that is, one containing the boundary of Σ and such that each face of G is a disk. e 1 , e 2 , e 3 , e 4 and taken to be outgoing edges. Let f : K n → C be a function of the edge variables of G and let a 1 , a 2 , a 3 , a 4 be the edge variables associated to e 1 , e 2 , e 3 , e 4 . Then f has extended gauge invariance at v if

f (a 1 , a 2 , a 3 , a 4 , b) = f (a 1 x, a 2 , a 3 x, a 4 , b) = f (a 1 , a 2 x, a 3 , a 4 x, b)
for all x ∈ K, where b is the tuple of all edge variables other than a 1 , a 2 , a 3 , a 4 .

With this definition in hand, we may formulate a general version of the Makeenko-Migdal equation for Σ, generalizing Proposition 6.22 in [Lévy3] in the plane case. The result applies to arbitrary structure groups K and to functions that are not necessarily given as the trace of a holonomy. In what follows, we allow the areas of the faces to be arbitrary positive real numbers; if we vary one area with the other areas fixed, we are changing the total area of the surface.

We consider a graph with four distinct edges e 1 , . . . , e 4 attached to a vertex v, and we label the four faces surrounding v as F 1 , . . . , F 4 , as in Figure 5, with the labeling chosen so that e 1 lies between F 4 and F 1 .

Theorem 2 (Abstract Makeenko-Migdal Equation for Σ). Following the notation of Definition 1, assume the four faces and four edges adjacent to v are distinct. Suppose f : K n → C is a smooth function with extended gauge invariance at v. If t 1 , . . . , t 4 denote the areas of the faces of G surrounding v, we have

∂ ∂t 1 - ∂ ∂t 2 + ∂ ∂t 3 - ∂ ∂t 4 K n f dµ = - K n ∇ a1 • ∇ a2 f dµ,
where µ is the normalized Yang-Mills measure, possibly with constraints on the boundary holonomies. The same result holds with µ replaced by the un-normalized measure μ. Using the arguments in Section 4 of [DHK2], it is possible to prove this result also when the faces are not distinct. It is also possible to formulate and prove a version of the result when the four edges emanating from v are not distinct, although the definition of extended gauge invariance needs some modification in this case. See Section 3.3 for more information.

In the theorem, the gradients are left-invariant gradients with respect to a 1 and a 2 with the other edge variables fixed. Explicitly,

(∇ a1 • ∇ a2 f )(a 1 , a 2 , a 3 , a 4 , b) = X ∂ 2 ∂s∂t f (a 1 e sX , a 2 e tX , a 3 , a 4 , b) s=t=0 ,
where X ranges over an orthonormal basis for the Lie algebra k of K and b represents the tuple of edge variables other than a 1 , . . . , a 4 . Using the extended gauge invariance of f, it is easy to show that

∇ a1 • ∇ a2 f = -∇ a2 • ∇ a3 f = ∇ a3 • ∇ a4 f = -∇ a4 • ∇ a1 f.
Suppose L is a closed curve traced out in G that has a crossing at v. Specifically, assume L starts at v, leaves v along e 1 , returns to v along e -1 4 , leaves v again along e 2 , and then finally returns to v along e -1 3 (with no visits to v besides those just mentioned). Then since holonomy is order-reversing, we will have tr(hol(L)) = tr(a -1 3 αa 2 a -1 4 βa 1 ), where α and β are words in the b variables. Any function of the this form is easily seen to have extended gauge invariance. If K = U (N ), we compute that

∇ a1 • ∇ a2 [tr(a -1 3 αa 2 a -1 4 βa 1 )] = X tr(a -1 3 αa 2 Xa -1 4 βa 1 X) = -tr(a -1 3 αa 2 )tr(a -1 4 βa 1 ) = -tr(L 2 )tr(L 1 ),
where L 1 and L 2 are as in Theorem 1, and where we used the elementary identity X XCX = -tr(C)I (e.g., [DHK1, Proposition 3.1]) in the second equality. This calculation shows that the abstract Makeenko-Migdal equation implies the Makeenko-Migdal equation for U (N ) (Theorem 1).

3.2. The generic case. Let us assume at first that our loop is traced out in an admissible graph G and that the vertex v is generic, meaning that the edges e 1 , . . . , e 4 and the faces F 1 , . . . , F 4 are distinct. We then make use of the following result, which was proven in [DHK2, Theorem 6].

Theorem 3 (Local Abstract Makeenko-Migdal Equation). Suppose f : K 4 → C is a smooth function satisfying the following "extended gauge invariance" property:

f (a 1 , a 2 , a 3 , a 4 ) = f (a 1 x, a 2 , a 3 x, a 4 ) = f (a 1 , a 2 x, a 3 , a 4 x)
for all a = (a 1 , a 2 , a 3 , a 4 ) in K 4 and all x in K. For each fixed α = (α 1 , α 2 , α 3 , α 4 ) in K 4 and t = (t 1 , t 2 , t 3 , t 4 ) in (R + ) 4 , define a measure µ α,t on K 4 by dµ α,t (a) =ρ t1 (a -1 2 α 1 a 1 )ρ t2 (a -1 3 α 2 a 2 )ρ t3 (a -1 4 α 3 a 3 )ρ t4 (a -1 1 α 4 a 4 ) da, where da is the normalized Haar measure on K 4 . Then for all α ∈ K 4 , we have

∂ ∂t 1 - ∂ ∂t 2 + ∂ ∂t 3 - ∂ ∂t 4 K 4 f dµ α,t = - K 4 ∇ a1 • ∇ a2 f dµ α,t .
We now come to the proof of the abstract Makeenko-Migdal equation in Theorem 2, in the generic case where the four edges e 1 , . . . , e 4 and the four faces F 1 , . . . , F 4 are distinct.

Proof (of Theorem 2). Let b denote the tuple of all edge variables other than a 1 , . . . , a 4 . The holonomies around the adjoining faces F i , i = 1, . . . , 4, will have the form h i = a -1 i+1 α i a i , where α i is a word in the b variables. Let us first consider integration with respect to the un-normalized Yang-Mills measure μ, with or without constraints on boundary holonomies. Since v lies in the interior of Σ, the edges e 1 , . . . , e 4 do not lie on the boundary. Thus, the holonomy around any boundary component will involve only the b variables. Integration with respect to μ therefore takes the form of integration over a 1 , . . . , a 4 with respect to µ α,t , where α is a function of the b variables, followed by integration in the b variables and possibly another layer of integration with respect to the constraint variables c j . In the un-normalized measure μ, the only dependence on t 1 , . . . , t 4 is in the inner layer of integration. Thus, we may push the time derivatives inside the outer layers of integration and allow them to hit on the integral over K 4 . If we then apply the local result in Theorem 3, Theorem 2 for μ will follow.

For the normalized measure, we must incorporate the partition function Z. Since Z depends only on the total area of the surface (i.e., the sum the areas of all the faces), we see that

∂ ∂t 1 - ∂ ∂t 2 + ∂ ∂t 3 - ∂ ∂t 4 Z = 0. v v
Fig. 6. A graph that is non-generic at v (left) and its generic counterpart Thus, Theorem 2 for the normalized measure easily follows from the corresponding result for the un-normalized measure.

3.3. The nongeneric case. Suppose G is an admissible graph and v is a vertex of G having four attached edges, where we count an edge twice if both ends of the edge are attached to v. We say that G is nongeneric at v if either the four edges are not distinct or the four faces surrounding v are not distinct. If G is not generic at v, we can embed G into another admissible graph G that is generic at v, as in Figure 6. If L is a loop traced out on G with a simple crossing at v, then "the same" loop can also be traced out on G . In that case, the expectation values of tr(hol(L)) and of tr(hol(L 1 ))tr(hol(L 2 ))-where L 1 and L 2 are as in Theorem 1-are the same whether we work over G or over G . This invariance result has two parts. First, there is invariance under subdividing an edge by adding a vertex in the middle of that edge, which is very easy to establish, as shown in Section 4.1 of [DHK2]. (The argument given there applies equally well in the surface case or the plane case.) Second, there is invariance under keeping the vertex set the same and adding a new edge. This invariance result is a consequence of Proposition 4.3.4 in [Lévy2], in the case that the Lévy process there is taken to be Brownian motion on K. Furthermore, it is not hard to see that area derivatives of expectation values give the same result whether computed over G or G . (See Section 4.3 of [DHK2].) Thus, the U (N ) version of the Makeenko-Migdal equation for the loop in G reduces to the corresponding result for the loop in G , which in turn follows from Theorem 2. In the graph on the left-hand side of Figure 6, for example, F 1 coincides with F 3 . Thus, t 3 is just another name for t 1 and the Makeenko-Migdal equation may be written as

2 ∂ ∂t 1 - ∂ ∂t 2 - ∂ ∂t 4
E{tr(hol(L))} = E{tr(hol(L 1 ))tr(hol(L 2 ))}.

It is also possible to formulate a version of Theorem 2 itself that holds in the nongeneric situation. If the edges e 1 , . . . , e 4 are distinct but the faces F 1 , . . . , F 4 are not distinct, Theorem 2 holds with no changes to the statement, and the arguments in Section 4 of [DHK2] show how this result can be reduced to the generic case. If the edges (and possibly also faces) are not distinct, the notion of extended gauge invariance needs some revision [DHK2, Section 4.2], after which one can reduce the result to the generic case. Since this process of reduction requires no changes from the arguments in [DHK2], we do not enter into the details here, but refer the interested reader to Sections 4.2 and 4.3 of [DHK2].

Fig. 2 .

 2 Fig.2. The loops L 1 (black) and L 2 (dashed)

3. 1 .Fig. 4 .

 14 Fig. 4. The holonomy z around the boundary of the disk is constrained to lie in C.

Fig. 5 .

 5 Fig. 5. The edges and faces at v

Supported by ERC grant, "Behaviour near criticality," held by M. Hairer Supported in part by NSF grant DMS-1301534 Supported in part by NSF CAREER award DMS-1254807


Camille Paloque-Bergès

Mapping A French Internet Experience: A Decade Of Unix

Introduction 1

France's developments in computer communication networks were largely determined by a governmental choice in 1978 to back up Transpac, a telecom virtual circuit model, to the detriment of Cyclades, a datagram-based research modelin order to stimulate French telecommunication engineering at an international and competitive model (Schafer 2012; Vedel 1984). After a decade-long institutional battle between the national and monopolistic company France Telecom and the National Institute for Research in Computer Science and Control (Inria), TRANSPAC was launched and set up to host Minitel online services a few years later. While Cyclades' chief researcher Louis Pouzin was exporting across the Atlantic his datagrams technology to the Arpanet (then to become Internet) team, French computer researchers and engineers were at home, deploying an alternative network based on the Unix operating system. From 1983 on, they developed Fnet, an informal infrastructure used as the local branch of EUNet, the European Unix-based data network run from the Netherlands.

Doing so, they "created a computer network social environment […] with the same ingredients that would make the Internet successful later" 1 (Bloch 2013). Open, decentralized,

collaborative, heterogeneous, and worldwide: its qualities appealed to the community of computer research and engineering in academic and private labs. Fnet became their unofficial Internet network provider until the success of the Web brought along commercial and official governmental-promoted new solutions such as Renater from 1993 on [START_REF] Schafer | Dans les Coulisses de l'Internet: RENATER, 20 Ans de Technologie[END_REF].

The history of Unixians' computer networking collaboration may have similarities to the Arpanet's "ideal scientific community" [START_REF] Flichy | The Internet Imaginaire[END_REF] 2007), but if we look at Usenet, ancestor to computerized social network initiated on Unix networks we might think differently. In France, starting and running the infrastructure was done in less than ideal conditions. I intend to show, beyond shedding light on a lesser-known part of computer networks history, that Fnet was a "shadow infrastructure": an informal, experimental, and unacknowledged network of machines annexed to the existing telecom network, as well as a network of peers using and rerouting public telecom resources from the academic world. I thus look into how Fnet was managed in terms of technical and administrative infrastructure, a difficult question given that it had no official existence. This can be read as a series of "tactical" practices (de Certeau 2011), as well as processes of negotiation, in a Science and Technology studies perspective, both of which providing basis for Actor-Network theory-type of mapping method [START_REF] Latour | Science in Action: How to Follow Scientists and Engineers Through Society[END_REF])which I use as an analytic metaphor, but will not apply literally. Although the networking side of Unix's history is documented [START_REF] Kelty | Two Bits: The Cultural Significance of Free Software[END_REF][START_REF] Salus | A Quarter Century of Unix[END_REF], its local role in paving the way for the way for the Internet internationally, like in France remains just a hint, although vivid in protagonists' memory [START_REF] Griset | Make the pig fly!": l'Inria, ses chercheurs et Internet des années 1970 aux années 1990[END_REF][START_REF] Huitema | Et Dieu créa l'Internet[END_REF]. Original sources in this chapter comprise archives from the Computer Lab at Cnam 2 , which hosted Fnet's administration from 1983 to 1986, as well as Usenet mail archives and oral interviews with the protagonists who supported and promoted Fnet up until Goggin and M. McLelland eds., 2017, New York, Routledge, pp. 153-170. 3 1993, when its missions shifted with the rise of the commercial Internet.

The Unixian Community and its French Members: Open Cooperation in the Margins

The Conservatoire national des arts et métiers [National Conservatory of Arts and Crafts, known as "Cnam"] is a higher education establishment based in Paris. It was created in the wake of the French Revolution in 1794 to stimulate industrial innovation, and has been since a privileged public institution for applied research and continuous education in the technological field. A computing center, called Laboratoire d'Informatique [Computer Lab] was created in the 1960s with support from the department of Applied Mathematics and Computing. 3 Its purpose was to support and manage the needs in computing resources for the teaching department, as well as general IT support for the administration. In the late 1970's and early 1980's, there was no official computer science research activity at Cnam, but a small, informal team of about four researchers calling themselves the "Systems lab" implemented and experimented with the Unix system within the laboratory. Led by Claude Kaiser, a professor at Cnam since 1974 with a research background from INRIA 4 , and Gérard Florin, head of the Computing Lab, who had defended a thesis in computer networking in 1975, they "played a major part in the implementation Unix, but also similar experiments with Ethernet, VAX-VMS, Unix-BSD, UUCP and then TCP/IP." 5 This early interest in versions of the Unix operating systems (Unix-BSD), Unix compatible machines (VAX), and network protocols (Unix's UUCP and the Internet's TCP-IP), set the stage for the deployment of Internet networks in France.

A Unixian Case for Collaboration: Standardization and Negotiation

Unix was then a paradigm for operating systems, chosen by a myriad of academic and private computer departments and research laboratories internationally [START_REF] Kelty | Two Bits: The Cultural Significance of Free Software[END_REF]. Ken Thompson designed at his own initiative the system in 1969 along with Denis Ritchie at the Bell Labs, the R&D department of American telecommunication company AT&T, to foster the reuse of software on incompatible machines. AT&T was forbidden to sell the operating system by a 1956 anti-trust decree restricting the company to exploit anything other than telecommunication equipment. From 1975 on, it was thus distributed to universities for a small licensing fee, along with its full source code, and was largely adopted by academia as a predecessor of the free software movement [START_REF] Kelty | Two Bits: The Cultural Significance of Free Software[END_REF][START_REF] Paloque-Berges | Histoires et Cultures du Libre: Des Logiciels Partagés aux Licences Échangées[END_REF]).

Unix's success is firstly rooted in then-emerging hacker ethics, a combination of informal cooperation, autonomy from traditional hierarchies placed under the banner of computer hardware, and software experimentalism. Victor A. Vyssotsky, then a Bell Labs' computer engineer, sums it up:

When Unix evolved within Bell Laboratories, it was not a result of some deliberate management initiative. It spread through channels of technical need and technical contact. This was typical of the way Unix spread around Bell Laboratories. […] it was used, modified, and tinkered up in a whole variety of organizations (Vyssotsky, quoted in Hauben and Hauben 1997: 89). 6 Yves Devillers 7 recalls how Unix traveled to France in a similar fashion: "It was driven by demand. If technology allowed to do it, then it was done. Everything that was doable or appropriable was." At a time when the governmental Plan Calcul [Planning for Computing, 1966Computing, -1975] ] was still influential, requiring public institutions to use domestic technological equipment [START_REF] Schafer | Dans les Coulisses de l'Internet: RENATER, 20 Ans de Technologie[END_REF][START_REF] Schafer | La France en Réseaux (Tome 1) La Rencontre des Télécommunications et de l'Informatique[END_REF]Mounier-Khun 2010a;2010b), Unix was an alternative system for hands-on experiments in software, and accessing remote computers [START_REF] Cornwall | The Hacker's Handbook[END_REF]. Moreover, Unix offered solutions in software portability and system standardizations to the 1970s computer industry problems [START_REF] Campbell-Kelly | Computer: A History of the Information Machine[END_REF]. Doing so, it gave a sense of community beyond "user clubs", traditionally restricted to proprietary computer systems (Mounier-Kuhn 2010a;2010b). It thus changed the face of software development, giving hacking culture a leading role in software cooperation for establishing norms and standards [START_REF] Raymond | A Brief History of Hackerdom, version 1.24[END_REF]. At Cnam, Kaiser was one of the main advocate for Unix as an international normative system: 8

From the point of view of software, we must avoid anarchy and incommunicability; an effort towards standardization must be made. [Unix is] a norm […] being established currently, and a lot of micro constructors offer or announce it.

Resort to Unix systems in France was complicated by the end of Plan Calcul policy, fueled by two decades of a strong-willed reorganization of technological research and development, as well as a significant drop in credit for technology equipment in the science field, and political and economic problems within Cnam: all this made the acquisition of Unix-compatible machines from American constructors difficult. Another factor was an American embargo in 1979, preventing French computer users to buy American equipment for fear it would be used to build nuclear weapons. However, the Computer lab needed to renew its computer parkhaving relied for a decade on the IBM 360 (Florin 2016 9 ). The research team had been familiarized with Unix machines thanks to a visiting engineer from the Netherlands and was looking to acquire a mini-computer compatible with the system, that would meet their budget and power needs (no French machine would be eligible, as testified by [START_REF] Florin | [END_REF]. They acquired a DEC PDP 11/70 machine in 1979, and in 1980 a DEC VAX 11/780, PDP's successor, both "workhorse machines of the early Unix culture" [START_REF] Raymond | A Brief History of Hackerdom, version 1.24[END_REF], but this occurred only through a lengthy negotiation with the Cnam's administration (requiring the computer be used for administration and accounting needs) and a formal and legally signed declaration that the computer would only be used for research and education purposes. Unix was implemented in the summer of 1980, running "outside of working hours" 10 . Eventually, a third Unix-compatible machine was acquired with another bypass method. Indeed, a VAX 11/780 clone was produced by a British manufacturer specially to sell to countries under embargo, like France and South Africa: entitled Systime-8750, it was bought in 1981 and constructed by the Computer Lab one VAX rack at the time, "importing it piece by piece" under the radar [START_REF] Florin | [END_REF]. This was the machine on which the Unix network was implemented, and coincidentally the only historic computer used Cnam preserved by the Museum of Arts and Crafts in its reserves [Cnam's Musée des arts et métiers, the French national museum of science and technology].

Protocols for Machine and Human Communication

At first, Unix was dedicated to sharing resources on time-sharing mini-computers. At the Bell Labs, Mike Lesk and David Nowitz had developed in 1976 a file transmission command into a distant machine communication protocol (Hauben and Hauben 1997: 206). Called UUCP (Unix-to-Unix Copy Protocol), it copied a file from one terminal to the other, or to a printer, but was soon appropriated to send messages between user accounts, a makeshift tactic similar to the birth of email a few years before [START_REF] Hafner | Where Wizards Stay Up Late[END_REF]. Implemented into a distant Unix machines network under a "store and forward" model, this point-to-point technique sent messages to a series of connected machine so as to reach its destination. It first ran on dial-up connectivity, thus needing a telecommunication network to dial into.

The UUCP protocol was encompassed with a hacker aura and do-it-yourself technical practice: "low-speed and unreliable, but cheap […] over ordinary phone lines" [START_REF] Raymond | A Brief History of Hackerdom, version 1.24[END_REF]. Like the Arpanet, and the subsequent TCP-IP protocol that led to the Internet in 1983,

UUCP was an open system running on an end-to-end model, meaning that a machine administrator could make a liaison without any authorization from a central authority. This determined the ease with which UUCP liaisons were created and spread all over the Unixian international community.

UUCP networks first entered Continental Europe thanks to cooperation between an academic lab and a corporate lab. The Centrum Wiskunde & Informatica (CWI 11 ) of Amsterdam University had a partnership with Armando P. Strettner from the DEC Lab at Maynard (Massachusetts, USA), a founding member of DEC Unix Engineering Group who played a major role in creating UUCPnet in the US as well as in Europe and Asia. As DEC machines were Unix-compatible, a transatlantic connection was extended between DEC labs in Maynard and their counterparts in the computing center of CWI Amsterdam, at the instigation of Teus Hagen and Piet Berteema, a computer science researcher and an engineer.

UUCP experiments appeared at Cnam in 1982, with, in the works, the implementation of an Ethernet local network with extensions to institutional partners. While Unix networks experiments were not qualified as research per se, they did fall under the category of distributed networks, process and data communication research. 12 Opening communication routes was aimed at (1) finding standards for computer networking, (2) experimenting critically with "file transfer software between connected systems on a permanent line or virtual circuit" and "certain distant commands, among which the 'Mail' command for electronic mail", and, last but not least, (3) building a network between French and American research centers. 13

International Cooperation: A Social Network of Peers

The spreading of Unix relied on a socio-professional networks of peers, supported mainly by non-profit associations due to the non-commercial nature of Unix. Indeed the Computer Lab was strongly involved with the world of computer research 14 and user organizations, such as US-based Unix user group Usenix, or its European and French counterparts, the European Unix Users Group (EEUG), or AFUU (French Association for Unix Users)the latter born in 1982. Thus, communication between distant peers was crucial in a small international community in which everyone knew everyone [START_REF] Bloch | [END_REF], fostered by events like the 11th "European Unix Systems User Group OPEN Meeting" held at Cnam in April 1982 and supported by EEUG. 15 Organized by Humberto Lucas and Bernard Martin, two engineers from the Computing Lab active in the "Systems lab", it was nicknamed "Spring in Paris". A focus of the meeting was the use of Unix for purposes of telecommunication and computer networks, with several presenting applications for Unix networking protocols (UUCP) and software (among which the news software called Usenet). 16 At this occasion, Teus Hagen and Piet Berteema from CWI showcased their new UUCPnet node implemented in Amsterdam, and announced they had created an organization for supporting the spread of Unix networks in Europe, called EUnet, for European UNIX Network. T. Hagen also presented and distributed software for group discussions (then called "electronic conferences") on UUCP networks, a system called Usenet that he brought back to Europe on a tape from a Usenix conference in San Francisco. This stimulated the creation of local branches in several countries in Europe, among which Fnet a year later, with the Computer Lab at Cnam used as server backbone and administrative support. As Unixians from other establishments or organizations flocked to connect to the Cnam UUCP hub or open their own, a "hive phenomenon" triggered by the circulation of computer engineers and scientists in France and abroad: "Everyone leaving a workplace for another wanted to connect their new workplace so as to maintain the means of communications" [START_REF] Devillers | [END_REF].

The Unixians' Internet Experience amidst Other Technical and Social Networks

Unix networks emerged in cooperation but also competition with other networks of computers. In the 1970s, they were mostly centralized and proprietary. While IBM's BITNET (EARN in Europe), UUCPnet's main rival, used a similar point and forward technique, and was widely used and tinkered with within academia, it was not born as a heterogeneous network, but relied on IBM machines only, contrary to UUCPnet, compatible with any machines with the Unix system installed. Nor was DECnet, relying solely on DEC's PDP machinesalthough Unix could be installed on PDPs, thus allowing UUCP connection. Unix networks were based on an open, distributed, and collaborative model, keen on including new users from many professional backgrounds. This principle materialized through the implementation of UUCP-based Usenet software, the open electronic conference system it hosted, effectively supporting "global conversations" from people beyond the computer science world up to the 1990s (Rheingold, 1993). Accordingly, at this point I will look more closely on the technical, but also social, constraints that accompanied the deployment of Unix networks.

Opposition and Compromise: Negotiating with "Authoritarian" Norms

The loose, informal structure of the Unixian community is key to understanding the spread of a technical network based on the social values of openness and cooperation. In a January 1984 AFUU meeting report, Humberto Lucas 17 recalls the initial project: to establish an "informal network of Unix systems with the UUCP/UUCICO protocols." 18 This informality was also political: the Unix community worked in the margins of government science and technology policies, "far from the decisions made in Bruxelles" [START_REF] Schafer | Dans les Coulisses de l'Internet: RENATER, 20 Ans de Technologie[END_REF]. [START_REF] Devillers | [END_REF] stresses the importance of the human factor: "one person only in the hierarchy was needed to add a node to the network", contrary to a complex vertical process of decision-making imposed by institutional hierarchy. Resorting to other system and network standards such as Unix and UUCP was thus considered as against the general technological policies in France: "In France, this spirit doesn't exist, at least for now", underlines Lucas in the report, in regard to using foreign hardware equipment or software standards. This oppositional stance doesn't mention, however, that the problem with acquiring Unix-compatible machines relied on a United States government embargo on shipping computer equipment to certain countries (as decided in 1979); nor that the deployment of UUUCP would need to run on protocol layer in theory rejected by Unixians, the X25 protocol. Indeed, X25 was the official protocol used on European networks like Transpac, under the umbrella of the OSI normative model recommended by the International Telecommunication Union (ITU) 19 after many negotiations and compromise [START_REF] Schafer | La France en Réseaux (Tome 1) La Rencontre des Télécommunications et de l'Informatique[END_REF]. Compromise in technological negotiations is thus an important notion that is often shadowed by protagonists, amidst their strong emphasis on resisting and bypassing institutional constraints.

The ease of installation for UUCPnet was apparent only. One of the first obstacles was an auto-dialing modem necessary to "call" a peer's number and create a new node in the network: it was strictly limited by European telecom standards, and forbidden by national telecom companies. Such modems were distributed by Teus Hagen, initiator of EUnet at CWI in Amsterdam, at the occasion of Spring in Paris in 1982 to participants who took them home to their lab upon their return, allowing several places in Europe to connect to UUCPnet (Hauben and Hauben 1997: 128-130), but a step behind than in the United States as underlined by Lucas in the 1984 AFUU reunion report.

Following the makeshift spirit of the Unixians, the deployment of a UUCP network was to rely at first on the existing infrastructure of telephone lines, in an act of derouting considered as a misappropriation, just like a few years before at the Bell Labs (Hauben and Hauben 1997: 128). As a result, much of Usenet's news distribution was implemented without explicit approval wherever a UUCP site was born, and was mostly invisible to the organizations' authorities and accounting. French Unixians justified their derouting with a criticism of the archetypal model of hierarchical and central authority implemented in France's technological policy (and especially in the new field of computer-based telecom networks). According to Devillers and Bloch, Transpac was a "bureaucratic creation" based on a principle of "descending pyramid", which Unixians fought against. By derouting the national telecom network, they also challenged post-war engineering work models, which applied new management and marketing techniques to shape the social (Vedel 1984) and the user itself with appropriate and closed technologiesthe Transpac-based Minitel was an archetype in that regards [START_REF] Thierry | Révolution 0.1 ». Utilisateurs et Communautés d'Utilisateurs au Premier Age de l'Informatique Personnelle et des Réseaux Grand Public (1978-1990[END_REF]. By contrast, Unixians defended an open model where users could shape back the technology they appropriated, and were doing so themselves by replacing Minitel terminal with computers.

Moreover, an effort was made to make EUnet and Fnet compatible with other norms and not a standalone network, as testified by memos from the Computer Lab. This challenge would prove acute for running messaging applications, one of the main uses of the UUCP networks.

Transpac's use of the official mail protocol X400 supported by the European OSI norm, was mocked as too normative by Unixians, but was necessary nonetheless to allow dialog between different servers to establish the international mail transfer protocol SMTP. 20 The distributed nature of UUCP networks allowed the different terminals to connect to different types of liaisons, but international liaisons could only be made through the EUnet backbone in Amsterdam. Thus, one of the main challenges in the following years would be to maintain the use of the UUCP protocols while, at the same time, complying with French and European norms.

Openness and Exclusiveness: The Intricate Relationship between Unix and the Internet

1982 was also the year when the term "Internet" was officially released to name the state of computer networks interconnection made possible by the TCP-IP protocol (implemented on the Arpanet in January 1983). The convergence between Usenet and the Internet takes root within a common technical ethics involving autonomy to install software from the hardware infrastructure, and "no global control at the operations level" (Leiner et al. 1997: 104). The Arpanet and Unix appeared complementary for the spread of computer internetworking technologies throughout the 1980's, because the system was compatible with many machines present on the international network. Internet protocols were integrated in the Berkeley version of Unix (BSD) in 1976: this gave a running start to computer network technologies in the Unixian world, "a key element in the Internet's successful widespread adoption" (Leiner et al. 1997: 105). As a result, an estimated 98 per cent of US computer science departments had adopted the code in the early 1990s [START_REF] Kelty | Two Bits: The Cultural Significance of Free Software[END_REF]. (Renard 2013). INRIA financed the transatlantic liaison for the profit of public research centers in France. Every public organism benefiting from the liaison had to be certified to NSF, while private clients had to continue using the Fnet network through the EUnet backbone in Amsterdam [START_REF] Huitema | Qui Etaient les Premiers? (Re: Au Sujet du Forum Internet[END_REF].

Despite bridges to the

There was nonetheless a running rivalry between the Arpanet/Internet social world, considered the high-class of computer network research, and UUCP/Usenet's, described by its protagonists as the "Poor Man's Arpanet" for the "commonfolk of the computer community" (Hauben and Hauben 1997: 122-124;[START_REF] Hafner | Where Wizards Stay Up Late[END_REF]. Hacker folklore likes to describe it as a network for "upstarts" with "primitive tools" [START_REF] Raymond | A Brief History of Hackerdom, version 1.24[END_REF] keen to perform hacker exploits [START_REF] Cornwall | The Hacker's Handbook[END_REF]. There was only a faint echo of this rivalry during our interviews, alluding to the fact that great French researchers in network computing, like Cyclade' Louis Pouzin and Gilles Khan were more interested in theoretical network models and the Internet than in using UUCP/Usenet. Nonetheless, a parallel can be drawn between American Unix users' defiance towards the Arpanet/Internet, and Fnet users' criticism of Transpac: while UUCP networks were completely decentralized and self-regulated at the servers' level, Arpanet answered to the US Defense Communications Agency; in turn, French Unixians were critical of Transpac's agreements with the French Defense department, creating a fear of general network surveillance [START_REF] Devillers | [END_REF], a decade after the SAFARI affair 21 .

The network class struggle postulated by the first generation of Usenet users was also a way to create an identity based on social and technical differentiation towards other computer networks, in particular, to rethink the limits of the computer networks research and engineering to be more inclusive of general computer users outside of the Unix-oriented computer science community. Usenet was firstly aimed at Unix toolbuilders, a social network "for and by the users" (Rheingold 1993: 105, 126). The focus on "users" shows an acute and early understanding of what computer networks were bound to become, a communication media reinvented by users who are less and less specialized in computer technologies, and looking to expand applications and uses outside of the computer-centric world [START_REF] Abbate | Inventing the Internet[END_REF]).

As such, from the start Usenet prided itself to openness and inclusivity, as claimed by the first Usenet manual "Invitation to a General Access UNIX Network", distributed as a handout to Usenix conferences participants in 1979, and circulating further into the many Unixian meetings [START_REF] Hauben | Netizens: On the History and Impact of Usenet and the Internet[END_REF]. The Usenet software itself was an exception to other "electronic conferences" systems, usually closed and limiting the number of participants [START_REF] Quarterman | The Matrix: Computer Networks and Conferencing Systems Worldwide[END_REF]). The network thus comprised diverse participants, mostly academic but also business-oriented, connecting from within their organization or, with the advances of micro-computers in the 1980s, from outside, as users could dial in the Usenet server at a distance.

French "Administratrivia" Networks: Beyond the Ideal "Commmonfolk of the Computer Community"

Principles of openness did not provide actual solutions to manage UUCP networks locally between the academia and corporate world. The history of Fnet as an informal organization shows the practical complexities of a human network running a technical network without formal support, and in a social environment where the organizational side of managing a network could be regarded as "administrativia"a pun made by Devillers when reporting management problems on the Usenet Fnet group in the early 1990s.

The Computer Lab at Cnam handled Fnet from 1983 to 1986, with Humberto Lucas presiding over the ad hoc loose organization, running the backbone and managing subscriptions with the help of fellow engineer Daniel Lippman. Distributing different volumes of data to members and billing them was the main issue. Because of the UUCPnet decentralized model, some sites were bigger as they were servers rerouting data from the European EUnet hub in Amsterdam to end-user terminals (Fnet members). Unix machines at Cnam played that role; soon, Usenet data in particular became a burden, as the social computer network was expanding rapidly and internationally, adding many newsgroups each year. The Computer Lab had to cover the cost of high-capacity servers and transmission lines on a tight budget, and kept Usenet groups subscription to a minimum (restricted to groups in the comp.* and sci.*, hierarchies, hosting discussions about computers and sciences). Moreover, because of administrative constraints, it could not bill Fnet members outside of a fixed rate rigid template. Lucas issued as soon as 1984 a call for participation (a financial supplement to cover the rise in traffic data), especially directed towards member organizations with looser accounting rules (industrial labs), but to no avail. This was a source of discord among the Fnet communitysimilar to the high bills problem produced by UUCP connections at the Bell Labs a decade earlier (Hauben and Hauben 1997: 128). This, as well as the lack of assistance and acknowledgment the Computer Lab and their employees received from within the establishment, drove Lucas to resign and leave the public sector. Fnet's administration was then transferred at INRIA, handled by Devillers' team, with about 40 machines in about 27 organizations connected to the network. 22 This is far more than the alleged 15 connections initially described in the official history of Inria, which also disregards the contribution of Cnam's team prior to 1986 [START_REF] Beltran | Histoire d'un pionnier de l'informatique : 40 ans de recherche à l'Inria[END_REF].

The resource and billing management problems continued after 1986, despite the INRIA team's choice to bill Fnet members according to their actual use. The high rise in subscription demands, yet scant effort shown by the members to participate in the technical management of Usenet, accentuated during the late 1980s and constituted a new administration problem. In 1985, the UUCP American network was requalified as a "commercial network" (like Transpac), while a new dedicated Usenet and mail service was opened to offer access to users outside of academia [START_REF] Salus | A Quarter Century of Unix[END_REF][START_REF] Quarterman | The Matrix: Computer Networks and Conferencing Systems Worldwide[END_REF]). This seems to have changed users' perceptions of the network, less as a collaborative project and more like a service for users.

Fnet itself had become the first semi-commercial, although non-lucrative provider for the French academia and beyond, and its managers were disappointed to see the Fnet members turning into clients. Recursively, members, considering the cost of subscription was high, were expecting better service without connection delays of bugs. Usenet's primary ethics of "open participation" was at stake: it was born as a toolbox, an unfinished code full of bugs tackled by users (Pfaffenberg 2003: 24), with a focus on informal use and tinkering rather than formal committees and arguments [START_REF] Hauben | Netizens: On the History and Impact of Usenet and the Internet[END_REF]. But the more the user base grew, the less participation was offered, which proved an acute problem as Fnet remained informal and experimental. Between 1986 and 1993, Fnet was administered by the small team of "Research and Development Expertise Committee" 23 at the INRIA headquarters, in addition to taking care of new functions since 1988: Internet's IP addressing, routing, and domain name systems. Moreover, supporting organizations like AFUU started to focus on developing services for commercial users, and started to move away from Fnet 24 .

Eventually, with the growing success of the Internet and the general professional and popular turn towards TCP-IP Internet networks, UUCP networks were less and less used into the 1990s. Fnet's difficult management, with IT overload and high bills, was considered an abusive provider having taken advantage of their virtual monopoly in the pre-Web era (traces of these discussions and conflicts can be found in the fnet.* newsgroups on Usenet). This reputation increased as new providers (commercial like Oléane or non-profit like FDN -French Data Network) started to emerge in the early 1990s. 25 When the newly created RIPE organization (Europeans IP Networks) relieved the Fnet team of their IP attribution and routing functions in 1992, Fnet had just become a legal non-profit organization ("association loi 1901"). When INRIA decided to back up officially another, IP-based, academia network called RENATER in 1993 [START_REF] Schafer | Dans les Coulisses de l'Internet: RENATER, 20 Ans de Technologie[END_REF], and after "five years of lengthy discussions" with the Fnet team (Renard 2013), Unixians' "Internet experience" had started a steep decline.

Mapping Unix Networks: Several Electronic Frontiers

The ethics of participative effort put forward by Unixians as a social and technical identity giving reason to the spread of their networks is in relative accordance to what will be formulated in the mid-1990s as the "electronic frontier", built by "pioneers": the few hardy technologists who can tolerate the austerity of its savage computer interfaces, incompatible communication protocols, proprietary barricades, cultural and legal ambiguities, and general lack of useful maps or metaphors [START_REF] Kapor | Across the Electronic Frontier[END_REF].

While the autonomous, anti-institutional cyberspace utopia has been reviewed critically by

Internet studies researchers, it might useful to analyze how aspects of the so-called "electronic frontiers" actually did serve as "useful maps or metaphors" for organizing communication networks and pushing them towards institutional forms. To sum up, how did the shadow infrastructure of Fnet and Unix networks in France produce elements useful for mapping and reflecting on the new computerized communication networks outside of the state telecomm monopoly?

Early Usenet Maps: Keeping Up a Public Network Directory

The Google archive of Usenet, going back to 1981 26 , reveals how Unix networks were spread in the United states and in Europe through a network of Unix machines, a spread documented in the form of UUCP and Usenet logical maps (the network represented as links between machines) featured in a special "20 Year Usenet Timeline" compiled by Google at the moment of the archive release on the Web in 2001. 27 The first logical maps were sent to Usenet users via the group net.general and net.news (general news about Usenet) as a way to track the progression of the network, and then through a dedicated group called net.news.map from [START_REF] Shannon | Usenet Logical Map[END_REF]to January 1985(net.news.map 1982[START_REF] Cornwall | The Hacker's Handbook[END_REF]. They functioned as a utility tool, a directory to map out Usenet links for sites wishing to be grafted to the network, but they also reinforced Usenet as a public network: Public at all times, and so any site which is on Usenet is expected to make public the fact that they are on Usenet, their Usenet connection and their name, phone number and electronic address of the contact for that site for the Usenet directory (Hauben and Hauben 1997: 125).

The first Usenet map was produced and released by Bill and Karen Shannon from the Department of Zoology at University of Toronto in 1982 28 : "this is a USENET map, NOT a UUCP map. It shows only NEWS connections AS WE EXPECT THEM TO BE ON FEB. 1." [START_REF] Shannon | Usenet Logical Map[END_REF]. In July that year, an update was provided by Mark Horton, offering the same reminder with instructions for reading the maps and using them as a probe to contact Usenet sites unsure of their status: Remember, USENET is defined as all sites getting net.all. If you are reading this, the machine you are reading it on should probably be on the map. But sites that only speak uucp for mail (not news) or that only get local groups or a select set of groups from net.all are not on the map.

The Shannons' and Horton's clarifications revel how intricate the network can be as a pile of protocols with common functionalities while operating through different systems and applications. The Usenet map message was used as a tool to verify if the logical map covered the territory of effective Usenet sites and users. Thus it was not only referential, but also pragmatically recursive, through recurrence and expansion every time the message was posted in the newsgroup.

The first European Usenet address appeared in a 1982 September update, a site called "ukc" for the University of Kent at Canterbury in England. In late December, the first official "USENET map for Europe" was sent, showing twelve nodes: one in the UK (Kent), one in Denmark (Copenhagen), and the other ten distributed all around the Netherlands, as CWI Amsterdam was becoming the central Usenet European backbone for EUnet. The February 1983 map showed a new site at the University of Edinburgh, a partner of the Computing Lab.

Thanks to these maps, it is possible to identify exactly when the French UUCP-Usenet nodes were opened at Cnam in March 1983, at a moment when the European penetration of Usenet accelerated with 41 Usenet sites, featuring among then a cluster of 5 Unix machines at Cnam [START_REF] Berteema | Europe Map[END_REF]. Figure 1 The map details the coordinates of the networked sites. The first coordinate is the name of the node through which point-to-point connection will go through. From here on are displayed the connection links for two different services of communication to the other machines on the network: UUCP-supported Usenet services (newsnet), and SMTP-supported email services (Mailnet), the first protocol bridge opened by the UUCP networks. The names of the nodes often include an abbreviation of the organization hosting the site ("mc" for Mathematical Center at CWI) that can be followed by the machine hardware type ("vax" for the Unixcompatible VAX 11), but variants also apply when there are several users from the same organization and the same machines. The now five Cnam machines (vmuCnam, sol11, lsiCnam, iieCnam, and sm0Cnam) connect to both newsnet (via mcvax) and Mailnet (via mcvax in Amsterdam or their own local node at Cnam, vmuCnam).

There were only a handful of machines on Usenet addresses, among which one named after Francis Martin, 29 a collaborator from INRIA, and located on a machine at the Computer lab. This shows the intricacy of Usenet accounts that are shared and rerouted though the UUCP networkand another one testifying of a node at IRCAM (the French Institute for Research and Coordination Accoustic/Music). Martin, along with Devillers at INRIA, Patrick Sinz of IRCAM, Pascal Levasseur, and Sylvain Langlois at their respective computer technology companies METROLOGIE and BULL, were the initiators of the expansion of Usenet through Fnet, as accounted in AFUU's reports that compile the same information as in the logical maps. A year later, in 1984, Usenet French sites had effectively sprung from public organizations (CNET Issy Moulineau for instance) as well as corporate (CGE, Bull, Métrologie for instance). It is, though, possible to speculate that far more UUCP users would access to Usenet data through bulk transfer from the main sites.

A Territory of Confusing Routes

In the early 1990s, just before the Web started to attracted a mass of new users from the outside of the computer science and engineering world, mapping UUCP and Usenet networks was still a current practice, for instance, in the Usenet group comp.mail.maps, coordinated by the University of Rutgers, which listed the UUCP maps by country (UUCP Mapping Project) (comp.mail.maps n.d.). A search for Fnet maps yields long lists of French Usenet and UUCP subdomains with an international link outside Europe, but also bridges to other computer French networks, like EARN, running on IBM's native protocol as well as X25. Since Usenet had a new protocol replacing UUCP with the Internet friendly NNTP, and since France was directly connected to the Internet in 1988, mapping Unix network routes could be done via following the IP addresses as well. Celebrating a "routing folklore", users would analyze electronic messages routing metadata generated (a "traceroute" automatically inscribed in the message header) in which IP addresses could be read. Some cases of twisted routes were considered an absurd logic that users liked to qualify as a "routing game": messages emitted from France to France would take a detour through foreign countries via so called "exotic routes"; some were stuck in a loop between two IP addresses, thus entering a "ping pong game" (see the discussions on the topic of routing at [START_REF] Michot | UUCP Map[END_REF]. Michel Fingerhut, a pioneer Fnet user at IRCAM, started to track in 1994 such traceroute absurdities 30 when he observed between other oddities that one message he sent to Ministère de la Culture would go out to the US, back to the Netherlands, and then to France. He jokingly qualified this routing with the code name "22 à Asnières", derived from a famous comical skit describing the intricate administrative and technical complexities of the old French telephone commuted system [START_REF] Raynaud | Fernand Raynaud "Le 22 à Asnières[END_REF].

The absurd routing games revealed a major issue of the computer network landscape. Indeed, public organizations did not have a common Internet provider: for instance, INRIA was provided by Renater (from 1993), while Ministère de la Culture was provided by Fnet. A Recourse to a simplification of Internet routage through a Commercial Internet Exchange hub (CIX), a precursor for internetwork connection architecture and business models for Internet providers already in use in the USA, was difficult and always delayed by the fear of a "cultural choc" 31 between the French national and centralized model, and the American commercial and decentralized model. 1993-1994 saw an active cooperation for a French CIX and the release of a first draft under the name "F-GIX", thanks to a formal agreement between Oléane and RENATER 32 , but was not immediately implemented. In 1993, French early Fnet and Internet adopters had an insight view of the difficult construction of "the wonderful information highway that is bound to lead to us towards a Radiant Future" 33 , as France was entering its own Information Highways decade 34 . But Fnet administrators themselves were lagging behind in terms of infrastructure building. At the same time, Usenet was going through an international crisis, known as the "Eternal September", which symbolized the popularization of Internet considered a risk for users of the pre-Web computer networks golden age (Schafer, Paloque-Berges and Georges 2014).

Epilogue: The Electronic Frontier as a Political Metaphor

From the start, Unixians were part of the self-perceived digital avant-garde willing to create new institutions supported by an "electronic democracy" [START_REF] Huitema | Et Dieu créa l'Internet[END_REF], Lévy 1997).

Expanding computer network by means of human and technical skills became synonymous for progress in a globalized world. Usenet aficionados defended a "netizen" identity [START_REF] Hauben | Netizens: On the History and Impact of Usenet and the Internet[END_REF], considering the social network a "network nation" on its own [START_REF] Raymond | A Brief History of Hackerdom, version 1.24[END_REF], or even more radically a "worldnet" for "a society that hopes to progress" through the general access to technological tools and a cooperation helping to best wield these tools (Hauben and Hauben 1997: 133). Similar to the probing function of messages showing logical maps described earlier, Usenet news were thought of as a means to see how far and wide and to whom the network reached. A student experiment recounted by the Haubens show how this probing function was accompanied by a postulate: the more difficult electronic messages were to reach a destination, the less democratic the country was. Thus Usenet and the Internet were seen as an enlightenment process for parts of the world still under un-democratic darknessa rationale vivid in the last years of the Soviet block, during which this experimentation was conducted. Within this experiment, a French perspective was given:

A response from a French user explained how the government charged a lot of money for an Internet connection in France and thus discouraged usage: "It's cheaper to send a 'hello' to someone in the US than to someone 5 kilometers from my desk!," the French user wrote, "If you have a 'stupidity chapter' in your paper, this could fill a few lines" (quoted in Hauben and Hauben 1997: 38). This government charge, stemming from Transpac's unwilling support of Fnet and the Internet, is one of the reasons for the "routing game" evoked earlier, and was a flaming topic of discussion on the Fnet Usenet groups. According to Christian Huitema, one of the instigators of the first TCP-IP connection within the Fnet network, France's access to the Internet was twice as low than Germany's, three time than the UK's, and twenty times than the USA: It's a shame. To be honest, I think this is proof of an underdevelopment, even of a reluctance again the modernization of the world and the huge openness provided by computers [START_REF] Huitema | Et Dieu créa l'Internet[END_REF]. This techno-enlightenment ideology is present and diffused all through the Unixian networks.

But it doesn't take into account "internal borders" re-created by Unixians and their peers on Usenet that have to do with actual inequality in terms of access, participation in arguments, and decision-taking [START_REF] Johnson | Law and Borders: The Rise of Law in Cyberspace[END_REF][START_REF] Healy | Cyberspace and Place: Middle Landscape on the Electronic Frontier[END_REF]. If Usenet was considered a virtual agora that would allow the Internet to open an electronic public sphere, its culture of flaming polemical debates makes the "electronic conference" network unreducible to a rationalistic, procedure-driven public discourse. Among "net.wars", power dominance of servers' administrator was strengthened, and old users were using experience and technical skills to ward off new users (Grossman 1998). In our interviews, genuine interest for the non-specialist user proved scarce, indeed; the UUCP / TCP-IP networks were seen first and foremost as a liberating tool from the constraints of the French institutional powers, to the benefit of the computer science (and computer businesses) community. Finally, following Fred Turner's study of the evolution of computer libertarian hobbyists and hackers and his analysis of their participation in the rise of a new "virtual class" (Turner 2006a;2006b), it would be important to follow the trajectories of Unixians to understand how the ideals of progress through the electronic frontier evolved in the 1990s and the 2000s Internet economy.

Conclusion

It is notable that the user figure has so much importance in Unix's own worldview on humancomputer interaction and was a principle for founding Usenet, the biggest pre-Web computerized social network. Meanwhile, the Arpanet development processes were much more hesitant towards integrating inventions stemming from the user point of view, despite the great part it actually played in the success of its innovations [START_REF] Abbate | Inventing the Internet[END_REF][START_REF] King | The Rise and Fall of Netville: The Saga of a Cyberspace Construction Boomtown in the Great Divide[END_REF]. This is one of the main reasons why Unix networks were at the avantgarde of the Internet experience that we all share today. But the example of French Unix users connecting their community to an international communication computer network reveals how the idea of using the network remains a specialist perspective, relying on active participation and to the benefit of a professional community. However, this definition of network user engagement was political. Striving for autonomy and independence from national and local technological policies while engaging in smart tinkering and negotiation with hardware and software norms meant believing in bottom-up user innovation instead of relying on top-down planning.

The remapping of France with UUCP networks is one of these useful metaphors that show the empowerment of new technological expertise, original because they did not start from a power standpoint. Unixians themselves didn't "take power" as the Internet "founding fathers" did, if we believe the power positions held currently by key figure Vinton Cerf, Vice President and Chief Internet Evangelist at Google. But they did, by diverting institutional authorities, contribute to create a new place for new power centralities. From the "Science:

The Endless Frontier" (which gave, in 1945, a running start for the development of technoscience, among which global communication networks) 35 to the 1990s Information Highways plan, do computer networks give technological progress a new meaning, supported by an "endless electronic frontier"? Previous logics of domination (social, economical, political) did not disappear, but a space was open for information organization to thrive on a supra-national level, leading to the rise of organizations for Internet standards (such as RIPE, ICANN, and IETF), which independence from established institutional power is still being questioned. If it's tempting to call these organizations, or proto-institutions, a ground for technical democracy, we see them, through the prism of Unixians networked organizations, as ad-hocracies far from being independent from power relationships.

  Internet created from the EUnet node, allowing French Unix users to retrieve Internet-run STMP mail on Fnet, the first direct liaison from a French computer to the Internet waited until 1988. It was set up at INRIA, where the Fnet administration had migrated in 1986, through a partnership with the American National Science Foundation Network (NSFNet), the main Internet backbone from that year until the mid-1990's. One of the first non-US countries to benefit from a direct liaison, France's access was made possible in big part thanks to the reputation INRIA's Cyclades had acquired with the datagram model a decade earlier

  shows the Usenet post announcing on March 10 the surge in European Usenet sites (we extracted the very first lines, with the announcement and confirmation of CWI Amsterdam as the EUnet backbone, and the very last lines, showing the entrance of Cnam into the network).

Figure 1 :

 1 Figure 1: Two extracts of an 18 March 1983 Usenet message, entitled "Europe map"

20

  Interview with Annie Renard (2013), an engineer at Inria in the late 1980 who took part in Fnet team and was particularly in charge of Domain Name System addressing after 1988. 21 SAFARI (Système Automatisé pour les fichiers administratifs et le repertoire des individus) was a project released in 1974 by the French government but abandoned soon after after much criticism for created interconnecting indexes of nominative files for French citizens. The SAFARI affair prompted the creation of the Commission nationale informatique et libertés (Cnil) in 1978. 22 A 1986 logical Fnet map shows 27 organizations connecting to Fneta lot of them having multiple connections in different labs or services, making the number of direct connection to Fnet higherand not counting the bridged connections from other networks like EARN. 14 public sector research and education organisms: Centre d'Etudes et de Recherches de Toulouse (Deri -Département d'Etudes et de Recherches Informatique); CNET (Centre Structures et Logiciels pour la Commutation -Centre de Lannion, division Télématique Informatique Mathématique (TIM); Dpt. ATL Assistance Technique et Logicielle -Division Outils de gestion de l'exploitation OGE, Départements Systèmes Microprocesseurs et Logiciels); CNRS (LASS -Laboratoire d'Automatique et d'Analyse des Systèmes); Ecole nationale supérieure des mines de Paris (Centre de Mathématiques Appliquées); Ecole des mines de Saint Etienne; Ecole nationale supérieure des télécommunications; IMAG (Institut IMAG, LIFTIA -Lab. d'Informatique Fondamentale et d'Intelligence Artificielle; TIM3; INPG Architecture Group); Inria (IRISA -Institut de Recherche en Informatique et Systèmes aléatoires; Centre de Rocquencourt; Centre de Sophia-Antipolis); IRCAM; Supelec; Université de Nancy 1 (Lab. CRIN); Université de Paris (LITP -Lab. d'Informatique Théorétique et Pratique); Université de Paris 6 (Centre de calcul et de recherche); Université Paris Sud (LRI -Lab. de Recherche en Informatique). 4 public sector enterprises: Agence de l'informatique; EDF (Direction des études et Recherches); Compagnie Générale d'Electricité

This is the pre-print version of a chapter published in Routledge Companion to Global Internet Histories, G.

Administrative archives of « Laboratoire d'Informatique », 1981Informatique », -1986 (34-02.11 B11-B12) (34-02.11 B11-B12) and « Fonds Lippman » (34-02.03 1), donated by Daniel Lippman, engineer at the « Laboratoire d'Informatique » during the 1980's and 1990's. Accessible at the archives department of Conservatoire national des arts et métiers, Paris, France.

First called "Computing Center", then "Computing lab" and "Computer lab", it evolved into an IT department in the 1990's, still in activity today.

Kaiser was to become "Chair of Computing and Programming" in 1984, in line with predecessor François-Henri Raymond, an inventor and industrialist close to the research world who started the first French computer manufacturer. The informal "Systems lab" became official in 1988 with the creation of CEDRIC laboratory, still in activity today.

According to an unpublished and un-dated article by Florin entitled "La première installation d'UNIX au Conservatoire National des Arts et Métiers", sent to the author by Kaiser. Florin also mentions the help brought by Kyrian O'Donnell, from Inria, that would start a long

Studies: Current Terrains, Future Directions, New York: New York University Press.

Vedel, T. (1984) "Les Ingénieurs des Telecommunications: Formation d'un Grand Corps", in

Culture Technique, 12: 63-75.

1 Interview with Laurent Bloch (2013), a computer engineer actively involved in Unix networks and the Fnet organization, in which he was vice-president between 1991 and 1992.

Bloch coined the idea of Unix networks as a "first French Internet experience" arguing that they predated the Internet in France and had similar purposes (email and online discussion groups) despite their technical differencesa feeling confirmed by all the other interviewees.

All quotes and extracts translated from French to English by the author.

Printing Office, Washington: 1945), hosted on the American National Science Foundation website, available at: http://www.nsf.gov/od/lpa/nsf50/vbush1945.htm.