%!PS-Adobe-3.0 %%Creator: dvips 5.51 Copyright 1986, 1993 Radical Eye Software %%Title: contact.dvi %%CreationDate: Tue Nov 1 16:26:57 1994 %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: ./dvips -o contact.ps contact.dvi %DVIPSSource: TeX output 1994.09.28:1334 %%BeginProlog %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N} B /TR{translate}N /isls false N /vsize 11 72 mul N /@rigin{isls{[0 -1 1 0 0 0] concat}if 72 Resolution div 72 VResolution div neg scale isls{Resolution hsize -72 div mul 0 TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X 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b(when)g(punctures)f(coincide)i(with)g(that)f(of)h(a)h(con)n(tribution)f(to)g (the)f(string)h(measure)f(whic)n(h)g(is)0 918 y(\\)p Fk(\016)s Fj(-function-concen)n(trated)g(on)i(the)g(b)r(oundary)g(of)h(mo)r(duli)f (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 (supp)r(osed)g(to)g(b)r(e)f(real-analytic)0 1237 y(functions)g(of)h(the)e(mo) 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 b(.)29 b(What)22 b(app)r(ear)f(to)g(b)r(e)g(con)n(tact)g(in)n(teractions)g 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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 (oth)0 1048 y(and,)e(aside)f(from)g(ph)n(ysical)h(on-shell)g(p)r(oles,)g(v)l (anishes)g(as)f(one)g(approac)n(hes)f(the)h(b)r(oundary)f(of)i(mo)r(duli)0 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 b(The)32 b(\\in)n(tegrated")g(form)g(of)h(a)f(NS)g(v)n(ertex)f(op)r(erator)0 1486 y(\()p Fl(i.e.)20 b Fr(when)e(one)h(thinks)g(of)h(the)e(v)n(ertex)h(op)r (erator)g(as)g(a)g(2-form)h(to)f(b)r(e)f(in)n(tegrated)h(o)n(v)n(er)g(the)g (Riemann)0 1595 y(surface\))h(is)h(naturally)g(in)g(the)e(\(0\)-picture.)29 b(Ho)n(w)n(ev)n(er,)21 b(if)g(one)f(na)-7 b(\177)-26 b(\020v)n(ely)21 b(tak)n(es)f(the)f(OPE)i(of)g(t)n(w)n(o)f(suc)n(h)0 1704 y(v)n(ertex)h(op)r 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y(o\013-shell)d(features)f(of)h(the)f (theory)-6 b(,)31 b(one)d(is)i Fl(p)m(otential)s(ly)f Fr(op)r(ening)f(a)h(P)n (andora's)h(b)r(o)n(x)f(of)g(violations)0 3342 y(of)d(sup)r(erconformal)e(in) n(v)l(ariance.)39 b(F)-6 b(or)26 b(free)f(theories,)h(it)f(app)r(ears)g(that) g(one)f(can)h(easily)h(snap)f(sh)n(ut)0 3451 y(the)d(lid)h(after)f (extracting)g(the)f(desired)h(con)n(tact)f(terms.)30 b(F)-6 b(or)23 b(more)e(general)h(theories,)h(ev)n(en)e(if)i(their)0 3560 y(auxiliary)h(\014eld)e(structure)f(is)h(kno)n(wn,)h(one)f(ma)n(y)g(not) g(b)r(e)f(so)i(luc)n(ky)-6 b(.)1283 3719 y(2)p eop %%Page: 3 4 3 3 bop 133 66 a Fr(In)25 b(the)g(op)r(erator)g(formalism,)i(a)e(v)n(ertex)g (op)r(erator)g(is)h(naturally)g(though)n(t)g(of)g(as)f(a)h(0-form)g(in-)0 179 y(serted)17 b(at)h(a)h(puncture;)f(it)g(alw)n(a)n(ys)i(app)r(ears)e(in)g (the)g(\()p Fi(\000)p Fr(1\)-picture.)27 b(The)17 b(picture-c)n(hanging)h(op) r(erator)0 291 y(whic)n(h)27 b(mo)n(v)n(es)g(it)g(in)n(to)g(the)g 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y(automatically)i(tak)n(en)f(in)n(to)h(accoun)n(t.) 35 b(All)25 b(of)g(this)f(can)g(b)r(e)g(carried)g(through)g(for)g(an)h (arbitrary)g(su-)0 1188 y(p)r(erconformal)d(\014eld)f(theory)-6 b(.)30 b(Indeed,)22 b(w)n(e)f(w)n(on't)i(sp)r(ecify)f(precisely)g(what)g(sup) r(erconformal)f(theory)0 1300 y(w)n(e)h(are)g(w)n(orking)h(with.)0 1597 y Fg(1.)34 b(Preliminaries)0 1758 y Fl(1.1.)d(The)23 b(Me)m(asur)m(e)133 1919 y Fr(As)f(discussed)g(ab)r(o)n(v)n(e,)h(the)f(de\014nition)h(of)f(the)g (normal-ordered)g(v)n(ertex)f(op)r(erators,)i(and)g(hence)0 2031 y(of)31 b(the)f(string)h(measure,)g(dep)r(ends)e(on)i(a)f(c)n(hoice)g (of)h(lo)r(cal)g(co)r(ordinate)f(at)h(the)e(insertion)i(p)r(oin)n(ts.)0 2143 y(The)26 b(op)r(erator)h(formalism)g(pro)n(vides)g(a)g(nice)f(framew)n (ork)h(for)g(k)n(eeping)f(trac)n(k)h(of)g(this)g(dep)r(endence)0 2255 y(and)i(constructing)f(explicit)h(expressions)g(for)h(the)e(sup)r (erstring)h(measure.)49 b(The)28 b(extra)g(data)h(of)h(a)0 2367 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2903 y Fr(\(1)p Fk(:)p Fr(1)q(\))0 3221 y(where)24 b Fk(\033)233 3239 y Fd(\003)263 3221 y Fk(v)295 3239 y Fh(i)342 3221 y Fr(is)i(the)d(push-forw)n(ard)j(of)f (a)g(tangen)n(t)f(v)n(ector)g(to)1616 3204 y Ff(c)1600 3221 y Fi(M)1680 3239 y Fh(g)r(;n)1784 3221 y Fr(to)h(a)g(tangen)n(t)f(v)n(ector)g (to)2469 3204 y Ff(b)2456 3221 y Fi(P)2502 3239 y Fh(g)r(;n)2582 3221 y Fr(.)0 3333 y(This)f(construction,)f(and)g(the)f(ab)r(o)n(v)n(e)h (notation,)i(are)d(describ)r(ed)g(in)i(great)f(detail)g(in)h([13)p (#reference.13) [[468 118 480 130] [1 1 1 [3 3]] [0 0 1]] pdfm (,5)p (#reference.5) [[483 118 489 130] [1 1 1 [3 3]] [0 0 1]] pdfm (])h(and)e([7) p (#reference.7) [[523 118 529 130] [1 1 1 [3 3]] [0 0 1]] pdfm (].)133 3448 y(The)f(tangen)n(t)g(space)f(to)780 3431 y Ff(b)767 3448 y Fi(P)813 3466 y Fh(g)r(;n)914 3448 y Fr(has)i(an)f(elegan)n(t)g(presen)n (tation)g(in)h(terms)e(of)i(Sc)n(hi\013er)f(v)l(ariations.)0 3560 y(It)f(has)g(the)f(structure)f(of)i(a)g(direct)f(sum)h(of)g Fk(n)g Fr(copies)f(of)h(the)g(Nev)n(eu-Sc)n(h)n(w)n(arz)e(algebra)i(\(one)f (for)i(eac)n(h)1283 3719 y(3)p eop %%Page: 4 5 4 4 bop 0 66 a Fr(puncture\))22 b(mo)r(dulo)h(a)h(certain)f(\\Borel")h (subalgebra.)35 b(The)22 b(generators)i(of)g(this)f(algebra)h(are)g(giv)n(en) 0 172 y(in)e(terms)g(of)g(the)g(lo)r(cal)g(co)r(ordinates)g(at)g(the)g (puncture)e(b)n(y)826 360 y Fk(l)846 378 y Fh(n)900 360 y Fi($)f(\000)p Fk(z)1071 332 y Fh(n)p Fe(+1)1199 315 y Fk(@)p 1182 345 73 3 v 1182 405 a(@)t(z)1277 360 y Fi(\000)1352 334 y Fe(1)p 1352 345 27 3 v 1352 383 a(2)1386 360 y Fr(\()p Fk(n)14 b Fr(+)h(1\))p Fk(z)1626 332 y Fh(n)1662 360 y Fk(\022)1719 315 y(@)p 1703 345 72 3 v 1703 405 a(@)t(\022)817 523 y(g)849 541 y Fh(k)900 523 y Fi($)993 497 y Fe(1)p 993 508 27 3 v 993 546 a(2)1027 523 y Fk(z)1061 496 y Fh(k)q Fe(+)1139 478 y Fc(1)p 1139 486 23 3 v 1139 512 a(2)1173 469 y Ff(\000)1228 478 y Fk(@)p 1211 508 72 3 v 1211 569 a(@)t(\022)1306 523 y Fi(\000)g Fk(\022)1430 478 y(@)p 1414 508 73 3 v 1414 569 a(@)t(z)1494 469 y Ff(\001)1525 523 y Fk(:)2463 435 y Fr(\(1)p Fk(:)p Fr(2)q(\))0 698 y(An)21 b(in\014nitesimal)i(c)n(hange)e(in)h(the)f(co)r(ordinate)g(used)h(at)f(a)h (puncture)e(is)j(giv)n(en)f(b)n(y)g(the)f(action)h(of)g(the)0 804 y(\\v)n(ertical")g(tangen)n(t)f(v)n(ectors)g Fk(l)788 822 y Fh(n)824 804 y Fk(;)32 b(g)906 822 y Fh(n)p Fe(+)988 804 y Fc(1)p 988 812 23 3 v 988 838 a(2)1021 804 y Fk(;)h(n)19 b Fi(\025)f Fr(0.)30 b(These)20 b(induce)g(a)i(c)n(hange)f(of)g(the)g(state)g (asso)r(ciated)0 910 y(to)h(the)g(punctured)e(surface)i(giv)n(en)g(b)n(y)480 1093 y Fi(h)p Fr(\006)p Fk(;)11 b(z)18 b Fi(\000)d Fk(\017z)760 1065 y Fh(n)p Fe(+1)878 1093 y Fi(\000)952 1067 y Fe(1)p 952 1078 27 3 v 952 1116 a(2)987 1093 y Fr(^)-34 b Fk(\017\022)r(z)1080 1065 y Fh(k)q Fe(+)1158 1048 y Fc(1)p 1158 1056 23 3 v 1158 1081 a(2)1192 1093 y Fi(j)19 b Fr(=)f Fi(h)p Fr(\006)p Fk(;)11 b(z)s Fi(j)1454 1039 y Ff(\000)1485 1093 y Fr(1)k(+)g Fk(\017L)1672 1111 y Fh(n)1723 1093 y Fr(+)g(^)-34 b Fk(\017G)1868 1111 y Fh(k)1901 1039 y Ff(\001)1946 1093 y Fr(+)15 b Fi(\001)c(\001)g(\001)h Fk(:)344 b Fr(\(1)p Fk(:)p Fr(3)q(\))0 1276 y(Again,)23 b(details)g(ma)n(y)f (b)r(e)f(found)i(in)f([13)p (#reference.13) [[241 488 253 500] [1 1 1 [3 3]] [0 0 1]] pdfm (,5)p (#reference.5) [[256 488 262 500] [1 1 1 [3 3]] [0 0 1]] pdfm (,7)p (#reference.7) [[265 488 271 500] [1 1 1 [3 3]] [0 0 1]] pdfm (].)0 1460 y Fl(1.2.)31 b(The)23 b(Co)m(or)m(dinate)g(F)-5 b(amily)133 1605 y Fr(The)23 b(geometry)g(appropriate)h(for)g(computing)g(con)n(tact)f (terms)g(in)h(the)f(op)r(erator)h(formalism)g(is)0 1711 y(no)n(w)19 b(w)n(ell-established.)29 b(The)18 b(heterotic)g(case)g(w)n(as)h(treated)e (in)i(detail)g(in)g([7)p (#reference.7) [[409 410 415 422] [1 1 1 [3 3]] [0 0 1]] pdfm (])g(and)g(w)n (e)f(will)i(b)r(orro)n(w)f(the)0 1817 y(results)j(of)g(some)e(of)i(the)f (calculations)i(presen)n(ted)c(there.)29 b(Here)20 b(w)n(e)i(will)g(b)r(e)f (con)n(ten)n(t)g(to)g(giv)n(e)h(a)g(brief)0 1923 y(accoun)n(t)f(of)i(the)e (geometry)g(and)h(the)g(reader)f(should)i(consult)f([7)p (#reference.7) [[365 372 371 384] [1 1 1 [3 3]] [0 0 1]] pdfm (])h(for)g(a)f (more)f(complete)g(accoun)n(t.)133 2029 y(When)i(the)h(p)r(oin)n(ts)g Fk(P)33 b Fr(and)24 b Fk(Q)f Fr(are)h(widely)g(separated,)h(it)f(is)g (appropriate)h(to)f(use)f(co)r(ordinates)0 2135 y(at)29 b(one)f(p)r(oin)n(t)h (that)f(are)g(indep)r(enden)n(t)f(of)i(the)f(lo)r(cation)h(of)g(the)f(other.) 49 b(A)28 b(con)n(v)n(enien)n(t)g(c)n(hoice)f(of)0 2241 y(co)r(ordinate)k(is) h(the)f(so-called)h(sup)r(erconformal)f(normal-ordered)g(\(SCNO\))f(co)r (ordinate)i([5)p (#reference.5) [[502 315 508 327] [1 1 1 [3 3]] [0 0 1]] pdfm (],)j(and)0 2347 y(letting)22 b(\()q Ff(e)-38 b Fk(r)r(;)16 b Ff(e)-42 b Fk(\032)q Fr(\))22 b(b)r(e)f(the)h(lo)r(cation)g(of)h Fk(P)9 b Fr(,)22 b(w)n(e)g(tak)n(e)g(as)h(co)r(ordinates)617 2530 y Fk(\020)646 2548 y Fh(P)691 2530 y Fr(\()p Fi(\001)p Fr(\))18 b(=)g Fk(z)s Fr(\()p Fi(\001)p Fr(\))d Fi(\000)h Ff(e)-39 b Fk(r)17 b Fi(\000)i Ff(e)-42 b Fk(\032)q(\022)r Fr(\()p Fi(\001)p Fr(\))p Fk(;)1453 2512 y Fr(\024)1447 2530 y Fk(\020)1476 2548 y Fh(P)1521 2530 y Fr(\()p Fi(\001)p Fr(\))18 b(=)g Fk(\022)r Fr(\()p Fi(\001)p Fr(\))d Fi(\000)k Ff(e)-41 b Fk(\032)66 b(:)481 b Fr(\(1)p Fk(:)p Fr(4)q(\))0 2712 y(The)25 b(p)r(oin)n(t)i Fk(P)34 b Fr(should)27 b(b)r(e)e(though)n(t)h(of)g(as)h(the)e(place)h(where)e Fk(\020)1638 2730 y Fh(P)1683 2712 y Fr(\()p Fi(\001)p Fr(\))i(and)1918 2695 y(\024)1912 2712 y Fk(\020)1941 2730 y Fh(P)1985 2712 y Fr(\()p Fi(\001)p Fr(\))g(v)l(anish.)43 b Fk(\020)2353 2730 y Fh(P)2397 2712 y Fr(\()p Fi(\001)p Fr(\))26 b(and)6 2801 y(\024)0 2818 y Fk(\020)29 2836 y Fh(P)73 2818 y Fr(\()p Fi(\001)p Fr(\))31 b(are)f(co)r(ordinates)h(that)f(are)g(sup)r(erconformally)h(related) f(to)h Fk(z)s Fr(\()p Fi(\001)p Fr(\))f(and)h Fk(\022)r Fr(\()p Fi(\001)p Fr(\).)55 b(F)-6 b(or)32 b(a)f(SCNO)0 2924 y(co)r(ordinate,)350 2907 y(\024)344 2924 y Fk(\020)373 2942 y Fh(P)417 2924 y Fr(\()p Fi(\001)p Fr(\))21 b(is)f(just)h Fk(D)r(\020)788 2942 y Fh(P)832 2924 y Fr(\()p Fi(\001)p Fr(\),)g(where)f Fk(D)i Fr(is)f(the)e(sup)r (er-deriv)l(ativ)n(e)1890 2898 y Fh(@)p 1877 2909 58 3 v 1877 2947 a(@)s(\022)1954 2924 y Fr(+)11 b Fk(\022)2071 2898 y Fh(@)p 2058 2909 59 3 v 2058 2947 a(@)s(z)2124 2924 y Fr(.)29 b(Co)r(ordinates)21 b(at)0 3030 y Fk(Q)p Fr(,)j(lo)r(cated)f(at,)h(sa)n(y)-6 b(,)26 b(\()p Fk(r)n(;)11 b(\032)p Fr(\),)25 b(can)e(b)r(e)g(giv)n(en)h(b)n(y)g (similar)h(expressions,)f(with)h Ff(e)-38 b Fk(r)25 b Fr(and)j Ff(e)-41 b Fk(\032)24 b Fr(replaced)e(b)n(y)i Fk(r)0 3136 y Fr(and)e Fk(\032)p Fr(.)133 3242 y(In)15 b(conformal)h(\014eld)f(theories,)i (the)d(collision)j(of)f(t)n(w)n(o)g(punctures)e(is)i(replaced)e(b)n(y)i(the)e (conformally)0 3348 y(equiv)l(alen)n(t)25 b(picture)e(of)i(the)f(t)n(w)n(o)h (punctures)e(lo)r(cated)h(on)h(a)f(sphere)g(pinc)n(hing)h(o\013)f(from)h(the) e(rest)h(of)0 3454 y(the)17 b(surface.)28 b(In)17 b(this)h(region)g(SCNO)f (co)r(ordinates)g(are)g(not)h(appropriate.)28 b(Instead,)18 b(the)f(co)r(ordinates)0 3560 y(of)22 b(the)e(p)r(oin)n(t)i Fk(P)30 b Fr(will)23 b(dep)r(end)c(on)j(the)e(lo)r(cation)i(of)g Fk(Q)p Fr(,)f(and)g(they)g(are)g(giv)n(en)g(b)n(y)h(a)f(plum)n(bing)h (\014xture)1283 3719 y(4)p eop %%Page: 5 6 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 b(This)34 b(construction)e(is)h(obtained)g(b)n(y)g(b)r(eginning)g(with)g(a)g (three-punctured)0 171 y(sup)r(er-sphere.)g(W)-6 b(e)23 b(can)h(use)f Fk(O)r(sp)p Fr(\(2)p Fk(;)11 b Fr(1\))24 b(symmetry)e(to)h(\014x)h(the)e (three)h(b)r(osonic)g(co)r(ordinates)h(of)g(the)0 275 y(punctures)16 b(at)i(0,)h(1,)h(and)e Fi(1)p Fr(.)28 b(Only)18 b(t)n(w)n(o)g(of)h(the)e (three)f(fermionic)i(co)r(ordinates)f(can)g(b)r(e)g(\014xed)h(though;)0 379 y(the)k(remaining)g(one)g(is)h(the)f(one)g(o)r(dd)g(mo)r(dulus)g(of)h (the)f(three-punctured)e(sup)r(er-sphere,)h(whic)n(h)h(w)n(e)0 483 y(denote)e Fk(\034)8 b Fr(.)29 b(Th)n(us)21 b(w)n(e)f(place)g(our)h (three)f(punctures)f(at)i(\(0,0\),)h(\(1,)p Fk(\034)8 b Fr(\))21 b(and)g(\()p Fi(1)p Fr(,0\).)30 b(The)20 b(p)r(oin)n(t)h(at)g Fi(1)g Fr(is)0 587 y(sewn)i(on)n(to)g(the)f(rest)h(of)g(the)f(surface)h(at)g (the)f(old)i(lo)r(cation)f(of)h Fk(Q)p Fr(.)32 b(If)23 b Fk(w)h Fr(and)f Fk(\030)j Fr(are)d(the)f(co)r(ordinates)0 692 y(on)e(the)g(sup)r (er-sphere,)g(w)n(e)g(can)g(write)g(the)g(most)g(general)g(co)r(ordinates)g (that)g(v)l(anish)i(at)e Fk(P)29 b Fr(and)21 b Fk(Q)f Fr(as)804 857 y Fk(w)d Fi(\000)c Ff(e)-36 b Fk(a)970 875 y Fe(1)1000 857 y Fk(\034)8 b(w)r(\030)17 b Fr(+)c Ff(e)-36 b Fk(a)1234 875 y Fe(2)1264 857 y Fk(w)1314 829 y Fe(2)1358 857 y Fi(\000)13 b Ff(e)-36 b Fk(a)1459 875 y Fe(3)1489 857 y Fk(\034)8 b(w)1576 829 y Fe(2)1605 857 y Fk(\030)18 b Fr(+)c Fi(\001)d(\001)g(\001)669 b Fr(\(1)p Fk(:)p Fr(5)q(\))0 1022 y(at)22 b(\(0,0\))h(and)33 1187 y(\()p Fk(w)10 b Fi(\000)e Fr(1)g(+)g Fk(\034)g(\030)s Fr(\))g Fi(\000)g Fk(a)476 1205 y Fe(1)502 1187 y Fk(\034)g Fr(\()p Fk(w)i Fi(\000)e Fr(1)g(+)g Fk(\034)g(\030)s Fr(\)\()p Fk(\030)j Fi(\000)d Fk(\034)g Fr(\))f(+)h Fk(a)1170 1205 y Fe(2)1196 1187 y Fr(\()p Fk(w)i Fi(\000)e Fr(1)g(+)g Fk(\034)g(\030)s Fr(\))1536 1160 y Fe(2)1570 1187 y Fi(\000)g Fk(a)1665 1205 y Fe(3)1695 1187 y Fk(\034)g Fr(\()p Fk(w)i Fi(\000)e Fr(1)g(+)g Fk(\034)g(\030)s Fr(\))2072 1160 y Fe(2)2098 1187 y Fr(\()p Fk(\030)j Fi(\000)d Fk(\034)g Fr(\))g(+)g Fi(\001)j(\001)g(\001)32 b Fr(\(1)p Fk(:)p Fr(6)q(\))0 1352 y(at)26 b(\(1,)p Fk(\034)8 b Fr(\).)43 b(W)-6 b(e)26 b(ha)n(v)n(e)g(just)h(giv)n(en)f(the)g(ev)n(en)f (co)r(ordinates)i(here.)41 b(The)26 b(o)r(dd)g(co)r(ordinates)g(are)g(found)0 1457 y(b)n(y)c(demanding)g(that)g(the)f(co)r(ordinate)h(transformation)h (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 b(A)n(t)27 b Fi(1)h Fr(w)n(e)f(can)g(simply)h(use)f Fi(\000)p Fr(1)p Fk(=w)i Fr(and)f Fk(\030)s(=w)r Fr(.)46 b(These)0 1665 y(co)r(ordinates)22 b(are)g(sewn)g(to)g(the)f(co)r(ordinates)h Fk(\020)1204 1683 y Fh(Q)1249 1665 y Fr(\()p Fi(\001)p Fr(\))g(and)1477 1647 y(\024)1471 1665 y Fk(\020)1500 1683 y Fh(Q)1545 1665 y Fr(\()p Fi(\001)p Fr(\))g(b)n(y)g(the)g(iden)n(ti\014cations)795 1872 y Fk(w)d Fr(=)940 1819 y Fk(\020)969 1837 y Fh(Q)1015 1819 y Fr(\()p Fi(\001)p Fr(\))p 940 1856 145 3 v 986 1917 a Fk(t)1010 1898 y Fe(2)1159 1872 y Fk(;)145 b(\030)21 b Fr(=)d Fi(\000)1508 1801 y Fr(\024)1502 1819 y Fk(\020)1531 1837 y Fh(Q)1576 1819 y Fr(\()p Fi(\001)p Fr(\))p 1502 1856 V 1562 1917 a Fk(t)1787 1872 y(:)658 b Fr(\(1)p Fk(:)p Fr(7)q(\))0 2037 y(Th)n(us,)23 b(w)n(e)e(ha)n(v)n(e)i(as)f(co)r(ordinates)g(for)h Fk(P)31 b Fr(and)22 b Fk(Q)p Fr(,)g(when)f(they)h(are)g(close)g(together,)616 2230 y Fk(\036)656 2248 y Fh(Q)701 2230 y Fr(\()p Fi(\001)p Fr(\))d(=)849 2177 y Fk(\020)878 2195 y Fh(Q)924 2177 y Fr(\()p Fi(\001)p Fr(\))p 849 2214 V 895 2275 a Fk(t)919 2256 y Fe(2)1016 2230 y Fr(+)14 b Ff(e)-36 b Fk(a)1118 2248 y Fe(1)1148 2230 y Fk(\034)1192 2177 y(\020)1221 2195 y Fh(Q)1266 2177 y Fr(\()p Fi(\001)p Fr(\))1357 2159 y(\024)1336 2177 y Fk(\020)1365 2195 y Fh(Q)1411 2177 y Fr(\()p Fi(\001)p Fr(\))p 1192 2214 289 3 v 1310 2275 a Fk(t)1334 2256 y Fe(3)1504 2230 y Fr(+)13 b Ff(e)-36 b Fk(a)1605 2248 y Fe(2)1643 2177 y Fk(\020)1672 2195 y Fh(Q)1717 2177 y Fr(\()p Fi(\001)p Fr(\))1787 2153 y Fe(2)p 1643 2214 175 3 v 1703 2275 a Fk(t)1727 2256 y Fe(4)1840 2230 y Fr(+)14 b Fi(\001)d(\001)g(\001)617 2365 y Fk(\036)657 2383 y Fh(P)701 2365 y Fr(\()p Fi(\001)p Fr(\))19 b(=)o(\006\()p Fi(\001)p Fr(\))c(+)g Fk(a)1076 2383 y Fe(1)1105 2365 y Fk(\034)8 b Fr(\006\()p Fi(\001)p Fr(\))1267 2348 y(\024)1260 2365 y(\006\()p Fi(\001)p Fr(\))15 b(+)f Fk(a)1494 2383 y Fe(2)1524 2365 y Fr(\006\()p Fi(\001)p Fr(\))1642 2337 y Fe(2)1687 2365 y Fr(+)g Fi(\001)d(\001)g(\001)24 b Fk(;)2463 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3395 y Fr(\()p Fk(P)9 b Fr(\))18 b(=)g Fk(t)290 3371 y Fe(2)342 3395 y Fr(and)477 3377 y(\024)471 3395 y Fk(\020)500 3413 y Fh(Q)546 3395 y Fr(\()p Fk(P)9 b Fr(\))18 b(=)g Fi(\000)p Fk(t\034)8 b Fr(.)29 b(W)-6 b(e)22 b(\014nd)798 3560 y Ff(e)-38 b Fk(r)20 b Fr(=)f Fk(r)d Fr(+)f Fk(t)1055 3533 y Fe(2)1099 3560 y Fr(+)f Fk(t\032\034)t(;)149 b Ff(e)-41 b Fk(\032)19 b Fr(=)f Fk(\032)d Fi(\000)f Fk(t\034)74 b(:)628 b Fr(\(1)p Fk(:)p Fr(10)q(\))1283 3719 y(5)p eop %%Page: 6 7 6 6 bop 0 66 a Fr(The)27 b(non-split)h(nature)f(of)h(this)g(transformation)g (is)h(ultimately)e(resp)r(onsible)h(for)g(the)f(corrections)0 181 y(to)c(the)f(\\na)-7 b(\177)-26 b(\020v)n(e")23 b(calculation)g(that)g (restores)f(the)g(W)-6 b(ard)24 b(iden)n(tities)1724 157 y Fe(#)1771 181 y Fr(.)31 b(This)24 b(will)g(b)r(e)d(seen)h(explicitly)0 296 y(b)r(elo)n(w.)133 414 y(Finally)-6 b(,)21 b(to)d(get)f(a)h(co)r (ordinate)f(family)i(appropriate)f(for)g(b)r(oth)f(when)g(the)g(punctures)f (are)i(widely)0 529 y(separated)j(and)g(when)g(they)g(are)g(close)h (together,)f(one)g(should)h(in)n(terp)r(olate)f(smo)r(othly)h(b)r(et)n(w)n (een)e(the)0 644 y(t)n(w)n(o)27 b(b)r(eha)n(viors)h(outlined)f(ab)r(o)n(v)n (e.)46 b(W)-6 b(e)27 b(do)g(this)h(b)n(y)f(in)n(tro)r(ducing)h(a)f(smo)r(oth) g(function)g Fk(f)7 b Fr(\()p Fi(j)p Fk(t)p Fi(j)p Fr(\))27 b(that)0 758 y(go)r(es)i(from)h(0)g(to)f(1)h(as)g Fi(j)p Fk(t)p Fi(j)g Fr(go)r(es)g(from)f(0)h(to)f Fi(1)p Fr(,)j(and)e(using)g(a)g(linear)g (in)n(terp)r(olation.)53 b(Th)n(us,)32 b(the)0 873 y(co)r(ordinates)22 b(that)g(w)n(e)g(will)h(use)f(are)741 1067 y Fk(\033)779 1085 y Fh(P)823 1067 y Fr(\()p Fi(\001)p Fr(\))d(=)972 1022 y Fk(f)7 b Fr(\()p Fi(j)p Fk(t)p Fi(j)p Fr(\))p 972 1051 153 3 v 1021 1112 a Fk(t)1045 1093 y Fe(2)1132 1067 y Fk(\020)1161 1085 y Fh(P)1205 1067 y Fr(\()p Fi(\001)p Fr(\))15 b(+)f(\(1)h Fi(\000)g Fk(f)7 b Fr(\()p Fi(j)p Fk(t)p Fi(j)p Fr(\)\))p Fk(\036)1715 1085 y Fh(P)1758 1067 y Fr(\()p Fi(\001)p Fr(\))740 1233 y Fk(\033)778 1251 y Fh(Q)823 1233 y Fr(\()p Fi(\001)p Fr(\))19 b(=)972 1189 y Fk(f)7 b Fr(\()p Fi(j)p Fk(t)p Fi(j)p Fr(\))p 972 1218 V 1021 1279 a Fk(t)1045 1260 y Fe(2)1132 1233 y Fk(\020)1161 1251 y Fh(Q)1206 1233 y Fr(\()p Fi(\001)p Fr(\))15 b(+)f(\(1)h Fi(\000)f Fk(f)7 b Fr(\()p Fi(j)p Fk(t)p Fi(j)p Fr(\)\))p Fk(\036)1715 1251 y Fh(Q)1760 1233 y Fr(\()p Fi(\001)p Fr(\))k Fk(:)2430 1142 y Fr(\(1)p Fk(:)p Fr(11)q(\))0 1424 y(The)18 b(precise)f(shap)r(e)h(of)h (the)f(function)g Fk(f)25 b Fr(will)20 b(drop)f(out)f(of)h(the)e (calculation.)30 b(An)n(y)18 b(co)r(ordinate)g(family)0 1539 y(whic)n(h)i(in)n(terp)r(olates)h(b)r(et)n(w)n(een)d(the)i(required)g(b)r (eha)n(vior)g(at)g(large)h Fi(j)p Fk(t)p Fi(j)g Fr(and)g(at)f(small)h Fi(j)p Fk(t)p Fi(j)g Fr(will)h(yield)f(the)0 1653 y(same)c(v)l(alue)i(for)g (the)e(measure.)27 b(\(In)18 b(the)g(case)f(of)i(dilaton)g(correlation)f (functions,)i(the)d(measure)g(itself)0 1768 y(do)r(es)22 b(dep)r(end)e(on)j (the)e(in)n(terp)r(olating)i(function)f Fk(f)7 b Fr(;)22 b(only)h(its)g(in)n (tegral)g(is)f(indep)r(enden)n(t)f(of)h Fk(f)7 b Fr(.\))0 2080 y Fg(2.)34 b(The)26 b(calculation)133 2247 y Fr(The)21 b(state)h Fi(h)p Fr(\006)p Fk(;)11 b(\033)573 2265 y Fh(P)618 2247 y Fk(;)g(\033)685 2265 y Fh(Q)730 2247 y Fi(j)23 b Fr(at)f Fk(r)e Fr(=)p 972 2203 32 3 v 18 w Fk(r)h Fr(=)d(0)k(can)g(b)r(e)f(expanded)g(as)101 2460 y Fi(h)p Fr(\006)p Fk(;)11 b(\033)242 2478 y Fh(P)287 2460 y Fk(;)g(\033)354 2478 y Fh(Q)399 2460 y Fi(j)19 b Fr(=)p Fi(h)p Fr(\006)p Fk(;)11 b(\033)629 2478 y Fh(P)663 2485 y Fc(0)692 2460 y Fk(;)g(\033)759 2478 y Fh(Q)801 2485 y Fc(0)831 2460 y Fi(j)849 2406 y Ff(\000)880 2460 y Fr(1)k(+)f Fk(\032\034)8 b(tL)1134 2426 y Fe(\()p Fh(P)f Fe(\))1134 2478 y Fd(\000)p Fe(1)1235 2460 y Fr(+)14 b Fk(\032\034)1372 2406 y Ff(\000)1410 2415 y Fr(2)p Fk(p)1476 2433 y Fe(2)1521 2415 y Fi(\000)h Fk(p)1621 2433 y Fe(1)p 1410 2445 241 3 v 1519 2506 a Fk(t)1659 2406 y Ff(\001)1690 2460 y Fk(L)1735 2426 y Fe(\()p Fh(P)7 b Fe(\))1735 2478 y(0)1835 2460 y Fr(+)15 b(2)1935 2406 y Ff(\000)1965 2460 y Fk(\032)g Fi(\000)g Fk(\034)8 b(t)2142 2406 y Ff(\001)2172 2460 y Fk(G)2224 2426 y Fe(\()p Fh(P)f Fe(\))2224 2487 y Fd(\000)2274 2469 y Fc(1)p 2274 2477 23 3 v 2274 2503 a(2)2325 2460 y Fr(+)14 b Fi(\001)d(\001)g(\001)2469 2406 y Ff(\001)503 2625 y Fi(\002)569 2571 y Ff(\000)599 2625 y Fr(1)k Fi(\000)722 2580 y Fk(\032\034)c Ff(e)-40 b Fk(p)826 2598 y Fe(1)p 722 2610 135 3 v 777 2671 a Fk(t)864 2625 y(L)909 2591 y Fe(\()p Fh(Q)p Fe(\))909 2643 y(0)1010 2625 y Fr(+)15 b(2)p Fk(\032G)1196 2591 y Fe(\()p Fh(Q)p Fe(\))1196 2652 y Fd(\000)1246 2634 y Fc(1)p 1246 2642 23 3 v 1246 2668 a(2)1298 2625 y Fr(+)f Fi(\001)d(\001)g(\001)1442 2571 y Ff(\001)1472 2625 y Fk(t)1496 2598 y Fe(2\()p Fh(L)1579 2573 y Fc(\()p Fb(P)c Fc(\))1579 2614 y(0)1654 2598 y Fe(+)p Fh(L)1731 2573 y Fc(\()p Fb(Q)p Fc(\))1731 2614 y(0)1807 2598 y Fe(\))p 1831 2576 24 3 v 1831 2625 a Fk(t)1855 2587 y Fe(2\()1908 2576 y(\026)1902 2587 y Fh(L)1938 2563 y Fc(\()p Fb(P)f Fc(\))1938 2604 y(0)2012 2587 y Fe(+)2059 2576 y(\026)2053 2587 y Fh(L)2089 2563 y Fc(\()p Fb(Q)p Fc(\))2089 2604 y(0)2165 2587 y Fe(\))2463 2727 y Fr(\(2)p Fk(:)p Fr(1)q(\))0 2842 y(where)18 b Fk(p)222 2860 y Fe(1)271 2842 y Fr(=)g Fk(a)376 2860 y Fe(1)406 2842 y Fr(\(1)9 b Fi(\000)g Fk(f)e Fr(\),)23 b Ff(e)-40 b Fk(p)672 2860 y Fe(1)720 2842 y Fr(=)17 b Ff(e)-36 b Fk(a)825 2860 y Fe(1)855 2842 y Fr(\(1)9 b Fi(\000)g Fk(f)e Fr(\),)20 b(and)f Fk(p)1247 2860 y Fe(2)1296 2842 y Fr(=)f Fk(a)1401 2860 y Fe(2)1431 2842 y Fr(\(1)9 b Fi(\000)g Fk(f)e Fr(\).)28 b Fk(\033)1710 2860 y Fh(P)1744 2867 y Fc(0)1793 2842 y Fr(and)19 b Fk(\033)1957 2860 y Fh(Q)1999 2867 y Fc(0)2047 2842 y Fr(are)g(the)g(equal)g(to)h Fk(\033)2556 2860 y Fh(P)0 2956 y Fr(and)i Fk(\033)167 2974 y Fh(Q)234 2956 y Fr(with)g Fk(\032)g Fr(and)g Fk(\034)29 b 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 (ersymmetry)-6 b(.)34 b(The)23 b(k)n(ey)h(p)r(oin)n(t)h(is)g(to)f(appreciate) 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 %%Page: 10 11 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) h(to)e(use)h(the)e(co)r(ordinates)i(appropriate)f(to)h(this)f(sup)r (erconformal)0 266 y(geometry)-6 b(.)28 b(In)18 b(particular,)i(to)e(see)g (that)g(the)g(state)g(inserted)f(at)i(the)e(nec)n(k)h(is)h(the)e(highest)i (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 %%Page: 11 12 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 (in)g(the)g(sup)r(er)g(case.)29 b(Nev)n(ertheless,)20 b(w)n(e)h(ha)n(v)n(e)g (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 (case)f(to)h(remo)n(v)n(e)f(all)0 1403 y(dep)r(endence)13 b(on)j(the)f (\\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 (the)f(string)0 1942 y(measure)d(on)g(the)h(co)r(ordinate)f(family)i(is)f (quite)g(distinct)g(from)f(the)g(\\in)n(tegration)i(am)n(biguit)n(y")g (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 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b(Near)27 b(eac)n(h)e(of)1267 3719 y(11)p eop %%Page: 12 13 12 12 bop 0 66 a Fr(the)29 b(b)r(oundaries)g(of)h(sup)r(ermo)r(duli)e(space,) j(ho)n(w)n(ev)n(er,)g(these)e(co)r(ordinates)g(degenerate,)g(and)h(m)n(ust)0 173 y(b)r(e)25 b(replaced)f(b)n(y)i(those)f(asso)r(ciated)g(to)g(the)g(pinc)n (hing)h(geometry)-6 b(.)39 b(The)24 b(transformations)j(b)r(et)n(w)n(een)0 279 y(the)e(co)r(ordinates)h(\()p Fk(r)n(;)11 b(\032)592 297 y Fe(1)623 279 y Fk(;)g(\032)686 297 y Fe(2)717 279 y Fr(\))26 b(and)g(those)g(appropriate)g(near)g(the)f(b)r(oundary)i(m)n(ust)e(in)n(v)n (olv)n(e)i(a)g(non-)0 385 y(split)h(transformation)f(for)g(at)g(least)g(one)f (of)h(the)f(comp)r(onen)n(ts)f(of)i(the)f(b)r(oundary)-6 b(.)44 b(In)27 b(the)f(case)g(at)0 491 y(hand,)34 b(it)e(is)g Fk(r)k Fi(!)f Fr(0)c(whic)n(h)h(in)n(v)n(olv)n(es)h(the)e(nonsplit)h(transformation) g(but,)i(ho)n(w)n(ev)n(er)d(w)n(e)g(c)n(ho)r(ose)0 598 y(to)i(co)r (ordinatize)f(the)g(in)n(terior)i(of)f(sup)r(ermo)r(duli)f(space,)k(it)d(is)g 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Fr(on-shell)h(p)r(oles.)1000 2856 y Fg(Ac)n(kno)n(wledgmen)n(ts)133 2962 y Fr(W)-6 b(e)16 b(w)n(ould)h(lik)n(e)g(to)g(thank)f(D.)g(Kutaso)n(v,)h(P)-6 b(.)18 b(Nelson)e(and)h(H.)f(V)-6 b(erlinde)17 b(for)f(useful)h(con)n(v)n (ersations.)0 3068 y(This)23 b(w)n(ork)f(w)n(as)h(supp)r(orted)e(b)n(y)h(NSF) h(gran)n(t)f(PHY90-21984.)1267 3719 y(12)p eop %%Page: 13 14 13 13 bop 1121 66 a Fg(References)7 223 y Fr([1])57 b(P)-6 b(.)19 b(Deligne)f(and)g(D.)h(Mumford,)f(\\The)f(irreducibilit)n(y)j(of)e (the)g(space)f(of)i(curv)n(es)e(of)i(giv)n(en)f(gen)n(us,")133 313 y(IHES)k(Publ.)h(Math.)f Fg(36)h Fr(\(1969\),)g(75.)7 402 y([2])57 b(F.)41 b(Kn)n(udsen)e(and)g(D.)i(Mumford,)e(\\The)h(pro)t(jectivit) n(y)g(of)g(the)f(mo)r(duli)h(space)f(of)i(stable)133 492 y(curv)n(es,)22 b(I,")h(Math.)f(Scand.)g Fg(39)h Fr(\(1976\))f(19.)7 582 y([3])57 b(J.)22 b(P)n(olc)n(hinski,)h(\\F)-6 b(actorization)23 b(of)f(b)r(osonic)g (string)g(amplitudes,")g(Nucl.)g(Ph)n(ys.)g Fg(B307)h Fr(\(1988\))133 672 y(61.)7 761 y([4])57 b(P)-6 b(.)31 b(Nelson,)g(\\Co)n(v)l(arian)n(t)g (insertions)g(of)g(general)f(v)n(ertex)f(op)r(erators,")h(Ph)n(ys.)h(Rev.)f (Lett.)g Fg(62)133 851 y Fr(\(1989\))22 b(993.)7 941 y([5])57 b(H.S.)26 b(La)f(and)h(P)-6 b(.)26 b(Nelson,)g(\\E\013ectiv)n(e)f(\014eld)g (equations)h(for)g(fermionic)f(strings,")i(Nucl.)f(Ph)n(ys.)133 1031 y Fg(B332)d Fr(\(1990\))g(83.)7 1120 y([6])57 b(J.)19 b(Distler)g(and)g(P)-6 b(.)20 b(Nelson,)f(\\T)-6 b(op)r(ological)22 b(couplings)d(and)g(con)n(tact)f(terms)g(in)h(2d)g(\014eld)f(theory)-6 b(,")133 1210 y(Comm)n(un.)21 b(Math.)i(Ph)n(ys.)g Fg(138)f Fr(\(1991\))h(273.)7 1300 y([7])57 b(M.)30 b(Do)n(yle,)g(\\Dilaton)h(con)n (tact)e(terms)g(in)h(the)f(b)r(osonic)h(and)g(heterotic)e(strings,")j (Princeton)133 1390 y(preprin)n(t)22 b(PUPT-1296,)h(hepth)f(9201076,)i (submitted)d(to)h(Nucl.)g(Ph)n(ys.)h Fg(B)p Fr(.)7 1479 y([8])57 b(J.)17 b(Distler)f(and)h(P)-6 b(.)17 b(Nelson,)g(\\The)f(dilaton)h(equation) f(in)h(semirigid)g(string)g(theory)-6 b(,")17 b(Nucl.)f(Ph)n(ys.)133 1569 y Fg(B366)23 b Fr(\(1991\))g(255.)7 1659 y([9])57 b(E.)32 b(W)-6 b(ong,)33 b(\\Recursion)e(relations)h(in)g(semirigid)g(top)r(ological) h(gra)n(vit)n(y",)g(P)n(enn)f(U.)f(preprin)n(t)133 1749 y(UPR-0491T,)23 b(No)n(v.)g(1991.)-26 1838 y([10])57 b(R.)25 b(Dijkgraaf)i(and)e(E.)h (Witten,)f(\\Mean)g(\014eld)g(theory)-6 b(,)26 b(top)r(ological)g(\014eld)f (theory)-6 b(,)26 b(and)f(m)n(ulti-)133 1928 y(matrix)d(mo)r(dels,")g(Nucl.)g (Ph)n(ys.)i Fg(B342)f Fr(\(1990\))f(486.)-26 2018 y([11])57 b(E.)17 b(V)-6 b(erlinde)16 b(and)g(H.)g(V)-6 b(erlinde,)17 b(\\A)f(solution)h(of)g(t)n(w)n(o-dimensional)g(top)r(ological)h(gra)n(vit)n (y)-6 b(,")18 b(Nucl.)133 2108 y(Ph)n(ys.)23 b Fg(B348)g Fr(\(1991\))g(457.) -26 2197 y([12])57 b(M.)26 b(Green)e(and)i(N.)g(Seib)r(erg,)f(\\Con)n(tact)g (in)n(teractions)h(in)g(sup)r(erstring)f(theory)-6 b(,")26 b(Nucl.)g(Ph)n(ys.)133 2287 y Fg(B299)d Fr(\(1988\))g(559.)-26 2377 y([13])57 b(L.)24 b(Alv)l(arez-Gaum)n(\023)-31 b(e,)24 b(C.)h(Gomez,)f(P)-6 b(.)25 b(Nelson,)g(G.)g(Sierra,)g(and)g(C.)f(V)-6 b(afa,)26 b(\\F)-6 b(ermionic)25 b(strings)133 2466 y(in)d(the)g(op)r(erator) g(formalism,")h(Nucl.)f(Ph)n(ys.)h Fg(B311)g Fr(\(1988\))g(333.)-26 2556 y([14])57 b(J.)16 b(Cohn,)h(\\Mo)r(dular)f(geometry)f(of)h(sup)r (erconformal)g(\014eld)f(theory)-6 b(,")17 b(Nucl.)f(Ph)n(ys.)h Fg(B306)g Fr(\(1988\))133 2646 y(239.)-26 2736 y([15])57 b(H.)22 b(V)-6 b(erlinde,)23 b(priv)l(ate)f(comm)n(unication.)-26 2825 y([16])57 b(H.)22 b(Sono)r(da)g(and)g(B.)h(Zw)n(eibac)n(h,)f(\\Co)n(v)l (arian)n(t)h(closed)f(string)h(theory)e(cannot)h(b)r(e)f(cubic,")h(Nucl.)133 2915 y(Ph)n(ys.)h Fg(B336)g Fr(\(1990\))g(185.)-26 3005 y([17])57 b(M.)22 b(Rothstein,)h(T)-6 b(rans.)23 b(AMS.)f Fg(299)h Fr(\(1987\))f(387.) -26 3095 y([18])57 b(J.)23 b(A)n(tic)n(k,)f(J.)h(Rabin)f(and)g(A.)h(Sen,)f (Nucl.)g(Ph)n(ys.)h Fg(B299)g Fr(\(1988\))g(279;)133 3184 y(H.)16 b(V)-6 b(erlinde,)16 b(\\A)f(note)g(on)h(the)f(in)n(tegral)h(o)n(v)n(er)g 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