Two dimensional Yang-Mills theory via stochastic differential equations

5. CONNECTION OPERATORS In order to evaluate the expectation of the product of the Wilson loop variables associated, for example, to two concentric circles it will be necessary to extend the previous formalism. Consider the configuration of curves shown in Figs. 5.la and b. In both cases the number of curves lying above a point x on the x axis changes

b

c

ab

(a)

04

FIGURE 5.1

84

GROSS, KING, AND SENGUPTA

at b as x moves from left to right. Denote by 99k the algebra of operators on (@“)k which commute with g UC)for all g in G. This was denoted by 2? in Sections 3 and 4. But we want to use a different k over different intervals. Let m
Bin &?,,,,

(5.1)

where Ike-m is the identity operator on (CN)k-m. Note that JB is indeed in ak since g~k~(B~Zk_,)(g-‘)~k~=g~“)B(g-l)~”~~Ik_,=B~Zk_, for all gin G. We define q:=

cp(~,b)Jq(b,c):a,,,+a~,

where cp(b, c): g,,, + @,, is defined as in Section 3 using the graphs of ur, .... y,,, over [b, c] and cp(a, b): 9Yk+ gk is defined using the graphs of (or, .... yk over [a, b]. We assert that just as in Theorem 3.6 the map cp determines the expectation of products of all the Wilson loops obtainable from cl, .... yk as follows. We refer to the curves which are the graphs of ol, .... (T,, yt, .... ym over [a, c] as the long curves and the graphs of orn + 1, ***, ok, Ym+ 1, ***,yk over the subinterval [a, b] as the short curves. Parallel translation along oj will always be taken from left to right and denoted U(aj) while parallel translation along yj will always be taken from right to left and denoted J’(Yj). For j= m + 1, .... k we join the short curve aj to the short curve yj by a vertical line segment above x = b from (b, aj(b)) to (b, y,(b)) as in Figure 5.la. Above the endpoints a and c we join the curves by vertical line segments in a manner similar to that of Theorem 3.6. Specifically, we choose permutations P in S, and Q in Sk to index the joining procedure. For j= 1, .... m we join a,(i) to yj by a vertical line segment above c. These two curves will then contribute a factor v(Yj) U(aP(j) ) to our final Wilson loops. For j= 1, . ... k we join yeci, to crj by a vertical line segment above a. These two curves will then contribute a factor U(crj) V(yeCj,) to our final Wilson loops. As in the discussion preceding Theorem 3.6 we obtain a finite number of closed curves rj, i= 1, .... r, for which h

trace U(rj)

j=l

= trace[&( V(y,)@

‘..

@

V(yk))(~~zk-m)(u((TL)O

...

@

wk))i.

(5.2)

Note that the factor P@I Zk-,,, is the operator needed for connecting the long curves by P over c and connecting the short curves by the identity permutation over b. LEMMA

5.1 (cf. Fig. 5.la).

1

= trace[&cp(a, b) Jq(b, c)f].

(5.3)

YANG-MILLS

85

THEORY

ProoJ For j= 1, .... k write Uj for parallel translation along aj from c1to b and Vi for parallel translation along yj from b to a. For j= 1, .... m write U,’ (respectively, Vi) for parallel translation along aj from b to c (respectively, along yj from c to 6). Then for the long curves U(Oj) = Ui Uj and P’(rj) = Vi Vi’, j= 1, .... m and for the short curves U(aj) = U, and V(rj) = Vj, j = m + 1, .... k. All primed operators are independent of all unprimed operators. Substitute these products into the right side of (5.2) and take the expectation first with respect to the primed operators to get

= E[trace &( V, 0 . . . @ Vk) x(E(F’;@

... 0 v~)fi(u;o

... @KJ)@~,-,)(u,@

.‘. OU,,]

= trace &E[ ( V, 0 . ..oV.)({cp(b,c)Ei}OZ,-,)U,O = trace &(a,

. ..OU.l

b) Jcp(b, c)p

(5.4)

which proves (5.3). The machinery for representing the situation of Fig. 5.lb is similar. The long curves are again the graphs of continuous functions (rl, .... o,,,, yl, .... y, on the whole interval [a, c], but this time short curves are the graphs of continuous functions (Tm+ 1, ..-, Ok, Y,,,+ 1, .... yk on the other subinterval [b, c]. Moreover the right hand connection permutation P is in Sk this time while Q is in S,. We join the short curves by a vertical line segment over b which joins (b, y,(b)) to (b, oj(b)), j=m+ 1, .... k. As before the configuration decomposes into closed curves t,, .... 7,. and we have this time fi

trace U(T,)

j=l =trace[(&Ozk~rn)(V(Y1)O

.”

8

v(yk))~(u(ol)@

‘.’

@

U(ok))l

(5.5)

wherein the f&Or zkp, effects the connections above b. As before we decompose the parallel translation operators along the long curves into independent factors and take first the expectation with respect to the operators over [b, c] to get

I

= E[trace{(Q@Z,P,)(V, x(U,O

...

@

0 ... 0 V,OZ,-,)(cp(b,

c)P)

um@zk-m)}l

= E[trace{ Q( I/, @ . . . 0 ~,)(J*cp(b, where .Z*: Bk + L8m is the adjoint

cPw10

... 0 Urn)}],

of J relative to the trace inner products. Here J

86

GROSS, KING, AND SENGUPTA

should be regarded as operating on all of L#((C~)~) into g((C”)“) by JB = B@ Zk-,,,, but this does not affect the value of J* on gk which is easily seen to lie in L!&. Upon taking the final expectation we obtain the identity which reduces Fig. 5.lb to Bralic maps, namely LEMMA

5.2 (cf. Fig. 5.lb). E

fi j=

trace U(r,)] = trace[&cp(a, b) J*cp(b, c)P I

1

(5.6)

We shall refer to the operators J and J* as connection operators since they contain the algebraic information which effects the connection of the short curves along vertical line segments at x = b in Figs. 5.la and 5.lb. In addition to these connection operators we shall need another type of connection operator. Suppose that in Fig. 5.la the curves c2 and yz continue on to the left of a over some interval [d, a] while 0, and y1 stop at a as shown. Over the interval [d, a] we have k = 1. The subscripts that label our curves are important because they determine the position of the corresponding parallel translation operators Uj and Vi in Eq. (3.5). Thus the curves c2 and yz should be labeled rrr and y1 over [d, a] but cannot be so labeled over [a, b] without causing labeling inconsistency for the other two curves over [a, c] which are already shown in Fig. 5.la. We define therefore a relabeling operator as follows. With a < b < c as before we suppose that we are given 2k continuous functions whose restrictions to [b, c] we label c;, .... ob, r;, .... 7; and whose corresponding restrictions to [a, b] we label y . Thus (Tagand ai are restrictions of the same continuous function ~~~d~a~&?&t&~ls. Write u(n) = j, and j?(r) = i,. Then ct and /I are in S,. Define RB=ai-‘B/j

(5.7)

for B in z2&. Clearly RB is also in ~47~. LEMMA 5.3. The map cp:= cp(a,6) Rq(b, c) determines the expectation of Wilson loops in the canonical way: tr(&cpP) is the expectation of the product of Wilson loops determined by connecting the continuous curves at a and c by Q and P, respectively.

Proof

Write Uj= U(aj), Vi= V(y,), Ui = V(ai), and Vi = V(yj) and observe the

identity tr[$(V,@

~~~OV,)B-‘((V~@

= tr[/$cV’((

. ..@V.)B(V;@

Vi, Vi)@ ..’ O(vj~v~))~((u~ui~)O

By independence of primed from is trace(&?). But by Corollary operators loops in which the Vj, Vi U&,, Uip,,) and 17; U, V, Vi

. . . @ZJ;))$(U,@ *‘* O(uLuik))].

. . . @U,)] (5.8)

unprimed operators the expectation of the left side 3.9 the right side is a product of traces of disjoint typical sequences that appear in a loop are where j, = ja-~88(r) = tLa- ‘Q/?(r) = Qir. Thus Vj, is

87

YANGMILLSTHEORY

preceded by V,! as desired, V: is preceded by U&, by VQt, as desired.

as desired, and U, is preceded

EXAMPLE 5.4. Let Ci and C2 be two intersecting circles with centers on the x axis and oriented clockwise. Denote by a < b < c < d the intersection points with the x axis. Let (T, and y1 be the functions on [a, c] whose graphs give the upper and lower halves of the leftmost circle C, while e2 and y2 are the functions on [b, d] whose graphs give the upper and lower halves of C2. Let a = /I = r be the transposition in S, and denote by R: B2 + g2 the relabeling operator; RB = t^C’BQ. Write JB = B@ I, for B in 93r and Zr = identity operator on CN as in (5.1) with k = 2, m = 1. Pick a point c’ with b < c’ < c. Then the Bralic map for the two circles is

q = da, b) J*cp(b, c’) Rq(c’, c) Jdc, d), where cp(a, b) is determined by or and y I over [a, b], cp(b, c’) is determined by or, c2, yl, yz (in this order) over [b, c’], cp(c’, c) is determined by Do, cI, y2, y1 (in this order) over [c’, c], and cp(c, d) is determined by crz, y2 over [c, d]. The right hand factor .Z closes the circle C, on the right (as in Figure 5.la) by a vertical segment (of length zero) and at the same time connects c2 1 [c’, c] to (TV1 [c, d]. It also connects y2 1 Cc’, c] to y2 1 [c, d]. The relabeling operator R reverses the order of factors at c’ so that .Z* may close the circle C2 at b and properly connect rrr 1 [a, b] to or 1 [b, c’] andy, 1 [a, b] toy1 1 [b, c’]. Wemaysimplifycp by observing that (p(c’, c) converges to Zso2as c’ ? c while cp(b,c’) converges to cp(b,c). Hence we have q=cp(a, 6) J*cp(b, c) RJq$c, d). Here RJ closes the circle C, at c and simultaneously makes the correct lateral linking of c2 on the right and left of c. Alternatively we could have put c’ = b obtaining q = cp(a,6) J*Rq(b, c) Jq(c, d) in which cp(b,c) now has a different meaning since it is determined by 02, cr, , y2, y I instead of o,, cr2, y,, y2. To get the expectation of the product of the two Wilson loops take P = Q = Z (which is the only choice since k = 1 on the end intervals). The expectation is then trace[qZ,]. It follows from Corollary 4.8 and the equation cp= cp(a,b) J*cp(b, c) RJq$c, d) that the expectation is a function of the seven finite areas determined by the two circles and the two inner vertical tangent lines. But in the next section we shall show that it depends only on the areas of the three bounded connected components determined by the two circles. Remark 5.5. If two or more curves should close on the left over some point b (cf. Fig. 5.1 b) while two or more other curves close on the right over the same point b (cf. Fig. la) then the overall Bralic map for this configuration can be constructed as before in either of the forms J15; or JT J4 where the Jj are injections as in (5.1). This can be derived directly as in the previous cases or by simply applying the results for Figs, 5.la and 5.lb consecutively in either order. Remark 5.6. As noted in Remark 4.10 parallel translation along the graphs of piecewise continuous functions is a well defined notion. Moreover one need only look at Eqs. (5.4), (5.5), and (5.8) to see that the linking operators J, J*, and R

88

GROSS, KING, AND SENGUPTA

over a point b implement the lateral connection of two graphs even if there is a jump discontinuity of the combined graphs at b. We summarize the results of this section in the following theorem. THEOREM 5.7. Supposethat a, < a, < . . . < a,, are real numbers and k, , .... k, are strictly positive integers. Let a;, .... ai,, y;, .... y;, be continuous real functions on [a,- ,, a,.]. There exist operators Ji: 9#r+, + ak,, i= 1, .... n - 1 such that the operator cp:6A3k. + gk, defined by

determines the expectation of all Wilson loops constructed from the curves of, yi by spectfied interior lateral and vertical connections at a,, .... a,- 1 as in the preceding cases.For vertical connections at the endpoints a, and a,, spectjiiedby permutations Q in Sk, and P in Sk. the expectation is trace(&pP). Each Ji is a product of an injection J as in (5.1), the adjoint of such an injection, and a relabeling operator R such as in (5.7). One or more of these three factors could be the identity operator. Remark 5.8. The vertical connections at the endpoints a, and a,, can be represented by connection operators also. Define 9&, to be @ and extend the definition (5.1) to m = 0 by putting J@‘z = zl,, for z in C. Choose a = I and fl= P in (5.7) to get the relabeling operator R,B = BP. Choose cl-l = Q and fl= I to get the relabeling operator L,B= QB. Then, in the notation of Theorem 5.7 we see that J’kl)*@& J’kn’ is an operator on @ consisting of multiplication by the number (J’ki“LocpR,J(kn)l, 1) = (&cpP, J@l)l) = ($qP, Ik,) = trace(&cpP), which is the expectation of the desired products of Wilson loops.

6. A STANDARD FORM FOR BRALIC'S MAP functions on .[a, b] while a;, .... ah,, Let Ol, . . . . ckl, yl, . . . . yk, be continuous 24 >aa.7y;, are continuous functions on [b, c]. We suppose specified lateral and vertical connections among the graphs of these functions at x = b as in Section 5, so that some curves extend continuously across b while others are joined by vertical line segments at b to curves on the same side of b but going in the opposite direction (i.e., a cri to a yj). The linking operator at b that accomplishes this may be chosen of the form I= R, J, R, J:R3 where the Ri are relabeling operators. Let W be the union of the graphs of all these functions, cri, yi, oi, yj’, along with the vertical segments. If K, is a connected component of ((a, b) x R) - W and K2 is a connected component of ((b, c) x R) - W we shall say that K, and K2 are adjacent (at b) if there is a curve in ((a, c) x R) - W joining some point p1 in K, to some point p2 in K,. In Fig. 6.1, K,, K, is an adjacent pair. K,, K; is another adjacent pair.

89

YANG-MILLSTHEORY

a

b

c

FIGURE 6.1

THEOREM 6.1. Suppose that K, and K2 are adjacent components at b and that E: LAYS, -+ Bk, and P. Bk2 + L??lkl are the associated area operators. Then

El = JF.

(6.1)

The proof relies on the following lemmas. We shall use this theorem in Section 6 and Section 7 to show that the expectation of products of Wilson loops depends in a certain sense on the areas of the bounded connected components of the complement of the loops. LEMMA 6.2. Supposethat .f is simply an injection J as in Eq. (5.1) and K, and K, are bounded. Then Theorem 6.1 holds.

ProoJ: This is the situation of Fig. 5.la which also illustrates a pair pl, pz. Write k, = k and k, =m < k for this case and put pi= (xi, zi) for i= 1, 2. Using

(4.9) we have 2k-

H(XI)=

1

C j=l

(Yj+,-Yj)Ej

and 2mWx2)=

c n=l

1

(Y:+, -Y;)%,

where Ej acts on Bk, E,!, acts on L&,,, the numbers yj are the ordinates {oi(xl), yi(xI)}~,I labeled in some nondecreasing order, and yk are obtained the same way from {cri(x2), y,.(~~)};=~. Since p1 and p2 do not lie on any graph there is a unique j and a unique n such that yj
and

Y~,
I

(6.2)

90

GROSS,

KING,

AND

SENGUPTA

Then the area operator for K, is Ei and the area operator for K, is EL. We must show that EjJ=

JE;.

(6.3)

As in Section 5, for i= 1, .... m, we refer to 0; v ai and yi v yi as the long curves. The remaining graphs, over [a, b], will be called the short curves. Since any long curve disconnects the vertical strip (a, c) x R each long curve lies above both points pi and pz or lies below both. Hence the set of long curves lying below pi is the same as the set lying below pz. By (4.10) we have -2E,=

i

Si,,iP.

(6.4)

r,p=l

Here S,,, acts on ~8~ and p and r run over a subset of { 1, .... 2k). Let L = {r : i, E Then L consists of the indices r for which i, refers to ( 4 ***,m, k + 1, .... k+m}}. a long curve in the notation uk+ r = yI. We write x; for the sum over (1, .... j} n L and CT for the sum over { 1, .... j} -L. Define -24

= y

C’ s,,,.

r

P

It follows from (6.4), from the similar definition of EA, from the definition of Si,,iP) and from the fact that the set of long curves lying below p1 is the same set as that for p2, that for B in 9?,,,.

F,(BOZ,-,)=(E~B)OZ,_,

It suffices therefore to show that (E,-F,)(BOZ,-,)=O. any p in (1, .... 2k) we have

Now

if m
(S,p+S,+,p)(BOZk-m)=O

then for

(4.5)

because Tz) commutes with BQ I,-,,, so that the left side of (6.5) is (e.g., for 1
C”+C” P

LEMMA

6.3.

Suppose that j=

r

C’+C” P

r

x”) P

S,~ip(BOZk-~)=O.

r

J as in Lemma 6.1 but that K, is unbounded. Then

EJ=O.

(6.6)

YANGMILLS

91

THEORY

Proof We continue the notation of the proof of Lemma 6.2. We must show that Ej J = 0. Since p 1 is in an unbounded component of ((a, c) x R) - W either there are no long curves which lie below p, (as, e.g., pi in Fig. S.la) or all long curves lie below p1 (as, e.g., p;‘). In the former case we have in the notation of Lemma 6.2, -2E,=C: Cl: S,,, and since p, lies above both short curves rri and yi or above neither we have, as before, that C:’ S,,,(B @ Zk _ ,) = 0. Hence EjJ= 0. In the latter case pI lies above all curves with the exception of an even number of short curves which come in matched pairs oi, yi. If we denote by C:” a sum over these matched pairs then C:” S,,JB@ ZkPm) = 0 because of (6.5). Hence

= i F si,,(B@zk~,)=o p=1

by (3.10).

i=l

Remark 6.4. By (4.9) EZk =0 and hence the area operator of an unbounded component is zero. Thus by Lemma 6.3, Theorem 6.1 holds if j = J and K2 is unbounded. It clearly holds if both K, and K, are unbounded. LEMMA 6.5. Suppose that 3 is the adjoint J* of an injection. Then Theorem 6.1 holds. In particular if K, is unbounded then J* F= 0.

Proof To take advantage of the extensive notation developed in Lemma 6.2 we interchange the left and right intervals in this proof. Thus we take the unprimed functions cir yi, i= 1, .... k to be defined over [b, c] while ai, ri, j= 1, .... m are defined over [a, b]. Moreover cri is connected to yi at b for i= m + 1, .... k. The short curves therefore lie over [b, c] now. This is the situation for which the linking operator is J* where J: g,,, + ak is given again by (5.1) as in Lemma 6.2. Assume p, lies over (6, c) now while p2 lies over (a, b). We let Ej: .G@~ -+ & be the area operator associated to the component K2 of ((b, c) x R) - W containing p1 and let EL: g,,, + a,,, be that associated to the component K, of ((a, b) x R) - W containing p2. As before pI and pz are assumed to lie in the same connected component of ((a, c) x R) - W. Then we must show J*Ej=

E;J*

(6.7)

if K, is bounded and J*Ej=O,

(6.8)

if K, is unbounded. The algebraic definitions of J, Ej, and EA are the same as in Lemma 6.2 even though the associated picture is different (the short curves now lie over [b, c] ). Hence by Lemma 6.2, EjJ= JEA. By Corollary 4.4 E, and EA are Hermitian in the trace inner products on gk and gm, respectively. Hence the adjoint of (6.3) gives (6.7) while the adjoint of (6.6) gives (6.8).

92

GROSS, KING, AND SENGUPTA LEMMA

6.6. If .? is simply a relabeling operator R then Theorem 6.1 holds.

Proof: In this case the number, k, of curves over [a, b] is the same as the number over [b, c]. We use the notation of Lemma 5.3. p1 lies over the interval (a, b) on the x-axis and pz lies over (b, c). Ej is the area operator for K, and EA is the area operator for K,. Both act on gk as in Lemma 6.2. We wish to show that E,R= RE:,.

We may assume that the connected components in question are bounded since otherwise Ej = Ei = 0. Let Ifi be the permutation in Slk given by $(i) = p(i) and $(k+i)=k+a(i) for i=l, .... k. Using the definitions (3.6) in Section 3 it is straightforward to verify that di-‘Ty’oi = TF(‘)) and hence RTr)(R-‘B) = TF(‘))B while R{(R-‘B) Tf’} = BT~?‘” for i= 1, .... k where RB= oi-‘Bb for all B in End(CN)k. Consequently RS,R-‘= S$Cij,tiCjj for i, j= 1, .... 2k in view of the definitions (3.7), (3.8), and (3.9). We write a,(x) =yiek(x) and ai =Y:-~ for i=k+l , .... 2k in order to apply the notation of Section 4. By (4.10) we have -2E~=~,,p.BS,pwhereA={s~{1,...,2k}:o,(x,)
YANGMILLS

THEORY

93

For r = 1, .... n - 1 we assume that some of the curves c$ are connected on the right by vertical line segments above a, to corresponding curves yi in the manner described in Section 5 while some of the curves al + 1 are connected on the left above a, to some corresponding y; + ‘. The other curves are connected laterally to curves over the adjacent intervals, cf: Fig. 6.1. All of this information is encoded in a connection operator J,.: ak, -+ ak,-, for each r = 1, .... n - 1 as described in Section 5. Let be the Bralic map for [a,- 1, a,] and write cpr = rp(f,- 1y 4): gk, + %, . .. cp,, for the overall Bralic map for [a, b]. cp=4”1JOP2-4 Denote by W the union of the graphs of all the functions al, yl together with the vertical connecting line segmentsabove the points a,, .... a,- ,. We assumethat the graphs intersect in only finitely many points and that the vertical line segmentsare disjoint. Denote by K, , .... Kr the bounded connected componentsof ((a, b) x R) - W. For each component K, pick a vertical strip (a,- 1, a,) x R which intersects K, and let E’“’ denote the area operator on %Y,.,associatedto any one of the nonempty connected components Kz of K, n ((a,- ,, a,) x R). Let A, be the area of K,. Then cp is a product in some order of the linking operators .?, , .... j,-, and the operators exp[ - A,E’“‘], s = 1, .... p. Proof: The projection of a component K, onto the x-axis is an interval which intersects one or more adjacent intervals (aj- 1, aj). For such a j, K, n ((ai_, , aj) x R) is a nonempty open set. Let K’ be one of its components and let E: S&, + S&, be the area operator associated to K’ by Corollary 4.8. qj is a product of exponentials one of which is exp[ - IK’I E]. Moreover if K’ borders the left (respectively, right) edge of the strip (aj- r, ai) x R then this exponential factor may be placed on the left (respectively, right) side of the product by Corollary 4.8. Pick a point in KY and a point in K’ and choose a piecewise linear curve in K, which connects these two points, which has no vertical segments and has no changes of slope above the points a,,, .... a,. This is possible because K, is open and connected. If K’ #KY then the furthest point along the curve which is not in K’ lies above ajp 1 or aj and therefore the curve enters K’ from the left or right boundary of the strip, respectively. In the former case K’ borders the left edge of the strip (aj- 1, a,) x R and the curve enters K’ from a connected component “K” of K, n ((ai+,, aj- I) x R) whose area operator F: 9$-, + g,+, satisfies FJj- I = Jj- 1E by Theorem 6.1. Hence exp[-IK’I F]j,-, =jj-i exp[-[K’l E]. Thus the factor exp[-IK’I E], which can be placed on the left of qj, can be commuted to the left of jj- 1 if one replaces E by F. If the curve enters K’ from the right edge of the strip (a,- I, aj) x R we can similarly commute this factor in qj past jj upon replacing E by some area operator on %,+,. By following backwards along the piecewise linear curve to KY one may, in this manner, commute the factor e-IrI E in q past the connection operators and Bralic maps vi by Theorem 6.1 and Corollary 4.8, changing E when necessary, till one arrives at KY. The factor e-lR” E has then changed to e-lK” da) which may be combined with the factor e-IKzI E’S’in cpI to give a factor exp[ -(lK’l + IKZI) EC”‘]. If we carry out this procedure for each connected component K’ of every nonempty intersection K, n ((aj- i, aj) x R) we may combine all the exponential factors

94

GROSS,

KING,

AND

SENGUPTA

associated to KS in this way into one factor exp[ - IK,( E’“‘]. Upon carrying this out for all s all the exponential factors in all the Bralic maps are exhausted by Corollary 4.8.

7. INVARIANCE

PROPERTIES

OF EXPECTATIONS

Using the results established in the previous sections, we will now analyze the expectation of a general product of Wilson loops. Theorem 6.8 allows us to write this expectation as a product (in some order) of connection operators {si} and exponential operators (exp - A.E'"'}, where {A,} are the areas of the connected components of the complement of the loops. This expression superficially depends on our gauge choice; a different choice of directions for the coordinate axes in R* leads (in general) to different cornmutant algebras {L#~,}, different connection operators, and different area operators (see the statement of Theorem 6.8). Our task in this section is to ‘prove that the expectation does not depend on the coordinate directions chosen. We do this by rewriting the expectation in a standard representation (see Definition 7.21) which displays the expected invariance properties. The result we get implies, in particular, Euclidean invariance of expectations, and factorization over disjoint components. Our first task is to describe the type of curves to which our results apply. DEFINITION 7.1. A curve is a continuous piecewise C1 function cr: [0, 1] = Z-t R*. The image of .c will be denoted o(Z). The curve c is closed if a(O) = a(l), and (T is simple if a(tr) # a(t,), for all 0 < tr < t2 < 1 unless tr = 0 and t2 = 1. We will consider only curves with tangent vectors defined and nonzero at all but a finite number of points in 03’. The tangent vector to cr will be denoted a’(t).

DEFINITION 7.2. Given an interval A c 03, denote by A x R the cylinder {(x, y) 1 x E A}. The curve 0 is a finite union of graphs over the interval A if

set

a(Z) n (A x R) = fi a(Z,), i=l

where {Zi}, 1 < i < n, is a collection of disjoint intervals in [0, 11, and each set a(Z,) is the graph of a continuous, piecewise C’ function on A. The curve (T is a finite union of graphs if there exist a,,
YANGMILLS

FIG.

95

THEORY

7.1. A closed curve which is a finite union of graphs.

7.3. The inverse of the curve (r is the curve CJ-’ defined by

DEFINITION

a-‘(t)=a(l-t),

O
7.4. (a) Let oi(t) = (xi(t), y,(t)), i = 1, 2, be two curves satisfying We define the extended product of 0, and c2 as the curve

DEFINITION

x1( 1) =x,(O).

0gt&

al(4t), (x,(l),

fJ,*o,(t)=

$
(2t-~)y2(0)-(2?-5)y,(l),

$
{ fl,(4t - 3),

(b) Let a(t) = (x(t), v(t)) satisfy x( 1) = x(0). We define the closing of u to be the closed curve

42th *a(t)=

O
i (x(1),(2t-l)y(O)-(2t-2)y(l)),

$
Remark 7.5. In this last definition, we consider a curve whose endpoints have the same x-coordinate. By adding on a vertical curve connecting these endpoints, we obtain a closed curve. This construction will be useful later when we define the standard representation of an expectation. In Section 2, we defined parallel transport along any curve whose image is the graph of a continuous function. Any curve 0 which is a ‘finite union of graphs can be written (up to change of parametrization) as the product of a finite number of curves,

where ai is either the graph of a function, or else has constant x-coordinate, for 1 < i < n. Parallel transport is trivial in the latter case. We then define parallel transport along c as U(a) = U(a,) U(U”-,)

It is important 595/194/l-7

to notice that this definition

‘*. U(a,).

is independent

of the parametrization

96

GROSS,

KING,

AND

SENGUPTA

of the curve cr. Accordingly, when discussing parallel transport we wll not mention parametrization. We will denote by W= { nl, .... o,} a collection of curves, and by W(Z) the union of the images of these curves. DEFINITION 7.6. Let W= {or, .... on} be a collection of curves. The point p E W(Z) is an intersection point of W if any of the following equations is satisfied:

p = cri(t) = ai( t’)

someO
some l
p = Oi( t) = a,( t’)

some O
some l
DEFINITION 7.7. The finite collection of closed curves W is admissible if W has a finite number of intersection points, and if each curve in W is a finite union of graphs. The finite collection W is weakly admissible if W(Z)’ has a finite number of connected components, and if each curve in W is a finite union of graphs.

Remark 7.8. The complement of the image of an admissible collection of closed curves is a finite union of disjoint, connected open sets, one of which is unbounded, and the rest of which are bounded. Thus an admissible collection is weakly admissible (the converse need not be true). DEFINITION 7.9. Let W= {a,, .... a,} be a weakly admissible collection of closed curves. We define the expectation of W to be

Remark 7.10. Since W is weakly admissible, there exist a0 < a, c ... c a4 such that each curve in W is a finite union of graphs over each interval (a,- r, u,), for 1 < r < q, and W(Z) is a subset of [a,,, a41 x R. Correspondingly, there is a Bralic map cpr for each interval (a, _, , a,), and a connection operator 1, for each endpoint a,. Theorem 5.7 and Remark 5.8 imply that the expectation of W is given by

This expression implies invariance of ( W) under translations in the x-direction, since the operators { cp,} and {jr} are invariant under translations in the x-direction. Furthermore, suppose that W= W, v W,, where W, and W, are collections of curves whose images W,(Z) and W,(Z) are disjoint, and suppose that there is a vertical hne which is disjoint from W(Z), such that W,(Z) and W,(Z) lie on different sides of the line. Let (a, 0) be the point where this line intersects the x-axis, and suppose that a E (a,- r, a,). Then the connection operators at the left

YANGMILLS

97

THEORY

and right hand ends of the interval (a,- r, a,), namely I,- 1 and jr, have the algebra &, = @ as domain and range, respectively. Remark 5.8 implies that =jo(P*~l...(P,-lQr-l w2>=.t(Pr+r(P4~q. Since cpr is equal to one, this implies

We will now analyze the expectation of an admissible collection W, and show that it equals the expectation of another collection, called a standard representation. The following definitions will provide an inductive procedure for finding this standard representation. 7.11. Let W be an admissible collection of closed curves, and let Rk} be the connected components of W(Z)c, where R, is the unbounded {Ro, R,, .... component. We associate to this a graph V(W) on k + 1 vertices (uO, ur, .... u,}, with edges connecting each pair of vertices vi and vi for which aRi n aRj contains a nonempty open subset of W(Z). DEFINITION

Remark 7.12. For an admissible collection W, the associated graph V( W) is connected. Fig. 7.2 shows several examples of collections with their associated graphs. DEFINITION 7.13. Let V be a connected graph. A tree on V is a minimal connected subset of V which contains every vertex. The distance between two vertices v and O’ in T is the number of edges in the smallest connected subset of T containing o and u’. With the choice of a base vertex, a tree becomes a partially ordered set, in the

w FIG.

7.2.

Collections

vw of curves with their associated graphs.

98

GROSS,

KING,

AND

SENGUPTA

following way. Suppose T is a tree on V( IV), and the base vertex is u,,. Then there is a unique sequence in T from u0 to u,, for each 1 is an ordered set, and we define v* as the (unique) maximum element of this set, for l vi, call this point pi; otherwise call it pi. Define p. to be the origin in the plane. We will call pi the gate for the region Ri, and we will denote by Ti the tangent vector to the parametrized curve at pF Note that each region R, has at least one gate on its boundary, namely pm, for 1 for 0 c t < 1. We also require that at each gate pm, the vectors T,,,, f;(O) and {r;(l) lj= m} should be non-zero, and no two of these vectors should be parallel. Let f be the collection of these curves. Then we say that r is a tree on W if r has no intersection points other than the gates { po, pl, .... pk}, and if each curve r,,, is a finite union of graphs. DEFINITION

{R,}.

Remark 7.15. The existence of these curves (r, > follows from our assumptions on W, as explained in Lemma Al in the Appendix. Fig. 7.3 shows an example of a tree r on a collection W. We will use r to extract from W some information about how the regions { Ro, R, , .... R, > are arranged relative to the curves in W. It will turn out that this information, along with the areas of the components (R,}, will completely determine ( W). First we recall the following definition. DEFINITION 7.16. Let v = (vi, v2) and w = (wi, wJ be two vectors given by their Cartesian coordinates. We define

sgn(0, w) = sgn(v, w2 - v2 wi). DEFINITION

7.17.

For each m, j = 1, .... k, we define

4,, = sgn(UO), T,)

C(l)) W, j) = sfimK(l)~ 0

if fi=j, m#j#O otherwise.

YANGMILLS

THEORY

99

(4

(e) (4 W (b) VW (cl T Cd)gates k-3 r FIG. 7.3. Constructing a tree f as a collection of curves IV.

DEFINITION 7.18. We shall say that the 2k distinct numbers {a,, .... a,; b , , .... bk} are ordered according to Z if the following conditions are satisfied:

0) a,
a,-cu,
ldm
if if

v,
Remark 7.19. These conditions are suffkient to determine the ordering of the 2k numbers {a,, .... a,; bl, .... bk}. The relevance of Definition 7.18 will become clear when we define the standard representation for ( W). Our standard representation uses k non-intersecting curves in the right half-plane with endpoints lying on the y-axis. The curves will enclose regions corresponding to the bounded components of W(Z)‘, and their endpoints on the y-axis will be ordered according to Z. As an example, we present below the ordering obtained from the example in Fig. 7.3.

100

GROSS,

EXAMPLE

7.20.

KING,

AND

SENGUPTA

From Fig. 7.3, we get the following: f&=6,=&=1;

6,=&=&=&=

-1

6(2,3)=6(1,5)=6(1,7)=6(5,7)=

1

6(3,2)=6(5,

-1

6(m, j) = 0

1)=6(7,

1)=6(7,5)=

otherwise.

This gives the following ordering on {a,, .... a,; b,, .... b,}: a,
The final piece of information we need concerns the relative positions of the gates on the curves. Each gate belongs to a curve (TVin W, and the curve oi gives an ordering for the gates belonging to it. If 0 < ti < I, < 1, we will write o(ti) < a(tz) to indicate that a(ti) precedes a(tz). Let g(i) be the number of gates in ai( We list the gates in a,(Z) in order along the curve, as piI
l
(7.1)

(here we have reparametrized cri so that it starts at pi,). We can now define the standard representation, using the information described above. We will denote by int(a) the union of the bounded connected components of a(Z)‘, for any closed curve cr. DEFINITION 7.21. Let {a,, .... a,; b,, .... bk} be ordered according to Z, and let (wm = (0, a,); z, = (0, b,)}, 1 O} for all 0 < t < 1. Define n closed curves Ym(l)=zm as follows (see Definitions 7.3, 7.4):

ei = *yfkE;)* . . . * 8

yi2

ai2* yf;,

1
(7.2)

(where ys = y *’ is either y or its inverse). We will call the collection of curves w= (cT1, ...) 8n} a standard representation for W if the following conditions hold: (i) (ii) (iii)

each curve I?~is a finite union of graphs the only intersection points of IV lie on the y-axis and u V,~wn

inf(*yi)

= IR,I,

l
(7.3)

YANG-MILLS

101

THEORY

Remark 7.22. Our standard representation should be seen as resulting from a particular deformation of the curves in W which preserves the areas of the components of the complement. Each curve oi is deformed into cfi, and all the intersection points of the curves end up on the y-axis. The condition (7.3) ensures that the areas of components of W(Z)’ are preserved. Figure 7.4 illustrates the construction of a standard representation for the collection of curves in Fig. 7.3. We now have our main invariance result, which states that ( W) can be computed from the information recorded in the tree r, the ordering of gates around curves, and the areas of the components (R,, .... Rk}. THEOREM 7.23. Let W be an admissiblecollection of closed curves, and let w be a standard representation of W. Then

(W)=(Rf).

In order to prove Theorem 7.23, we will construct a family of collections of curves { m(a)} which interpolate between W and IV, for 0 < 01d 1. This will involve the following definition of a squashed curve. DEFINITION 7.24. Let a(t) = (x(t), y(t)) be a curve, and let u satisfy 0 < tl < 1. Then we define the squashed curve a(~) by

da; t) = (x(t), q(t)). Zl

z2 w2 z3 z4 w4 w3 Wl

25 z6 w6 w5 z7 w7 FIG. 7.4. been denoted

Standard representation 8,. for m = 1, . ... 7.

for the curves

in Fig. 7.3. The

region

corresponding

to R,

has

102

GROSS,

KING,

AND

SENCUPTA

Note that if c is the graph of a function over the interval (a, b), then so also is a(a), for O
W.

k.

The reader should compare Eqs. (7.4) and (7.1). As an example. Fig. 7.5 illustrates the case of two curves g1 and (TV, along with their extensions 5i and 5,. The curves have been separated so that they can be distinguished easily. In fact, we will prove our results by approximating the curves {c?~} in just this way, so that the collections will be admissible. By taking a limit where the width of the “tubes” goes to zero, we will recover the result for the curves defined by Eq. (7.4). Furthermore, the reader can check that the vertical ordering of the loops y,, y2, and y3 in Fig. 7.5 is determined by the tree Z in accordance with Definition 7.18. This should serve as a motivation for our definitions, and should help the interested reader. to construct a standard representation for any other collection of curves.

(4

(b)

FIG.

7.5.

Construction

of interpolating

curves.

103

YANGMILLSTHEORY

Now we use Definition 7.24 to squash this curve in the following 0 < cx< 1, define the curve

way. For

5i-a = e,g(j)( 1 - a) * Flsb( 1 --01) * y2;)(GI) * pjg(,j( l - a) *~~~*tr,,,(l-a)*~~;‘(1-a)*~~~~(a)*~i,(1-2).

(7.5 1

Note that every curve to the left of the y-axis has been squashed by a factor 1 -a, while every curve to the right of the y-axis has been squashed by a factor LY. This is designed to keep fixed the areas of the connected components of the complement of the curves. For example in Fig. 7.5, if we neglect the areas contained in the “tubes”, each connected component (there are only three) has equal areas on the right and left of the y-axis. When the curves are squashed according to Eq. (7.5), the total area of each connected component does not change as a function of a. In the limiting case u = 0, the reader can check that parallel transport around the curves 8,,, and d,,, is the same as for the original curves crl and u2. Similarly, when CI= 1, we get the parallel transport for the curves 6, and cYz.We will prove that the expectation does not change as the curves are squashed in this way, and this will imply Theorem 7.23. Now let @(a) be the collection of closed curves {a,,,, .... a,,}, for each 0 < a 6 1. Note that #‘(a) is a weakly admissible collection of closed curves (it is not admissible since the intersection points form an open set). PROPOSITION

7.25.

( w(a))

is independent of a, for 0 < tl d 1.

Before proving Proposition 7.25, we use it to deduce Theorem 7.23, and we also deduce our main results about invariance properties of expectations. Proof of Theorem 7.23.

By Proposition oww=

7.25 we have mw.

When a =O, the squashed curve yi(a) lies entirely on the x-axis, where parallel transport is trivial. Therefore (7.1) and (7.5) imply that parallel transport along C’i.O is given by u(5i,0)=

u(ei,g(i)) =

.‘.

u(ei,l)

U(a,).

Hence we deduce
(W>.

when CL= 1, (7.2) and (7.5) imply that u(ai,l) = U(yfy = U(Si)

. . . Uoq’)

104

GROSS, KING, AND SENGUPTA

and hence (lV(l))=(W). If w= (cr, ...) a,,] is a collection of curves and cp: IV + [w2is a C’ mapping, will denote by W, the collection of curves { cpo ai, .... cpo a,,}.

we

THEOREM 7.26. Let W= {a,, .... a,,} be an admissiblecollection of closed curves, and let qr: [w2-+ Iw2be an orientation-preserving diffeomorphism. Let (R,, .... Rk) be the bounded connected components of W(Z)‘, and let {cp(R,), .... rp(R,)} be the corresponding components of W,(Z)‘. Suppose that W, is an admissible collection and that

Then

Proof Let Z be a tree on W (see Definition 7.14), and let cp0 Z be the image of Z under cp.By translation in the x-direction, if necessary, we can assume that W,(Z) and cp0 T(Z) lie to the left of the y-axis (see Remark 7.10). By adding on a C1 curve connecting the translated endpoint of cp0 Z to the origin, and applying Lemma A3 from the Appendix, we obtain a tree Zcp on W,. Since cp is orientation preserving, the values of the coefficients (6,) and {s(m, j)} are preserved under rp, for each m, j = 1, .... k. Therefore Z and r, define the same ordering for 2k points {a,, .... a,; b i, .... bk}. Furthermore the ordering of gates on each curve ai is preserved under cp. Therefore since the areas of the corresponding connected components in W(Z)’ and W,(Z)’ are equal, it follows that Wand W, share the same standard representations. Hence Theorem 7.23 implies that ( W) = ( W,). COROLLARY 7.27 (Euclidean invariance). Let cp: [w2+ Iw2be a rotation or translation. Let W be an admissiblecollection, and supposeW, is also admissible.Then

COROLLARY 7.28 (Factorization of components). Let W be an admissiblecollection of closed curves. Let rp: [w + Iw2 be a continuous, simple curve satisfying q(t) = (t, 0) for 1tl > T, for some T> 0, such that the image of cp is disjoint from W(Z). Let W, and W2 be the subcollections of curves in W whoseimages W,(Z) and W,(Z) belong to different componentsof the complement of the image of cp. Then

Proof The proof follows by choosing an appropriate tree Z on Wand applying Remark 7.10. By translation invariance, we can translate Wand cpso that W(Z) lies

YANGMILLS

10.5

THEORY

to the left of the y-axis, and so that the translation of the curve cp passes through the origin in the plane. The translated curve cp divides the plane into two comand Nz lies ponents N, and N,, where N1 lies above the x-axis asymptotically, below. Without loss of generality, we can assume W,(I) c N,, and so we can choose the curves in r attached to W, so that their images all lie in N, also (excluding endpoints at the origin). Similarly the curves in f attached to W, can be chosen to lie in N2, except for the endpoints at the origin. With this choice of r, we can find a standard representation r for which the curves associated with W, all lie above the x-axis, and the curves associated with W, all lie below. By rotation invariance, we can rotate the curves in P through 90”, in which case the y-axis separates the curves into disjoint sets. Since the expectation factorizes in this case (see Remark 7.10), it also factorizes for wand hence for W. Proof of Proposition 7.25. We will construct a sequence of admissible collections of curves @(a; E) depending on a parameter E, which converge pointwise to @(a) as E + 0. For each value of E, we will prove that ( w(a; E) ) is independent of a, and therefore (m(a)) also will be independent of a (see Remark 4.10). Figure 7.5 illustrates how these approximations are constructed, namely by replacing the curves in the tree r by non-intersecting “tubes”. Each region R, in W(l)’ is connected by a tube to a region of equal area on the right of the y-axis. This new collection of curves is admissible, and after squashing by a factor 1 - a on the left of the y-axis and by a on the right, we obtain the collection @(a; a). Except for some overlap of lines on the y-axis, the collection @(a; E) is also admissible. Therefore by the result of Section 5, we can write (i?‘(a; a)) as a product of Bralic maps and connection operators. First we will show that the dependence on a occurs only in the Bralic maps (cp,]. Then we will show that these Bralic maps may be combined in such a way that the dependence on a disappears (this uses the results of Section 6). It is important to notice that the vertical ordering of points on the y-axis produced by r is exactly that which allows us to construct disjoint tubes connecting W with the standard representation. Now let E> 0, and let B,(p) be the disk of radius E centered at p. We will assume E is sufficiently small so that the disks about the gates {B,(p,)}, 0
Xi,1 *

@i,,

1
(7.6)

We now use the following result, which makes concrete the “tubes” illustrated in Fig. 7.5. The result says that each curve p,,, (which connects pm to po, the origin) can be replaced by a tube bounded by two curves P,,, and t,, in such a way that all the new curves {p,, z,> are disjoint.

106

GROSS,

KING,

AND

SENGUPTA

LEMMA 7.29. There are 2k distinct points { wL, zk}, m = 1, .... k, on the y-axis within distance E of the origin, whose y-coordinates are ordered according to Z, and 2k curves {p,, z,}, m = l,..., k, satisfying the following conditions:

0)

(ii) (iii)

P,(~)=w&, p,(O)=q,,

r,(l)=.& Zm(0)=rm

ifcS,=l;

pm(0)=rm,

Tm(0)=qm

ifs,=

-1

excluding endpoints, the curves {p,, z,.; xigj} are disjoint, for all 1
(iv) the images p,(I) for each 1
and z,(Z)

belong to the e-neighborhood of the set r,,,(Z),

This lemma is proved in the Appendix; we will now use it to establish Proposition 7.25. Let R be a standard representation for W, so that @ is a collection of n closed curves {a,, .... a,} built out of the k curves {yl, .... yk}. We will consider a different set of curves (/I,, .... Bk} which closely approximate {rI, .... yk}. They are constructed according to Definition 7.21, but with the area condition (iii) slightly modified. We shall specify the corresponding condition shortly. Then for each m = 1, .... k we define the curve Cm=T;l*pm*pm.

(7.7)

Using these curves, we construct n closed curves {p,, .... pL,} as follows: pi=&&)

* q-y

* c;;

* Xi,g(i)-I * “‘*

* xi,* * c$

Xi,2

l
(7.8)

Again C6 = C +I is either the curve C or its inverse. We will denote by R(E) the weakly admissible collection of curves {p,, . ... ,u,}. Next we define squashed curves as before. First we have c,,, = r,‘(l

-a) * IL(a) * PA1 -a),

l
(7.9)

for 0
* &J( 1 - a) * c?11.a

1
(7.10)

We will denote by m(cr; E) the weakly admissible collection of curves {P 1,m,.... pu,,,>, Consider the part of @(a; E)(Z)~ lying to the left of the y-axis. Each bounded component of this set contains on its boundary an interval on the y-axis, lying inside the disk of radius E about the origin. The endpoints of these intervals belong to the set of points { wL, zk}, m = 1, .... k. We will collect these components into sets {K,}, m = 1, .... k, according to the intervals on their boundaries. For brevity, we will denote by [a, b] the interval on the y-axis with endpoints a and 6;

YANG-MILLS

107

THEORY

where a lies below b. For each M = 1, .... k, let K,,, be the union of the connected components whose boundaries contain one of the intervals [wk, z;], [wk, wi], {zj, zL] or [z;, wj] with j= i= m. These intervals have been chosen so that in the limit E + 0, the set K,,, will converge to R,, along with some part of the tree f. For each m, the set K,,, contains (say) d, connected components {K,,,, 1, .... Km,dm}. We can now define the area constraint for the curve pm:

u int(*Bi)= i=ffI lKmJ, 1W*B,)I f+>%l

l
(7.11)

In the limit E+ 0, the area of the set K, converges to IR,I. It is straightforward to show that we can find curves {pi, .... /lk} which converge to the original curves b 1, .... yk} as E + 0. In this case the collection of curves w(a; E) converges to w(a). The boundary of the connected set int(*/?,)/u,, “, int(Qj) contains intervals on the y-axis whose endpoints belong to the set { wl, .... w,; z,, ,.., zk}. These intervals are precisely those described above for K,,,, but with each wh, z& replaced by w,, z,, for a = i, j, m. We now consider the expectation (m(s)). There is a set of numbers c,, < c,< ...
=~ocp~~~cp*~~~~p-~l(pp(Pp+l~p+l

-(Pp+q.fp+q

where {.?,, cp,} are the corresponding connection operators and Bralic maps. Note that the connection operator at x=0 is the identity, since the vertical ordering of the curves arriving from both sides is given by the tree K In any interval (c, _ 1, c,), the Bralic map is a product of exponentials

where { A,,i} are the areas of connected components, and ( E,,i} are the area operators. If each curve in this interval is squashed by a factor a, the corresponding Bralic map is

Furthermore, the positions of the points {co, ci, .... cP+,} and the connection operators (jr> do not change when curves are squashed. Therefore we also have, for O
. . . ~pp+qbVp+q. The limiting cases a = 0, 1 follow from Corollary 4.9 and Remark 4.10 on convergence of expectations. Note that the dependence on a occurs only in the Bralic maps, and not the connection operators.

108

GROSS,

KING,

AND

SENGUPTA

Now consider one of the sets K,,, lying to the left of the y-axis (these were introduced after Eq. (7.10)). It is composed of d,,, connected sets (K,,,,, .... Km,,), and by construction each component K,,,i intersects the vertical strip (cP, 0) x R. For each Km,i, we choose one connected component of Kmqin ((c,, 0) x R), and we let Em,i be the area operator associated to this component. We define a new Bralic map for the interval (c,, 0), namely, @,(l -a)=exp

[

-(l-a)

5 m=l

g

IL,J

Em,i .

i=l

I

Since the complement of all curves lying to the left of the y-axis is the union of the sets {K,), it follows from Corollary 4.8 and Theorem 6.8 that ( W(a; E)) = .lJ,

...Jp-I$p(l

-a)(Pp+,(a)j,+,...cp,+,(a)~~+,,.

Corresponding

to the set Km, there is the connected component int(*bi) lying on the right of the y-axis; we will denote this set by W*LMJ,,, L,, for convenience. Equation (7.11) implies that K,,, and L, have equal areas. Furthermore, there is a l-l correspondence between the bounded components of (c,,O)xR and of (O,c,+,)xR, namely that given by the vertical ordering of the lines. It follows that the area operator for a component of (cP, 0) x R which contains the interval [wL, zk] on its boundary is equal to the area operator for the component of (0, cP+ i) x 03 which contains [w,, z,] on its boundary (see the discussion before and after Eq. (7.11) for this notation), and similarly for the other intervals [wh, wj], [zj, z&] and [zj’, w:]. For each value of m, we have already chosen d,,, components of (c,, 0) x R. Let us pick one of these components, and call its area operator E,!,,. Then EI, is also the area operator of the corresponding component of (0, cP+ ,) x R, and by construction this component belongs to L,. Since L, is connected, we can again apply Corollary 4.8 and Theorem 6.8. We get a new Bralic map over (0, cP+ ,)

and we then have (@(a;

E))

^ A = foJl

~~.~p-l~.p(l-a)~p+,(a)~~+f.~-~~++---~jp+q.

Finally we make use of the relation (7.11). In the proof of Theorem 6.8, we established the result by moving operators exp[ - lK[ E] from one interval to another. The end result was that each connected component was represented by one single exponential operator exp[ - A,E’“)], which contained the total area A, of the component. However, we could now undo this process, and split up the operator exp[ - A,E(“‘] among different intervals, in each of which it would appear as another exponential operator. But there is no reason why the coefficients of the

YANG-MILLS

109

THEORY

area operators in these exponentials should be the same as in the original expression. Indeed as long as their sum is exactly A,, so that they can be recombined into the single area operator exp[ - A,E’“‘], these individual coefficients can be chosen freely. Consider now the d,,, components of (c,, 0) x R which were chosen previously. Their corresponding components in (0, cP+ 1) x IR all belong to L,, and their area operators are {E,,i}. Therefore if we define

-a

,f m=l

1

5 IK,J Em.; , i=l

it follows from (7.11) and the previous discussion that (@(a;

E)) =I0 . ..jp-I~p(l-a)~.+,(a)~~+,.,.j,+,.

This expression is independent is also independent of a.

of a, for any E sufficiently small. Therefore ( #‘(a))

APPENDIX

We state below three lemmas which imply the results used in Definition 7.14, Theorem 7.26 and Lemma 7.29. In each case, W is an admissible collection of closed curves, and p, p’ are points in the image W(Z). We will assume that p, p’ are not intersection points of W, and that the tangent vectors are defined at these points. LEMMA Al. Let T be the tangent vector at p, and let v be any other vector satisfying sgn( T, v) # 0. Then there is a C’ curve y satisfying y(O) = p, y’(O) = v and y(t) e W(Z)‘for all 0 < t < 1.

The proof of this lemma is illustrated in Fig. Al. Since ail curves in W Remark. are continuous, and since the tangent vector is defined at p, there is a

FIGURE

Al

110

GROSS, KING, AND SENGUPTA

(b)

(4 FIGURE A2

neighbourhood of p in which the curve looks like the picture in Fig. Al. Lemma Al implies the existence of the curves {r,,,} in Definition 7.14. LEMMA A2. Let (yl, .... y,} be C1 curues satisfying ri( 1) = p, sgn(y:( l), T) > 0 for all 1~ i < m, where T is the tangent vector at p. We assumefurthermore that the curves are disjoint, and do not intersect W(Z), except at the point p. Then the curves can be deformed in a neighborhood of p so that they becomecompletely disjoint, with distinct endpoints on the curve near p,

Remark. The lemma is illustrated in Fig. A2, which also serves as the proof. Note that the condition sgn(y;(l), T) >O, 1
Remark.

Once again the proof of this lemma is illustrated

P

r .

a

b

FIGURE A3

P’

in Fig. A3.

YANGMILLS

THEORY

111

FIGUREA4

We can now address the proof of Lemma 7.29. We have a tree Z on the set W(Z), connecting the gates by C ’ curves. At each gate pm, there are a number of incoming curves and one outgoing curve. By Lemma A2, the incoming curves can be deformed so that their endpoints are disjoint. Since they originate from different gates, these deformed curves are now entirely disjoint. Let Zi* be one such curve, so that Zi*(0) = pi. The points qi, ri lie on the same curve as pi, within a distance E. Therefore by Lemmas A3 and A2, we can find two disjoint curves within the s-neighborhood of Zi* (I) which connect qi and ri to points on the curve near p,. This procedure is followed for every incoming curve at pm, in such a way that all the resulting curves are disjoint, and so that their endpoints all lie between qm and I,. The resulting situation at p, is illustrated in Figure A4. It is clear that the ordering of the endpoints between q,,, and r,,, is determined by the coefficients (XL 4). F ur th ermore, suppose that ti = j, so that the two outgoing curves shown in Fig. A4, whose initial points are qm and r,,,, go through the region Rj and terminate on the curve near pi, between qj and rj. Then by Lemmas A3 and A2 all the incoming curves near pm can be extended through the region Rj so that their endpoints lie on the curve near pi, and so that they remain disjoint. The result is illustrated in Fig. A5 By iterating this construction at every gate in the tree Z, we generate 2k disjoint curves which connect the points {qm, r,} to distinct points on the y-axis near p,,. These are the curves (p,, r,}.

FIGURE AS 595119411.8

112

GROSS,

KING,

AND

SENGUPTA

REFERENCES

R. HOEGH-KROHN, AND H. HOLDEN, J. Funct. Anal. 78 (1988) 154. R. HOEGH-KROHN, AND H. HOLDEN, “Stochastic processes, mathematics and physics, in “Lecture Notes in Mathematics,” Vol. 1158 (S. Albeverio and P. Blanchard, Eds.), SpringerVerlag, New York, 1986. S. ALBEVERIO, R. HOEGH-KROHN, AND H. HOLDEN, “Stochastic processes in classical and quantum systems, in “Lecture Notes in Physics, Vol. 262 (S. Albeverio, et al., Eds.), Springer-Verlag, New York, 1986. S. ALBEVERFO, R. HOEGH-KROHN, AND H. HOLDEN, in “Stochastic Space-Time Models and Limit Theorems (Arnold and Kotelenez, Eds.), Reidel, Boston, 1985. S. ALBEVERIO, R. HOEGH-KROHN, AND H. HOLDEN, “Stochastic Methods and Computer Techniques in Quantum Dynamics, in “Acta Physica Austriaca Supl.” Vol. 26, Springer-Verlag, New York, 1984. W. AMBROSE AND I. M. SINGER, Trans. Amer. Math. Sot. 75 (1953), 428. I. YA. AREF’EVA, Theor. Math. Phys. 43 (1980), 353. R. BALIAN, J. M. DROUFFE,C. ITZYKSON, Phys. Rev. D 10 (1974) 3376; 11 (1975) 2098 and 2104. A. A. BELAVIN AND V. G. DRINFELD, Functional Anal. Appl. 16 (1982), 159. [English]

1. S. ALBEVERIO, 2. S. ALBEVERIO,

3.

4. 5. 6. 7.

8.

9. 10. I. BIALYNICKI-BIRULA, Bull. Acad. Pal. Sci. 11 (1963), 135. Il. N. BRALIC, Phys. Rev. D 22 (1980), 3090. 12. CHAN HONG-MO, PETER SCHARBACH AND Tsou SHEUNGTSUN, Ann. Phys. 166 (1986), 396. 13. H. G. DUTCH AND V. F. MILLER, Fortschr. Phys. 27 (1979), 547. 14. B. DRIVER, J. Funcf. Anal. 83 (1989), 185. 15. B. DRIVER, Commun. Math. Phys., in press. 16. P. FISHBANE AND S. GASIOROWICZ, P. KAUS, Phys. Reo. D 24 (1981), 2324. 17. L. GROSS,J. Funct. Anal. 63 (1985), 1. 18. L. GROSS, Canad. Math. Sot. Conf: Proc. 9 (1988), 193. 19. N. IKEDA AND S. WATANABE, “Stochastic Differential Equations and Diffusion Processes,”

p. 464,

North-Holland, New York, 1981. 20. V. KAZAKOV, Nucl. Phys. B 179 (1981), 283. 21. V. KAZAKOV, AND J. KOSTOV, Phys. Lett. B 105 (1981), 453. 22. 23. 24. 25. 26.

V. KAZAKOV, Nucl. Phys. B 176 (1980), 199. S. KLIMEK AND W. KONDRACKI, Commun. Math. Phys. 113 (1987), 389. T. KOHNO, Ann. Inst. Fourier Univ. Grenoble 4, No. 37(1987), 139. C. N. KOZAMEH AND E. T. NEWMAN, Phys. Rev. D. 31 (1985), 801. A. LICHNEROWICZ, Theorie globale des connexions et des groupes

d’holonomie, in “Cons&ho Nazionale delle Ricerche Monogratie Matematiche 2,” Edizioni Cremonese, Roma, 1962.

27. S. MANDELSTAM, Ann. Phys. 19 (1962), 1. 28. S. MANDELSTAM, Ann. Phys. (1962), 25. 29. S. MANDELSTAM, Phys. Reu. 175 (1968), 1580. 30. H. P. MCKEAN JR., “Stochastic Integrals,” p. 140; Academic Press, New York, 1969. 31. A. A. MIGDAL, Ann. Phys. 126 (1980), 279. 32. Y. NAMBU, Phys. L&t. B 80 (1979), 372. 33. A. M.POLYAKO~‘, Nucl. Phys. B 164 (1979), 171. 34. D. STR~~~K AND S. VARADHAN, “Multidimensional Diffusion Processes,” p. 338, Springer-Verlag,

New York, 1979. 35. TUNG-MOW

YAN, Phys. Reo. D 22

(1980), 1652.