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%%Title: arXiv:hep-th/9901001v3  10 May 1999
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m(h)h(cuts)-127 1362 y(whic)m(h)e(giv)m(e)f Fs(S)6 b(L)p
FB(\(2)p Fs(;)17 b Ft(Z)p FB(\))36 b(transformations)e(comp)s(ensating)
i(for)f(the)i(mono)s(dromies.)52 b(It)36 b(w)m(as)h(suggested)-127
1511 y(b)m(y)44 b(Zwiebac)m(h)g(et)g(al.)705 1475 y Fr(1\))14
b FB(-)g Fr(6\))938 1511 y FB(that)43 b(when)i(7-branes)e(of)g(a)g(sp)s
(eci\014c)h(com)m(bination)e(are)h(on)g(top)h(of)f(eac)m(h)-127
1661 y(other,)35 b(v)-5 b(arious)34 b(groups)i(con)m(taining)d
(exceptional)h(groups,)i(a\016ne)f(groups,)h(and)f(other)g(exotic)f
(groups)-127 1810 y(app)s(ear)k(as)h(gauge)f(groups)g(on)g(the)h
(7-branes.)61 b(T)-8 b(o)38 b(construct)i Fs(A)2347 1825
y Fk(n)2394 1810 y FB(,)g Fs(D)2542 1825 y Fk(n)2627
1810 y FB(and)e Fs(E)2894 1825 y Fk(n)2980 1810 y FB(gauge)g(theories,)
i(w)m(e)-127 1960 y(need)47 b(three)h(t)m(yp)s(es)g(of)e(7-branes)g
(with)g(lab)s(els)f([1)p Fs(;)17 b FB(0],)50 b([1)p Fs(;)17
b Fm(\000)p FB(1])46 b(and)h([1)p Fs(;)17 b FB(1].)84
b(F)-8 b(ollo)m(wing)3229 1923 y Fr(1\))p Fk(;)12 b Fr(2\))3390
1960 y FB(,)50 b(w)m(e)d(call)-127 2109 y(them)35 b(A-,)h(B-)f(and)h
(C-branes,)h(resp)s(ectiv)m(ely)-8 b(.)54 b(The)37 b(degrees)g(of)e
(freedom)g(con)m(tributing)f(to)i(the)g(gauge)-127 2258
y(symmetry)d(enhancemen)m(t)i(arise)e(from)f(string)g(junctions)h
(connecting)h(some)f(of)g(these)h(7-branes.)46 b(The)-127
2408 y(\()p Fs(p;)17 b(q)t FB(\))32 b(c)m(harges)i(of)e(strings)g
(attac)m(hed)h(on)g(a)f(7-brane)g(should)h(b)s(e)g(equal)f(to)g(the)h
(lab)s(el)e(of)h(the)h(7-branes.)19 2557 y(It)27 b(is)g(kno)m(wn)i
(that)d(t)m(yp)s(e)i(I)s(IB)g(theory)f(compacti\014ed)g(on)g
Fs(S)2165 2521 y Fr(2)2204 2557 y FB(,)h(whic)m(h)g(can)f(also)g(b)s(e)
g(expressed)j(as)d(a)g(K3)-127 2707 y(compacti\014cation)34
b(of)i(F-theory)-8 b(,)37 b(is)f(dual)g(to)f(heterotic)i(string)e
(theory)i(compacti\014ed)f(on)g Fs(T)3378 2671 y Fr(2)3417
2707 y FB(.)55 b(On)36 b Fs(S)3731 2671 y Fr(2)-127 2856
y FB(w)m(e)31 b(ha)m(v)m(e)g(24)f(7-branes)g(corresp)s(onding)g(to)f
(singular)g(\014b)s(ers)i(of)e(elliptically)d(\014b)s(ered)31
b(K3.)42 b(As)30 b(is)g(p)s(oin)m(ted)-127 3006 y(out)f(in)140
2970 y Fr(5\))206 3006 y FB(,)g(the)g(junction)g(lattice)e(of)i(the)g
(t)m(yp)s(e)h(I)s(IB)f(theory)g(on)g Fs(S)2258 2970 y
Fr(2)2326 3006 y FB(has)g(the)h(signature)e(\(+)3194
2970 y Fr(18)3269 3006 y Fs(;)17 b Fm(\000)3390 2970
y Fr(2)3429 3006 y FB(\),)30 b(whic)m(h)-127 3155 y(is)j(the)h(same)g
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(string)g(theory)-8 b(.)48 b(Because)35 b(there)-127
3305 y(is)f(unique)h(18)24 b(+)f(2-dimensional)32 b(Loren)m(tzian)i
(self-dual)f(ev)m(en)k(lattice,)c(it)h(is)g(trivial)e(to)j(sho)m(w)h
(that)e(the)-127 3454 y(junction)24 b(lattice)g(is)g(iden)m(tical)g(to)
g(the)i(Narain)d(lattice)h(on)h(heterotic)f(side.)41
b(The)26 b(purp)s(ose)g(of)f(the)g(presen)m(t)-127 3603
y(pap)s(er)j(is,)g(b)m(y)h(means)f(of)f(dualities,)g(to)h(establish)f
(the)h(one)h(to)e(one)h(corresp)s(ondence)i(b)s(et)m(w)m(een)g(in)m(v)
-5 b(arian)m(t)-127 3753 y(c)m(harges)39 b(of)e(junctions)h(in)f(t)m
(yp)s(e)i(I)s(IB)f(theory)g(and)g(the)g(quan)m(tum)g(n)m(um)m(b)s(ers)h
(sp)s(ecifying)e(p)s(erturbativ)m(e)-127 3902 y(states)c(of)g(a)f
(heterotic)g(string.)19 4052 y(These)48 b(t)m(w)m(o)e(theories)g(are)f
(dual)g(in)g(the)h(manner)f(w)m(e)i(no)m(w)f(describ)s(e.)84
b(On)45 b Fs(S)3036 4016 y Fr(2)3076 4052 y FB(,)k(w)m(e)d(ha)m(v)m(e)h
(24)e(7-)-127 4201 y(branes.)73 b(Although)41 b(there)i(are)f(man)m(y)h
(represen)m(tations)g(of)f(these)h(7-branes,)i(w)m(e)e(restrict)f
(ourselv)m(es)-127 4351 y(to)c(the)h(case)h(that)f(they)g(are)g
(represen)m(ted)i(as)e(four)f(sets)i(of)e Fs(A)2243 4314
y Fr(4)2283 4351 y Fs(B)5 b(C)i FB(.)62 b(Once)39 b(w)m(e)h(obtain)e
(some)g(results)-127 4500 y(for)d(this)f(bac)m(kground,)j(w)m(e)g(can)e
(extend)i(them)e(to)f(other)i(regions)e(of)h(mo)s(duli)e(space)j(b)m(y)
g(a)f(con)m(tin)m(uous)-127 4649 y(deformation)41 b(of)h(the)h(bac)m
(kground.)76 b(In)43 b(the)g(w)m(eak)h(coupling)d(limit)f(\()p
Fs(g)2615 4664 y Fr(str)2746 4649 y Fm(!)45 b FB(0\),)g(a)e(pair)e
(consisting)-127 4799 y(of)36 b(a)g(B-brane)g(and)g(a)g(C-brane)h(is)e
(reduced)j(to)e(an)g(orien)m(tifold)e(7-plane.)2657 4763
y Fr(7\))2758 4799 y FB(Therefore,)39 b(in)c(this)h(limit,)-127
4948 y(w)m(e)41 b(ha)m(v)m(e)g(four)e(orien)m(tifold)e(7-planes,)k(and)
e(the)h(compacti\014cation)e(manifold)f Fs(S)2942 4912
y Fr(2)3020 4948 y FB(b)s(ecomes)j(a)g Fs(T)3575 4912
y Fr(2)3614 4948 y Fs(=)p Ft(Z)3731 4963 y Fr(2)-127
5098 y FB(orien)m(tifold.)73 b(By)44 b(means)g(of)f(a)g(T)g(dualit)m(y)
g(transformation)e(along)h(t)m(w)m(o)i(directions)f(on)g
Fs(T)3366 5062 y Fr(2)3405 5098 y Fs(=)p Ft(Z)3522 5113
y Fr(2)3562 5098 y FB(,)j(the)-127 5247 y(theory)c(is)f(transformed)g
(in)m(to)g(t)m(yp)s(e)h(I)g(theory)g(on)f Fs(T)1891 5211
y Fr(2)1930 5247 y FB(,)j(and)d(it)g(is)g(S)g(dual)g(to)g(the)h
Fs(S)6 b(O)s FB(\(32\))40 b(heterotic)-127 5397 y(string)32
b(theory)h(on)g Fs(T)663 5361 y Fr(2)702 5397 y FB(.)1797
5687 y(2)p eop
%%Page: 3 3
3 2 bop 137 158 a FB(Through)45 b(this)f(series)i(of)e(dualit)m(y)f
(transformations,)j(w)m(e)g(can)e(relate)g(BPS)h(states)h(in)e
(heterotic)-9 307 y(string)c(theory)h(to)f(those)h(in)e(t)m(yp)s(e)j(I)
s(IB)e(theory)-8 b(.)68 b(Let)40 b(us)h(assume)g(the)g(t)m(w)m(o)g
(compacti\014ed)f(directions)-9 457 y(to)e(b)s(e)h Fs(x)310
420 y Fr(8)388 457 y FB(and)g Fs(x)639 420 y Fr(9)679
457 y FB(,)h(and)e(call)f(momen)m(ta)g(and)i(winding)f(n)m(um)m(b)s
(ers)h(along)e(them)h Fs(m)3133 420 y Fi(0)3133 481 y
Fr(8)3173 457 y FB(,)i Fs(m)3325 420 y Fi(0)3325 481
y Fr(9)3365 457 y FB(,)g Fs(n)3490 472 y Fr(8)3568 457
y FB(and)f Fs(n)3822 472 y Fr(9)3861 457 y FB(.)-9 606
y(\(Generically)-8 b(,)25 b(momen)m(ta)g Fs(m)1059 570
y Fi(0)1059 631 y Fr(8)1124 606 y FB(and)h Fs(m)1392
570 y Fi(0)1392 631 y Fr(9)1457 606 y FB(are)g(not)f(in)m(tegers,)j
(due)e(to)f(the)i(presence)g(of)f(Wilson)e(lines,)i(while)-9
756 y(winding)34 b(n)m(um)m(b)s(ers)i Fs(n)816 771 y
Fr(8)891 756 y FB(and)g Fs(n)1142 771 y Fr(9)1217 756
y FB(are)f(alw)m(a)m(ys)h(in)m(tegers.\))53 b(In)35 b(T)-8
b(able)35 b(I,)i(it)d(is)h(displa)m(y)m(ed)h(ho)m(w)g(the)g(states)-9
905 y(carrying)26 b(these)j(c)m(harges)f(are)f(transformed)f(b)m(y)i
(the)f(dualities.)40 b(As)28 b(is)e(kno)m(wn)j(from)c(this)i(table,)g
(c)m(harges)316 1086 y Fh(T)-8 b(able)32 b(I.)73 b(Ob)5
b(jects)30 b(carrying)g(c)m(harges)h Fg(m)1793 1053 y
Fi(0)1793 1108 y Fk(\026)1870 1086 y Fh(and)f Fg(n)2102
1100 y Fk(\026)2178 1086 y Fh(in)f(eac)m(h)j(theory)e(related)h(b)m(y)f
(dualities.)p 978 1221 1924 5 v 978 1237 V 1570 1342
a Fs(m)1655 1306 y Fi(0)1655 1367 y Fr(8)1949 1342 y
Fs(m)2034 1306 y Fi(0)2034 1367 y Fr(9)2327 1342 y Fs(n)2385
1357 y Fr(8)2677 1342 y Fs(n)2735 1357 y Fr(9)p 978 1391
V 1078 1495 a FB(hetero)150 b Fs(x)1548 1459 y Fr(8)1621
1495 y FB(KK)99 b Fs(x)1927 1459 y Fr(9)1999 1495 y FB(KK)105
b Fs(x)2311 1459 y Fr(8)2383 1495 y FB(F1)110 b Fs(x)2661
1459 y Fr(9)2734 1495 y FB(F1)1083 1645 y(t)m(yp)s(e)34
b(I)155 b Fs(x)1548 1609 y Fr(8)1621 1645 y FB(KK)99
b Fs(x)1927 1609 y Fr(9)1999 1645 y FB(KK)h Fs(x)2306
1609 y Fr(8)2378 1645 y FB(D1)f Fs(x)2656 1609 y Fr(9)2728
1645 y FB(D1)1028 1794 y(t)m(yp)s(e)33 b(I)s(IA)120 b
Fs(x)1568 1758 y Fr(8)1640 1794 y FB(F1)f Fs(x)1927 1758
y Fr(9)1999 1794 y FB(KK)163 b(D0)227 b(D2)1030 1944
y(t)m(yp)s(e)33 b(I)s(IB)122 b Fs(x)1568 1908 y Fr(8)1640
1944 y FB(F1)138 b Fs(x)1946 1908 y Fr(9)2019 1944 y
FB(F1)119 b Fs(x)2306 1908 y Fr(9)2378 1944 y FB(D1)99
b Fs(x)2656 1908 y Fr(8)2728 1944 y FB(D1)p 978 1993
V -9 2264 a(\()p Fs(P)92 2279 y Fk(\026)138 2264 y Fs(;)17
b(Q)259 2279 y Fk(\026)306 2264 y FB(\))33 b(\()p Fs(\026)c
FB(=)g(8)p Fs(;)17 b FB(9\))33 b(of)f(strings)i(winding)e(along)g(the)i
Fs(x)2103 2228 y Fr(8)2176 2264 y FB(and)g Fs(x)2422
2228 y Fr(9)2495 2264 y FB(directions)f(in)f(t)m(yp)s(e)j(I)s(IB)e
(theory)h(are)-9 2414 y(represen)m(ted)h(b)m(y)e(quan)m(tum)g(n)m(um)m
(b)s(ers)h(of)e(heterotic)g(string)g(as)h(follo)m(ws:)903
2644 y Fs(P)966 2659 y Fr(8)1033 2644 y FB(=)27 b Fs(m)1221
2603 y Fi(0)1221 2669 y Fr(8)1261 2644 y Fs(;)114 b(Q)1479
2659 y Fr(8)1547 2644 y FB(=)27 b Fm(\000)1737 2577 y
FB(1)p 1737 2621 49 4 v 1737 2712 a(2)1797 2644 y Fs(n)1855
2659 y Fr(9)1894 2644 y Fs(;)115 b(P)2099 2659 y Fr(9)2165
2644 y FB(=)28 b Fs(m)2354 2603 y Fi(0)2354 2669 y Fr(9)2394
2644 y Fs(;)114 b(Q)2612 2659 y Fr(9)2679 2644 y FB(=)2793
2577 y(1)p 2793 2621 V 2793 2712 a(2)2852 2644 y Fs(n)2910
2659 y Fr(8)2949 2644 y Fs(:)614 b FB(\(1)3677 2623 y(.)3704
2644 y(2\))-9 2865 y(The)41 b(factor)e(1)p Fs(=)p FB(2)g(on)g(the)i
(righ)m(t-hand)d(side)i(of)f(the)h(second)h(and)f(the)g(fourth)g
(equation)f(in)g(\(1)3581 2843 y(.)3608 2865 y(2\))g(is)h(a)-9
3014 y(consequence)h(of)c(the)h(fact)f(that)h(the)g(T)g(dual)e(of)i(a)f
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g(I)s(IA)-9 3163 y(D-particle)30 b(on)j(an)f(orien)m(tifold)e(8-plane.)
137 3313 y(W)-8 b(e)23 b(should)g(note)g(that)f(in)g(generic)h(7-brane)
g(bac)m(kgrounds,)j(there)e(is)e(a)g(certain)h(subtlet)m(y)g
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(b)s(ers)i(of)e(strings)h(on)g(the)h(t)m(yp)s(e)g(I)s(IB)f(side.)42
b(W)-8 b(e)29 b(explain)g(this)f(in)h(Section)-9 3612
y(3)j(b)s(efore)h(giving)e(the)i(\014nal)f(result.)1029
3945 y Fl(x)p Fy(2.)93 b(Sp)s(ectrum)37 b(of)h(a)h(Heterotic)d(String)
137 4167 y FB(If)30 b(w)m(e)i(assume)f(the)f(left)g(mo)m(v)m(er)h
(\(without)e(tilde\))g(is)h(sup)s(ersymmetric)g(and)h(the)g(righ)m(t)e
(mo)m(v)m(er)i(\(with)-9 4316 y(tilde\))41 b(is)h(b)s(osonic,)j(the)e
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(for)h Fs(T)3269 4280 y Fk(d)3352 4316 y FB(compacti\014ed)-9
4466 y(heterotic)32 b(string)g(theory)h(are)g(giv)m(en)g(b)m(y)698
4644 y Fs(M)802 4607 y Fr(2)p 644 4688 252 4 v 644 4779
a FB(8)p Fs(\031)t(T)809 4794 y Fr(str)934 4711 y FB(=)1047
4644 y(1)p 1047 4688 49 4 v 1047 4779 a(2)1106 4711 y(\()p
Fs(a)1195 4664 y Fk(\026)1195 4733 y Fr(0)1242 4711 y
FB(\))1280 4670 y Fr(2)1341 4711 y FB(+)22 b(\()p Fs(N)1555
4726 y Fr(osc)1676 4711 y FB(+)g Fs(a)p FB(\))28 b(=)2005
4644 y(1)p 2005 4688 V 2005 4779 a(2)2063 4711 y(\()2104
4704 y Fj(e)2101 4711 y Fs(a)2152 4664 y Fk(\026)2152
4733 y Fr(0)2199 4711 y FB(\))2237 4670 y Fr(2)2299 4711
y FB(+)2407 4644 y(1)p 2407 4688 V 2407 4779 a(2)2465
4711 y(\()2507 4702 y Fj(e)2503 4711 y Ft(a)2557 4726
y Fr(0)2597 4711 y FB(\))2635 4670 y Fr(2)2697 4711 y
FB(+)22 b(\()2844 4679 y Fj(f)2833 4711 y Fs(N)2911 4726
y Fr(osc)3031 4711 y Fm(\000)h FB(1\))p Fs(;)345 b FB(\(2)3677
4690 y(.)3704 4711 y(1\))-9 4948 y(where)38 b Fs(\026)c
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b(lab)s(els)f(the)i(compacti\014ed)f(directions,)h(and)g(the)g(constan)
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5098 y(the)h(NS)g(sector)h(and)e(0)h(for)f(the)h(R)g(sector.)57
b(The)37 b(v)-5 b(ariables)36 b Fs(a)2324 5051 y Fk(\026)2324
5120 y Fr(0)2407 5098 y FB(and)2604 5091 y Fj(e)2601
5098 y Fs(a)2652 5051 y Fk(\026)2652 5120 y Fr(0)2736
5098 y FB(represen)m(t)j(the)e(zero)g(mo)s(de)f(of)-9
5247 y(the)29 b(compacti\014ed)g(co)s(ordinates)f Fs(X)1324
5211 y Fk(\026)1371 5247 y FB(,)i(and)f(the)g(v)m(ector)2071
5238 y Fj(e)2067 5247 y Ft(a)2121 5262 y Fr(0)2190 5247
y FB(is)g(the)g(zero-mo)s(de)f(asso)s(ciated)h(with)g(the)g(16-)-9
5397 y(dimensional)f(in)m(ternal)i(space)i(of)f(the)g(righ)m(t)g(mo)m
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%%Page: 4 4
4 3 bop -127 158 a FB(wrapping)28 b(n)m(um)m(b)s(ers,)i(the)f(2)p
Fs(d)13 b FB(+)g(16-dimensional)24 b(v)m(ector)30 b(\()p
Fs(a)2120 111 y Fk(\026)2120 180 y Fr(0)2167 158 y Fs(;)2213
151 y Fj(e)2210 158 y Fs(a)2261 111 y Fk(\026)2261 180
y Fr(0)2308 158 y Fs(;)2356 149 y Fj(e)2352 158 y Ft(a)2406
173 y Fr(0)2446 158 y FB(\))e(tak)m(es)h(v)-5 b(alues)29
b(in)e(the)i(Loren)m(tzian)-127 307 y(self-dual)35 b(ev)m(en)j
(lattice,)e(the)g(so-called)f(`Narain)h(lattice'.)53
b(F)-8 b(or)35 b(simplicit)m(y)-8 b(,)35 b(let)h(us)h(assume)g(the)f
(torus)-127 457 y Fs(T)-56 420 y Fk(d)17 457 y FB(is)c(rectangular.)43
b(Then,)33 b(the)g(zero)g(mo)s(des)g(are)f(quan)m(tized)i(as)524
592 y Fj(0)524 738 y(B)524 788 y(B)524 838 y(B)524 891
y(@)647 693 y Fs(a)698 646 y Fk(\026)698 715 y Fr(0)649
836 y Fj(e)647 843 y Fs(a)698 796 y Fk(\026)698 865 y
Fr(0)653 983 y Fj(e)649 992 y Ft(a)703 1007 y Fr(0)795
592 y Fj(1)795 738 y(C)795 788 y(C)795 838 y(C)795 891
y(A)895 838 y FB(=)998 592 y Fj(0)998 738 y(B)998 788
y(B)998 838 y(B)998 891 y(@)1198 640 y Fk(l)1219 648
y Ff(s)p 1131 656 189 4 v 1131 665 a Fi(p)p 1190 665
36 3 v 55 x Fr(2)p Fk(R)1278 728 y Ff(\026)1346 583 y
Fj(\020)1396 679 y Fs(m)1481 694 y Fk(\026)1550 679 y
FB(+)22 b Ft(w)1729 694 y Fk(\026)1797 679 y Fm(\001)g
Ft(k)h Fm(\000)2038 640 y Fr(1)p 2038 656 36 4 v 2038
713 a(2)2083 679 y Ft(w)2164 694 y Fk(\026)2233 679 y
Fm(\001)f Ft(w)2364 694 y Fk(\027)2407 679 y Fs(n)2465
694 y Fk(\027)2508 583 y Fj(\021)2580 679 y Fm(\000)2717
631 y Fk(R)2770 639 y Ff(\026)p 2690 656 149 4 v 2690
665 a Fi(p)p 2749 665 36 3 v 55 x Fr(2)p Fk(l)2805 728
y Ff(s)2848 679 y Fs(n)2906 694 y Fk(\026)1199 804 y(l)1220
812 y Ff(s)p 1132 820 189 4 v 1132 829 a Fi(p)p 1190
829 36 3 v 1190 884 a Fr(2)q Fk(R)1279 892 y Ff(\026)1347
747 y Fj(\020)1397 843 y Fs(m)1482 858 y Fk(\026)1551
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2842 2434 V 2842 2526 a(2)2942 2350 y Fr(16)2918 2375
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eop
%%Page: 5 5
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1864 y Fs(N)1740 1879 y Fr(osc)1861 1864 y FB(+)22 b
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%%Page: 6 6
6 5 bop 19 158 a FB(When)40 b(w)m(e)g(compactify)e(heterotic)h(string)f
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%%Page: 9 9
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1479 y(shrink)k(to)f(a)g(p)s(oin)m(t.)43 b(They)34 b(ha)m(v)m(e)g(the)f
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1724 y Ft(N)238 1739 y Fr(1)305 1724 y FB(:)d(\(0)447
1683 y Fr(16)521 1724 y Fs(;)17 b FB(1)p Fs(;)g FB(1)p
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FB(1)p Fs(;)g FB(1)p Fs(;)g Fm(\000)p FB(1)p Fs(;)g Fm(\000)p
FB(1\))p Fs(;)112 b Ft(N)1838 1739 y Fr(2)1905 1724 y
FB(:)28 b(\(1)2047 1683 y Fr(16)2121 1724 y Fs(;)17 b
Fm(\000)p FB(15)p Fs(;)g Fm(\000)p FB(13)p Fs(;)g FB(11)p
Fs(;)g FB(9)p Fs(;)g Fm(\000)p FB(7)p Fs(;)g Fm(\000)p
FB(5)p Fs(;)g FB(3)p Fs(;)g FB(1\))p Fs(:)157 b FB(\(3)3629
1702 y(.)3655 1724 y(17\))-9 1969 y(If)28 b(all)f Fs(Q)293
1984 y Fe(a)333 1994 y Ff(i)392 1969 y FB(of)h(a)g(junction)h(are)f
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(junction)h(\()p Ft(N)3140 1984 y Fr(1)3194 1969 y FB(+)14
b Ft(N)3372 1984 y Fr(2)3411 1969 y FB(\))p Fs(=)p FB(2,)29
b(w)m(e)g(can)-9 2119 y(mak)m(e)h(it)f(a)h(prop)s(er)g(junction.)42
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y Fe(c)104 2293 y Ff(i)159 2268 y FB(to)24 b(an)m(y)h(v)-5
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y(.)1704 2268 y(7\)-\(3)1911 2247 y(.)1937 2268 y(15\))h(w)m(e)h
(adopted)g(the)f(con)m(v)m(en)m(tion)i Fs(Q)3314 2283
y Fe(b)3359 2292 y Fd(1)3426 2268 y FB(=)h Fs(Q)3606
2283 y Fe(c)3642 2292 y Fd(1)3709 2268 y FB(=)h(0.)137
2418 y(Finally)-8 b(,)35 b(let)g(us)i(con\014rm)e(that)h(the)h(in)m
(tersection)f(n)m(um)m(b)s(er)h(de\014ned)g(in)e(Ref.)3037
2381 y Fr(2\))3140 2418 y FB(is)h(consisten)m(t)h(with)-9
2567 y(the)k(inner)f(pro)s(duct)h(of)f(the)h(v)m(ector)h
Ft(K)p FB(.)67 b(According)40 b(to)h(Ref.)2362 2531 y
Fr(2\))2428 2567 y FB(,)i(the)e(in)m(tersection)f(n)m(um)m(b)s(ers)i
(among)-9 2717 y(unit)32 b(junctions)g(are)267 2972 y(\()p
Ft(a)359 2987 y Fk(i)410 2972 y Fm(\001)21 b Ft(a)513
2987 y Fk(j)550 2972 y FB(\))28 b(=)g(\()p Ft(b)820 2987
y Fk(i)870 2972 y Fm(\001)22 b Ft(b)982 2987 y Fk(j)1019
2972 y FB(\))27 b(=)h(\()p Ft(c)1276 2987 y Fk(i)1326
2972 y Fm(\001)22 b Ft(c)1426 2987 y Fk(j)1462 2972 y
FB(\))28 b(=)f Fs(\016)1674 2987 y Fk(ij)1735 2972 y
Fs(;)114 b FB(\()p Ft(a)1968 2987 y Fk(i)2019 2972 y
Fm(\001)22 b Ft(b)2131 2987 y Fk(j)2168 2972 y FB(\))27
b(=)h Fm(\000)2424 2904 y FB(1)p 2424 2949 V 2424 3040
a(2)2483 2972 y Fs(;)114 b FB(\()p Ft(a)2716 2987 y Fk(i)2767
2972 y Fm(\001)22 b Ft(c)2867 2987 y Fk(j)2903 2972 y
FB(\))28 b(=)3082 2904 y(1)p 3082 2949 V 3082 3040 a(2)3141
2972 y Fs(;)374 b FB(\(3)3629 2950 y(.)3655 2972 y(18\))1350
3319 y(\()p Ft(b)1450 3334 y Fk(i)1501 3319 y Fm(\001)22
b Ft(c)1601 3334 y Fk(j)1637 3319 y FB(\))28 b(=)1806
3144 y Fj(8)1806 3219 y(<)1806 3369 y(:)1930 3249 y Fm(\000)p
FB(1)112 b(\()p Fs(i)28 b Fm(\024)h Fs(j)6 b FB(\))1969
3398 y(1)138 b(\()p Fs(i)28 b(>)f(j)6 b FB(\))p Fs(:)3542
3319 y FB(\(3)3629 3297 y(.)3655 3319 y(19\))-9 3590
y(\(In)28 b(our)f(con)m(v)m(en)m(tion,)j(in)m(tersection)d(n)m(um)m(b)s
(ers)i(ha)m(v)m(e)f(signs)g(opp)s(osite)f(to)g(those)h(in)f(Ref.)3229
3554 y Fr(2\))3295 3590 y FB(.\))42 b(Using)27 b(these)-9
3739 y(equations,)33 b(w)m(e)g(obtain)1288 3889 y(\()p
Ft(J)22 b Fm(\001)g Ft(J)p FB(\))27 b(=)h Ft(k)1742 3848
y Fr(2)1804 3889 y FB(+)22 b(2)p Fs(m)2036 3904 y Fr(8)2075
3889 y Fs(n)2133 3904 y Fr(8)2195 3889 y FB(+)g(2)p Fs(m)2427
3904 y Fr(9)2467 3889 y Fs(n)2525 3904 y Fr(9)2564 3889
y Fs(:)951 b FB(\(3)3629 3867 y(.)3655 3889 y(20\))-9
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4427 y Fl(x)p Fy(4.)93 b(Conclusion)137 4649 y FB(W)-8
b(e)51 b(ha)m(v)m(e)g(established)f(the)h(corresp)s(ondence)h(b)s(et)m
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4799 y(strings)30 b(compacti\014ed)f(on)h Fs(T)1074 4763
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b(c)m(harges)k(of)d(junctions)h(on)g Fs(S)2805 4763 y
Fr(2)2875 4799 y FB(with)f(24)h(7-branes.)43 b(In)30
b(or-)-9 4948 y(der)e(to)f(\014nd)i(the)f(necessary)i(relations,)d(w)m
(e)i(used)g(a)e(\015at)h(bac)m(kground)h(on)e(the)i(t)m(yp)s(e)f(I)s
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b(w)m(e)h(ha)m(v)m(e)h(obtained)d(the)i(relations)d(\(3)1606
5076 y(.)1633 5098 y(7\)-\(3)1840 5076 y(.)1866 5098
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b(arian)m(t)30 b(under)j(con)m(tin)m(uous)g(deformations)d(of)i(mo)s
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y(9)p eop
%%Page: 10 10
10 9 bop -127 158 a FB(mo)s(duli)30 b(space)k(of)e(elliptic)d(K3)k(is)
1339 341 y Fs(S)6 b(O)s FB(\(18)p Fs(;)17 b FB(2\))p
869 385 1349 4 v 869 477 a Fs(S)6 b(O)s FB(\(18\))21
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b(O)s FB(\(18)p Fs(;)17 b FB(2;)g Ft(Z)p FB(\))2250 408
y Fm(\002)23 b Ft(R)2434 367 y Fr(+)2515 408 y Fm(\002)f
Ft(R)2698 367 y Fr(+)2757 408 y Fs(;)689 b FB(\(4)3560
387 y(.)3586 408 y(1\))-127 667 y(where)40 b(the)f(t)m(w)m(o)h(copies)f
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b(ahler)38 b(structure)i(of)f(the)g(base)h(manifold)c(and)j(that)-127
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b(.)49 b(In)35 b(the)f(case)h(of)f(the)h(compacti\014cation)d(of)h
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b(Hence,)33 b(the)f(mo)s(duli)d(space)k(of)e(t)m(yp)s(e)h(I)s(IB)g
(theory)-127 1115 y(compacti\014ed)27 b(on)g Fs(S)633
1079 y Fr(2)700 1115 y FB(is)g(iden)m(tical)f(to)h(that)g(of)h
Fs(T)1679 1079 y Fr(2)1718 1115 y FB(-compacti\014ed)e(heterotic)i
(string)e(theory)-8 b(,)30 b(\(2)3396 1094 y(.)3422 1115
y(10\),)e(and)-127 1265 y(it)c(is)h(exp)s(ected)i(that)e(at)g(an)m(y)g
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b(applying)e(these)j(form)m(ulae)e(to)g(a)h(v)m(ector)g
Ft(W)1956 1429 y Fr(8)2030 1414 y FB(with)g(the)g(Wilson)e(line)h(\(2)
3029 1393 y(.)3056 1414 y(17\),)h(w)m(e)g(correctly)-127
1564 y(obtain)c(the)i(junction)g Ft(\006)p FB(,)f(although)g(the)h(bac)
m(kground)g(is)f(not)h(\015at.)1339 1898 y Fy(Ac)m(kno)m(wledgemen)m
(ts)19 2120 y FB(I)e(w)m(ould)g(lik)m(e)f(to)h(thank)g(B.Zwiebac)m(h,)h
(K.Murak)-5 b(ami)29 b(and)i(I.Kishimoto)d(for)i(v)-5
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(t)g(on)h(the)g(mo)s(duli)d(space)j(of)g(K3.)1560 2589
y Ft(References)68 2707 y FB(1\))49 b(M.R.Gab)s(erdiel)31
b(and)h(B.Zwiebac)m(h,)i(Nucl.Ph)m(ys.)p Ft(B518)p FB(\(1998\)151,)e
FA(hep-th/9709013)p FB(,)204 2856 y Fc(\\Exc)-5 b(eptional)33
b(gr)-5 b(oups)35 b(fr)-5 b(om)34 b(op)-5 b(en)34 b(strings.")68
3006 y FB(2\))49 b(O.DeW)-8 b(olfe)31 b(and)i(B.Zwiebac)m(h,)g
FA(hep-th/9804210)p FB(,)204 3155 y Fc(\\String)h(Junctions)h(for)f(A)n
(rbitr)-5 b(ary)36 b(Lie)e(A)n(lgebr)-5 b(a)35 b(R)-5
b(epr)g(esentations.")68 3305 y FB(3\))49 b(O.DeW)-8
b(olfe,)31 b FA(hep-th/9809026)p FB(,)204 3454 y Fc(\\A\016ne)j(Lie)h
(A)n(lgebr)-5 b(as,)34 b(String)g(Junctions)h(and)f(7-Br)-5
b(anes.")68 3603 y FB(4\))49 b(O.DeW)-8 b(olfe,)31 b(T.Hauer,)j
(A.Iqbal)e(and)h(B.Zwiebac)m(h,)g FA(hep-th/9812028)p
FB(,)204 3753 y Fc(\\Unc)-5 b(overing)33 b(the)i(Symmetries)f(on)g
([p,q])g(7-br)-5 b(anes:)44 b(Beyond)34 b(the)h(Ko)-5
b(dair)g(a)34 b(Classi\014c)-5 b(ation.")68 3902 y FB(5\))49
b(O.DeW)-8 b(olfe,)31 b(T.Hauer,)j(A.Iqbal)e(and)h(B.Zwiebac)m(h,)g
FA(hep-th/9812209)p FB(,)204 4052 y Fc(\\Unc)-5 b(overing)44
b(In\014nite)g(Symmetries)g(on)h([p,q])g(7-br)-5 b(anes:)64
b(Kac-Mo)-5 b(o)g(dy)45 b(A)n(lgebr)-5 b(as)44 b(and)g(Be-)204
4201 y(yond.")68 4351 y FB(6\))49 b(Y.Imam)m(ura,)31
b(Ph)m(ys.Rev.)p Ft(D58)p FB(\(1998\)106005,)i FA(hep-th/9802189)p
FB(,)204 4500 y Fc(\\)p Fs(E)326 4515 y Fr(8)400 4500
y Fc(\015avor)i(multiplets.")68 4649 y FB(7\))49 b(A.Sen,)33
b(Nucl.Ph)m(ys.)p Ft(B475)p FB(\(1996\)562,)g FA(hep-th/9605150)p
FB(,)204 4799 y Fc(\\F-The)-5 b(ory)33 b(and)i(Orientifolds.")68
4948 y FB(8\))49 b(O.DeW)-8 b(olfe,)31 b(T.Hauer,)j(A.Iqbal)e(and)h
(B.Zwiebac)m(h,)204 5098 y(Nucl.Ph)m(ys.)p Ft(B534)p
FB(\(1998\)261,)f FA(hep-th/9805220)p FB(,)204 5247 y
Fc(\\Constr)-5 b(aints)40 b(on)g(the)h(BPS)f(Sp)-5 b(e)g(ctrum)41
b(of)g(N)g(=)g(2,)h(D)e(=)h(4)f(The)-5 b(ories)40 b(with)h(A-D-E)g
(Flavor)204 5397 y(Symmetry.")1773 5687 y FB(10)p eop
%%Page: 11 11
11 10 bop 186 158 a FB(9\))48 b(A.Mikhailo)m(v,)29 b(N.Nekraso)m(v)i
(and)e(S.Sethi,)h(Nucl.Ph)m(ys.)p Ft(B531)p FB(\(1998\)345,)g
FA(hep-th/9803142)p FB(,)321 307 y Fc(\\Ge)-5 b(ometric)35
b(R)-5 b(e)g(alizations)34 b(of)g(BPS)h(States)g(in)f(N=2)h(The)-5
b(ories.")1891 5687 y FB(11)p eop
%%Page: 12 12
12 11 bop 2434 465 a Ft(\012)2515 429 y Fr(0)2442 590
y Ft(\006)1687 856 y Fs(\016)1106 274 y(A)1179 238 y
Fk(n)1226 274 y Fs(B)5 b(C)i(B)e(C)1604 112 y Fj(\022)1679
181 y Fr(1)1679 272 y(0)1729 112 y Fj(\023)-20 b(\022)1896
178 y Fr(0)1899 242 y Fd(1)p 1855 254 117 4 v 1855 295
a(8)p Fb(\000)p Ff(n)1997 112 y Fj(\023)2060 236 y(\022)2136
305 y Fr(0)2136 397 y(1)2186 236 y Fj(\023)2060 568 y(\022)2136
637 y Fr(1)2136 729 y(0)2186 568 y Fj(\023)1230 814 y
@beginspecial 289 @llx 319 @lly 431 @urx 381 @ury 1420
@rwi @setspecial
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