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path: root/support/hypertex/tanmoy/ghostview-1.5-hacked/test.ps
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0 0 bop 2246 66 a Fr(PUPT{1312)-72 653 y Fq(W)-10 b(orld-Sheet)39
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611 y Fp(?)787 999 y Fr(Jacques)21 b(Distler)i(and)f(Mark)g(D.)g(Do)n(yle)908
1125 y(Joseph)g(Henry)g(Lab)r(oratories)999 1231 y(Princeton)g(Univ)n(ersit)n
(y)901 1338 y(Princeton,)h(NJ)44 b(08544)h(USA)133 1603 y(Green)27
b(and)h(Seib)r(erg)g(sho)n(w)n(ed)g(that,)i(in)f(simple)f(treatmen)n(ts)f(of)
i(fermionic)f(string)g(theory)-6 b(,)31 b(it)0 1710 y(is)c(necessary)f(to)h
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(erators)h(collide.)44 b(Otherwise,)0 1816 y(certain)18 b(sup)r(erconformal)g
(W)-6 b(ard)19 b(iden)n(tities)g(w)n(ould)g(b)r(e)f(violated.)29
b(In)18 b(this)h(note,)g(w)n(e)f(sho)n(w)h(ho)n(w)g(these)0
1922 y(con)n(tact)d(terms)g(arise)h(naturally)h(when)e(prop)r(er)g(accoun)n
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r(oundary)h(of)h(mo)r(duli)f(space)0 2241 y(is)f(describ)r(ed)e(correctly)-6
b(.)0 3230 y(3/92)p 0 3309 800 3 v -66 3482 a Fo(?)20 b Fn(Email:)27
b Fm(distler@puhep1.princeton.edu,)k(mdd@puhep1.princeton.edu)17
b Fn(.)-15 3575 y(Researc)n(h)22 b(supp)r(orted)f(b)n(y)g(NSF)g(gran)n(t)f
(PHY90-21984.)p eop
%%Page: 1 2
1 1 bop 133 66 a Fr(The)24 b(existence)f(of)h(con)n(tact)g(terms)f(in)i
(string)g(theory)f(seems,)g(after)h(a)f(momen)n(t's)g(though)n(t,)h(to)0
173 y(b)r(e)e(somewhat)h(parado)n(xical.)37 b(In)24 b(the)f
Fl(stable)i(c)m(omp)m(acti\014c)m(ation)e Fr(of)i(mo)r(duli)f(space)f([1)p
 (#reference.1) [[462 687 468 699] [1 1 1 [3 3]] [0 0 1]] pdfm (,2)p
 (#reference.2) [[471 687 477 699] [1 1 1 [3 3]] [0 0 1]] pdfm (],)k
(punctures)0 279 y(\()p Fl(i.e.)22 b Fr(the)e(lo)r(cations)i(at)g(whic)n(h)f
(v)n(ertex)f(op)r(erators)h(are)g(inserted\))g(are,)g Fl(by)i(de\014nition)g
Fr(not)e(allo)n(w)n(ed)h(to)0 386 y(collide.)30 b(Rather,)20
b(as)g(t)n(w)n(o)g(punctures)e(approac)n(h)i(eac)n(h)f(other,)h(w)n(e)g(can)f
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(surface)h(surrounding)g(them)0 599 y(pinc)n(hes)h(o\013)h(\(Fig.)h(1\).)30
b(If)21 b(the)g(punctures)g(nev)n(er)f(collide,)j(there)e(w)n(ould)h(seem)e
(to)i(b)r(e)f(little)h(role)g(for)g(a)0 705 y Fk(\016)s Fr(-function)16
b(con)n(tact)g(term.)27 b(A)17 b(more)f(sophisticated)i(p)r(erson)e(migh)n(t)
h(replace)f(the)h(notion)g(of)h(a)f(con)n(tact)0 812 y(term)23
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(the)f(string)h(measure)f(whic)n(h)g(is)0 918 y(\\)p Fk(\016)s
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(space)g(corresp)r(onding)g(to)g(the)g(pinc)n(hed)0 1024 y(surface)p
Fr(.")i(But)g(this)g(form)n(ulation)h(seems)e(rather)g(strained)h(and)g(hard)
g(to)g(deriv)n(e)f(from)h(conformal)0 1131 y(\014eld)c(theory)-6
b(.)33 b(After)22 b(all,)j(conformal)e(\014eld)g(theory)f(amplitudes)h(are)g
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r(duli,)i(whic)n(h)f(a)g Fk(\016)s Fr(-function)g(certainly)g(is)h(not.)266
1883 y(Fig.)g(1:Colliding)i(punctures.)133 2070 y(F)-6 b(ortunately)g(,)21
b(it)e(no)n(w)h(app)r(ears)f(that)f(\\con)n(tact)h(in)n(teractions")g(ha)n(v)
n(e)g(a)g(p)r(erfectly)f(natural)i(home)0 2176 y(in)i(string)g(theory)-6
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b(The)22 b(imp)r(ortan)n(t)h(p)r(oin)n(t)0 2389 y(is)h(that)f(in)h(order)e
(to)i(de\014ne)e(the)g(op)r(erators)h(whic)n(h)h(go)f(in)n(to)h(a)f
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(need,)g(implicitly)-6 b(,)21 b(a)d(normal-ordering)h(prescription)g(or,)g
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h(a)g(lo)r(cal)g(co)r(ordinate)f(at)h(the)f(insertion)h(p)r(oin)n(t.)34
b(Actually)-6 b(,)25 b(w)n(e)e(need)f(a)i(family)g(of)0 2708
y(suc)n(h)k(lo)r(cal)i(co)r(ordinates)f(whic)n(h)f(v)l(ary)i(as)f(w)n(e)f(v)l
(ary)i(the)e(mo)r(duli)h(of)g(the)f(surface.)49 b(Our)29 b(standard)0
2815 y(b)r(eliefs)21 b(ab)r(out)g(the)f(prop)r(erties)h(of)g(string)h
(amplitudes)f(obtain)g(when)g(the)f(co)r(ordinate)g(families)i(v)l(ary)0
2921 y(holomorphically)29 b(with)f(the)f(mo)r(duli.)47 b(Unfortunately)-6
b(,)30 b(there)c(are)i(obstructions)g(to)f(\014nding)h(suc)n(h)0
3028 y(a)h(holomorphic)f(co)r(ordinate)g(family)i(globally)g(on)f(mo)r(duli)f
(space.)49 b(W)-6 b(e)28 b(are)g(forced)g(to)h(allo)n(w)h(our)0
3134 y(normal-ordering)23 b(prescription)f(to)g(v)l(ary)h
(non-holomorphically)g(with)f(the)g(mo)r(duli.)133 3241 y(One)i(consequence)f
(of)i(this)h(phenomenon)d(is)j(that)f(the)f(one-p)r(oin)n(t)h(function)g(of)h
(the)e(zero)g(mo-)0 3347 y(men)n(tum)e(dilaton)j(v)n(ertex)f(op)r(erator)g
(\(whic)n(h)g(na)-7 b(\177)-26 b(\020v)n(ely)25 b(decouples\))e(is)i(not)f
(zero,)g(but)g(rather)g(is)h(pro-)0 3454 y(p)r(ortional)30
b(to)e(the)g(Euler)h(c)n(haracteristic)f(of)h(the)f(Riemann)g(surface)g([3)p
 (#reference.3) [[403 96 409 108] [1 1 1 [3 3]] [0 0 1]] pdfm (,4)p
 (#reference.4) [[412 96 418 108] [1 1 1 [3 3]] [0 0 1]] pdfm (,5)p
 (#reference.5) [[422 96 428 108] [1 1 1 [3 3]] [0 0 1]] pdfm (].)52
b(In)29 b(the)f(presence)e(of)0 3560 y(other)h(v)n(ertex)g(op)r(erator)g
(insertions,)k(one)c(needs)g(a)g(more)g(re\014ned)g(treatmen)n(t)e(of)k(the)e
(b)r(eha)n(vior)g(of)1283 3719 y(1)p eop
%%Page: 2 3
2 2 bop 0 66 a Fr(co)r(ordinate)22 b(families)h(as)g(punctures)e(collide.)31
b(This)23 b(leads)g(to)f(the)g(dilaton)h(insertion)g(b)r(eing)f(prop)r(or-)0
176 y(tional)e(to)f(the)f(Euler)g(c)n(haracteristic)g(of)i(the)e
Fl(punctur)m(e)m(d)e Fr(Riemann)i(surface)h([6)p
 (#reference.6) [[424 686 429 698] [1 1 1 [3 3]] [0 0 1]] pdfm (].)30
b(In)18 b(other)g(w)n(ords,)j(the)0 285 y(dilaton)26 b(insertion)f(coun)n(ts)
f(the)g(p)r(o)n(w)n(ers)g(of)i(the)d(string)j(coupling)f(in)g(the)f
(correlation)h(function,)g(or,)0 394 y(if)h(y)n(ou)f(wish,)h(there)e(is)h(a)g
(lo)n(w-energy)g(theorem)f Fk(D)g Fi(\030)f Fk(\025)1478 368
y Fh(@)p 1463 378 63 3 v 1463 417 a(@)s(\025)1533 394 y Fr(.)38
b(F)-6 b(or)26 b(the)e(b)r(osonic)h(string,)i(this)e(is)h(true)0
503 y(only)e(for)g(the)e(\014rst)i(v)l(ariation)h(with)e(resp)r(ect)f(to)h
Fk(\025)g Fr(\(one)g(dilaton)h(insertion\),)h(but)e(for)h(the)e(heterotic)0
612 y(string,)27 b(it)f(holds)g(to)g(all)h(orders)e(\(m)n(ultiple)g(dilaton)i
(insertions\))f([7)p
 (#reference.7) [[379 608 385 620] [1 1 1 [3 3]] [0 0 1]] pdfm (].)41
b(In)26 b(similar)g(fashion,)i(one)d(can)0 721 y(reco)n(v)n(er)19
b(the)g(\\dilaton")j([8)p
 (#reference.8) [[188 588 194 600] [1 1 1 [3 3]] [0 0 1]] pdfm (])f(and)f
(\\puncture")f([9)p
 (#reference.9) [[289 588 295 600] [1 1 1 [3 3]] [0 0 1]] pdfm (])i
(equations)f(of)h(top)r(ological)g(gra)n(vit)n(y)-6 b(,)23
b(p)r(ostulated)d(in)0 830 y([10)p
 (#reference.10) [[75 569 87 581] [1 1 1 [3 3]] [0 0 1]] pdfm (,11)p
 (#reference.11) [[90 569 102 581] [1 1 1 [3 3]] [0 0 1]] pdfm (].)32
b(The)20 b(k)n(ey)g(p)r(oin)n(t)h(to)f(b)r(e)g(emphasized)f(here)h(is)h(that)
f(none)g(of)h(these)f(results)h(in)n(v)n(olv)n(e)g(m)n(ysteri-)0
939 y(ous)i Fk(\016)s Fr(-functions)f(at)g(the)g(b)r(oundary)g(of)i(mo)r
(duli)e(space.)31 b(The)21 b(string)j(measure)d(is)i(p)r(erfectly)e(smo)r
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1157 y(space.)133 1268 y(Another)j(sort)h(of)g(con)n(tact)f(term)g(whic)n(h)g
(one)h(encoun)n(ters)e(in)j(fermionic)e(string)h(theory)g(is)g(that)0
1377 y(prop)r(osed)32 b(b)n(y)g(Green)g(and)g(Seib)r(erg)f([12)p
 (#reference.12) [[253 470 264 482] [1 1 1 [3 3]] [0 0 1]] pdfm (].)62
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%%Page: 3 4
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5 5 bop 0 66 a Fr(construction)32 b([14)p
 (#reference.14) [[146 706 158 718] [1 1 1 [3 3]] [0 0 1]] pdfm (].)63
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b Fr(\(1)p Fk(:)p Fr(6)q(\))0 1352 y(at)26 b(\(1,)p Fk(\034)8
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(from)f(\()p Fk(w)r(;)11 b(\030)s Fr(\))21 b(b)r(e)h(sup)r(erconformal)f([7)p
 (#reference.7) [[501 456 507 468] [1 1 1 [3 3]] [0 0 1]] pdfm (].)32
b(The)0 1561 y(min)n(us)c(signs)g(are)f(for)h(later)g(con)n(v)n(enience.)45
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Fr(set)22 b(to)g(zero,)f(and)h(rescaled)f(b)n(y)h(a)g(factor)g(of)h
Fk(t)1788 2932 y Fe(2)1817 2956 y Fr(.)30 b(The)22 b(ghost)g(insertions)g
(that)0 3071 y(result)j(from)g(computing)g(the)f(pushforw)n(ards)j(of)e(the)g
(tangen)n(t)g(v)n(ectors)g(asso)r(ciated)g(to)g(the)g(mo)r(duli)0
3186 y(corresp)r(onding)f(to)g(the)g(lo)r(cations)h(of)g(the)f(p)r(oin)n(ts)g
Fk(P)33 b Fr(and)25 b Fk(Q)f Fr(b)n(y)g(the)g(co)r(ordinate)g(family)h(are)f
(giv)n(en)p 0 3278 800 3 v -66 3434 a Fe(#)6 3460 y Fn(The)j(role)f(of)g
(non-split)g(co)r(ordinate)g(transformations)g(on)f(sup)r(ermo)r(duli)i
(space)f(in)h(this)f(regard)g(w)n(as)g(also)0 3557 y(noted)21
b(b)n(y)g(H.)g(V)-5 b(erlinde)21 b([15)p
 (#reference.15) [[180 78 191 90] [1 1 1 [3 3]] [0 0 1]] pdfm (].)1283
3719 y Fr(6)p eop
%%Page: 7 8
7 7 bop 0 66 a Fr(in)25 b([7)p
 (#reference.7) [[90 706 96 718] [1 1 1 [3 3]] [0 0 1]] pdfm (].)41
b(The)24 b(measure)g(is)i(formed)e(b)n(y)i(m)n(ultiplying)g(these)e(six)i
(con)n(tributions)f(together)f(\(including)0 167 y(delta)d(functions)g(for)g
(the)f(insertions)i(for)f Fk(\032)g Fr(and)g Fk(\034)8 b Fr(\).)29
b(The)21 b(insertions)g(are)g(\(ev)l(aluated)f(at)h Fk(r)f
Fr(=)p 2421 123 32 3 v 19 w Fk(r)g Fr(=)e(0\))189 310 y Fk(B)s
Fr([)p Fk(\033)298 328 y Fd(\003)330 310 y Fr(\()380 266 y
Fk(@)p 364 295 71 3 v 364 356 a(@)t(r)442 310 y Fr(\)])h(=)c
Fi(\000)638 257 y Ff(\000)687 266 y Fr(1)p 676 295 54 3 v 676
356 a Fk(t)700 337 y Fe(2)738 257 y Ff(\001)769 310 y Fk(b)798
276 y Fe(\()p Fh(P)7 b Fe(\))798 328 y Fd(\000)p Fe(1)898 310
y Fr(+)964 217 y Ff(\022)1021 266 y Fr(2)p Fk(p)1087 283 y
Fe(1)1117 266 y Fk(\034)p 1021 295 133 3 v 1061 356 a(t)1085
337 y Fe(2)1162 217 y Ff(\023)1211 310 y Fk(\014)1253 276 y
Fe(\()p Fh(P)g Fe(\))1249 337 y Fd(\000)1298 320 y Fc(1)p 1298
328 23 3 v 1298 353 a(2)1352 310 y Fi(\000)1419 257 y Ff(\000)1468
266 y Fr(1)p 1457 295 54 3 v 1457 356 a Fk(t)1481 337 y Fe(2)1519
257 y Ff(\001)1549 310 y Fk(b)1578 276 y Fe(\()p Fh(Q)p Fe(\))1578
328 y Fd(\000)p Fe(1)1679 310 y Fr(+)1746 217 y Ff(\022)1803
266 y Fr(2)t Ff(e)-41 b Fk(p)1869 283 y Fe(1)1899 266 y Fk(\034)p
1803 295 133 3 v 1842 356 a(t)1866 337 y Fe(2)1944 217 y Ff(\023)1993
310 y Fk(\014)2035 276 y Fe(\()p Fh(Q)p Fe(\))2031 337 y Fd(\000)2080
320 y Fc(1)p 2080 328 23 3 v 2080 353 a(2)2135 310 y Fr(+)14
b Fi(\001)d(\001)g(\001)189 491 y Fk(B)s Fr([)p Fk(\033)298
509 y Fd(\003)330 491 y Fr(\()380 446 y Fk(@)p 364 476 71 3
v 364 537 a(@)p 403 493 32 3 v 4 w(r)442 491 y Fr(\)])19 b(=)c
Fi(\000)638 437 y Ff(\000)687 446 y Fr(1)p 676 476 54 3 v 676
503 24 3 v 676 552 a Fk(t)700 514 y Fe(2)738 437 y Ff(\001)766
474 y Fr(\026)769 491 y Fk(b)798 457 y Fe(\()p Fh(P)7 b Fe(\))798
509 y Fd(\000)p Fe(1)898 491 y Fi(\000)964 437 y Ff(\000)1013
446 y Fr(1)p 1003 476 54 3 v 1003 503 24 3 v 1003 552 a Fk(t)1027
514 y Fe(2)1064 437 y Ff(\001)1092 474 y Fr(\026)1095 491 y
Fk(b)1124 457 y Fe(\()p Fh(Q)p Fe(\))1124 509 y Fd(\000)p Fe(1)1225
491 y Fr(+)14 b Fi(\001)d(\001)g(\001)187 677 y Fk(B)s Fr([)p
Fk(\033)296 695 y Fd(\003)327 677 y Fr(\()378 632 y Fk(@)p
361 662 74 3 v 361 723 a(@)t(\032)442 677 y Fr(\)])19 b(=)557
583 y Ff(\022)606 677 y Fi(\000)675 632 y Fk(\032)p 666 662
54 3 v 666 723 a(t)690 703 y Fe(2)742 677 y Fr(+)816 632 y(2)p
Fk(\034)p 816 662 70 3 v 839 723 a(t)894 583 y Ff(\023)943
677 y Fk(b)972 643 y Fe(\()p Fh(P)7 b Fe(\))972 695 y Fd(\000)p
Fe(1)1072 677 y Fr(+)1138 583 y Ff(\022)1195 632 y Fr(2)p 1195
662 34 3 v 1200 723 a Fk(t)1251 677 y Fr(+)1326 632 y(2)p Fk(p)1392
650 y Fe(1)1422 632 y Fk(\032\034)p 1326 662 168 3 v 1382 723
a(t)1406 703 y Fe(2)1501 583 y Ff(\023)1550 677 y Fk(\014)1592
643 y Fe(\()p Fh(P)g Fe(\))1588 704 y Fd(\000)1637 686 y Fc(1)p
1637 694 23 3 v 1637 720 a(2)1677 677 y Fi(\000)557 769 y Ff(\022)623
818 y Fk(\032)p 614 848 54 3 v 614 909 a(t)638 889 y Fe(2)675
769 y Ff(\023)724 863 y Fk(b)753 828 y Fe(\()p Fh(Q)p Fe(\))753
881 y Fd(\000)p Fe(1)854 863 y Fr(+)921 769 y Ff(\022)978 818
y Fr(2)p 978 848 34 3 v 983 909 a Fk(t)1034 863 y Fr(+)1108
818 y(2)t Ff(e)-41 b Fk(p)1174 836 y Fe(1)1204 818 y Fk(\032\034)p
1108 848 168 3 v 1165 909 a(t)1189 889 y Fe(2)1283 769 y Ff(\023)1332
863 y Fk(\014)1374 828 y Fe(\()p Fh(Q)p Fe(\))1370 890 y Fd(\000)1419
872 y Fc(1)p 1419 880 23 3 v 1419 906 a(2)1475 863 y Fr(+)14
b Fi(\001)d(\001)g(\001)197 1049 y Fk(B)s Fr([)p Fk(\033)306
1067 y Fd(\003)338 1049 y Fr(\()384 1004 y Fk(@)p 372 1034
63 3 v 372 1095 a(@)t(t)442 1049 y Fr(\)])19 b(=)c Fi(\000)638
955 y Ff(\022)695 1004 y Fr(2)p 695 1034 34 3 v 700 1095 a
Fk(t)736 955 y Ff(\023)785 1049 y Fk(b)814 1014 y Fe(\()p Fh(P)7
b Fe(\))814 1067 y Fd(\000)p Fe(1)914 1049 y Fr(+)980 955 y
Ff(\022)1037 1004 y Fr(2)p Fk(\034)p 1037 1034 70 3 v 1060
1095 a(t)1130 1049 y Fr(+)1204 1004 y(4)p Fk(p)1270 1022 y
Fe(1)1301 1004 y Fk(\034)p 1204 1034 133 3 v 1259 1095 a(t)1345
955 y Ff(\023)1394 1049 y Fk(\014)1436 1014 y Fe(\()p Fh(P)g
Fe(\))1432 1076 y Fd(\000)1481 1058 y Fc(1)p 1481 1066 23 3
v 1481 1092 a(2)1536 1049 y Fr(+)14 b Fi(\001)d(\001)g(\001)197
1235 y Fk(B)s Fr([)p Fk(\033)306 1253 y Fd(\003)338 1235 y
Fr(\()384 1190 y Fk(@)p 372 1220 63 3 v 372 1282 a(@)p 411
1233 24 3 v 4 w(t)442 1235 y Fr(\)])19 b(=)c Fi(\000)638 1141
y Ff(\022)695 1190 y Fr(2)p 695 1220 34 3 v 700 1233 24 3 v
700 1282 a Fk(t)736 1141 y Ff(\023)783 1217 y Fr(\026)785 1235
y Fk(b)814 1200 y Fe(\()p Fh(P)7 b Fe(\))814 1253 y Fd(\000)p
Fe(1)914 1235 y Fr(+)14 b Fi(\001)d(\001)g(\001)185 1416 y
Fk(B)s Fr([)p Fk(\033)294 1434 y Fd(\003)325 1416 y Fr(\()377
1371 y Fk(@)p 359 1400 76 3 v 359 1461 a(@)t(\034)442 1416
y Fr(\)])19 b(=)c Fi(\000)638 1362 y Ff(\000)668 1416 y Fk(\034)705
1362 y Ff(\001)735 1416 y Fk(b)764 1381 y Fe(\()p Fh(P)7 b
Fe(\))764 1433 y Fd(\000)p Fe(1)865 1416 y Fi(\000)931 1362
y Ff(\000)961 1416 y Fr(2)994 1362 y Ff(\001)1025 1416 y Fk(\014)1067
1381 y Fe(\()p Fh(P)g Fe(\))1063 1443 y Fd(\000)1112 1425 y
Fc(1)p 1112 1433 23 3 v 1112 1459 a(2)1167 1416 y Fr(+)14 b
Fi(\001)d(\001)g(\001)2463 854 y Fr(\(2)p Fk(:)p Fr(2)q(\))0
1548 y(where)26 b(the)g(dots)h(represen)n(t)e(higher)i(terms)f(\()p
Fk(b)1209 1566 y Fh(n)1245 1548 y Fk(;)11 b(n)26 b Fi(\025)h
Fr(0)g(and)g Fk(\014)1651 1566 y Fh(n)1686 1548 y Fk(;)11 b(n)27
b Fi(\025)1868 1522 y Fe(1)p 1868 1533 27 3 v 1868 1571 a(2)1903
1548 y Fr(\))f(that)h(annihilate)h(strong)0 1648 y(ph)n(ysical)23
b(states)f(\(SPSs\).)133 1749 y(T)-6 b(o)25 b(insert)f(a)h(SPS)f(at)h
Fk(P)33 b Fr(and)24 b(at)h Fk(Q)p Fr(,)f(there)g(are)g(t)n(w)n(o)g(p)r
(ossible)h(con)n(tributions)g(to)g(the)e(measure)0 1849 y(from)f(the)f(ab)r
(o)n(v)n(e)i(insertions:)195 1973 y Fk(b)224 1939 y Fe(\()p
Fh(P)7 b Fe(\))224 1991 y Fd(\000)p Fe(1)307 1956 y Fr(\026)309
1973 y Fk(b)338 1939 y Fe(\()p Fh(P)g Fe(\))338 1991 y Fd(\000)p
Fe(1)423 1973 y Fk(\016)s Fr([)p Fk(\014)516 1939 y Fe(\()p
Fh(P)g Fe(\))512 2000 y Fd(\000)561 1982 y Fc(1)p 561 1990
23 3 v 561 2016 a(2)601 1973 y Fr(])p Fk(b)648 1939 y Fe(\()p
Fh(Q)p Fe(\))648 1991 y Fd(\000)p Fe(1)732 1956 y Fr(\026)734
1973 y Fk(b)763 1939 y Fe(\()p Fh(Q)p Fe(\))763 1991 y Fd(\000)p
Fe(1)849 1973 y Fk(\016)s Fr([)p Fk(\014)942 1939 y Fe(\()p
Fh(Q)p Fe(\))938 2000 y Fd(\000)987 1982 y Fc(1)p 987 1990
V 987 2016 a(2)1028 1973 y Fr(])22 b Fk(;)145 b(b)1260 1939
y Fe(\()p Fh(P)7 b Fe(\))1260 1991 y Fd(\000)p Fe(1)1342 1956
y Fr(\026)1345 1973 y Fk(b)1374 1939 y Fe(\()p Fh(P)g Fe(\))1374
1991 y Fd(\000)p Fe(1)1459 1973 y Fk(\016)s Fr([)p Fk(\014)1552
1939 y Fe(\()p Fh(P)g Fe(\))1548 2000 y Fd(\000)1597 1982 y
Fc(1)p 1597 1990 V 1597 2016 a(2)1636 1973 y Fr(])p Fk(b)1683
1939 y Fe(\()p Fh(Q)p Fe(\))1683 1991 y Fd(\000)p Fe(1)1768
1956 y Fr(\026)1770 1973 y Fk(b)1799 1939 y Fe(\()p Fh(Q)p
Fe(\))1799 1991 y Fd(\000)p Fe(1)1885 1973 y Fk(\014)1927 1939
y Fe(\()p Fh(Q)p Fe(\))1923 2000 y Fd(\000)1972 1982 y Fc(1)p
1972 1990 V 1972 2016 a(2)2013 1973 y Fk(\016)2046 1946 y Fd(0)2064
1973 y Fr([)p Fk(\014)2124 1939 y Fe(\()p Fh(Q)p Fe(\))2120
2000 y Fd(\000)2169 1982 y Fc(1)p 2169 1990 V 2169 2016 a(2)2210
1973 y Fr(])22 b Fk(:)195 b Fr(\(2)p Fk(:)p Fr(3)q(\))0 2098
y(Using)24 b(the)f(formal)h(iden)n(ti\014cation)g(of)g Fk(\014)t(\016)1068
2074 y Fd(0)1085 2098 y Fr(\()p Fk(\014)t Fr(\))f(and)g Fi(\000)p
Fk(\016)s Fr(\()p Fk(\014)t Fr(\),)f(w)n(e)h(easily)i(\014nd)e(the)g(con)n
(tribution)h(from)0 2198 y(the)d(insertions)i(in)g(\(2.2)p
 (#equation.2.2) [[165 322 180 334] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f(to)h(b)
r(e)853 2277 y Fk(t)p 820 2307 91 3 v 820 2368 a Fi(j)p Fk(t)p
Fi(j)880 2349 y Fe(6)918 2322 y Fk(b)947 2288 y Fe(\()p Fh(P)7
b Fe(\))947 2340 y Fd(\000)p Fe(1)1030 2305 y Fr(\026)1033
2322 y Fk(b)1062 2288 y Fe(\()p Fh(P)g Fe(\))1062 2340 y Fd(\000)p
Fe(1)1147 2322 y Fk(\016)s Fr([)p Fk(\014)1240 2288 y Fe(\()p
Fh(P)g Fe(\))1236 2349 y Fd(\000)1285 2331 y Fc(1)p 1285 2339
23 3 v 1285 2365 a(2)1324 2322 y Fr(])p Fk(b)1371 2288 y Fe(\()p
Fh(Q)p Fe(\))1371 2340 y Fd(\000)p Fe(1)1456 2305 y Fr(\026)1458
2322 y Fk(b)1487 2288 y Fe(\()p Fh(Q)p Fe(\))1487 2340 y Fd(\000)p
Fe(1)1573 2322 y Fk(\016)s Fr([)p Fk(\014)1666 2288 y Fe(\()p
Fh(Q)p Fe(\))1662 2349 y Fd(\000)1711 2331 y Fc(1)p 1711 2339
V 1711 2365 a(2)1751 2322 y Fr(])p Fk(;)676 b Fr(\(2)p Fk(:)p
Fr(4)q(\))0 2465 y(where)21 b(there)g(has)h(b)r(een)f(some)h(cancellation)g
(b)r(et)n(w)n(een)e(the)i(t)n(w)n(o)g(di\013eren)n(t)f(con)n(tributions)i(in)
f(\(2.3)p  (#equation.2.3) [[510 274 525 286] [1 1 1 [3 3]] [0 0 1]] pdfm 
(\).)133 2565 y(Th)n(us)g(w)n(e)g(ha)n(v)n(e)117 2597 y Ff(Z)183
2634 y(\002)215 2687 y Fr(d)s Fk(r)16 b Fi(^)k Fr(d)p 405 2644
32 3 v 3 w Fk(r)c Fi(^)k Fr(d)s Fk(t)14 b Fi(^)20 b Fr(d)p
697 2639 24 3 v 3 w Fk(t)p Fi(j)5 b Fr(d)s Fk(\032)g Fr(d)s
Fk(\034)900 2634 y Ff(\003)948 2687 y Fi(h)p Fr(\006)p Fk(;)11
b(\033)1089 2705 y Fh(P)1134 2687 y Fk(;)g(\033)1201 2705 y
Fh(Q)1247 2687 y Fi(j)g(\001)g(\001)g(\001)i Fk(B)s Fr([)p
Fk(\033)1474 2705 y Fd(\003)1505 2687 y Fr(\()1555 2643 y Fk(@)p
1539 2672 71 3 v 1539 2733 a(@)t(r)1618 2687 y Fr(\)])p Fk(B)s
Fr([)p Fk(\033)1771 2705 y Fd(\003)1803 2687 y Fr(\()1852 2643
y Fk(@)p 1837 2672 V 1837 2733 a(@)p 1876 2689 32 3 v 4 w(r)1915
2687 y Fr(\)])p Fk(\016)1992 2634 y Ff(\002)2019 2687 y Fk(B)s
Fr([)p Fk(\033)2128 2705 y Fd(\003)2160 2687 y Fr(\()2211 2643
y Fk(@)p 2194 2672 74 3 v 2194 2733 a(@)t(\032)2275 2687 y
Fr(\)])2319 2634 y Ff(\003)331 2864 y Fi(\002)h Fk(B)s Fr([)p
Fk(\033)506 2882 y Fd(\003)538 2864 y Fr(\()583 2819 y Fk(@)p
572 2848 63 3 v 572 2909 a(@)t(t)642 2864 y Fr(\)])p Fk(B)s
Fr([)p Fk(\033)795 2882 y Fd(\003)827 2864 y Fr(\()873 2819
y Fk(@)p 861 2848 V 861 2910 a(@)p 900 2862 24 3 v 4 w(t)932
2864 y Fr(\)])p Fk(\016)1009 2810 y Ff(\002)1036 2864 y Fk(B)s
Fr([)p Fk(\033)1145 2882 y Fd(\003)1177 2864 y Fr(\()1229 2819
y Fk(@)p 1211 2848 76 3 v 1211 2909 a(@)t(\034)1294 2864 y
Fr(\)])1338 2810 y Ff(\003)1366 2864 y Fi(j)p Fk( )r Fi(i)1455
2836 y Fh(P)1515 2864 y Fi(\012)g(j)p Fk( )r Fi(i)1670 2836
y Fh(Q)1735 2864 y Fr(=)117 2940 y Ff(Z)183 2976 y(\002)215
3030 y Fr(d)s Fk(r)i Fi(^)k Fr(d)p 405 2986 32 3 v 3 w Fk(r)c
Fi(^)k Fr(d)s Fk(t)14 b Fi(^)20 b Fr(d)p 697 2981 24 3 v 3
w Fk(t)p Fi(j)5 b Fr(d)s Fk(\032)g Fr(d)s Fk(\034)900 2976
y Ff(\003)948 3030 y Fi(h)p Fr(\006)p Fk(;)11 b(\033)1089 3048
y Fh(P)1123 3055 y Fc(0)1153 3030 y Fk(;)g(\033)1220 3048 y
Fh(Q)1262 3055 y Fc(0)1291 3030 y Fi(j)331 3201 y(\002)397
3148 y Ff(\000)427 3201 y Fr(1)k(+)g Fk(\032\034)8 b(tL)682
3167 y Fe(\()p Fh(P)f Fe(\))682 3219 y Fd(\000)p Fe(1)782 3201
y Fr(+)15 b Fk(\032\034)920 3148 y Ff(\000)958 3157 y Fr(2)p
Fk(p)1024 3174 y Fe(2)1069 3157 y Fi(\000)g Fk(p)1169 3174
y Fe(1)p 958 3186 241 3 v 1067 3247 a Fk(t)1207 3148 y Ff(\001)1237
3201 y Fk(L)1282 3167 y Fe(\()p Fh(P)7 b Fe(\))1282 3219 y(0)1383
3201 y Fr(+)15 b(2)1483 3148 y Ff(\000)1513 3201 y Fk(\032)g
Fi(\000)g Fk(\034)8 b(t)1690 3148 y Ff(\001)1720 3201 y Fk(G)1772
3167 y Fe(\()p Fh(P)f Fe(\))1772 3228 y Fd(\000)1821 3211 y
Fc(1)p 1821 3219 23 3 v 1821 3244 a(2)1858 3148 y Ff(\001)331
3366 y Fi(\002)397 3313 y Ff(\000)427 3366 y Fr(1)15 b Fi(\000)550
3322 y Fk(\032\034)c Ff(e)-40 b Fk(p)654 3339 y Fe(1)p 550
3351 135 3 v 605 3412 a Fk(t)692 3366 y(L)737 3332 y Fe(\()p
Fh(Q)p Fe(\))737 3384 y(0)838 3366 y Fr(+)15 b(2)p Fk(\032G)1024
3332 y Fe(\()p Fh(Q)p Fe(\))1024 3393 y Fd(\000)1074 3376 y
Fc(1)p 1074 3384 23 3 v 1074 3409 a(2)1111 3313 y Ff(\001)1142
3366 y Fk(t)1166 3339 y Fe(2\()p Fh(L)1249 3315 y Fc(\()p Fb(P)6
b Fc(\))1249 3355 y(0)1324 3339 y Fe(+)p Fh(L)1401 3315 y Fc(\()p
Fb(Q)p Fc(\))1401 3355 y(0)1476 3339 y Fe(\))p 1500 3318 24
3 v 1500 3366 a Fk(t)1524 3329 y Fe(2\()1577 3317 y(\026)1571
3329 y Fh(L)1607 3304 y Fc(\()p Fb(P)h Fc(\))1607 3345 y(0)1682
3329 y Fe(+)1729 3317 y(\026)1723 3329 y Fh(L)1759 3304 y Fc(\()p
Fb(Q)p Fc(\))1759 3345 y(0)1834 3329 y Fe(\))331 3524 y Fi(\002)397
3471 y Ff(\000)469 3479 y Fk(t)p 435 3509 91 3 v 435 3570 a
Fi(j)p Fk(t)p Fi(j)495 3551 y Fe(6)534 3471 y Ff(\001)564 3524
y Fk(b)593 3490 y Fe(\()p Fh(P)g Fe(\))593 3542 y Fd(\000)p
Fe(1)676 3507 y Fr(\026)679 3524 y Fk(b)708 3490 y Fe(\()p
Fh(P)g Fe(\))708 3542 y Fd(\000)p Fe(1)793 3524 y Fk(\016)s
Fr([)p Fk(\014)886 3490 y Fe(\()p Fh(P)g Fe(\))882 3551 y Fd(\000)931
3534 y Fc(1)p 931 3542 23 3 v 931 3567 a(2)970 3524 y Fr(])p
Fk(b)1017 3490 y Fe(\()p Fh(Q)p Fe(\))1017 3542 y Fd(\000)p
Fe(1)1102 3507 y Fr(\026)1104 3524 y Fk(b)1133 3490 y Fe(\()p
Fh(Q)p Fe(\))1133 3542 y Fd(\000)p Fe(1)1219 3524 y Fk(\016)s
Fr([)p Fk(\014)1312 3490 y Fe(\()p Fh(Q)p Fe(\))1308 3551 y
Fd(\000)1357 3534 y Fc(1)p 1357 3542 V 1357 3567 a(2)1397 3524
y Fr(])p Fi(j)p Fk( )r Fi(i)1504 3497 y Fh(P)1565 3524 y Fi(\012)14
b(j)p Fk( )r Fi(i)1720 3497 y Fh(Q)1766 3524 y Fk(:)2463 3108
y Fr(\(2)p Fk(:)p Fr(5)q(\))1283 3719 y(7)p eop
%%Page: 8 9
8 8 bop 0 66 a Fr(The)21 b(state)h Fk(b)328 84 y Fd(\000)p
Fe(1)396 49 y Fr(\026)399 66 y Fk(b)428 84 y Fd(\000)p Fe(1)499
66 y Fk(\016)s Fr([)p Fk(\014)588 84 y Fd(\000)636 67 y Fc(1)p
636 75 23 3 v 636 100 a(2)670 66 y Fr(])p Fi(j)p Fk( )r Fi(i)h
Fr(has)g Fk(L)964 84 y Fe(0)1012 66 y Fr(=)1090 40 y Fe(1)p
1090 51 27 3 v 1090 89 a(2)1147 66 y Fr(and)1284 50 y(\026)1276
66 y Fk(L)1321 84 y Fe(0)1369 66 y Fr(=)18 b(1)23 b(and)f(th)n(us,)g(pic)n
(king)h(o\013)f(the)f(term)g(prop)r(or-)0 171 y(tional)i Fk(\032\034)8
b Fr(,)23 b(\(2.5)p
 (#equation.2.5) [[131 687 146 699] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f(b)r
(ecomes)672 254 y Ff(Z)739 291 y(\002)771 345 y Fr(d)s Fk(r)16
b Fi(^)k Fr(d)p 961 301 32 3 v 3 w Fk(r)c Fi(^)k Fr(d)s Fk(t)14
b Fi(^)20 b Fr(d)p 1253 296 24 3 v 3 w Fk(t)p Fi(j)5 b Fr(d)s
Fk(\032)g Fr(d)s Fk(\034)1456 291 y Ff(\003)377 525 y Fi(h)p
Fr(\006)p Fk(;)11 b(\033)518 543 y Fh(P)552 550 y Fc(0)582
525 y Fk(;)g(\033)649 543 y Fh(Q)691 550 y Fc(0)720 525 y Fi(j)q
Fk(\032\034)810 431 y Ff(\022)859 525 y Fk(tL)928 490 y Fe(\()p
Fh(P)c Fe(\))928 542 y Fd(\000)p Fe(1)1028 525 y Fr(+)1103
480 y(\(2)p Fk(a)1197 498 y Fe(2)1241 480 y Fi(\000)15 b Fk(a)1343
498 y Fe(1)1388 480 y Fi(\000)e Ff(e)-36 b Fk(a)1489 498 y
Fe(1)1519 480 y Fr(\)\(1)14 b Fi(\000)h Fk(f)7 b Fr(\))p 1103
509 648 3 v 1398 570 a(2)p Fk(t)1773 525 y Fr(+)15 b(4)p Fk(tG)1949
490 y Fe(\()p Fh(Q)p Fe(\))1949 552 y Fd(\000)1998 534 y Fc(1)p
1998 542 23 3 v 1998 568 a(2)2036 525 y Fk(G)2088 490 y Fe(\()p
Fh(P)7 b Fe(\))2088 552 y Fd(\000)2137 534 y Fc(1)p 2137 542
V 2137 568 a(2)2174 431 y Ff(\023)754 713 y Fi(\002)820 659
y Ff(\000)858 668 y Fi(j)p Fk(t)p Fi(j)918 644 y Fe(2)p 858
698 91 3 v 892 759 a Fk(t)957 659 y Ff(\001)987 713 y Fk(b)1016
679 y Fe(\()p Fh(P)g Fe(\))1016 731 y Fd(\000)p Fe(1)1099 696
y Fr(\026)1102 713 y Fk(b)1131 679 y Fe(\()p Fh(P)g Fe(\))1131
731 y Fd(\000)p Fe(1)1216 713 y Fk(\016)s Fr([)p Fk(\014)1309
679 y Fe(\()p Fh(P)g Fe(\))1305 740 y Fd(\000)1354 722 y Fc(1)p
1354 730 23 3 v 1354 756 a(2)1393 713 y Fr(])p Fk(b)1440 679
y Fe(\()p Fh(Q)p Fe(\))1440 731 y Fd(\000)p Fe(1)1525 696 y
Fr(\026)1527 713 y Fk(b)1556 679 y Fe(\()p Fh(Q)p Fe(\))1556
731 y Fd(\000)p Fe(1)1642 713 y Fk(\016)s Fr([)p Fk(\014)1735
679 y Fe(\()p Fh(Q)p Fe(\))1731 740 y Fd(\000)1780 722 y Fc(1)p
1780 730 V 1780 756 a(2)1820 713 y Fr(])p Fi(j)p Fk( )r Fi(i)1927
686 y Fh(P)1988 713 y Fi(\012)14 b(j)p Fk( )r Fi(i)2143 686
y Fh(Q)2189 713 y Fk(:)2463 523 y Fr(\(2)p Fk(:)p Fr(6)q(\))0
876 y(A)n(t)26 b(\014rst)g(sigh)n(t,)i(the)d(term)g(prop)r(ortional)i(to)f
(\(1)17 b Fi(\000)g Fk(f)7 b Fr(\))26 b(lo)r(oks)h(lik)n(e)f(bad)g(news,)h
(since)f(it)g(dep)r(ends)f(on)0 980 y(the)k(shap)r(e)f(of)i(the)f(function)g
Fk(f)7 b Fr(,)31 b(and)e(hence)f(on)h(the)g(details)h(of)g(the)e(co)r
(ordinate)h(family)h(c)n(hosen.)0 1084 y(Actually)-6 b(,)29
b(the)d(co)r(e\016cien)n(ts)f Fk(a)782 1102 y Fh(i)831 1084
y Fr(and)h Ff(e)-36 b Fk(a)1000 1102 y Fh(i)1048 1084 y Fr(are)27
b(not)f(to)h(b)r(e)f(tak)n(en)g(to)g(b)r(e)g(arbitrary)-6 b(.)45
b(As)26 b(discussed)h(in)0 1188 y([7)p
 (#reference.7) [[75 504 81 516] [1 1 1 [3 3]] [0 0 1]] pdfm (],)c(a)e
(unique)g(c)n(hoice)f(for)h(the)f(co)r(e\016cien)n(ts)g(is)i(singled)g(out)e
(b)n(y)h(demanding)g(that)f(dilaton)i(insertions)0 1292 y(b)r(eha)n(v)n(e)j
(correctly)-6 b(.)40 b(There)25 b(one)g(found)h(that)f Fk(a)1224
1310 y Fe(1)1279 1292 y Fr(=)f Fk(a)1390 1310 y Fe(2)1444 1292
y Fr(=)f Ff(e)-36 b Fk(a)1555 1310 y Fe(1)1609 1292 y Fr(=)24
b Fi(\000)o Ff(e)-36 b Fk(a)1772 1310 y Fe(2)1826 1292 y Fr(=)24
b Fi(\000)p Fr(1)p Fk(=)p Fr(4.)41 b(\(Actually)-6 b(,)28 b(the)c
Ff(e)-36 b Fk(a)2578 1310 y Fh(i)0 1396 y Fr(w)n(ere)18 b(not)h(determined)e
(in)i([7)p  (#reference.7) [[198 467 204 479] [1 1 1 [3 3]] [0 0 1]] pdfm 
(],)i(but)e(a)g(calculation)h(exactly)e(analogous)j(to)e(the)f(ones)h(presen)
n(ted)e(there)0 1500 y(\014xes)24 b(these)f(co)r(e\016cien)n(ts)g(as)h(w)n
(ell.\))36 b(With)25 b(these)e(v)l(alues)h(for)h(the)e(co)r(e\016cien)n(ts,)h
(the)f(o\013ending)h(term)0 1604 y(v)l(anishes,)e(and)e(all)h(dep)r(endence)d
(on)i(the)f(shap)r(e)h(of)h(the)e(co)r(ordinate)h(family)h(c)n(hosen)f(drops)
g(out.)30 b Fl(A)n(ny)0 1709 y Fr(co)r(ordinate)19 b(family)i(whic)n(h)f(in)n
(terp)r(olates)g(to)f(the)h(asymptotic)f(b)r(eha)n(vior)h(\(1.5)p
 (#equation.1.5) [[414 410 429 422] [1 1 1 [3 3]] [0 0 1]] pdfm (\),\(1.6)p
 (#equation.1.6) [[441 410 456 422] [1 1 1 [3 3]] [0 0 1]] pdfm (\))h(giv)n
(es)g(the)e(same)0 1813 y(result)j(for)h(the)e(measure)131
1880 y Ff(Z)197 1916 y(\002)230 1970 y Fr(d)s Fk(r)16 b Fi(^)k
Fr(d)p 420 1926 32 3 v 3 w Fk(r)c Fi(^)j Fr(d)s Fk(t)c Fi(^)k
Fr(d)p 711 1921 24 3 v 3 w Fk(t)p Fi(j)5 b Fr(d)s Fk(\032)g
Fr(d)s Fk(\034)914 1916 y Ff(\003)963 1970 y Fi(h)p Fr(\006)p
Fk(;)11 b(\033)1104 1988 y Fh(P)1138 1995 y Fc(0)1167 1970
y Fk(;)g(\033)1234 1988 y Fh(Q)1276 1995 y Fc(0)1306 1970 y
Fi(j)p Fk(\032\034)1395 1916 y Ff(\000)1425 1970 y Fk(L)1470
1936 y Fe(\()p Fh(P)c Fe(\))1470 1988 y Fd(\000)p Fe(1)1571
1970 y Fr(+)14 b(4)p Fk(G)1722 1936 y Fe(\()p Fh(Q)p Fe(\))1722
1997 y Fd(\000)1772 1979 y Fc(1)p 1772 1987 23 3 v 1772 2013
a(2)1810 1970 y Fk(G)1862 1936 y Fe(\()p Fh(P)7 b Fe(\))1862
1997 y Fd(\000)1911 1979 y Fc(1)p 1911 1987 V 1911 2013 a(2)1948
1916 y Ff(\001)955 2123 y Fi(\002)15 b(j)p Fk(t)p Fi(j)1082
2096 y Fe(2)1112 2123 y Fk(b)1141 2089 y Fe(\()p Fh(P)7 b Fe(\))1141
2141 y Fd(\000)p Fe(1)1224 2106 y Fr(\026)1227 2123 y Fk(b)1256
2089 y Fe(\()p Fh(P)g Fe(\))1256 2141 y Fd(\000)p Fe(1)1341
2123 y Fk(\016)s Fr([)p Fk(\014)1434 2089 y Fe(\()p Fh(P)g
Fe(\))1430 2150 y Fd(\000)1479 2132 y Fc(1)p 1479 2140 V 1479
2166 a(2)1518 2123 y Fr(])p Fk(b)1565 2089 y Fe(\()p Fh(Q)p
Fe(\))1565 2141 y Fd(\000)p Fe(1)1650 2106 y Fr(\026)1652 2123
y Fk(b)1681 2089 y Fe(\()p Fh(Q)p Fe(\))1681 2141 y Fd(\000)p
Fe(1)1767 2123 y Fk(\016)s Fr([)p Fk(\014)1860 2089 y Fe(\()p
Fh(Q)p Fe(\))1856 2150 y Fd(\000)1905 2132 y Fc(1)p 1905 2140
V 1905 2166 a(2)1945 2123 y Fr(])p Fi(j)p Fk( )r Fi(i)2052
2096 y Fh(P)2113 2123 y Fi(\012)14 b(j)p Fk( )r Fi(i)2268 2096
y Fh(Q)2314 2123 y Fk(:)2463 2039 y Fr(\(2)p Fk(:)p Fr(7)q(\))133
2279 y(Our)23 b(demand)g(that)h(the)f(co)r(ordinate)g(family)i(exhibit)f(a)g
(certain)f(asymptotic)h(b)r(eha)n(vior)g(as)g(w)n(e)0 2383
y(approac)n(h)18 b(the)e(b)r(oundary)i(of)g(mo)r(duli)f(space)g(has)h(b)r
(oth)f(a)h(ph)n(ysical)h(motiv)l(ation)f(\(it)f(giv)n(es)i(the)d(correct)0
2487 y(b)r(eha)n(vior)i(for)g(dilaton)g(correlation)g(functions\))f(and)h(a)f
(mathematical)g(motiv)l(ation.)29 b(When)17 b(w)n(e)g(think)0
2592 y(ab)r(out)26 b(the)f(plum)n(bing)h(construction)g(as)g(gluing)h(a)f
(certain)g Fl(\014xtur)m(e)f Fr({)h(a)g(3-punctured)e(sphere)h
Fl(with)0 2696 y Fr(co)r(ordinates)j({)g(on)n(to)g(the)f(rest)h(of)g(the)f
(surface,)i(it)g(mak)n(es)e(sense)g(to)h(demand)f(that)g(the)h(resulting)0
2800 y(sewn)h(surface)g(b)r(e)f(the)h(same,)h(regardless)g(of)f(whic)n(h)g
(puncture)f(w)n(e)h(c)n(hose)g(to)g(attac)n(h)g(to)g(the)f(rest)0
2904 y(of)k(the)f(surface.)57 b(In)32 b(other)f(w)n(ords,)j(w)n(e)d(demand)g
(in)n(v)l(ariance)h(under)e(\(a)i(subgroup)g(of)5 b(\))32 b(m\033bius)0
3008 y(transformations)24 b(of)f(the)f(3-punctures)g(sphere.)32
b(As)23 b(in)g(the)f(b)r(osonic)h(string)h([6)p
 (#reference.6) [[434 177 440 189] [1 1 1 [3 3]] [0 0 1]] pdfm (,16)p
 (#reference.16) [[443 177 455 189] [1 1 1 [3 3]] [0 0 1]] pdfm (],)h(this)f
(leads)f(to)g(a)0 3112 y(de\014nite)f(c)n(hoice)h(for)h(the)e(\014rst)h(few)g
(co)r(e\016cien)n(ts)f(in)i(\(1.5)p
 (#equation.1.5) [[316 158 331 170] [1 1 1 [3 3]] [0 0 1]] pdfm (\),\(1.6)p
 (#equation.1.6) [[344 158 359 170] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)35
b(Still,)25 b(one)e(migh)n(t)g(b)r(e)f(puzzled)g(that)0 3216
y(the)d(measure)g(formed)g(with)h(SPSs)g(should)h(b)r(e)e(sensitiv)n(e)h(at)g
(all)h(to)f(the)g(asymptotic)f(b)r(eha)n(vior)h(of)h(the)0
3320 y(co)r(ordinate)h(family)-6 b(.)31 b(W)-6 b(e)22 b(will)i(defer)d(a)i
(discussion)g(of)f(this)h(p)r(oin)n(t)f(to)g(section)g(3.)133
3425 y(Of)k(the)g(t)n(w)n(o)h(terms)f(in)h(\(2.7)p
 (#equation.2.7) [[212 102 227 114] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)h(the)e
Fk(G)1105 3390 y Fe(\()p Fh(Q)p Fe(\))1105 3452 y Fd(\000)1155
3434 y Fc(1)p 1155 3442 V 1155 3468 a(2)1192 3425 y Fk(G)1244
3390 y Fe(\()p Fh(P)7 b Fe(\))1244 3452 y Fd(\000)1294 3434
y Fc(1)p 1294 3442 V 1294 3468 a(2)1357 3425 y Fr(is)27 b(precisely)f(what)h
(is)g(exp)r(ected)d(from)j(a)g(na)-7 b(\177)-26 b(\020v)n(e)0
3544 y(analysis)18 b(of)f(the)f(collision)i(of)f(t)n(w)n(o)g(v)n(ertex)f(op)r
(erators.)28 b(Indeed,)17 b(the)e Fk(G)1766 3509 y Fe(\()p
Fh(P)7 b Fe(\))1766 3571 y Fd(\000)1816 3553 y Fc(1)p 1816
3561 V 1816 3587 a(2)1869 3544 y Fr(com)n(bines)16 b(with)g(the)g
Fk(\016)s Fr([)p Fk(\014)2497 3509 y Fe(\()p Fh(P)7 b Fe(\))2493
3571 y Fd(\000)2542 3553 y Fc(1)p 2542 3561 V 2542 3587 a(2)2582
3544 y Fr(])1283 3719 y(8)p eop
%%Page: 9 10
9 9 bop 0 66 a Fr(to)24 b(form)g(a)g(picture-c)n(hanging)g(op)r(erator,)h
(whic)n(h)f(con)n(v)n(erts)g(the)f(state)h(at)g Fk(P)33 b Fr(from)24
b(the)f(\()p Fi(\000)p Fr(1\))h(to)g(the)0 172 y(\(0\))f(picture)f(\(and)h
(similarly)h(for)g(the)e(state)h(at)g Fk(Q)p Fr(\).)31 b(The)23
b Fk(L)1537 138 y Fe(\()p Fh(P)7 b Fe(\))1537 190 y Fd(\000)p
Fe(1)1646 172 y Fr(term)21 b(is)j(new)e(and,)i(as)g(w)n(e)e(shall)i(see,)0
278 y(pla)n(ys)f(the)e(role)i(of)f(the)g(\\con)n(tact)f(term")g(p)r
(ostulated)h(b)n(y)h(Green)e(and)h(Seib)r(erg.)133 384 y(First,)i(let)f(us)g
(see)g(that)f(it)i(mak)n(es)e(sense)g(b)n(y)i(examining)f(the)f
(transformation)i(of)f(the)f(measure)0 490 y(under)16 b(the)f(co)r(ordinate)h
(transformation)h(in)f(\(1.10)p
 (#equation.1.10) [[280 630 301 642] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)29
b(There)15 b(are)h(three)g(con)n(tributions)g(to)h(the)e(c)n(hange.)0
596 y(The)21 b(\014rst)h(is)g(the)f(ho)n(w)h(the)f(in)n(tegration)i(measure)d
([)5 b(d)t Ff(e)-38 b Fk(r)16 b Fi(^)j Fr(d)p 1513 540 32 3
v 4 w Ff(e)-38 b Fk(r)q Fi(j)5 b Fr(d)i Ff(e)-41 b Fk(\032)p
Fr(])22 b(transforms)g(to)g([)5 b(d)s Fk(t)14 b Fi(^)19 b Fr(d)p
2297 547 24 3 v 3 w Fk(t)o Fi(j)5 b Fr(d)s Fk(\034)j Fr(].)30
b(The)0 702 y(c)n(hange)22 b(of)g(v)l(ariables)i(requires)d(the)h(in)n(tro)r
(duction)g(of)h(a)f(Jacobian)h(\(Berezinian\),)514 917 y([)5
b(d)t Ff(e)-38 b Fk(r)17 b Fi(^)j Fr(d)p 728 861 32 3 v 4 w
Ff(e)-38 b Fk(r)q Fi(j)5 b Fr(d)i Ff(e)-41 b Fk(\032)p Fr(])19
b(=)f(sdet)1092 823 y Ff(\022)1162 847 y Fh(@)p Ff(e)-34 b
Fh(r)p 1162 858 58 3 v 1165 896 a(@)s(t)1305 844 y(@)r Ff(e)e
Fh(\032)p 1305 858 59 3 v 1309 896 a(@)s(t)1162 955 y(@)p Ff(e)i
Fh(r)p 1160 965 62 3 v 1160 1004 a(@)s(\034)1305 951 y(@)r
Ff(e)e Fh(\032)p 1304 965 V 1304 1004 a(@)s(\034)1384 823 y
Ff(\023)1433 917 y Fr(det)1536 823 y Ff(\022)1593 872 y Fk(@)p
1632 816 32 3 v 5 w Ff(e)e Fk(r)p 1593 901 71 3 v 1597 963
a(@)p 1636 915 24 3 v 4 w(t)1672 823 y Ff(\023)1721 917 y Fr([)5
b(d)s Fk(t)14 b Fi(^)20 b Fr(d)p 1926 868 V 3 w Fk(t)o Fi(j)5
b Fr(d)s Fk(\034)j Fr(])p Fk(:)378 b Fr(\(2)p Fk(:)p Fr(8)q(\))0
1131 y(Using)23 b(\(1.10)p
 (#equation.1.10) [[110 514 131 526] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)g(this)
g(results)f(in)692 1330 y([)5 b(d)t Ff(e)-38 b Fk(r)17 b Fi(^)i
Fr(d)p 905 1274 32 3 v 4 w Ff(e)-38 b Fk(r)r Fi(j)5 b Fr(d)i
Ff(e)-41 b Fk(\032)p Fr(])19 b(=)1149 1285 y(4)p Fi(j)p Fk(t)p
Fi(j)1242 1261 y Fe(2)p 1149 1315 124 3 v 1199 1376 a Fk(t)1281
1276 y Ff(\000)1311 1330 y Fr(1)c(+)1433 1285 y Fk(\032\034)p
1433 1315 71 3 v 1440 1376 a Fr(2)p Fk(t)1512 1276 y Ff(\001)1543
1330 y Fr([)5 b(d)s Fk(t)14 b Fi(^)20 b Fr(d)p 1748 1281 24
3 v 3 w Fk(t)p Fi(j)5 b Fr(d)s Fk(\034)j Fr(])p Fk(:)555 b
Fr(\(2)p Fk(:)p Fr(9)q(\))0 1513 y(But)31 b(the)f Fk(B)35 b
Fr(insertions)d(\(2.2)p
 (#equation.2.2) [[195 446 210 458] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f(also)h
(c)n(hange,)h(in)n(tro)r(ducing)f(a)f(comp)r(ensating)f(Jacobian)i(in)f
(\(2.4)p  (#equation.2.4) [[517 446 532 458] [1 1 1 [3 3]] [0 0 1]] pdfm 
(\).)0 1619 y(Finally)-6 b(,)25 b(c)n(hanging)e(v)l(ariables)g(to)f
Fk(t)g Fr(and)g Fk(\034)30 b Fr(also)23 b(c)n(hanges)f(the)f(state)h
Fi(h)p Fr(\006)p Fi(j)p Fr(.)30 b(Equation)23 b(\(1.3)p
 (#equation.1.3) [[471 427 486 439] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g(giv)n
(es)117 1783 y Fi(h)p Fr(\006)p Fk(;)11 b(z)18 b Fi(\000)e
Ff(e)-39 b Fk(r)17 b Fr(+)i Ff(e)-42 b Fk(\032)q(\022)r Fi(j)19
b Fr(=)o Fi(h)p Fr(\006)p Fk(;)11 b(z)18 b Fi(\000)e Ff(e)-38
b Fk(r)r Fi(j)873 1729 y Ff(\000)903 1783 y Fr(1)15 b Fi(\000)g
Fr(2)t Ff(e)-41 b Fk(\032G)1137 1801 y Fd(\000)1187 1783 y
Fc(1)p 1187 1791 23 3 v 1187 1817 a(2)1221 1729 y Ff(\001)1270
1783 y Fr(=)18 b Fi(h)p Fr(\006)p Fk(;)11 b(z)18 b Fi(\000)c
Fr(\()p Fk(r)j Fr(+)d Fk(t)1721 1755 y Fe(2)1751 1783 y Fr(\))g
Fi(\000)h Fk(t\032\034)22 b Fi(\000)15 b Fr(\()p Fk(\032)g
Fi(\000)f Fk(t\034)8 b Fr(\))p Fk(\022)r Fi(j)553 1909 y Fr(=)o
Fi(h)p Fr(\006)p Fk(;)j(z)18 b Fi(\000)d Fr(\()p Fk(r)h Fr(+)f
Fk(t)986 1881 y Fe(2)1015 1909 y Fr(\))p Fi(j)1059 1855 y Ff(\000)1090
1909 y Fr(1)g(+)f Fk(t\032\034)8 b(L)1344 1927 y Fd(\000)p
Fe(1)1431 1909 y Fr(+)14 b(2\()p Fk(\032)h Fi(\000)g Fk(t\034)8
b Fr(\))p Fk(G)1811 1927 y Fd(\000)1860 1909 y Fc(1)p 1860
1917 V 1860 1943 a(2)1893 1855 y Ff(\001)1924 1909 y Fk(:)2430
1848 y Fr(\(2)p Fk(:)p Fr(10)q(\))0 2076 y(In)23 b(this)h(form)n(ula,)h(w)n
(e)e(see)f(the)h(consequences)e(of)j(the)f(non-split)h(nature)f(of)h(\(1.10)p
 (#equation.1.10) [[430 344 451 356] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)35
b(The)22 b Fk(L)2368 2094 y Fd(\000)p Fe(1)2463 2076 y Fr(term)0
2182 y(w)n(ould)30 b(not)f(b)r(e)f(there)g(for)h(some)f(split)i
(transformation.)51 b(So,)32 b(though)c(the)h(measure)e(is)j(p)r(erfectly)0
2288 y(con)n(tin)n(uous,)24 b(the)f(app)r(earance)e(of)j(the)f(extra)f(term)g
(in)i(\(2.7)p  (#equation.2.7) [[332 306 348 318] [1 1 1 [3 3]] [0 0 1]] pdfm 
(\))f(is)h(a)f(consequence)e(of)j(expanding)f(it)h(in)0 2394
y(co)r(ordinates)i(appropriate)g(to)g(the)f(neigh)n(b)r(orho)r(o)r(d)h(of)g
(the)f(b)r(oundary)h(of)g(mo)r(duli)g(space)g(when)f(the)0
2500 y(t)n(w)n(o)d(punctures)f(collide.)133 2606 y(Recall)16
b(that)g(our)g(goal)i(is)e(to)h(p)r(erform)e(an)h(op)r(erator)g(pro)r(duct)g
(expansion,)i(whic)n(h)e(in)g(this)h(con)n(text)0 2712 y(amoun)n(ts)e(to)g
(rewriting)h(the)e(state)h(that)f(arises)i(from)f(the)g(collision)h(of)g(t)n
(w)n(o)g(op)r(erators)f(as)g(the)g(insertion)0 2818 y(of)25
b(a)g(single)g(op)r(erator)f(at)h(the)f(collision)i(p)r(oin)n(t.)38
b(The)24 b(result)g(in)h(Green)f(and)g(Seib)r(erg)g(w)n(as)h(that)g(the)0
2924 y(na)-7 b(\177)-26 b(\020v)n(e)29 b(OPE)g(resulted)g(in)g(a)g(state)g
(that)g(w)n(as)g(not)g(the)g(highest)g(comp)r(onen)n(t)e(of)j(a)f(sup)r
(er\014eld)f(and)0 3030 y(hence)18 b(violated)j(sup)r(ersymmetry)-6
b(.)28 b(Restoring)20 b(sup)r(ersymmetry)e(required)h(adding)i(a)g(con)n
(tact)e(term.)0 3136 y(Ho)n(w)n(ev)n(er,)33 b(as)f(w)n(e)e(will)j(shortly)e
(see,)i(the)d(state)h(that)g(w)n(e)f(\014nd)h Fl(is)h Fr(the)e(highest)h
(comp)r(onen)n(t)e(of)j(a)0 3242 y(sup)r(er\014eld)16 b(inserted)g(at)g(the)g
(collision)j(p)r(oin)n(t,)f(and)f(th)n(us,)h(no)f(con)n(tact)f(term)f(is)i
(needed.)27 b(It)16 b(is)h(precisely)0 3348 y(the)23 b(\\extra")i
Fk(L)402 3366 y Fd(\000)p Fe(1)497 3348 y Fr(term)e(that)h(restores)g(sup)r
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f(what)0 3454 y(\\taking)31 b(the)e(OPE")h(means)f(in)i(the)e(op)r(erator)g
(formalism:)46 b(tak)n(e)30 b(the)f(t)n(w)n(o)i(states)e(at)h
Fk(P)39 b Fr(and)30 b Fk(Q)p Fr(,)0 3560 y(con)n(tract)c(them)f(with)i(the)f
(state)h(asso)r(ciated)g(to)g(the)f(3-punctured)f(sphere,)j(and)f(insert)f
(the)g(state)1283 3719 y(9)p eop
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10 10 bop 0 66 a Fr(that)20 b(results)f(at)h(the)f(nec)n(k)g(where)g(the)g
(3-punctured)f(sphere)h(is)i(joined)f(on)n(to)g(the)f(rest)g(of)h(the)f
(surface)0 166 y(\(Fig.)24 b(1\).)31 b(In)22 b(doing)h(so,)h(it)e(is)h(vital)
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(erconformal)0 266 y(geometry)-6 b(.)28 b(In)18 b(particular,)i(to)e(see)g
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(comp)r(onen)n(t)0 366 y(of)25 b(a)f(sup)r(er\014eld,)h(w)n(e)f(need)f(to)h
(use)g(a)g(con)n(tour-deformation,)h(or)g(more)e(p)r(edan)n(tically)-6
b(,)26 b(to)f(mak)n(e)e(use)0 466 y(of)k(the)f(Borel-am)n(biguit)n(y)i(asso)r
(ciated)e(to)h(the)f(3-punctured)f(sphere.)43 b(F)-6 b(rom)27
b(the)f(Nev)n(eu-Sc)n(h)n(w)n(artz)0 566 y(algebra,)d(w)n(e)f(ha)n(v)n(e)859
666 y Fk(L)904 684 y Fd(\000)p Fe(1)994 666 y Fr(=)c(2)p Fi(f)p
Fk(G)1182 684 y Fd(\000)1233 666 y Fc(1)p 1233 674 23 3 v 1233
700 a(2)1266 666 y Fk(;)11 b(G)1347 684 y Fd(\000)1398 666
y Fc(1)p 1398 674 V 1398 700 a(2)1432 666 y Fi(g)18 b Fr(=)g(4)p
Fk(G)1638 639 y Fe(2)1638 688 y Fd(\000)1688 671 y Fc(1)p 1688
679 V 1688 704 a(2)1722 666 y Fk(:)0 787 y Fr(Th)n(us,)23 b(\(2.7)p
 (#equation.2.7) [[110 576 125 588] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f(b)r
(ecomes)160 833 y Ff(Z)226 870 y(\002)258 924 y Fr(d)s Fk(r)16
b Fi(^)k Fr(d)p 448 880 32 3 v 3 w Fk(r)c Fi(^)k Fr(d)s Fk(t)14
b Fi(^)20 b Fr(d)p 740 875 24 3 v 3 w Fk(t)p Fi(j)5 b Fr(d)s
Fk(\032)g Fr(d)s Fk(\034)943 870 y Ff(\003)991 924 y Fr(4)p
Fk(\032\034)j Fi(j)p Fk(t)p Fi(j)1155 896 y Fe(2)1186 924 y
Fi(h)p Fr(\006)p Fk(;)j(\033)1327 941 y Fh(P)1361 948 y Fc(0)1391
924 y Fk(;)g(\033)1458 941 y Fh(Q)1500 948 y Fc(0)1529 924
y Fi(j)1547 870 y Ff(\000)1578 924 y Fk(G)1630 889 y Fe(\()p
Fh(P)c Fe(\))1630 951 y Fd(\000)1680 933 y Fc(1)p 1680 941
23 3 v 1680 967 a(2)1731 924 y Fr(+)14 b Fk(G)1849 889 y Fe(\()p
Fh(Q)p Fe(\))1849 951 y Fd(\000)1899 933 y Fc(1)p 1899 941
V 1899 967 a(2)1936 870 y Ff(\001)1966 924 y Fk(G)2018 889
y Fe(\()p Fh(P)7 b Fe(\))2018 951 y Fd(\000)2068 933 y Fc(1)p
2068 941 V 2068 967 a(2)984 1077 y Fi(\002)14 b Fk(b)1079 1042
y Fe(\()p Fh(P)7 b Fe(\))1079 1094 y Fd(\000)p Fe(1)1162 1059
y Fr(\026)1165 1077 y Fk(b)1194 1042 y Fe(\()p Fh(P)g Fe(\))1194
1094 y Fd(\000)p Fe(1)1279 1077 y Fk(\016)s Fr([)p Fk(\014)1372
1042 y Fe(\()p Fh(P)g Fe(\))1368 1104 y Fd(\000)1417 1086 y
Fc(1)p 1417 1094 V 1417 1120 a(2)1456 1077 y Fr(])p Fk(b)1503
1042 y Fe(\()p Fh(Q)p Fe(\))1503 1094 y Fd(\000)p Fe(1)1588
1059 y Fr(\026)1590 1077 y Fk(b)1619 1042 y Fe(\()p Fh(Q)p
Fe(\))1619 1094 y Fd(\000)p Fe(1)1705 1077 y Fk(\016)s Fr([)p
Fk(\014)1798 1042 y Fe(\()p Fh(Q)p Fe(\))1794 1104 y Fd(\000)1843
1086 y Fc(1)p 1843 1094 V 1843 1120 a(2)1883 1077 y Fr(])p
Fi(j)p Fk( )r Fi(i)1990 1049 y Fh(P)2051 1077 y Fi(\012)14
b(j)p Fk( )r Fi(i)2206 1049 y Fh(Q)2252 1077 y Fk(:)2430 993
y Fr(\(2)p Fk(:)p Fr(11)q(\))0 1218 y(The)28 b(factor)h(of)g(\()p
Fk(G)500 1184 y Fe(\()p Fh(P)7 b Fe(\))500 1245 y Fd(\000)550
1228 y Fc(1)p 550 1236 V 550 1261 a(2)605 1218 y Fr(+)19 b
Fk(G)728 1184 y Fe(\()p Fh(Q)p Fe(\))728 1245 y Fd(\000)778
1228 y Fc(1)p 778 1236 V 778 1261 a(2)815 1218 y Fr(\))29 b(is)g(equiv)l
(alen)n(t)g(\(up)f(to)h(the)f(Borel-am)n(biguit)n(y)i(asso)r(ciated)f(to)g
(the)0 1337 y(3-punctured)23 b(sphere\))h(to)h(an)f(insertion)i
Fk(G)1118 1303 y Fe(\()p Fh(N)5 b Fe(\))1118 1364 y Fd(\000)1167
1347 y Fc(1)p 1167 1355 V 1167 1380 a(2)1210 1337 y Fr(,)25
b(on)g(the)f(nec)n(k)g(of)h(the)f(sphere)g(that)g(is)i(pinc)n(hing)f(o\013.)0
1437 y(E\013ectiv)n(ely)-6 b(,)20 b(w)n(e)f(can)f(replace)g(the)g(sum)h(of)g
(t)n(w)n(o)g(con)n(tours)g(surrounding)h Fk(P)27 b Fr(and)19
b Fk(Q)g Fr(with)g(one)f(con)n(tour)0 1537 y(surrounding)23
b(b)r(oth)f(and)g(deformed)f(this)h(con)n(tour)g(on)n(to)g(the)g(nec)n(k)f
(\(Fig.)i(2\).)266 2161 y(Fig.)g(2:Con)n(tour)g(manipulations.)133
2315 y(The)c Fk(b)g Fr(and)g Fk(\016)s Fr(\()p Fk(\014)t Fr(\))e(insertions)k
(ha)n(v)n(e)e(remo)n(v)n(ed)f(the)h(ghost)h(dep)r(endence)c(in)k(our)g(SPSs)g
(and)f(w)n(e)g(are)0 2415 y(left)j(with)h(a)f(state)g Fi(j)516
2398 y Ff(e)504 2415 y Fk( )s Fi(i)576 2433 y Fh(M)657 2415
y Fr(in)g(the)f(matter)g(theory)-6 b(,)23 b(inserted)e(on)i(the)e(nec)n(k,)h
(giv)n(en)g(b)n(y)622 2538 y Fi(j)653 2520 y Ff(e)640 2538
y Fk( )t Fi(i)713 2556 y Fh(M)790 2538 y Fr(=)c Fk(G)912 2503
y Fe(\()p Fh(N)5 b Fe(\))912 2565 y Fd(\000)962 2547 y Fc(1)p
962 2555 V 962 2581 a(2)1004 2538 y Fi(j)p Fk(\033)1060 2556
y Fh(N)1111 2538 y Fi(ih)p Fk(\033)1201 2556 y Fh(P)1245 2538
y Fk(;)11 b(\033)1312 2556 y Fh(Q)1357 2538 y Fi(j)p Fk(G)1427
2503 y Fe(\()p Fh(P)c Fe(\))1427 2565 y Fd(\000)1477 2547 y
Fc(1)p 1477 2555 V 1477 2581 a(2)1514 2538 y Fi(j)p Fk( )r
Fi(i)1603 2510 y Fh(P)1603 2556 y(M)1677 2538 y Fi(\012)15
b(j)p Fk( )r Fi(i)1833 2506 y Fh(Q)1833 2557 y(M)1959 2538
y Fk(:)453 b Fr(\(2)p Fk(:)p Fr(12)q(\))0 2661 y(Here)19 b
Fi(j)p Fk(\033)211 2678 y Fh(N)262 2661 y Fi(ih)p Fk(\033)352
2678 y Fh(P)396 2661 y Fk(;)11 b(\033)463 2678 y Fh(Q)509 2661
y Fi(j)21 b Fr(is)g(the)f(state)g(asso)r(ciated)h(to)f(the)g(3-punctured)g
(sphere,)g(view)n(ed)g(as)h(an)g(elemen)n(t)0 2760 y(of)f Fi(H)11
b(\012)f(H)259 2736 y Fd(\003)299 2760 y Fi(\012)g(H)418 2736
y Fd(\003)449 2760 y Fr(.)29 b(Since)19 b Fi(j)697 2743 y Ff(e)685
2760 y Fk( )s Fi(i)757 2778 y Fh(M)836 2760 y Fr(is)h(the)f(highest)h(comp)r
(onen)n(t)e(of)j(a)f(sup)r(er\014eld)f(\()p Fl(i.e.)h Fr(it)g(is)g
Fk(G)2319 2778 y Fd(\000)2369 2761 y Fc(1)p 2369 2769 V 2369
2794 a(2)2423 2760 y Fr(acting)0 2860 y(on)i(the)g(lo)n(w)n(est)g(comp)r
(onen)n(t)f(of)h(a)h(sup)r(er\014eld\),)e(it)i(is)f(sup)r(ersymmetric.)133
2960 y(Th)n(us)e(w)n(e)g(ha)n(v)n(e)h(main)n(tained)f(w)n(orld-sheet)g(sup)r
(ersymmetry)e Fl(without)i Fr(adding)h(an)g(explicit)f(con-)0
3060 y(tact)26 b(term)g(when)g(the)g(t)n(w)n(o)h(v)n(ertex)f(op)r(erators)h
(coincide.)43 b(The)27 b(correction)f(to)g(the)h(\\na)-7 b(\177)-26
b(\020v)n(e")27 b(result)0 3160 y(\(of)20 b(taking)g(the)f(OPE)h(of)g(t)n(w)n
(o)g(\(0\)-picture)f(v)n(ertex)g(op)r(erators\))g(arises)h(not)g(from)f(an)h
(explicit)g(con)n(tact)0 3260 y(term,)h(but)g(rather)h(from)f(a)h(prop)r(er)g
(accoun)n(t)f(of)h(the)f(sup)r(erconformal)g(geometry)g(asso)r(ciated)h(to)g
(the)0 3360 y(collision)31 b(of)f(t)n(w)n(o)g(punctures.)51
b(Since)29 b(the)g(correction)f(is)i(prop)r(ortional)h(to)f
Fk(L)2051 3326 y Fe(\()p Fh(P)7 b Fe(\))2051 3378 y Fd(\000)p
Fe(1)2137 3360 y Fr(,)31 b(it)f(lo)r(oks)h(lik)n(e)f(a)0 3460
y(total)c(deriv)l(ativ)n(e)g(in)g(the)e(mo)r(dulus)h(asso)r(ciated)h(to)f
(the)g(lo)r(cation)h(of)g Fk(P)9 b Fr(.)39 b(Hence)24 b(its)i(e\013ects)e
(can)h(b)r(e)0 3560 y Fl(simulate)m(d)d Fr(b)n(y)g(adding)h(a)f
Fk(\016)s Fr(-function)f(con)n(tact)h(term)e(at)j(the)e(b)r(oundary)h(of)h
(mo)r(duli)f(space.)1267 3719 y(10)p eop
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11 11 bop 0 66 a Fg(3.)34 b(Concluding)28 b(Remarks)133 216
y Fr(One)20 b(sligh)n(tly)i(puzzling)e(feature)g(of)h(our)g(calculation)h(is)
f(that)f(the)g(dep)r(endence)e(of)j(\(2.6)p
 (#equation.2.6) [[483 679 499 691] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g(on)g
(the)0 324 y(co)r(ordinate)h(family)h(used)e(to)h(construct)f(the)h(measure)f
(dropp)r(ed)g(out)h(only)h(for)f(co)r(ordinate)g(families)0
432 y(with)16 b(the)e(correct)h(asymptotic)g(b)r(eha)n(vior.)28
b(In)15 b(the)g(b)r(osonic)h(string,)h(one)f(can)f(pro)n(v)n(e)g(that)h(the)e
(measure)0 540 y(for)20 b(SPSs)g(is)g(completely)e(insensitiv)n(e)i(to)g(the)
f(co)r(ordinate)g(family)h(c)n(hosen.)29 b(But)19 b(the)g(argumen)n(t)f
(rests)0 648 y(on)24 b(the)f(fact)h(that)g(\012)g(is)h(the)e(pullbac)n(k)h
(of)h(a)f(di\013eren)n(tial)g(form)g(on)g Fi(P)1781 666 y Fh(g)r(;n)1885
648 y Fr(whic)n(h,)h(for)f(SPSs,)h(has)g(the)0 756 y(prop)r(ert)n(y)f(that)f
(a\))i(it)f(is)h(constan)n(t)f(along)h(the)e(\014b)r(ers)h(of)g
Fi(P)1510 774 y Fh(g)r(;n)1612 756 y Fi(!)e(M)1780 774 y Fh(g)r(;n)1883
756 y Fr(and)i(b\))g(it)h(annihilates)g(the)0 863 y(v)n(ertical)c
(directions.)30 b(In)21 b(the)g(sup)r(er)f(case,)i(ho)n(w)n(ev)n(er,)f(\012)g
(is)h(in)g(no)f(sense)g(a)g(pullbac)n(k)h(of)f(a)h(di\013eren)n(tial)0
971 y(form)h(on)264 954 y Ff(b)251 971 y Fi(P)297 989 y Fh(g)r(;n)376
971 y Fr(.)33 b(Rather)22 b([17)p
 (#reference.17) [[192 543 204 555] [1 1 1 [3 3]] [0 0 1]] pdfm (],)j(it)e
(is)g(a)g(section)g(of)g(the)f(sheaf)h Fa(ber)34 b Fr(of)23
b(Berezin)f(forms)h(on)2327 954 y Ff(c)2311 971 y Fi(M)2391
989 y Fh(g)r(;n)2493 971 y Fr(and)0 1079 y(has)j(no)g(in)n(v)l(arian)n
(tly-de\014ned)g(precursor)g(\\upstairs")g(on)1509 1062 y Ff(b)1496
1079 y Fi(P)1542 1097 y Fh(g)r(;n)1622 1079 y Fr(.)41 b(So)25
b(the)g(corresp)r(onding)h(argumen)n(t)0 1187 y(cannot)21 b(b)r(e)f(made)h
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(seen)g(that)g(merely)f(demanding)h(the)0 1295 y(correct)h(asymptotic)i(b)r
(eha)n(vior)g(of)g(the)f(co)r(ordinate)h(family)g(is)h(enough)e(in)i(this)f
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(\\shap)r(e")h(of)h(the)e(co)r(ordinate)h(family)g(c)n(hosen.)28
b(P)n(erhaps)16 b(a)g(generalization)g(of)h(the)0 1510 y(argumen)n(ts)22
b(of)h([7)p  (#reference.7) [[147 446 153 458] [1 1 1 [3 3]] [0 0 1]] pdfm 
(])h(w)n(ould)f(allo)n(w)i(this)e(to)f(b)r(e)g(extended)f(globally)k(o)n(v)n
(er)e(mo)r(duli)g(space.)31 b(If)23 b(not,)g(then)0 1618 y(it)h(migh)n(t)f
(still)i(b)r(e)e(the)g(case)g(that)g(these)g(un)n(w)n(an)n(ted)g(con)n
(tributions)h(are)f(actually)h(total)g(deriv)l(ativ)n(es)0
1726 y(in)32 b(the)e(mo)r(duli)i(and,)i(hence,)e(decouple)f(an)n(yw)n(a)n(y)
-6 b(,)35 b(although)d(that)f(is)h(hard)g(to)f(see)g(explicitly)h(in)0
1834 y(this)23 b(particular)h(calculation.)33 b(Finally)-6
b(,)26 b(let)d(us)g(note)g(that)g(the)f(p)r(ossible)i(dep)r(endence)c(of)j
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(raised)0 2050 y(in)k(the)f(literature)h(\(see)f([18)p
 (#reference.18) [[188 349 200 361] [1 1 1 [3 3]] [0 0 1]] pdfm (,19)p
 (#reference.19) [[203 349 215 361] [1 1 1 [3 3]] [0 0 1]] pdfm (])j(and)e
(references)e(therein\))h(a)h(few)g(y)n(ears)g(ago.)39 b(F)-6
b(or)26 b(the)e(in)n(tegration)0 2157 y(am)n(biguit)n(y)j(w)n(as)g(not)f
(really)h(an)g(am)n(biguit)n(y)g(in)g(the)e(measure)g([20)p
 (#reference.20) [[369 330 381 342] [1 1 1 [3 3]] [0 0 1]] pdfm (],)30
b(but)c(rather,)h(an)g(am)n(biguit)n(y)g(in)0 2265 y(the)21
b(region)h(of)g(in)n(tegration.)31 b(Here)20 b(the)h(region)h(of)g(in)n
(tegration)h(\(de\014ned)d(b)n(y)i(some)f(cuto\013)f Fi(j)p
Fk(t)p Fi(j)g Fk(>)e(\017)p Fr(\))j(is)0 2373 y(the)f(same;)h(it)g(is)h
(really)f(the)f(measure)g(that)g(su\013ers)h(from)f(a)h(p)r(oten)n(tial)g(am)
n(biguit)n(y)-6 b(.)31 b(W)-6 b(e)21 b(b)r(eliev)n(e)f(that)0
2481 y(there)c(is,)j(in)e(fact,)i(no)e(am)n(biguit)n(y)g(so)h(long)f(as)h
(one)e(restricts)h(oneself)g(to)g(co)r(ordinate)g(families)h(with)f(the)0
2589 y(correct)i(asymptotic)g(b)r(eha)n(vior)h(at)g(in\014nit)n(y)h(\()p
Fl(i.e.)f Fr(co)r(ordinate)f(families)i(whic)n(h)f(also)h(giv)n(e)f(the)f
(correct)0 2697 y(dilaton)k(insertions\).)31 b(A)21 b(more)h(careful)g(in)n
(v)n(estigation)h(of)g(this)g(question)f(w)n(ould)g(b)r(e)g(desirable.)133
2805 y(What)g(are)f(the)g(implications)i(of)f(our)g(results)g(for)g
(calculating)g(string)h(scattering)e(amplitudes?)0 2913 y(The)j(simplest)g
(scattering)f(amplitude)h(where)f(these)g(issues)i(arise)f(is)h(the)f(4-p)r
(oin)n(t)g(function)g(on)h(the)0 3021 y(sphere.)259 3004 y
Ff(c)243 3021 y Fi(M)323 3039 y Fe(0)p Fh(;)p Fe(4)421 3021
y Fr(has)h(dimension)f(1)p Fi(j)p Fr(2,)j(and)e(the)f(b)r(oundary)g(of)1638
3004 y Ff(c)1622 3021 y Fi(M)1702 3039 y Fe(0)p Fh(;)p Fe(4)1800
3021 y Fr(consists)h(of)g(three)e(divisors)j(of)0 3129 y(dimension)19
b(0)p Fi(j)p Fr(2.)30 b(Eac)n(h)18 b(b)r(oundary)h(comp)r(onen)n(t)f(corresp)
r(onds)g(to)h(the)f(sphere)g(pinc)n(hing)h(in)h(t)n(w)n(o,)g(with)0
3237 y(t)n(w)n(o)f(punctures)f(landing)i(on)g(either)e(side)h(of)h(the)e
(pinc)n(h.)29 b(When)19 b(all)h(of)g(the)e(punctures)g(are)h(far)g(apart,)0
3344 y(w)n(e)27 b(can)g(c)n(ho)r(ose)g(co)r(ordinates)g(for)h(sup)r(ermo)r
(duli)f(space)g(b)n(y)h(sa)n(ying)g(that)g(the)e(four)i(punctures)e(are)0
3452 y(lo)r(cated)j(at)h(\(0)p Fk(;)11 b Fr(0\))31 b(\(1)p
Fk(;)11 b(\032)627 3470 y Fe(1)658 3452 y Fr(\))p Fk(;)41 b
Fr(\()p Fk(r)n(;)11 b Fr(0\))30 b(and)g(\()p Fi(1)p Fk(;)11
b(\032)1207 3470 y Fe(2)1238 3452 y Fr(\).)53 b(W)-6 b(e)30
b(also)h(demand)e(that)h(the)f(lo)r(cal)h(co)r(ordinate)0 3560
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(somewhere)e([20)p
 (#reference.20) [[220 591 232 603] [1 1 1 [3 3]] [0 0 1]] pdfm (].)52
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810 y(nonsplit)i(transformation)f(manifests)g(itself)h(as)f(a)g(correction)f
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b(T)-6 b(o)26 b(it,)i(Green)d(and)h(Seib)r(erg)f(w)n(ere)0
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3454 y([20])57 b(P)-6 b(.)22 b(Nelson,)f(\\Lectures)f(on)h(sup)r(ermanifolds)
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eop
%%Trailer
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userdict /end-hook known{end-hook}if
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end