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%%Title: arXiv:hep-th/9705076 v1   12 May 1997
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(coupling)h Fl(\000)p Ft(1)p Fm(=\034)5 b Ft(.)21 b(This)15
b(is)g(the)h(EM-dualit)o(y)957 1939 y(\\symmetry".)30
b(It)18 b(is)h(not)f(really)g(a)h(symmetry)e(since)957
1989 y(it)d(acts)h(on)f(the)g(couplings|it)f(is)i(an)f(equiv)n(alence)g
(b)q(e-)957 2038 y(t)o(w)o(een)i(t)o(w)o(o)g(descriptions)g(of)f(the)i
(ph)o(ysics.)24 b(The)16 b(EM-)957 2088 y(dualit)o(y)d(transformation)g
(\()p Fm(S)r Ft(\))k(and)d(the)i(in)o(v)n(ariance)e(of)957
2138 y(the)g(ph)o(ysics)h(under)f(2)p Fm(\031)h Ft(shifts)f(of)f(the)i
Fm(#)p Ft(-angle)e(\()p Fm(T)6 b Ft(\),)957 2214 y Fm(S)14
b Ft(:)25 b Fm(\034)16 b Fl(!)11 b(\000)p Ft(1)p Fm(=\034)r(;)90
b(T)18 b Ft(:)25 b Fm(\034)16 b Fl(!)11 b Fm(\034)5 b
Ft(+1)p Fm(;)219 b Ft(\(7\))957 2289 y(generate)17 b(the)h(group)e
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(transfor-)957 2339 y(mations:)957 2429 y Fm(\034)h Fl(!)1049
2401 y Fm(a\034)e Ft(+)c Fm(b)p 1049 2419 114 2 v 1049
2458 a(c\034)k Ft(+)c Fm(d)1167 2429 y(;)1269 2370 y
Fj(\022)1306 2404 y Fm(a)44 b(b)1308 2454 y(c)g(d)1398
2370 y Fj(\023)1440 2429 y Fl(2)12 b Fm(S)r(L)p Ft(\(2)p
Fm(;)7 b Fn(Z)p Ft(\))p Fm(:)141 b Ft(\(8\))998 2529
y(Because)27 b(the)e(dual)f Fm(U)5 b Ft(\(1\))24 b(p)q(oten)o(tial)g
(couples)i(to)957 2579 y(monop)q(oles)19 b(with)h(unit)g(strength,)j(w)
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384 y(with)21 b(electric)i(and)e(magnetic)f(c)o(harges)i(\()p
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 (#equation.2) [[87 577 92 589] [1 1 1 [3 3]] [0 0 1]] pdfm (\))26
b(is)f(c)o(harge)i(conjugation.)52 b(The)26 b(e\013ect)h(of)e(the)47
633 y Fm(T)g Ft(transformation)17 b(on)i(the)g(c)o(harges)h(is)f(\()p
Fm(n)763 639 y Fk(e)781 633 y Fm(;)7 b(n)825 639 y Fk(m)856
633 y Ft(\))20 b Fl(!)47 683 y Ft(\()p Fm(n)88 689 y
Fk(e)106 683 y Ft(+)p Fm(n)163 689 y Fk(m)195 683 y Fm(;)7
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89 783 y(This)e(description)h(of)e(EM-dualit)o(y)h(migh)o(t)e(mak)o(e)g
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b(Its)23 b(imp)q(ortance)f(stems)h(from)47 982 y(the)15
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1381 y(w)o(e)k(will)e(see)j(in)e Fm(N)5 b Ft(=2)16 b(sup)q(ersymmetric)
f(examples)f(in)47 1431 y(the)h(next)f(lecture.)47 1533
y Fn(1.2.)24 b(S-dualit)o(y)89 1583 y Ft(S-dualit)o(y)15
b(is)h(the)i(exact)f(quan)o(tum)e(equiv)n(alence)i(of)47
1633 y(t)o(w)o(o)i(apparen)o(tly)h(\(classically\))f(di\013eren)o(t)h
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b(of)h(a)f(theory)i(with)e(sp)q(eci\014c)j(deforma-)47
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2031 y(the)14 b(dualit)o(y)e(group)g(are)i(supp)q(osed)g(to)f(connect)h
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y(A)10 b(famous)f(example)g([2)p
 (#cite.OliveMontonen) [[178 182 183 194] [1 1 1 [3 3]] [0 0 1]] pdfm 
({4)p  (#cite.Osborn) [[188 182 193 194] [1 1 1 [3 3]] [0 0 1]] pdfm 
(])f(is)j(the)g(strong-w)o(eak)f(cou-)47 2280 y(pling)i(dualit)o(y)f
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Fm(g)836 2315 y Fr(2)866 2330 y Ft(and)47 2380 y(1)p
Fm(=g)110 2365 y Fr(2)143 2380 y Ft(are)15 b(iden)o(ti\014ed.)20
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y(p)q(elling)k(evidence)j(for)e(this)g(dualit)o(y)f(in)g(\014eld)h
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 (#cite.VafaWitten) [[244 98 249 110] [1 1 1 [3 3]] [0 0 1]] pdfm (,6)p
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(].)1093 235 y(In)f(this)h(case)h(the)g(exactly)e(marginal)e(op)q
(erator)k(cor-)1051 285 y(resp)q(onds)e(to)f(c)o(hanging)e(the)j
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b(w)o(e)e(see)h(that)f(the)g(top)q(ology)f(is)g(mo)q(di\014ed)1051
584 y(quan)o(tumly)12 b(to)i(that)g(of)f(a)g(half-closed)h(in)o(terv)n
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Fm(N)5 b Ft(=4)15 b(theory)f(is)h(the)g(angularit)o(y)e(of)g(the)1051
733 y(theta)h(angle:)j Fm(#)11 b Fl(')h Fm(#)6 b Ft(+)i(2)p
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783 y(space)h(is)e(then)h(the)g(upp)q(er-half)g(plane)f
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Fm(\034)17 b(>)1051 833 y Ft(0)p Fl(g)p Ft(,)i(where)h
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Ft(\))12 b(+)h Fm(i)p Ft(\(4)p Fm(\031)1600 818 y Fr(2)1619
833 y Fm(=g)1661 818 y Fr(2)1679 833 y Ft(\).)33 b(Com)o(bining)1051
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932 y Ft(giv)o(es)d(the)g(quan)o(tum)e(cou-)1051 982
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1032 y(acts)18 b(on)e(the)i(coupling)e(as)h(in)f(\(8)p
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b(The)17 b(top)q(ology)f(of)1051 1082 y Fl(M)p Ft(|the)21
b(fundamen)o(tal)f(domain)f(of)h Fm(S)r(L)p Ft(\(2)p
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b(that)h(of)f(an)g(op)q(en)h(disk.)20 b(The)15 b(du-)1051
1181 y(alit)o(y)20 b(group,)i Fm(S)r(L)p Ft(\(2)p Fm(;)7
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1231 y(group)12 b(with)g(t)o(w)o(o)f(generators)j Fm(S)g
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b(This)19 b(group)g(clearly)1051 1331 y(enco)q(des)c(more)d
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1486 y Fr(2)1303 1480 y Ft(and)g Fn(Z)1414 1486 y Fr(3)1447
1480 y Ft(orbifold)e(p)q(oin)o(ts)i(of)f Fl(M)p Ft(,)g(if)g(w)o(e)1051
1530 y(assign)i Fl(M)1226 1536 y Fk(cl)1269 1530 y Ft(a)g(\015at)g
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y(b)o(y)i(these)h(generators)h(then)e(describ)q(e)i(the)f(holonom)o(y)
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1729 y Ft(,)g(though)f(the)i(top)q(ology)d(of)h Fl(M)h
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1978 y(A)f(natural)g(connection)i(on)e(the)h(coupling)f(space)h
Fl(M)1051 2028 y Ft(do)q(es)j(exist)f([7)p
 (#cite.JackOsborn) [[374 230 379 242] [1 1 1 [3 3]] [0 0 1]] pdfm ({9)
p  (#cite.Dolan) [[384 230 389 242] [1 1 1 [3 3]] [0 0 1]] pdfm (].)j
(The)d(idea)g(is)f(simply)f(that)i Fm(n)p Ft(-p)q(oin)o(t)1051
2078 y(correlators)j(of)e(the)h(op)q(erators)h Fl(O)1605
2063 y Fk(i)1635 2078 y Ft(conjugate)f(to)f(the)1051
2128 y(couplings)g Fm(\034)1254 2134 y Fk(i)1284 2128
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%%Page: 3 3
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Fl(O)91 302 y Fk(j)108 319 y Ft(\(0\))f(=)g Fm(c)235
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1052 y(necessitating)f(enlarging)f Fl(M)g Ft(to)g(include)g(all)f(of)h
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b(sup)q(ersymmetric)i(\014eld)h(theories)h(with)-47 1201
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(of)g(their)i(scaling)e(dimensions)g(to)-47 1550 y(their)19
b Fm(R)p Ft(-c)o(harges)g([10)p
 (#cite.Seiberg9411) [[136 345 146 357] [1 1 1 [3 3]] [0 0 1]] pdfm 
(].)31 b(Th)o(us)19 b(the)g(connection)g(v)n(an-)-47
1599 y(ishes,)14 b(and)g Fl(M)f Ft(is)h(lo)q(cally)f(\015at.)-6
1649 y(The)18 b(dualit)o(y)e(group)i(elemen)o(ts)f(whic)o(h)h(do)f(not)
g(act)-47 1699 y(freely)c(on)f Fl(M)171 1705 y Fk(cl)212
1699 y Ft(therefore)i(do)e(indeed)i(enco)q(de)g(a)e(ph)o(ys-)-47
1749 y(ical)g(holonom)o(y)f(of)h(op)q(erators)j(parallelly)c(transp)q
(orted)-47 1799 y(in)g Fl(M)g Ft(around)h(their)g(\014xed)g(p)q(oin)o
(ts.)17 b(The)12 b(kno)o(wn)f(\(con-)-47 1849 y(jectured\))24
b(S-dualit)o(y)e(groups)h(for)f Fm(N)5 b Ft(=2)23 b(sup)q(ersym-)-47
1898 y(metric)13 b(\014eld)g(theories)i(will)d(b)q(e)i(discussed)h(in)e
(the)h(next)-47 1948 y(lecture.)-47 2031 y Fn(1.3.)24
b(IR-dualit)o(y)-6 2081 y Ft(IR-dualit)o(y)11 b(refers)k(to)e(the)g
(equiv)n(alence)g(of)g(t)o(w)o(o)f(the-)-47 2131 y(ories)i(in)f(the)i
(infrared,)e Fe(i.e.)g Ft(at)h(long)f(distances)i(or)e(lo)o(w)-47
2181 y(energies.)18 b(In)12 b(the)h(con)o(text)g(of)e(sup)q
(ersymmetric)h(gauge)-47 2231 y(theories,)24 b(this)d(is)h(sometimes)d
(refered)k(to)f(as)f(\\non-)-47 2280 y(Ab)q(elian)g(dualit)o(y")e(or)i
(\\)p Fm(N)5 b Ft(=1)21 b(dualit)o(y".)39 b(Ho)o(w)o(ev)o(er,)-47
2330 y(more)12 b(generally)m(,)g(IR-dual)g(theories)i(are)g(simply)d
(theo-)-47 2380 y(ries)f(that)g(b)q(elong)f(to)g(the)i(same)d(R)o(G)h
(univ)o(ersalit)o(y)g(class.)-6 2430 y(The)16 b(reason)g(w)o(e)f(giv)o
(e)g(a)g(sp)q(ecial)g(name)f(to)i(theories)-47 2480 y(in)10
b(the)h(same)f(univ)o(ersalit)o(y)g(class)h(is)f(that)h(in)f(the)i
(case)f(of)-47 2529 y(sup)q(ersymmetric)i(gauge)h(theories,)g(the)g
(resulting)g(IR-)-47 2579 y(dual)d(pairs)h(ha)o(v)o(e)f(prop)q(erties)j
(reminiscen)o(t)d(of)g(the)i Fm(\034)j Fl($)957 235 y(\000)p
Ft(1)p Fm(=\034)21 b Ft(transformation)16 b(in)g(EM)i(and)f
(S-dualities.)26 b(In)957 285 y(particular,)13 b(it)g(is)g(found)g
(that)g(if)g(b)q(oth)g(theories)i(in)e(the)957 335 y(IR-dual)j(pair)i
(are)g(asymptotically)d(free)k(\(AF\),)f(then)957 384
y(their)e(strong-coupling)g(scales)h(\003)1501 390 y
Fr(1)1536 384 y Ft(and)f(\003)1648 390 y Fr(2)1683 384
y Ft(will)f(v)n(ary)957 434 y(in)o(v)o(ersely:)957 519
y(\003)986 500 y Fk(b)1001 504 y Fd(1)986 530 y Fr(1)1018
519 y Ft(\003)1047 500 y Fk(b)1062 504 y Fd(2)1047 530
y Fr(2)1092 519 y Fl(\030)c Fm(\026)1160 502 y Fk(b)1175
506 y Fd(1)1191 502 y Fr(+)p Fk(b)1231 506 y Fd(2)1250
519 y Fm(;)507 b Ft(\(10\))957 603 y(where)15 b Fm(b)1095
609 y Fr(1)1128 603 y Ft(and)f Fm(b)1227 609 y Fr(2)1260
603 y Ft(are)h(sp)q(eci\014ed)h(constan)o(ts,)f(and)f
Fm(\026)h Ft(is)957 653 y(an)e(arbtrary)i(matc)o(hing)d(scale.)998
703 y(More)j(strikingly)m(,)e(there)j(are)f(IR-dual)f(pairs)g(one)h(of)
957 753 y(whic)o(h)10 b(is)g(AF)g(and)g(the)g(other)h(IR-free,)g(whic)o
(h)f(ha)o(v)o(e)f(dif-)957 802 y(feren)o(t)18 b(gauge)g(groups.)29
b(Restated,)19 b(this)f(means)e(that)957 852 y(certain)j(AF)f(theories)
i(whic)o(h)e(are)h(w)o(ell-describ)q(ed)h(at)957 902
y(short)12 b(distances)h(in)f(terms)g(of)f(a)h(gauge)g(group)g
Fm(G)1741 908 y Fr(1)1771 902 y Ft(giv)o(e)957 952 y(rise)22
b(to)f(long)f(w)o(a)o(v)o(e-length)h(ph)o(ysics)h(whic)o(h)f(is)g(w)o
(ell-)957 1002 y(describ)q(ed)12 b(b)o(y)e(a)h(di\013eren)o(t)g(gauge)f
(group)h Fm(G)1645 1008 y Fr(2)1663 1002 y Ft(.)17 b(F)m(urther-)957
1052 y(more,)d(there)j(is)f(no)f(clear)h(sense)i(in)d(whic)o(h)g(the)i
(gauge)957 1101 y(b)q(osons)f(of)g Fm(G)1177 1107 y Fr(2)1211
1101 y Ft(are)h(related)g(to)f(those)h(of)f Fm(G)1671
1107 y Fr(1)1689 1101 y Ft(;)h(in)f(par-)957 1151 y(ticular,)d
Fm(G)1133 1157 y Fr(2)1165 1151 y Ft(is)h(not)f(simply)f(a)i(subgroup)g
(of)f Fm(G)1708 1157 y Fr(1)1727 1151 y Ft(.)998 1201
y(W)m(e)f(will)e(discuss)j(the)g(evidence)g(for)f(IR-dualit)o(y)e(and)
957 1251 y(its)17 b(p)q(ossible)h(connection)g(to)g(S-dualit)o(y)e(in)h
(the)h(third)957 1301 y(lecture.)957 1384 y Fn(1.4.)24
b(Extensions)998 1434 y Ft(So)d(far)g(w)o(e)g(ha)o(v)o(e)g(discussed)i
(EM-dualit)o(y)d(in)h(free)957 1483 y Fm(U)5 b Ft(\(1\))20
b(gauge)g(theories,)j(S-dualit)o(y)c(in)g(theories)j(with)957
1533 y(exactly)10 b(marginal)d(couplings,)i(and)h(IR-dualit)o(y)e(of)h
(the-)957 1583 y(ories)i(with)g(couplings)g(whic)o(h)g(undergo)h(R)o(G)
e(\015o)o(w.)17 b(The)957 1633 y(concepts)h(as)g(in)o(tro)q(duced)f(ha)
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b(so)h(migh)o(t)e(b)q(e)j(exp)q(ected)g(to)f(apply)f(more)g(broadly)m
(.)998 1732 y(In)18 b(particular,)g(none)h(of)e(these)i(dualities)e
(relies)i(in)957 1782 y(principle)11 b(on)h(sup)q(ersymmetry)m(.)k(W)m
(e)c(form)o(ulated)d(EM-)957 1832 y(dualit)o(y)k(in)h(Maxw)o(ell)g
(theory)m(.)20 b(The)15 b(description)h(of)e(S-)957 1882
y(dualit)o(y)c(as)h(the)h(quan)o(tum)e(mo)q(di\014cation)f(of)i(the)h
(top)q(ol-)957 1932 y(ogy)k(\(and)g(holonomies\))f(of)h(the)h
(parameter)f(space)i(of)957 1982 y(couplings)c(and)g(IR-dualities)g(as)
g(univ)o(ersalit)o(y)g(classes)957 2031 y(of)21 b(gauge)g(theories)i
(sho)o(w)f(them)f(to)g(b)q(e)i(p)q(oten)o(tially)957
2081 y(generic)15 b(phenomena)e(in)g(\014eld)h(theories.)20
b(Ho)o(w)o(ev)o(er,)14 b(at)957 2131 y(presen)o(t)i(only)e(sup)q
(ersymmetry)g(giv)o(es)h(us)g(the)h(con)o(trol)957 2181
y(o)o(v)o(er)f(the)i(strong)f(coupling)e(dynamics)h(of)g(gauge)g(the-)
957 2231 y(ories)20 b(necessary)i(to)d(giv)o(e)g(evidence)i(for)f(S-)g
(and)f(IR-)957 2280 y(dualities.)998 2330 y(Also,)h(nothing)f(in)g(the)
i(description)f(of)f(these)i(du-)957 2380 y(alities)e(limits)e(them)h
(to)i(four)f(dimensions.)34 b(Indeed,)957 2430 y(w)o(e)17
b(ha)o(v)o(e)h(seen)h(extensions)f(of)f(EM-dualit)o(y)f(to)i(other)957
2480 y(dimensions)g(in)h(F.)h(Quev)o(edo's)g(lectures.)37
b(Lik)o(ewise,)957 2529 y(analogs)19 b(of)h(S-dualit)o(y)g(in)g(sup)q
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(dimensions)g(are)i(the)f(sub)r(ject)i(of)p eop
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4 3 bop 47 106 a Ft(4)47 235 y(man)o(y)12 b(other)j(lectures)g(at)f
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564 y(p)q(erQCD)14 b(\(see)g(the)g(talks)f(of)f(A.)h(Klemm)d(for)j
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813 y(an)g Fm(N)5 b Ft(=1)17 b(v)o(ector)h(m)o(ultiplet)e
Fm(V)27 b Ft(and)17 b(an)g Fm(N)5 b Ft(=1)18 b(c)o(hiral)47
863 y(m)o(ultiplet)13 b(\010)h(in)h(the)g(adjoin)o(t.)k(W)m(e)c(refer)h
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g(of)g(\010)g(as)h(the)g(Higgs)47 962 y(\014eld.)i(An)11
b Fm(N)5 b Ft(=2)11 b(v)o(ector)h(m)o(ultiplet)e(is)h(made)f(up)i(of)e
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b(and)h(the)f(gauge)g(coupling)g Fm(\034)5 b Ft(.)25
b(The)17 b(one-lo)q(op)47 1311 y(b)q(eta)e(function)e(for)h
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1387 y Ft(=)12 b Fl(\000)p Fm(T)6 b Ft(\()p Fm(Ad)p Ft(\))j(+)357
1347 y Fj(X)381 1436 y Fk(i)424 1387 y Fm(T)d Ft(\()p
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Fm(R)313 1657 y Fk(i)326 1651 y Ft(.)28 b(Th)o(us,)17
b(for)g(example,)f(for)h Fm(S)r(U)5 b Ft(\()p Fm(n)899
1657 y Fk(c)917 1651 y Ft(\))47 1701 y(with)14 b Fm(n)167
1707 y Fk(f)203 1701 y Ft(h)o(yp)q(erm)o(ultiplets)f(in)h(the)g
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1756 y Fr(1)p Fk(`)340 1750 y Ft(=)e Fl(\000)p Ft(2)p
Fm(n)462 1756 y Fk(c)488 1750 y Ft(+)d Fm(n)554 1756
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1950 y Ft(=)f(2)p Fm(n)195 1956 y Fk(c)212 1950 y Ft(.)17
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b(marginal)e(op)q(erators)89 2131 y Ft(A)20 b(non-renormalization)d
(theorem)i([11)p
 (#cite.Seiberg88) [[241 206 251 218] [1 1 1 [3 3]] [0 0 1]] pdfm (,12)
p  (#cite.Seiberg93) [[254 206 264 218] [1 1 1 [3 3]] [0 0 1]] pdfm (])
f(states)47 2181 y(that)23 b(the)h(b)q(eta)g(function)e(is)h(giv)o(en)g
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Fm(N)5 b Ft(=2)17 b(sup)q(erQCD)h([13)p
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(].\))47 2480 y(Ho)o(w)o(ev)o(er,)23 b(since)f(b)q(eta)g(functions)f
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2579 y(non-renormalization)g(theorem)h(v)n(alid?)1093
235 y(The)k(sup)q(ersymmetric)f(non-renormalization)f(the-)1051
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b(is,)k(the)f(kinetic)f(terms)h(are)1051 534 y(not)16
b(rescaled)h(to)f(their)g(canonical)g(normalization)d(af-)1051
584 y(ter)20 b(an)e(R)o(G)g(transformation.)31 b(Ph)o(ysical)19
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b(the)h(anomalous)d(dimensions)h Fm(\015)1608 1038 y
Fk(i)1635 1032 y Ft(of)h(the)g(h)o(yp)q(erm)o(ul-)1051
1082 y(tiplets.)1093 1132 y(Reasoning)19 b(along)h(these)h(lines)g(w)o
(as)f(used)h(in)f([14)p
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1181 y(to)c(sho)o(w)g(that)g(the)h(exact)g(ph)o(ysical)f(b)q(eta)g
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1315 y Fm(\014)1074 1321 y Fk(\034)1107 1315 y Fl(/)f
Fm(\014)1174 1321 y Fr(1)p Fk(`)1216 1315 y Ft(+)1257
1276 y Fj(X)1282 1364 y Fk(i)1324 1315 y Fm(\015)1345
1321 y Fk(i)1360 1315 y Fm(T)6 b Ft(\()p Fm(R)1438 1321
y Fk(i)1451 1315 y Ft(\))p Fm(:)384 b Ft(\(12\))1051
1430 y(Similarly)l(,)11 b(for)i(an)h(op)q(erator)1051
1514 y Fm(\016)r Fl(W)h Ft(=)d Fm(\025)1202 1475 y Fj(Y)1223
1563 y Fk(i)1255 1514 y Ft(\(\010)1301 1520 y Fk(i)1315
1514 y Ft(\))1331 1497 y Fk(r)1346 1501 y Fc(i)1863 1514
y Ft(\(13\))1051 1631 y(in)h(the)h(sup)q(erp)q(oten)o(tial)g(\(in)f
Fm(N)5 b Ft(=1)14 b(language\),)e(the)i(ex-)1051 1681
y(act)g(b)q(eta-function)g(for)g(its)g(coupling)f Fm(\025)h
Ft(is)1051 1765 y Fm(\014)1074 1771 y Fk(\025)1108 1765
y Fl(/)e Ft(3)d Fl(\000)1223 1726 y Fj(X)1247 1814 y
Fk(i)1290 1765 y Fm(r)1309 1771 y Fk(i)1323 1765 y Ft(\(1)g(+)g
Fm(\015)1431 1771 y Fk(i)1446 1765 y Ft(\))p Fm(:)389
b Ft(\(14\))1093 1882 y(No)o(w,)10 b(follo)o(wing)f([15)p
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(],)h(w)o(e)h(lo)q(ok)f(for)h(\014xed)g(p)q(oin)o(t)g(the-)1051
1932 y(ories)16 b(b)o(y)g(solving)f(for)h(p)q(oin)o(ts)f(in)h
Fm(\034)5 b Ft(-)p Fm(\025)16 b Ft(coupling)f(space)1051
1982 y(where)20 b Fm(\014)1199 1988 y Fk(\034)1239 1982
y Ft(=)g Fm(\014)1314 1988 y Fk(\025)1355 1982 y Ft(=)f(0.)32
b(These)19 b(are)g(t)o(w)o(o)f(conditions)1051 2031 y(on)f(t)o(w)o(o)g
(unkno)o(wn)g(parameters,)h(and)f(so)g(generically)1051
2081 y(ha)o(v)o(e)i(isolated)f(\014xed-p)q(oin)o(t)h(solutions.)33
b(Ho)o(w)o(ev)o(er,)20 b(if)1051 2131 y(w)o(e)c(can)g(arrange)f(that)h
Fm(\014)1458 2137 y Fk(\034)1493 2131 y Fl(/)f Fm(\014)1563
2137 y Fk(\025)1585 2131 y Ft(,)g(then)i(there)f(is)g(only)1051
2181 y(one)c(indep)q(enden)o(t)g(condition)e(on)h(t)o(w)o(o)g(unkno)o
(wns,)g(and)1051 2231 y(so)g(there)i(will)c(t)o(ypically)h(b)q(e)i(a)f
(line)f(of)h(\014xed)g(p)q(oin)o(ts)g(and)1051 2280 y(th)o(us)16
b(an)f(exactly)g(marginal)e(op)q(erator)j(\(whic)o(h)f(mo)o(v)o(es)1051
2330 y(us)f(along)f(this)h(line\).)1093 2380 y(T)m(o)29
b(apply)g(this)h(strategy)h(for)e(\014nding)h(exactly)1051
2430 y(marginal)9 b(op)q(erators)k(to)e Fm(N)5 b Ft(=)q(2)11
b(theories,)i(consider)f(the)1051 2480 y Fm(N)5 b Ft(=1)12
b(sup)q(erQCD)h(theory)f(with)f(an)g(adjoin)o(t)g(c)o(hiral)g(su-)1051
2529 y(p)q(er\014eld)k(\010)f(and)f(pairs)h(of)f(c)o(hiral)g(\\quark")g
(sup)q(er\014elds)1051 2579 y Fm(Q)1084 2564 y Fk(i)1112
2579 y Ft(and)1202 2569 y Fj(e)1193 2579 y Fm(Q)1226
2585 y Fk(i)1254 2579 y Ft(in)h(represen)o(tations)i
Fm(R)1623 2585 y Fk(i)1651 2579 y Ft(and)p 1732 2546
V 14 w Fm(R)1764 2585 y Fk(i)1792 2579 y Ft(\()p Fe(i.e.)d
Ft(the)p eop
%%Page: 5 5
5 4 bop 1822 106 a Ft(5)-47 235 y(\014eld)17 b(con)o(ten)o(t)h(of)f(an)
g Fm(N)5 b Ft(=2)18 b(sup)q(erQCD)g(theory\).)30 b(In)-47
285 y(addition)14 b(to)i(the)g(gauge)f(coupling)g Fm(\034)5
b Ft(,)15 b(w)o(e)h(add)f(sup)q(er-)-47 335 y(p)q(oten)o(tial)d(terms)g
Fl(W)j Ft(=)341 303 y Fj(P)385 347 y Fk(i)405 335 y Fm(\025)429
341 y Fk(i)443 335 y Fm(Q)476 320 y Fk(i)490 335 y Ft(\010)528
324 y Fj(e)520 335 y Fm(Q)553 341 y Fk(i)567 335 y Ft(.)i(By)c(the)g
(global)-47 384 y(symmetries)f(of)h(this)h(theory)m(,)f(the)i
(anomalous)c(dimen-)-47 434 y(sions)h(of)g(the)i(quarks)e(and)h(an)o
(ti-quarks)f(are)h(the)g(same:)-47 484 y Fm(\015)-26
492 y Fk(Q)0 484 y Fc(i)j Ft(=)c Fm(\015)96 500 y Fj(e)92
502 y Fk(Q)118 506 y Fc(i)145 484 y Fl(\021)g Fm(\015)210
490 y Fk(i)224 484 y Ft(.)18 b(F)m(rom)13 b(\(12)p
 (#equation.12) [[163 601 173 613] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g
(and)h(\(13)p
 (#equation.13) [[204 601 214 613] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g(w)
o(e)g(then)g(get)-33 579 y Fm(\014)-10 585 y Fk(\034)53
579 y Fl(/)41 b Ft(\()p Fm(\015)163 585 y Fr(\010)190
579 y Fl(\000)p Ft(1\))p Fm(T)6 b Ft(\()p Fm(Ad)p Ft(\))j(+)425
540 y Fj(X)449 628 y Fk(i)485 579 y Ft(\(1+2)p Fm(\015)596
585 y Fk(i)610 579 y Ft(\))p Fm(T)d Ft(\()p Fm(R)704
585 y Fk(i)718 579 y Ft(\))p Fm(;)-47 680 y(\014)-24
686 y Fk(\025)-4 690 y Fc(i)53 680 y Fl(/)41 b Fm(\015)147
686 y Fr(\010)183 680 y Ft(+)9 b(2)p Fm(\015)266 686
y Fk(i)281 680 y Fm(;)472 b Ft(\(15\))-47 754 y(implying)12
b(the)k(existence)h(of)e(a)g(\014xed)g(line)g(and)g(an)g(ex-)-47
804 y(actly)h(marginal)e(op)q(erator)j(precisely)h(when)f(the)g(one-)
-47 854 y(lo)q(op)d(b)q(eta)i(function)f(v)n(anishes,)480
823 y Fj(P)524 866 y Fk(i)545 854 y Fm(T)6 b Ft(\()p
Fm(R)623 860 y Fk(i)636 854 y Ft(\))14 b(=)g Fm(T)6 b
Ft(\()p Fm(Ad)p Ft(\),)-47 904 y(since)18 b(then)h Fm(\014)180
910 y Fk(\034)219 904 y Fl(/)269 872 y Fj(P)313 916 y
Fk(i)333 904 y Fm(T)6 b Ft(\()p Fm(R)411 910 y Fk(i)425
904 y Ft(\))p Fm(\014)464 910 y Fk(\025)484 914 y Fc(i)499
904 y Ft(.)30 b(In)17 b(addition,)g(one)-47 953 y(can)22
b(c)o(hec)o(k)h(that)f(the)g(\014xed)h(line)e(is)h(appro)o(ximately)-47
1003 y Fm(\034)d Ft(=)14 b Fm(\025)60 1009 y Fk(i)90
1003 y Ft(at)h(w)o(eak)g(coupling)g(whic)o(h)g(are)h(the)g
Fm(N)5 b Ft(=2)15 b(su-)-47 1053 y(p)q(ersymmetric)e(v)n(alues)g(of)h
(the)g(couplings.)-47 1135 y Fn(2.2.)24 b(Lo)o(w-energy)j(e\013ectiv)o
(e)f(actions)h(and)g(S-)51 1185 y(dualit)o(y)13 b(groups)-6
1234 y Ft(Evidence)18 b(for)d(S-dualit)o(y)g(in)h(these)i(scale-in)o(v)
n(arian)o(t)-47 1284 y Fm(N)5 b Ft(=2)13 b(theories)i(can)f(b)q(e)g
(found)f(b)o(y)h(lo)q(oking)e(at)h(the)h(ex-)-47 1334
y(act)19 b(lo)o(w-energy)g(e\013ectiv)o(e)i(action)e(on)g(the)h(Coulom)
o(b)-47 1384 y(branc)o(h)14 b(of)f(their)i(mo)q(duli)c(spaces.)-6
1434 y(Classically)m(,)29 b(the)e(Coulom)o(b)e(branc)o(h)j
Fl(M)f Ft(is)g(the)-47 1483 y(manifold)16 b(of)i(degenerate)j(v)n(acua)
e(where)h(the)f(quarks)-47 1533 y(ha)o(v)o(e)h(zero)i(v)o(evs)f(while)f
(the)i(Higgs)e(\014eld)h(tak)o(es)g(v)n(al-)-47 1583
y(ues)g(in)g(the)g(\(complexi\014ed\))f(Cartan)g(subalgebra)h(of)-47
1633 y(the)e(gauge)g(group.)32 b(This)19 b(breaks)h(the)f(gauge)g
(group)-47 1683 y(as)14 b Fm(G)e Fl(!)f Fm(U)5 b Ft(\(1\))188
1668 y Fk(r)221 1683 y Ft(where)15 b Fm(r)e Ft(=rank\()p
Fm(G)p Ft(\).)19 b(Th)o(us)c(\(lo)q(cally\))-47 1732
y Fl(M)g(')h Fn(C)100 1717 y Fk(r)119 1732 y Ft(,)g(and)h(w)o(e)f(will)
f(denote)i(its)g(co)q(ordinates)g(b)o(y)-47 1782 y Fl(h)p
Fm(\036)p Fl(i)10 1795 y Fk(k)30 1782 y Ft(,)d Fm(k)e
Ft(=)g(1)p Fm(;)7 b(:)g(:)g(:)t(;)g(r)q Ft(.)-6 1832
y(Quan)o(tumly)m(,)h(a)h(non-renormalization)e(theorem)i(im-)-47
1882 y(plies)21 b(these)i(\015at)e(directions)i(are)e(not)h(lifted.)40
b(Note)-47 1932 y(that)11 b(the)h(p)q(oin)o(t)f Fl(h)p
Fm(\036)p Fl(i)272 1944 y Fk(k)304 1932 y Ft(=)h(0)f(will)f(b)q(e)i
(the)g(scale-in)o(v)n(arian)o(t)-47 1982 y(\(\014xed-p)q(oin)o(t\))e(v)
n(acuum,)g(but)g(that)h(mo)o(ving)d(a)o(w)o(a)o(y)h(from)-47
2031 y(it)26 b(along)g(the)i(Coulom)o(b)d(branc)o(h)i(to)g(p)q(oin)o
(ts)g(with)-47 2081 y Fl(h)p Fm(\036)p Fl(i)10 2094 y
Fk(k)48 2081 y Fl(6)p Ft(=)18 b(0)g(sp)q(on)o(taneously)g(breaks)h(the)
f(scale)g(in)o(v)n(ari-)-47 2131 y(ance)25 b(of)e(the)i(underlying)f
(theory)h(and)f(generically)-47 2181 y(\015o)o(ws)16
b(to)h(the)g(IR-free)g Fm(U)5 b Ft(\(1\))414 2166 y Fk(r)448
2181 y Ft(Maxw)o(ell)16 b(theory)m(.)26 b(The)-47 2231
y(couplings)19 b Fm(\034)159 2237 y Fk(j)r(k)215 2231
y Ft(of)g(this)h(lo)o(w-energy)f(e\013ectiv)o(e)i Fm(U)5
b Ft(\(1\))820 2216 y Fk(r)-47 2280 y Ft(theory)14 b(can)g(receiv)o(e)h
(quan)o(tum)e(corrections.)-6 2330 y(The)22 b Fm(U)5
b Ft(\(1\))22 b Fm(S)r(L)p Ft(\(2)p Fm(;)7 b Fn(Z)p Ft(\))23
b(EM-dualit)o(y)d(group)i(gener-)-47 2380 y(alizes)27
b(straigh)o(t-forw)o(ardly)f(to)h(an)f Fm(S)r(p)p Ft(\(2)p
Fm(r)o(;)7 b Fn(Z)p Ft(\))27 b(EM-)-47 2430 y(dualit)o(y)18
b(group)i(in)f(the)i Fm(U)5 b Ft(\(1\))441 2415 y Fk(r)479
2430 y Ft(theory)m(.)36 b(\(Note)20 b(that)-47 2480 y(the)11
b(app)q(earance)g(of)f(this)h(lo)o(w-energy)f(EM-dualit)o(y)f(has)-47
2529 y(nothing)15 b(to)i(do)f(with)g(the)h(S-dualit)o(y)e(w)o(e)h(are)h
(lo)q(oking)-47 2579 y(for!\))22 b(Seib)q(erg)17 b(and)e(Witten)g([13)p
 (#cite.SeibergWittenI) [[177 98 187 110] [1 1 1 [3 3]] [0 0 1]] pdfm 
(])f(observ)o(ed)j(that)e(this)957 235 y(EM-dualit)o(y)k(implies)g
(that)i(the)g(e\013ectiv)o(e)h(couplings)957 285 y Fm(\034)975
291 y Fk(j)r(k)1030 285 y Ft(need)d(not)g(b)q(e)h(a)e(single-v)n(alued)
g(function)h(on)g Fl(M)957 335 y Ft(\(the)12 b(Coulom)o(b)e(branc)o
(h\),)j(but)f(can)g(b)q(e)h(a)f(section)g(of)g(an)957
384 y Fm(S)r(p)p Ft(\(2)p Fm(r)o(;)7 b Fn(Z)p Ft(\))k(bundle)h(on)f
Fl(M)p Ft(.)17 b(W)m(e)11 b(enco)q(de)i(the)f Fm(S)r(p)p
Ft(\(2)p Fm(r)o(;)7 b Fn(Z)p Ft(\))957 434 y(\\p)q(erio)q(dicit)o(y")17
b(of)h Fm(\034)1280 440 y Fk(j)r(k)1334 434 y Ft(geometrically)f(b)o(y)
h(taking)g Fm(\034)1807 440 y Fk(j)r(k)957 484 y Ft(to)13
b(b)q(e)h(the)g(complex)e(structure)j(of)e(a)g Fm(T)1584
469 y Fr(2)p Fk(r)1632 484 y Ft(torus)h(whic)o(h)957
534 y(v)n(aries)k(o)o(v)o(er)g Fl(M)p Ft(.)30 b(It)18
b(is)f(sho)o(wn)h(in)g([16)p
 (#cite.DonagiWitten) [[444 589 455 601] [1 1 1 [3 3]] [0 0 1]] pdfm 
(])e(that)i(the)h(to-)957 584 y(tal)j(space)i(of)e Fm(T)1234
569 y Fr(2)p Fk(r)1292 584 y Ft(\014b)q(ered)j(o)o(v)o(er)e
Fl(M)f Ft(is)h(the)h(phase)957 633 y(space)19 b(of)f(an)g(in)o
(tegrable)g(system.)31 b(F)m(urthermore,)19 b(in)957
683 y(ev)o(ery)c(kno)o(wn)g(example)e(it)i(turns)g(out)g(that)g(this)g
(aux-)957 733 y(iliary)10 b(torus)j Fm(T)1197 718 y Fr(2)p
Fk(r)1232 733 y Ft(=Jac\(\006)1370 739 y Fk(r)1389 733
y Ft(\),)g(the)f(Jacobian)g(torus)h(of)f(a)957 783 y(gen)o(us-)p
Fm(r)i Ft(Reimann)d(surface)k(\006)1449 789 y Fk(r)1467
783 y Ft(,)e(though)g(no)g(explana-)957 833 y(tion)g(of)g(this)h(fact)g
(is)g(kno)o(wn.)998 883 y(The)19 b(solutions)g(for)f(the)i(lo)o
(w-energy)e(e\013ectiv)o(e)i(ac-)957 932 y(tion)9 b(on)h(the)h(Coulom)o
(b)d(branc)o(h)i(are)h(usually)e(presen)o(ted)957 982
y(in)16 b(terms)i(of)e(the)i(\006)1283 988 y Fk(r)1301
982 y Ft(.)28 b(No)18 b(systematic)e(pro)q(cedure)k(is)957
1032 y(kno)o(wn)d(for)g(generating)g(these)i(families)c(of)i(Riemann)
957 1082 y(surfaces.)h(The)13 b(solutions)e(ha)o(v)o(e)h(so)g(far)g(b)q
(een)h(found)f(ei-)957 1132 y(ther)h(b)o(y)e(making)f(simple)g(guesses)
k(for)e(the)g(form)e(of)i(the)957 1181 y(surfaces)g(or)e(for)g(the)h
(in)o(tegrable)f(system,)h(or)f(from)f(t)o(yp)q(e)957
1231 y(I)q(IB)23 b(string)g(compacti\014cations)e(as)i(review)o(ed)h
(in)e(A.)957 1281 y(Klemm's)10 b(lectures)k(in)e(this)g(sc)o(ho)q(ol)h
(or,)f(most)f(recen)o(tly)m(,)957 1331 y(from)g(brane)j
(con\014gurations)g(in)f(t)o(yp)q(e)h(I)q(IA)f(string)h(the-)957
1381 y(ory)f(or)h(M-theory)h([17)p
 (#cite.Witten9703) [[378 386 388 398] [1 1 1 [3 3]] [0 0 1]] pdfm (].)
998 1431 y(The)j(scale-in)o(v)n(arian)o(t)d Fm(N)5 b
Ft(=2)17 b(theories)h(whose)f(asso-)957 1480 y(ciated)11
b(Riemann)e(surfaces)j(w)o(ere)g(kno)o(wn)f(\(b)q(efore)h([17)p
 (#cite.Witten9703) [[498 362 508 374] [1 1 1 [3 3]] [0 0 1]] pdfm 
(]\))957 1530 y(are)d(the)h(theories)h(with)e(a)g(single)g
Fm(S)r(U)c Ft(,)10 b Fm(S)r(O)q Ft(,)g(or)g Fm(S)r(p)g
Ft(gauge)957 1580 y(group)16 b(and)h(quarks)g(in)g(the)g(fundamen)o
(tal)e(\(de\014ning\))957 1630 y(represen)o(tation)22
b([18)p  (#cite.APS) [[371 326 381 338] [1 1 1 [3 3]] [0 0 1]] pdfm 
(,19)p  (#cite.AS) [[384 326 394 338] [1 1 1 [3 3]] [0 0 1]] pdfm (].)
38 b(It)21 b(is)g(found)g(in)f(all)g(these)957 1680 y(cases)f(that)e
(these)i(curv)o(es)g(dep)q(end)g(on)e(the)h(marginal)957
1729 y(coupling)c Fm(\034)19 b Ft(in)c(suc)o(h)h(a)e(w)o(a)o(y)g(that)h
(they)h(are)f(in)o(v)n(arian)o(t)957 1779 y(under)h(iden)o
(ti\014cations)f(of)f Fm(\034)20 b Ft(under)c(a)f(discrete)i(group)957
1829 y(isomorphic)11 b(to)h(\000)1241 1814 y Fr(0)1260
1829 y Ft(\(2\))g Fl(\032)f Fm(S)r(L)p Ft(\(2)p Fm(;)c
Fn(Z)p Ft(\).)19 b(This)13 b(is)f(evidence)957 1879 y(that)18
b(the)h(S-dualit)o(y)e(group)h(of)g(all)f(these)i(theories)h(is)957
1929 y(isomorphic)12 b(to)i(\000)1244 1914 y Fr(0)1262
1929 y Ft(\(2\).)998 1978 y(\000)1024 1984 y Fr(0)1043
1978 y Ft(\(2\))k(is)f(the)i(subset)g(of)e Fm(S)r(L)p
Ft(\(2)p Fm(;)7 b Fn(Z)p Ft(\))19 b(matrices)e(with)957
2028 y(ev)o(en)f(upp)q(er)h(o\013-diagonal)d(en)o(try)m(,)i(or)g(equiv)
n(alen)o(tly)m(,)f(is)957 2078 y(isomorphic)9 b(to)h(the)h(subgroup)g
(of)f Fm(S)r(L)p Ft(\(2)p Fm(;)d Fn(Z)p Ft(\))k(generated)957
2128 y(b)o(y)h Fm(S)j Ft(and)e Fm(T)1163 2113 y Fr(2)1181
2128 y Ft(.)18 b(The)13 b(only)f(relation)g(satis\014ed)h(b)o(y)f
(these)957 2178 y(generators)19 b(is)g Fm(S)1236 2163
y Fr(2)1274 2178 y Ft(=)h(1,)f(th)o(us)g(c)o(haracterizing)g(\000)1771
2163 y Fr(0)1789 2178 y Ft(\(2\))957 2228 y(more)13 b(abstractly)h(as)g
(the)g(group)g(freely)g(generated)i(b)o(y)957 2277 y
Fm(S)g Ft(and)d Fm(T)1108 2262 y Ff(0)1133 2277 y Ft(with)g(the)h(one)f
(relation)g Fm(S)1552 2262 y Fr(2)1583 2277 y Ft(=)f(1.)17
b(Since)d(the)957 2327 y(S-dualit)o(y)e(group)h(is)g(the)h(group)f(w)o
(e)g(divide)g(the)h(classi-)957 2377 y(cal)k(coupling)g(space)i(b)o(y)m
(,)f(w)o(e)g(exp)q(ect)i(a)d(single)h(p)q(oin)o(t)957
2427 y(with)j(a)g Fn(Z)1132 2433 y Fr(2)1173 2427 y Ft(holonom)o(y)d
(in)j(the)h(quan)o(tum)d(coupling)957 2477 y(space)13
b(at)e(the)i(\014xed)f(p)q(oin)o(t)g(of)f(the)i Fm(S)r
Ft(-transformation's)957 2526 y(action.)k(Indeed,)d(the)f(fundamen)o
(tal)e(domain)g(of)h(\000)1771 2511 y Fr(0)1789 2526
y Ft(\(2\))957 2576 y(in)i(the)h Fm(\034)k Ft(upp)q(er)d(half-plane)d
(can)i(b)q(e)g(tak)o(en)g(to)g(b)q(e)g(the)p eop
%%Page: 6 6
6 5 bop 47 106 a Ft(6)47 235 y(region)16 b Fl(\000)p
Ft(1)p Fl(\024)p Ft(Re)p Fm(\034)5 b Fl(\024)p Ft(1)16
b(and)g Fl(j)p Fm(\034)5 b Fl(j)14 b(\025)h Ft(1.)24
b(The)17 b(b)q(oundaries)47 285 y(of)i(this)h(domain)d(are)j(iden)o
(ti\014ed)f(so)h(that)f(there)i(is)e(a)47 335 y Fn(Z)76
341 y Fr(2)112 335 y Ft(orbifold)c(p)q(oin)o(t)i(at)f
Fm(\034)22 b Ft(=)16 b Fm(i)h Ft(and)g(\\cusps")h(with)e(no)47
384 y(holonom)o(y)9 b(at)i Fm(\034)16 b Ft(=)c Fm(i)p
Fl(1)f Ft(\(w)o(eak)g(coupling\))g(and)g Fm(\034)16 b
Ft(=)c Fl(\006)p Ft(1)47 434 y(\(ultra-strong)i(coupling\).)89
484 y(An)j(imp)q(ortan)o(t)f(feature)j(of)e(this)g(dualit)o(y)g(group)g
(is)47 534 y(that)f(it)e(do)q(es)i(not)g(remo)o(v)o(e)e(all)g(the)i
(ultra-strong)f(cou-)47 584 y(pling)k(p)q(oin)o(ts)h(from)f(the)i(quan)
o(tum)d(coupling)h(space.)47 633 y(An)32 b(imp)q(ortan)o(t)f(exception)
h(is)g Fm(S)r(U)5 b Ft(\(2\))33 b Fm(N)5 b Ft(=2)32 b(su-)47
683 y(p)q(erQCD)19 b(with)e(four)h(fundmen)o(tal)d(\015a)o(v)o(ors.)30
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25 b(He)18 b(argues)f(that)47 1732 y(5-brane)e(con\014gurations)g(in)f
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320 y Ft(=)i Fm(f)1440 326 y Fk(i)1454 320 y Fm(;)89
b(k)1577 326 y Fr(0)1607 320 y Fl(\021)12 b Fm(k)1673
326 y Fk(n)p Fr(+1)1749 320 y Fl(\021)f Ft(0)p Fm(:)38
b Ft(\(16\))1051 404 y(In)15 b(these)h(mo)q(dels)e(the)h
Fm(x)1448 389 y Fr(6)1481 404 y Ft(co)q(ordinate)g(is)g(in\014nite)f
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1051 504 y(drical)20 b(transv)o(erse)i(space.)39 b(\006)21
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y Fr(10)1051 603 y Ft(cylinder)h(giv)o(e)e(the)i Fm(n)f
Ft(complex)f(coupling)h(parameters)1051 653 y(of)f(the)i(mo)q(del.)1093
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34 b(The)19 b(indistinguishablit)o(y)e(of)1051 902 y(the)g(punctures)h
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2579 y(lence)c(of)g(the)g(theories)h(under)g(a)e(large)h(set)g(of)f
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384 y(mo)q(duli)e(spaces)16 b(of)d(v)n(acua)g(of)h(the)g(t)o(w)o(o)f
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783 y(The)d(standard)h(strategy)f(to)g(do)g(this)g(is)g(to)g(\014nd)g
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1181 y(to)c(the)h(IR)f(\(so-called)g(\\dangerously)g(irrelev)n(an)o(t)g
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1530 y(try)c(un)o(brok)o(en)g(so)g(that)h(the)f(v)n(acuum)f(energy)i
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1829 y(v)n(acuum)16 b(state)j(at)f(the)h(coupling)e(is)h(v)n(aried.)30
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1978 y(IR.)998 2028 y(This)10 b(strategy)h(w)o(as)f(pursued)h(in)e(the)
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2178 y(related)23 b(AF)f(or)g(scale-in)o(v)n(arian)o(t)f
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2477 y(p)q(erturbation)14 b(used)g(to)f(\015o)o(w)g(do)o(wn)g(to)g(the)
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2028 y(plying)i(an)h(in)o(to)f(mapping)f(of)i(the)h(the)f
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2526 y(gauge)h(groups)f([17)p
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1051 676 y(2.)37 b(C.)16 b(Mon)o(tonen)h(and)f(D.)g(Oliv)o(e,)h
Fa(Ph)o(ys.)f(Lett.)h Ft(72B)p 1866 683 V 1121 726 a(\(1977\))c(117.)
1051 776 y(3.)37 b(E.)24 b(Witten)g(and)g(D.)g(Oliv)o(e,)i
Fa(Ph)o(ys.)e(Lett.)h Ft(78B)p 1866 783 V 1121 825 a(\(1978\))13
b(97.)1051 875 y(4.)37 b(H.)13 b(Osb)q(orn,)h Fa(Ph)o(ys.)g(Lett.)g
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975 y Fa(Nucl.)13 b(Ph)o(ys.)h Ft(B432)p 1344 982 92
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b(Ph)o(ys.)h Ft(B431)p 1344 1082 V 13 w(\(1994\))g(484.)1051
1124 y(7.)37 b(I.)23 b(Jac)o(k)g(and)h(H.)f(Osb)q(orn,)k
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a(\(1990\))13 b(647.)1051 1224 y(8.)37 b(H.)32 b(Sono)q(da,)k
(hep-th/9205085,)f Fa(Nucl.)d(Ph)o(ys.)1121 1274 y Ft(B383)p
1121 1281 V 32 w(\(1992\))g(173;)41 b(hep-th/9205084,)35
b Fa(Nucl.)1121 1324 y(Ph)o(ys.)13 b Ft(B394)p 1234 1331
V 14 w(\(1993\))g(302.)1051 1373 y(9.)37 b(B.)11 b(Dolan,)f
(hep-th/9403070,)g Fa(In)o(t.)h(J.)g(Mo)q(d.)g(Ph)o(ys.)1121
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1473 y(10.)j(N.)32 b(Seib)q(erg,)37 b(hep-th/9411149,)e
Fa(Nucl.)d(Ph)o(ys.)1121 1523 y Ft(B435)p 1121 1530 92
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1622 y(12.)i(N.)32 b(Seib)q(erg,)37 b(hep-ph/9309335,)e
Fa(Ph)o(ys.)e(Lett.)1121 1672 y Ft(318B)p 1121 1679 V
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(Witten,)i(hep-th/9407087,)1121 1772 y Fa(Nucl.)13 b(Ph)o(ys.)h
Ft(B462)p 1344 1779 V 13 w(\(1994\))g(19.)1051 1822 y(14.)i(V.)f(No)o
(vik)o(o)o(v,)f(M.)h(Shifman,)e(A.)i(V)m(ainsh)o(tein,)g(and)1121
1872 y(V.)20 b(Zakharo)o(v,)h Fa(Ph)o(ys.)f(Lett.)h Ft(166B)p
1616 1879 V 20 w(\(1986\))e(329;)1121 1921 y(M.)11 b(Shifman,)e(A.)j(V)
m(ainsh)o(tein,)f(and)g(V.)h(Zakharo)o(v,)1121 1971 y
Fa(Ph)o(ys.)f(Lett.)h Ft(166B)p 1332 1978 V 11 w(\(1986\))f(334;)g(M.)h
(Shifman)d(and)1121 2021 y(A.)k(V)m(ainsh)o(tein,)g Fa(Nucl.)g(Ph)o
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2071 y(15.)j(R.)i(Leigh)g(and)h(M.)g(Strassler,)i(hep-th/9503121,)1121
2121 y Fa(Nucl.)13 b(Ph)o(ys.)h Ft(B447)p 1344 2128 V
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Ft(B460)p 1344 2227 V 13 w(\(1996\))g(299.)1051 2270
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2370 y(Shap)q(ere,)11 b(hep-th/9505100,)f Fa(Ph)o(ys.)g(Rev.)g(Lett.)g
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2469 y(19.)j(P)m(.C.)39 b(Argyres)j(and)f(A.D.)e(Shap)q(ere,)49
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Ft(370B)p 463 342 V 12 w(\(1996\))g(76;)f(hep-)22 384
y(th/9602031,)g Fa(Ph)o(ys.)j(Lett.)g Ft(375B)p 468 391
V 13 w(\(1996\))f(175;)f(M.)22 434 y(Berk)o(o)q(oz,)h(P)m(.)e(Cho,)h(P)
m(.)g(Kraus,)g(M.)g(Strassler,)h(hep-)22 484 y(th/9705003.)-47
534 y(22.)i(P)m(.)h(Argyres,)h(R.)f(Plesser,)i(and)e(N.)g(Seib)q(erg,)h
(hep-)22 584 y(th/9603042,)12 b Fa(Nucl.)h(Ph)o(ys.)h
Ft(B471)p 476 591 V 14 w(\(1996\))f(159.)-47 633 y(23.)i(P)m(.)f
(Argyres,)i(R.)e(Plesser,)i(and)f(A.)f(Shap)q(ere,)i(hep-)22
683 y(th/9608129,)c Fa(Nucl.)h(Ph)o(ys.)h Ft(B483)p 476
690 V 14 w(\(1997\))f(172.)-47 733 y(24.)i(P)m(.)e(Argyres)i(and)f(A.)f
(Buc)o(hel,)i(unpublished.)p eop
%%Trailer
end 
end

userdict /end-hook known{end-hook}if
%%EOF