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%%Title: arXiv:hep-th/9901001v3 10 May 1999 |
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FB(\(2)p Fs(;)17 b Ft(Z)p FB(\))33 b(mono)s(drom)m(y)e |
|
Fs(M)2813 694 y Fr([)p Fk(p;q)r Fr(])2978 679 y FB(de\014ned)j(b)m(y) |
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1183 952 y Fs(M)1277 968 y Fr([)p Fk(p;q)r Fr(])1438 |
|
952 y FB(=)1541 781 y Fj(0)1541 930 y(@)1664 882 y FB(1)22 |
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b Fm(\000)g Fs(pq)192 b(p)2167 846 y Fr(2)1715 1032 y |
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Fm(\000)p Fs(q)1839 996 y Fr(2)2030 1032 y FB(1)21 b(+)h |
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Fs(pq)2344 781 y Fj(1)2344 930 y(A)2433 952 y Fs(;)1013 |
|
b FB(\(1)3560 931 y(.)3586 952 y(1\))-127 1212 y(and)38 |
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b(to)g(a)m(v)m(oid)f(m)m(ultiple)f(v)-5 b(alues)38 b(of)f(\014elds)h |
|
(and)g(string)g(c)m(harges,)i(w)m(e)f(should)f(in)m(tro)s(duce)g(branc) |
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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 |
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FB(\(2)p Fs(;)17 b Ft(Z)p FB(\))36 b(transformations)e(comp)s(ensating) |
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i(for)f(the)i(mono)s(dromies.)52 b(It)36 b(w)m(as)h(suggested)-127 |
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1511 y(b)m(y)44 b(Zwiebac)m(h)g(et)g(al.)705 1475 y Fr(1\))14 |
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b FB(-)g Fr(6\))938 1511 y FB(that)43 b(when)i(7-branes)e(of)g(a)g(sp)s |
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(eci\014c)h(com)m(bination)e(are)h(on)g(top)h(of)f(eac)m(h)-127 |
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1661 y(other,)35 b(v)-5 b(arious)34 b(groups)i(con)m(taining)d |
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(exceptional)h(groups,)i(a\016ne)f(groups,)h(and)f(other)g(exotic)f |
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(groups)-127 1810 y(app)s(ear)k(as)h(gauge)f(groups)g(on)g(the)h |
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(7-branes.)61 b(T)-8 b(o)38 b(construct)i Fs(A)2347 1825 |
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y Fk(n)2394 1810 y FB(,)g Fs(D)2542 1825 y Fk(n)2627 |
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1810 y FB(and)e Fs(E)2894 1825 y Fk(n)2980 1810 y FB(gauge)g(theories,) |
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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 |
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y(symmetry)d(enhancemen)m(t)i(arise)e(from)f(string)g(junctions)h |
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(connecting)h(some)f(of)g(these)h(7-branes.)46 b(The)-127 |
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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 |
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(lab)s(el)e(of)h(the)h(7-branes.)19 2557 y(It)27 b(is)g(kno)m(wn)i |
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(that)d(t)m(yp)s(e)i(I)s(IB)g(theory)f(compacti\014ed)g(on)g |
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Fs(S)2165 2521 y Fr(2)2204 2557 y FB(,)h(whic)m(h)g(can)f(also)g(b)s(e) |
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g(expressed)j(as)d(a)g(K3)-127 2707 y(compacti\014cation)34 |
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b(of)i(F-theory)-8 b(,)37 b(is)f(dual)g(to)f(heterotic)i(string)e |
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(theory)i(compacti\014ed)f(on)g Fs(T)3378 2671 y Fr(2)3417 |
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2707 y FB(.)55 b(On)36 b Fs(S)3731 2671 y Fr(2)-127 2856 |
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y FB(w)m(e)31 b(ha)m(v)m(e)g(24)f(7-branes)g(corresp)s(onding)g(to)f |
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(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 |
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2970 y Fr(5\))206 3006 y FB(,)g(the)g(junction)g(lattice)e(of)i(the)g |
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(t)m(yp)s(e)h(I)s(IB)f(theory)g(on)g Fs(S)2258 2970 y |
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Fr(2)2326 3006 y FB(has)g(the)h(signature)e(\(+)3194 |
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2970 y Fr(18)3269 3006 y Fs(;)17 b Fm(\000)3390 2970 |
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y Fr(2)3429 3006 y FB(\),)30 b(whic)m(h)-127 3155 y(is)j(the)h(same)g |
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(as)g(the)g(signature)f(for)g(the)h(Narain)f(lattice)f(of)h(heterotic)g |
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(string)g(theory)-8 b(.)48 b(Because)35 b(there)-127 |
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3305 y(is)f(unique)h(18)24 b(+)f(2-dimensional)32 b(Loren)m(tzian)i |
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(self-dual)f(ev)m(en)k(lattice,)c(it)h(is)g(trivial)e(to)j(sho)m(w)h |
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(that)e(the)-127 3454 y(junction)24 b(lattice)g(is)g(iden)m(tical)g(to) |
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g(the)i(Narain)d(lattice)h(on)h(heterotic)f(side.)41 |
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b(The)26 b(purp)s(ose)g(of)f(the)g(presen)m(t)-127 3603 |
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y(pap)s(er)j(is,)g(b)m(y)h(means)f(of)f(dualities,)g(to)h(establish)f |
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(the)h(one)h(to)e(one)h(corresp)s(ondence)i(b)s(et)m(w)m(een)g(in)m(v) |
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-5 b(arian)m(t)-127 3753 y(c)m(harges)39 b(of)e(junctions)h(in)f(t)m |
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(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 |
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(sp)s(ecifying)e(p)s(erturbativ)m(e)-127 3902 y(states)c(of)g(a)f |
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(heterotic)g(string.)19 4052 y(These)48 b(t)m(w)m(o)e(theories)g(are)f |
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(dual)g(in)g(the)h(manner)f(w)m(e)i(no)m(w)f(describ)s(e.)84 |
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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 |
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(24)e(7-)-127 4201 y(branes.)73 b(Although)41 b(there)i(are)f(man)m(y)h |
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(represen)m(tations)g(of)f(these)h(7-branes,)i(w)m(e)e(restrict)f |
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(ourselv)m(es)-127 4351 y(to)c(the)h(case)h(that)f(they)g(are)g |
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(represen)m(ted)i(as)e(four)f(sets)i(of)e Fs(A)2243 4314 |
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y Fr(4)2283 4351 y Fs(B)5 b(C)i FB(.)62 b(Once)39 b(w)m(e)h(obtain)e |
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(some)g(results)-127 4500 y(for)d(this)f(bac)m(kground,)j(w)m(e)g(can)e |
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(extend)i(them)e(to)f(other)i(regions)e(of)h(mo)s(duli)e(space)j(b)m(y) |
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g(a)f(con)m(tin)m(uous)-127 4649 y(deformation)41 b(of)h(the)h(bac)m |
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(kground.)76 b(In)43 b(the)g(w)m(eak)h(coupling)d(limit)f(\()p |
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Fs(g)2615 4664 y Fr(str)2746 4649 y Fm(!)45 b FB(0\),)g(a)e(pair)e |
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(consisting)-127 4799 y(of)36 b(a)g(B-brane)g(and)g(a)g(C-brane)h(is)e |
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(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 |
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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 |
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5098 y FB(orien)m(tifold.)73 b(By)44 b(means)g(of)f(a)g(T)g(dualit)m(y) |
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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 |
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(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 |
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Fs(S)6 b(O)s FB(\(32\))40 b(heterotic)-127 5397 y(string)32 |
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b(theory)h(on)g Fs(T)663 5361 y Fr(2)702 5397 y FB(.)1797 |
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5687 y(2)p eop |
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3 2 bop 137 158 a FB(Through)45 b(this)f(series)i(of)e(dualit)m(y)f |
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(transformations,)j(w)m(e)g(can)e(relate)g(BPS)h(states)h(in)e |
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(heterotic)-9 307 y(string)c(theory)h(to)f(those)h(in)e(t)m(yp)s(e)j(I) |
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s(IB)e(theory)-8 b(.)68 b(Let)40 b(us)h(assume)g(the)g(t)m(w)m(o)g |
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(compacti\014ed)f(directions)-9 457 y(to)e(b)s(e)h Fs(x)310 |
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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 |
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(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 |
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y FB(and)f Fs(n)3822 472 y Fr(9)3861 457 y FB(.)-9 606 |
|
y(\(Generically)-8 b(,)25 b(momen)m(ta)g Fs(m)1059 570 |
|
y Fi(0)1059 631 y Fr(8)1124 606 y FB(and)h Fs(m)1392 |
|
570 y Fi(0)1392 631 y Fr(9)1457 606 y FB(are)g(not)f(in)m(tegers,)j |
|
(due)e(to)f(the)i(presence)g(of)f(Wilson)e(lines,)i(while)-9 |
|
756 y(winding)34 b(n)m(um)m(b)s(ers)i Fs(n)816 771 y |
|
Fr(8)891 756 y FB(and)g Fs(n)1142 771 y Fr(9)1217 756 |
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y FB(are)f(alw)m(a)m(ys)h(in)m(tegers.\))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 |
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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 |
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3014 y(consequence)h(of)c(the)h(fact)f(that)h(the)g(T)g(dual)e(of)i(a)f |
|
(winding)f(t)m(yp)s(e)j(I)f(D-string)e(is)h(half)f(of)h(a)h(t)m(yp)s(e) |
|
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) |
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g(bac)m(kgrounds,)j(there)e(is)e(a)g(certain)h(subtlet)m(y)g |
|
(concerning)-9 3462 y(the)29 b(de\014nition)f(of)h(winding)f(n)m(um)m |
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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 |
|
(mass)g(form)m(ula)e(and)i(the)g(lev)m(el)f(matc)m(hing)g(condition)f |
|
(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 |
|
FB(=)g(10)24 b Fm(\000)i Fs(d;)17 b(:)g(:)g(:)e(;)i FB(9)36 |
|
b(lab)s(els)f(the)i(compacti\014ed)f(directions,)h(and)g(the)g(constan) |
|
m(t)g Fs(a)g FB(is)f Fm(\000)p FB(1)p Fs(=)p FB(2)g(for)-9 |
|
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 |
|
(v)m(er.)43 b(Due)31 b(to)g(the)g(quan)m(tization)f(of)h(the)g(momen)m |
|
(ta)f(and)1915 5687 y(3)p eop |
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4 3 bop -127 158 a FB(wrapping)28 b(n)m(um)m(b)s(ers,)i(the)f(2)p |
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Fs(d)13 b FB(+)g(16-dimensional)24 b(v)m(ector)30 b(\()p |
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Fs(a)2120 111 y Fk(\026)2120 180 y Fr(0)2167 158 y Fs(;)2213 |
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151 y Fj(e)2210 158 y Fs(a)2261 111 y Fk(\026)2261 180 |
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y Fr(0)2308 158 y Fs(;)2356 149 y Fj(e)2352 158 y Ft(a)2406 |
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173 y Fr(0)2446 158 y FB(\))e(tak)m(es)h(v)-5 b(alues)29 |
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b(in)e(the)i(Loren)m(tzian)-127 307 y(self-dual)35 b(ev)m(en)j |
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(lattice,)e(the)g(so-called)f(`Narain)h(lattice'.)53 |
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b(F)-8 b(or)35 b(simplicit)m(y)-8 b(,)35 b(let)h(us)h(assume)g(the)f |
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(torus)-127 457 y Fs(T)-56 420 y Fk(d)17 457 y FB(is)c(rectangular.)43 |
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b(Then,)33 b(the)g(zero)g(mo)s(des)g(are)f(quan)m(tized)i(as)524 |
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592 y Fj(0)524 738 y(B)524 788 y(B)524 838 y(B)524 891 |
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y(@)647 693 y Fs(a)698 646 y Fk(\026)698 715 y Fr(0)649 |
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836 y Fj(e)647 843 y Fs(a)698 796 y Fk(\026)698 865 y |
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Fr(0)653 983 y Fj(e)649 992 y Ft(a)703 1007 y Fr(0)795 |
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592 y Fj(1)795 738 y(C)795 788 y(C)795 838 y(C)795 891 |
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y(A)895 838 y FB(=)998 592 y Fj(0)998 738 y(B)998 788 |
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y(B)998 838 y(B)998 891 y(@)1198 640 y Fk(l)1219 648 |
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y Ff(s)p 1131 656 189 4 v 1131 665 a Fi(p)p 1190 665 |
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36 3 v 55 x Fr(2)p Fk(R)1278 728 y Ff(\026)1346 583 y |
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Fj(\020)1396 679 y Fs(m)1481 694 y Fk(\026)1550 679 y |
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FB(+)22 b Ft(w)1729 694 y Fk(\026)1797 679 y Fm(\001)g |
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Ft(k)h Fm(\000)2038 640 y Fr(1)p 2038 656 36 4 v 2038 |
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713 a(2)2083 679 y Ft(w)2164 694 y Fk(\026)2233 679 y |
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Fm(\001)f Ft(w)2364 694 y Fk(\027)2407 679 y Fs(n)2465 |
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694 y Fk(\027)2508 583 y Fj(\021)2580 679 y Fm(\000)2717 |
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631 y Fk(R)2770 639 y Ff(\026)p 2690 656 149 4 v 2690 |
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665 a Fi(p)p 2749 665 36 3 v 55 x Fr(2)p Fk(l)2805 728 |
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y Ff(s)2848 679 y Fs(n)2906 694 y Fk(\026)1199 804 y(l)1220 |
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812 y Ff(s)p 1132 820 189 4 v 1132 829 a Fi(p)p 1190 |
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829 36 3 v 1190 884 a Fr(2)q Fk(R)1279 892 y Ff(\026)1347 |
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747 y Fj(\020)1397 843 y Fs(m)1482 858 y Fk(\026)1551 |
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843 y FB(+)g Ft(w)1730 858 y Fk(\026)1798 843 y Fm(\001)g |
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Ft(k)g Fm(\000)2039 804 y Fr(1)p 2039 820 36 4 v 2039 |
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877 a(2)2084 843 y Ft(w)2165 858 y Fk(\026)2234 843 y |
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Fm(\001)g Ft(w)2365 858 y Fk(\027)2408 843 y Fs(n)2466 |
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858 y Fk(\027)2509 747 y Fj(\021)2581 843 y FB(+)2716 |
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795 y Fk(R)2769 803 y Ff(\026)p 2689 820 149 4 v 2689 |
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829 a Fi(p)p 2748 829 36 3 v 55 x Fr(2)p Fk(l)2804 892 |
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y Ff(s)2847 843 y Fs(n)2905 858 y Fk(\026)1830 1007 y |
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Ft(k)h Fm(\000)f Fs(n)2069 1022 y Fk(\026)2116 1007 y |
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Ft(w)2197 1022 y Fk(\026)3003 592 y Fj(1)3003 738 y(C)3003 |
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788 y(C)3003 838 y(C)3003 891 y(A)3092 838 y Fs(;)354 |
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b FB(\(2)3560 816 y(.)3586 838 y(2\))-127 1219 y(where)33 |
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b Fs(R)228 1234 y Fk(\026)307 1219 y FB(are)f(compacti\014cation)e |
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(radii,)g Ft(k)i FB(is)f(a)h(v)m(ector)g(on)g(a)g(16-dimensional)c |
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(self-dual)j(ev)m(en)i(lattice)-127 1368 y Fs(\000)14 |
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b FB(,)35 b(and)f Ft(w)281 1383 y Fk(\026)362 1368 y |
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FB(are)g(Wilson)f(lines)h(whic)m(h)h(tak)m(e)g(v)-5 b(alues)34 |
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b(on)g(the)h(torus)f Ft(R)2509 1332 y Fr(16)2584 1368 |
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y Fs(=\000)14 b FB(.)48 b(F)-8 b(rom)33 b(this)h(equation,)h(w)m(e)-127 |
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1518 y(can)d(read)g(mo)s(di\014ed)f(momen)m(ta)f Fs(m)1174 |
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1482 y Fi(0)1174 1543 y Fk(\026)1221 1518 y FB(,)i(whic)m(h)h(are)f |
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(momen)m(ta)e(con)m(taining)h(the)h(shift)f(due)i(to)e(the)i(Wilson) |
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-127 1667 y(line,)e(as)1114 1827 y Fs(m)1199 1786 y Fi(0)1199 |
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1851 y Fk(\026)1273 1827 y FB(=)d Fs(m)1462 1842 y Fk(\026)1531 |
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1827 y FB(+)22 b Ft(w)1710 1842 y Fk(\026)1778 1827 y |
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Fm(\001)g Ft(k)h Fm(\000)2019 1759 y FB(1)p 2019 1803 |
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49 4 v 2019 1895 a(2)2078 1827 y Ft(w)2159 1842 y Fk(\026)2228 |
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1827 y Fm(\001)e Ft(w)2358 1842 y Fk(\027)2402 1827 y |
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Fs(n)2460 1842 y Fk(\027)2503 1827 y Fs(;)943 b FB(\(2)3560 |
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1805 y(.)3586 1827 y(3\))-127 2026 y(where)32 b Fs(m)238 |
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2041 y Fk(\026)317 2026 y FB(and)f Fs(n)563 2041 y Fk(\026)641 |
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2026 y FB(are)g(in)m(tegers.)44 b(No)m(w,)32 b(w)m(e)g(are)f |
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(considering)g Fs(S)6 b(O)s FB(\(32\))30 b(heterotic)h(string)f(theory) |
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-8 b(,)32 b(and)-127 2175 y Fs(\000)46 b FB(should)33 |
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b(b)s(e)g(the)g Fs(S)6 b(O)s FB(\(32\))31 b(w)m(eigh)m(t)i(lattice,)e |
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(whic)m(h)i(is)f(generated)h(b)m(y)h(the)f(basis)483 |
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2458 y Fs(\013)545 2473 y Fk(i)601 2458 y FB(=)27 b Ft(e)755 |
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2473 y Fk(i)806 2458 y Fm(\000)22 b Ft(e)956 2473 y Fk(i)p |
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Fr(+1)1075 2458 y FB(\()p Fs(i)28 b FB(=)f(1)p Fs(;)17 |
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b(:)g(:)g(:)f(;)h FB(15\))p Fs(;)113 b(\013)1883 2473 |
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y Fr(16)1986 2458 y FB(=)27 b Ft(e)2140 2473 y Fr(15)2237 |
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2458 y FB(+)22 b Ft(e)2386 2473 y Fr(16)2461 2458 y Fs(;)114 |
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b(\013)2664 2473 y Fk(s)2729 2458 y FB(=)2842 2390 y(1)p |
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2842 2434 V 2842 2526 a(2)2942 2350 y Fr(16)2918 2375 |
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y Fj(X)2921 2557 y Fk(i)p Fr(=1)3054 2458 y Ft(e)3105 |
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2473 y Fk(i)3134 2458 y Fs(;)312 b FB(\(2)3560 2436 y(.)3586 |
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2458 y(4\))-127 2743 y(where)32 b(the)e Ft(e)369 2758 |
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y Fk(i)428 2743 y FB(form)f(an)i(orthonormal)d(basis)i(of)g |
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Ft(R)1776 2706 y Fr(16)1850 2743 y FB(.)43 b(F)-8 b(or)29 |
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b(the)i(follo)m(wing)d(argumen)m(t,)j(it)e(is)h(con)m(v)m(enien)m(t) |
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-127 2892 y(to)i(de\014ne)i(the)f(2)p Fs(d)22 b FB(+)g(16-dimensional) |
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29 b(v)m(ector)953 3132 y Ft(K)e FB(=)h(\()p Ft(k)p Fs(;)17 |
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b(m)1398 3147 y Fr(10)p Fi(\000)p Fk(d)1564 3132 y Fs(;)g(n)1666 |
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3147 y Fr(10)p Fi(\000)p Fk(d)1831 3132 y Fs(;)g(:)g(:)g(:)f(;)h(m)2135 |
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3147 y Fr(8)2175 3132 y Fs(;)g(n)2277 3147 y Fr(8)2316 |
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3132 y Fs(;)g(m)2445 3147 y Fr(9)2485 3132 y Fs(;)g(n)2587 |
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3147 y Fr(9)2626 3132 y FB(\))p Fs(:)782 b FB(\(2)3560 |
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3110 y(.)3586 3132 y(5\))-127 3371 y(F)-8 b(urthermore,)32 |
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b(w)m(e)i(in)m(tro)s(duce)e(v)m(ectors)i Ft(W)1489 3386 |
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y Fk(\026)1568 3371 y FB(that)f(satisfy)1533 3611 y Fs(a)1584 |
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3563 y Fk(\026)1584 3632 y Fr(0)1658 3611 y FB(=)28 b |
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Ft(W)1878 3626 y Fk(\026)1946 3611 y Fm(\001)22 b Ft(K)p |
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Fs(;)1362 b FB(\(2)3560 3589 y(.)3586 3611 y(6\))-127 |
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3850 y(where)34 b(the)f(inner)f(pro)s(duct)h(is)f(de\014ned)i(b)m(y)f |
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(using)g(the)g(metric)1048 4452 y Fs(g)1095 4467 y Fk(M)7 |
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b(N)1265 4452 y FB(=)1368 3982 y Fj(0)1368 4128 y(B)1368 |
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4178 y(B)1368 4228 y(B)1368 4277 y(B)1368 4327 y(B)1368 |
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4377 y(B)1368 4427 y(B)1368 4477 y(B)1368 4526 y(B)1368 |
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4576 y(B)1368 4626 y(B)1368 4676 y(B)1368 4729 y(@)1491 |
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4072 y Ft(1)1547 4087 y Fr(16)1869 4222 y FB(1)1721 4371 |
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y(1)2023 4484 y(.)2062 4509 y(.)2100 4534 y(.)2380 4691 |
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y(1)2232 4841 y(1)2479 3982 y Fj(1)2479 4128 y(C)2479 |
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4178 y(C)2479 4228 y(C)2479 4277 y(C)2479 4327 y(C)2479 |
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4377 y(C)2479 4427 y(C)2479 4477 y(C)2479 4526 y(C)2479 |
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4576 y(C)2479 4626 y(C)2479 4676 y(C)2479 4729 y(A)2568 |
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4452 y Fs(:)878 b FB(\(2)3560 4430 y(.)3586 4452 y(7\))-127 |
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5063 y(F)-8 b(or)32 b(example,)g(in)g(the)h Fs(S)805 |
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5027 y Fr(1)876 5063 y FB(compacti\014ed)g(case,)g(the)g(v)m(ector)h |
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Ft(W)2256 5078 y Fr(9)2327 5063 y FB(is)f(giv)m(en)f(b)m(y)1060 |
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5330 y Ft(W)1176 5345 y Fr(9)1242 5330 y FB(=)1446 5263 |
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y Fs(l)1475 5278 y Fk(s)p 1356 5307 246 4 v 1356 5325 |
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a Fm(p)p 1439 5325 49 4 v 82 x FB(2)p Fs(R)1562 5422 |
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y Fr(9)1628 5184 y Fj( )1694 5330 y Ft(w)1775 5345 y |
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Fr(9)1814 5330 y Fs(;)17 b Fm(\000)1945 5263 y FB(1)p |
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1945 5307 V 1945 5399 a(2)2004 5330 y Ft(w)2087 5289 |
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y Fr(2)2085 5355 y(9)2148 5330 y Fm(\000)2258 5263 y |
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Fs(R)2333 5227 y Fr(2)2332 5288 y(9)p 2258 5307 115 4 |
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v 2280 5399 a Fs(l)2311 5370 y Fr(2)2309 5423 y Fk(s)2382 |
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5330 y Fs(;)g FB(1)2475 5184 y Fj(!)2557 5330 y Fs(:)889 |
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b FB(\(2)3560 5309 y(.)3586 5330 y(8\))1797 5687 y(4)p |
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eop |
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5 4 bop -9 158 a FB(The)33 b(v)m(ectors)i Ft(W)639 173 |
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y Fk(\026)717 158 y FB(also)d(satisfy)h(the)g(equation)1561 |
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370 y Ft(W)1677 385 y Fk(\026)1745 370 y Fm(\001)22 b |
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Ft(W)1911 385 y Fk(\027)1982 370 y FB(=)27 b Fm(\000)p |
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Fs(\016)2205 385 y Fk(\026\027)2292 370 y Fs(:)1271 b |
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FB(\(2)3677 348 y(.)3704 370 y(9\))-9 582 y(Therefore,)27 |
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b(they)f(form)e(an)g(orthonormal)f(basis)h(of)h(the)g |
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Fs(d)p FB(-dimensional)c(sub-manifold)h(with)j(a)f(negativ)m(e)-9 |
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731 y(de\014nite)29 b(metric.)42 b(The)30 b(mo)s(duli)c(parameters)k |
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(of)e(the)i(heterotic)f(string)g(theory)h(are)f(sp)s(eci\014ed)h(b)m(y) |
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g(giving)-9 881 y(this)35 b(sub-manifold)d(and)j(the)h(string)e |
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(coupling)g(constan)m(t,)i(and)f(this)g(implies)e(that)i(the)g(mo)s |
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(duli)d(space)-9 1030 y(of)g(the)h(heterotic)f(string)g(compacti\014ed) |
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g(on)h Fs(T)1729 994 y Fk(d)1802 1030 y FB(is)1503 1195 |
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y Fs(S)6 b(O)s FB(\(16)21 b(+)h Fs(d;)17 b(d)p FB(\))p |
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947 1239 1695 4 v 947 1331 a Fs(S)6 b(O)s FB(\(16)20 |
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b(+)i Fs(d)p FB(\))g Fm(\002)h Fs(S)6 b(O)s FB(\()p Fs(d)p |
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FB(\))21 b Fm(\002)h Fs(S)6 b(O)s FB(\(16)21 b(+)h Fs(d;)17 |
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b(d)p FB(;)g Ft(Z)p FB(\))2673 1263 y Fm(\002)23 b Ft(R)2857 |
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1221 y Fr(+)2915 1263 y Fs(;)600 b FB(\(2)3629 1241 y(.)3655 |
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1263 y(10\))-9 1503 y(where)34 b Ft(R)357 1467 y Fr(+)448 |
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1503 y FB(is)e(asso)s(ciated)g(with)h(the)g(string)f(coupling)f |
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(constan)m(t)i Fs(g)2508 1518 y Fr(str)2595 1503 y FB(.)137 |
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1652 y(The)h(BPS)f(condition)e(for)h(a)g(heterotic)h(string)f(is)1662 |
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1864 y Fs(N)1740 1879 y Fr(osc)1861 1864 y FB(+)22 b |
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Fs(a)28 b FB(=)f(0)p Fs(:)1325 b FB(\(2)3629 1843 y(.)3655 |
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1864 y(11\))-9 2077 y(F)-8 b(or)32 b(BPS)h(states,)g(the)g(norm)f(of)g |
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Ft(K)g FB(should)h(b)s(e)g(smaller)d(than)j(2:)927 2289 |
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y Ft(K)1015 2247 y Fr(2)1082 2289 y FB(=)27 b(\()1227 |
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2280 y Fj(e)1223 2289 y Ft(a)1277 2304 y Fr(0)1317 2289 |
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y FB(\))1355 2247 y Fr(2)1416 2289 y FB(+)22 b(\()1555 |
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2282 y Fj(e)1552 2289 y Fs(a)1603 2241 y Fk(\026)1603 |
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2310 y Fr(0)1650 2289 y FB(\))1688 2247 y Fr(2)1750 2289 |
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y Fm(\000)g FB(\()p Fs(a)1938 2241 y Fk(\026)1938 2310 |
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y Fr(0)1985 2289 y FB(\))2023 2247 y Fr(2)2090 2289 y |
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FB(=)28 b Fm(\000)p FB(2\()2369 2256 y Fj(f)2358 2289 |
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y Fs(N)2436 2304 y Fr(osc)2557 2289 y FB(+)2657 2282 |
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y Fj(e)2655 2289 y Fs(a)p FB(\))g Fm(\024)g FB(2)p Fs(:)589 |
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b FB(\(2)3629 2267 y(.)3655 2289 y(12\))-9 2501 y(Equation)41 |
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b(\(2)508 2479 y(.)534 2501 y(12\))g(corresp)s(onds)h(to)f(the)g(BPS)h |
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(condition)e(for)g(junctions)h(\()p Ft(J)28 b Fm(\001)f |
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Ft(J)p FB(\))42 b Fm(\024)h FB(2)d(giv)m(en)i(in)3699 |
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2464 y Fr(8\))p Fk(;)14 b Fr(9\))3861 2501 y FB(.)-9 |
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2650 y(\(In)34 b(this)f(pap)s(er,)h(w)m(e)h(adopt)e(a)h(signature)f |
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(for)g(the)h(in)m(tersection)g(n)m(um)m(b)s(er)g(that)f(is)g(opp)s |
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(osite)g(to)h(that)f(in)-9 2800 y(Refs.)216 2763 y Fr(2\))p |
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Fk(;)15 b Fr(4\))380 2800 y FB(.\))43 b(The)34 b(mass)e(form)m(ula)f |
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(can)i(b)s(e)g(rewritten)f(as)1570 2974 y Fs(M)1674 2938 |
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y Fr(2)p 1516 3018 252 4 v 1516 3110 a FB(4)p Fs(\031)t(T)1681 |
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3125 y Fr(str)1805 3042 y FB(=)c(\()p Ft(W)2063 3057 |
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y Fk(\026)2131 3042 y Fm(\001)22 b Ft(K)p FB(\))2307 |
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3001 y Fr(2)2346 3042 y Fs(:)1169 b FB(\(2)3629 3020 |
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y(.)3655 3042 y(13\))-9 3265 y(In)33 b(particular,)e(massless)i(states) |
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g(should)g(satisfy)1675 3477 y Ft(W)1791 3492 y Fk(\026)1860 |
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3477 y Fm(\001)21 b Ft(K)28 b FB(=)f(0)p Fs(:)1338 b |
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FB(\(2)3629 3456 y(.)3655 3477 y(14\))-9 3689 y(Equation)35 |
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b(\(2)502 3668 y(.)528 3689 y(14\))g(de\014nes)h(a)f(subspace)i(of)d |
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(co)s(dimension)g Fs(d)p FB(.)50 b(Because)36 b(all)d(of)h(the)i |
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Ft(W)3228 3704 y Fk(\026)3309 3689 y FB(ha)m(v)m(e)g(negativ)m(e)-9 |
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3839 y(norms)f(and)i(the)f(whole)g(space)h(has)f(signature)g(\(+)1914 |
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3802 y Fk(d)p Fr(+16)2080 3839 y Fs(;)17 b Fm(\000)2201 |
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3802 y Fk(d)2241 3839 y FB(\),)37 b(this)f(subspace)i(has)e(a)g(p)s |
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(ositiv)m(e)f(de\014nite)-9 3988 y(metric.)55 b(Because)38 |
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b(the)f(lattice)f(on)g(the)i(subspace)g(corresp)s(onds)h(to)d(massless) |
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h(gauge)g(\014elds,)h(the)f(sub-)-9 4138 y(lattice)28 |
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b(is)h(nothing)g(but)h(a)f(ro)s(ot)g(lattice)f(of)h(the)h(gauge)f |
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(group.)43 b(Let)29 b(us)i(in)m(tro)s(duce)e(the)h(basis)g |
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Ft(U)3563 4153 y Fk(i)3621 4138 y FB(on)f(the)-9 4287 |
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y(sub-lattice)i(corresp)s(onding)i(to)f(the)h(fundamen)m(tal)f(w)m |
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(eigh)m(t.)44 b(Then,)34 b(an)m(y)g(v)m(ector)g Ft(K)e |
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FB(can)h(b)s(e)g(expanded)-9 4436 y(as)1534 4586 y Ft(K)28 |
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b FB(=)f Fs(a)1804 4601 y Fk(i)1833 4586 y Ft(U)1919 |
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4601 y Fk(i)1969 4586 y FB(+)22 b Fs(c)2109 4601 y Fk(\026)2156 |
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4586 y Ft(W)2272 4601 y Fk(\026)2318 4586 y Fs(:)1197 |
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b FB(\(2)3629 4564 y(.)3655 4586 y(15\))-9 4773 y(The)35 |
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b(co)s(e\016cien)m(ts)f Fs(a)738 4788 y Fk(i)800 4773 |
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y FB(giv)m(e)g(the)g(Dynkin)g(lab)s(el)e(of)h(the)h(state)g(asso)s |
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(ciated)g(with)f(the)h(v)m(ector)h Ft(K)p FB(.)46 b(On)34 |
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b(the)-9 4923 y(other)g(hand,)h(the)f(co)s(e\016cien)m(ts)h |
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Fs(c)1226 4938 y Fk(\026)1307 4923 y FB(giv)m(e)f(cen)m(tral)g(c)m |
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(harges)h(of)f(the)g(state,)h(b)s(ecause)g(the)g(mass)f(square)h(of)-9 |
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5072 y(eac)m(h)e(state)g(is)f(giv)m(en)h(as)g(the)g(sum)f(of)h(the)g |
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(square)g(of)f(the)h Fs(c)2203 5087 y Fk(\026)2250 5072 |
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y FB(:)1676 5250 y Fs(M)1780 5214 y Fr(2)p 1622 5294 |
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V 1622 5386 a FB(4)p Fs(\031)t(T)1787 5401 y Fr(str)1912 |
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5317 y FB(=)2015 5234 y Fj(X)2054 5409 y Fk(\026)2152 |
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5317 y Fs(c)2194 5276 y Fr(2)2194 5342 y Fk(\026)2240 |
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5317 y Fs(:)1275 b FB(\(2)3629 5296 y(.)3655 5317 y(16\))1915 |
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5687 y(5)p eop |
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6 5 bop 19 158 a FB(When)40 b(w)m(e)g(compactify)e(heterotic)h(string)f |
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(theory)i(on)f(a)g(torus)g Fs(T)2531 122 y Fk(d)2571 |
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158 y FB(,)i(new)f(2)p Fs(d)e(U)10 b FB(\(1\))39 b(\014elds)h(app)s |
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(ear)-127 307 y(from)f(the)i(metric)e(and)h(the)h(NS-NS)g(2-form)d |
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(\014eld,)k(and)f(at)f(a)g(generic)g(p)s(oin)m(t)f(in)h(mo)s(duli)e |
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(space,)43 b(the)-127 457 y(gauge)24 b(group)h(is)f(a)g(subgroup)h(of)f |
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Fs(S)6 b(O)s FB(\(32\))g Fm(\002)g Fs(U)k FB(\(1\))1699 |
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420 y Fr(2)p Fk(d)1773 457 y FB(.)41 b(F)-8 b(or)23 b(example,)j(if)d |
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(w)m(e)j(compactify)d(the)i Fs(x)3332 420 y Fr(8)3397 |
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457 y FB(direction)-127 606 y(and)33 b(in)m(tro)s(duce)f(the)h(Wilson)f |
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(line)1139 838 y Ft(w)1220 853 y Fr(8)1287 838 y FB(=)1391 |
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716 y Fj(\022\022)1523 770 y FB(1)p 1523 814 49 4 v 1523 |
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906 a(2)1582 716 y Fj(\023)1643 732 y Fk(n)1706 838 y |
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Fs(;)17 b FB(0)1799 796 y Fr(16)p Fi(\000)p Fk(n)1971 |
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716 y Fj(\023)2049 838 y Fs(;)114 b FB(\()p Fs(n)28 b(<)g |
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FB(8\))919 b(\(2)3511 816 y(.)3538 838 y(17\))-127 1082 |
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y(then)37 b(gauge)g(symmetry)g(is)f(brok)m(en)h(to)g |
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Fs(S)6 b(O)s FB(\(2)p Fs(n)p FB(\))24 b Fm(\002)h Fs(S)6 |
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b(O)s FB(\(32)23 b Fm(\000)j FB(2)p Fs(n)p FB(\))f Fm(\002)g |
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Fs(U)10 b FB(\(1\))2721 1045 y Fr(2)2761 1082 y FB(.)55 |
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b(If)37 b(w)m(e)h(adjust)f(the)g(com-)-127 1231 y(pacti\014cation)31 |
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b(radius)h(to)h(the)g(v)-5 b(alue)32 b(sp)s(eci\014ed)h(b)m(y)1506 |
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1420 y(1)p 1506 1464 V 1506 1555 a(2)1564 1487 y Ft(w)1647 |
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1446 y Fr(2)1645 1512 y(8)1708 1487 y FB(+)1816 1420 |
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y Fs(R)1891 1384 y Fr(2)1890 1444 y(8)p 1816 1464 115 |
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4 v 1838 1555 a Fs(l)1869 1527 y Fr(2)1867 1580 y Fk(s)1969 |
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1487 y FB(=)27 b(1)p Fs(;)1276 b FB(\(2)3511 1466 y(.)3538 |
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1487 y(18\))-127 1745 y(then)46 b(some)f(winding)f(mo)s(des)h(b)s |
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(ecome)h(massless,)j(and)c(the)h(gauge)f(symmetry)g Fs(S)6 |
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b(O)s FB(\(2)p Fs(n)p FB(\))30 b Fm(\002)h Fs(U)10 b |
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FB(\(1\))46 b(is)-127 1895 y(enhanced)34 b(to)e Fs(E)492 |
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1910 y Fk(n)p Fr(+1)630 1895 y FB(.)43 b(W)-8 b(e)33 |
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b(can)g(con\014rm)f(this)g(b)m(y)i(observing)f(the)g(sub-lattice)e(giv) |
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m(en)i(b)m(y)g(\(2)3284 1873 y(.)3311 1895 y(14\).)1065 |
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2229 y Fl(x)p Fy(3.)92 b(Corresp)s(ondence)38 b(of)g(states)19 |
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2451 y FB(In)g(Ref.)333 2415 y Fr(3\))400 2451 y FB(,)g(it)f(is)g(sho)m |
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(wn)i(that)e(the)h(junction)f(lattice)f(asso)s(ciated)h(with)g(the)h |
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(7-brane)f(bac)m(kground)-127 2601 y Fs(A)-54 2565 y |
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Fk(n)-7 2601 y Fs(B)5 b(C)i(B)e(C)38 b FB(\()p Fs(n)28 |
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b(<)g FB(8\))j(is)f(iden)m(tical)g(to)h(the)g(ro)s(ot)f(lattice)g(of)h |
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(the)g(a\016ne)h(Lie)f(algebra)3020 2569 y Fj(b)2996 |
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2601 y Fs(E)3068 2616 y Fk(n)p Fr(+1)3205 2601 y FB(.)43 |
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b(On)31 b(the)h(het-)-127 2750 y(erotic)h(side,)g(this)g(bac)m(kground) |
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h(corresp)s(onds)h(to)e Fs(T)1812 2714 y Fr(2)1884 2750 |
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y FB(compacti\014cation)e(with)i(the)g(Wilson)f(line)g(\(2)3608 |
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2729 y(.)3635 2750 y(17\))-127 2900 y(and)39 b(the)g |
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(compacti\014cation)e(radius)h Fs(R)1354 2915 y Fr(8)1433 |
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2900 y FB(satisfying)g(\(2)1961 2878 y(.)1988 2900 y(18\).)61 |
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b(In)39 b(the)h(7-brane)e(bac)m(kground,)k(in)c(addi-)-127 |
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3049 y(tion)31 b(to)i(junctions)f Ft(\012)694 3013 y |
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Fk(i)755 3049 y FB(asso)s(ciated)g(with)g(the)h(\014nite)f(Lie)g |
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(algebra)g Fs(E)2436 3064 y Fk(n)p Fr(+1)2573 3049 y |
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FB(,)h(w)m(e)g(ha)m(v)m(e)h(three)f(indep)s(enden)m(t)-127 |
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3199 y(junctions,)39 b Fs(\016)t FB(,)h Ft(\012)522 3162 |
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y Fr(0)561 3199 y FB(,)g(and)e Ft(\006)g FB(\(Fig.1\).)58 |
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b(An)38 b(arbitrary)f(junction)h(on)g(the)g(bac)m(kground)h(is)f |
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(expanded)h(as)-127 3348 y(follo)m(ws:)1229 3498 y Ft(J)28 |
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b FB(=)g Fs(a)1470 3513 y Fk(i)1498 3498 y Ft(\012)1579 |
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3456 y Fk(i)1629 3498 y FB(+)22 b Fs(k)s Ft(\012)1862 |
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3456 y Fr(0)1924 3498 y FB(+)2028 3491 y Fj(e)2022 3498 |
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y Fs(n\016)k FB(+)c Fs(\033)t Ft(\006)p Fs(:)1059 b FB(\(3)3560 |
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3476 y(.)3586 3498 y(1\))-127 3693 y(Co)s(e\016cien)m(ts)48 |
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b Fs(k)j FB(and)720 3686 y Fj(e)714 3693 y Fs(n)c FB(corresp)s(ond)h |
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(to)f(the)h(lev)m(el)e(and)i(the)f(grade,)k(while)c Fs(\033)k |
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FB(is)46 b(nothing)h(but)g(the)-127 3843 y(asymptotic)42 |
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b(NS-NS)h(c)m(harge.)74 b(The)44 b(in)m(tersection)e(n)m(um)m(b)s(ers)i |
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(among)d(these)j(junctions)f(are)g(giv)m(en)f(in)-127 |
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3992 y(T)-8 b(able)31 b(I)s(I.)44 b(Here)32 b Fs(A)586 |
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3956 y Fk(ij)678 3992 y FB(is)g(the)g(in)m(v)m(erse)h(matrix)d(of)i |
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(the)g(Cartan)f(matrix)g(of)g(the)h(\014nite)f(Lie)h(algebra)e |
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Fs(E)3606 4007 y Fk(n)p Fr(+1)3744 3992 y FB(.)80 4195 |
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y Fh(T)-8 b(able)31 b(I)s(I.)72 b(In)m(tersection)31 |
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b(n)m(um)m(b)s(ers)e(among)i(junctions)e(generating)h(the)h(a\016ne)f |
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(Lie)g(algebra)3380 4169 y Fj(b)3360 4195 y Fg(E)3427 |
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4209 y Fk(n)p Fr(+1)p 1464 4473 5 150 v 1528 4428 a Ft(\012)1609 |
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4392 y Fk(i)1749 4428 y Ft(\012)1830 4392 y Fr(0)1970 |
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4428 y Fs(\016)179 b Ft(\006)p 1246 4477 1152 5 v 1301 |
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4582 a(\012)1382 4546 y Fk(i)p 1464 4627 5 150 v 1516 |
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4582 a Fs(A)1589 4546 y Fk(ij)1296 4731 y Ft(\012)1377 |
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4695 y Fr(0)p 1464 4776 V 1784 4731 a Fm(\003)135 b FB(1)190 |
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b Fm(\003)1332 4881 y Fs(\016)p 1464 4925 V 410 w FB(1)1315 |
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5030 y Ft(\006)p 1464 5075 V 388 w Fm(\003)283 b(\000)2254 |
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4991 y Fr(1)p 2204 5007 133 4 v 2204 5064 a(8)p Fi(\000)p |
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Fk(n)-127 5397 y FB(The)32 b(comp)s(onen)m(ts)f(expressed)i(as)e(`)p |
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Fm(\003)p FB(')f(in)g(the)h(table)f(dep)s(end)i(on)e(the)h(con)m(v)m |
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(en)m(tion)h(used)g(in)e(de\014ning)g(the)1797 5687 y(6)p |
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eop |
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7 6 bop -9 158 a FB(junctions)30 b Ft(\012)490 122 y |
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Fr(0)560 158 y FB(and)h Ft(\006)p FB(.)43 b(By)31 b(adding)e |
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Fs(\016)35 b FB(to)30 b(them,)h(w)m(e)g(can)g(\014x)g(these)h(comp)s |
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(onen)m(ts)f(to)f(arbitrary)f(v)-5 b(alues.)-9 307 y(The)40 |
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b(corresp)s(ondence)g(of)f(the)g(v)m(ector)h Ft(U)1561 |
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322 y Fk(i)1628 307 y FB(in)e(\(2)1835 286 y(.)1861 307 |
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y(15\))g(and)h(junctions)g Ft(\012)2739 271 y Fk(i)2806 |
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307 y FB(is)f(trivial.)59 b(Here,)41 b(w)m(e)f(fo)s(cus)-9 |
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457 y(on)33 b(determining)e(ho)m(w)i(w)m(e)h(can)g(repro)s(duce)f(the)h |
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(in)m(tersection)f(\()p Ft(\006)22 b Fm(\001)g Ft(\006)p |
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FB(\),)33 b(whic)m(h)g(is)g(indep)s(enden)m(t)h(of)e(the)-9 |
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606 y(con)m(v)m(en)m(tion.)62 b(Because)40 b(the)f(three)g(junctions)f |
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Ft(\012)1849 570 y Fr(0)1889 606 y FB(,)i Fs(\016)i FB(and)c |
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Ft(\006)h FB(carry)f(cen)m(tral)h(c)m(harges)g(and)g |
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Fs(\016)j FB(and)d Fs(\012)3849 570 y Fr(0)-9 756 y FB(resp)s(ect)31 |
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b(the)f(same)g(sup)s(ersymmetry)-8 b(,)31 b(the)f(con)m(tribution)f(of) |
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g(these)i(three)g(c)m(harges)g(to)e(the)h(mass)g(can)g(b)s(e)-9 |
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905 y(expressed)35 b(as)1407 1014 y Fs(M)1511 978 y Fr(2)p |
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1354 1059 252 4 v 1354 1150 a FB(4)p Fs(\031)t(T)1519 |
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1165 y Fr(str)1643 1082 y FB(=)28 b(\()p Fs(xk)d FB(+)d |
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Fs(y)2072 1075 y Fj(e)2066 1082 y Fs(n)p FB(\))2162 1041 |
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y Fr(2)2223 1082 y FB(+)g Fs(z)2370 1041 y Fr(2)2410 |
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1082 y Fs(\033)2469 1041 y Fr(2)2509 1082 y Fs(;)1054 |
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b FB(\(3)3677 1060 y(.)3704 1082 y(2\))-9 1300 y(where)34 |
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b Fs(x)p FB(,)e Fs(y)k FB(and)d Fs(z)k FB(are)32 b(some)g(constan)m |
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(ts.)45 b(If)33 b(w)m(e)g(assume)g(the)g(comp)s(onen)m(ts)g(`)p |
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Fm(\003)p FB(')f(in)g(T)-8 b(able)32 b(I)s(I)h(are)f(zero,)-9 |
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1449 y(b)m(y)e(comparing)e(this)i(equation)f(with)h(Eq.\(2)1631 |
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1428 y(.)1658 1449 y(16\),)g(w)m(e)h(obtain)d(the)j(follo)m(wing)c |
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(corresp)s(ondence)k(b)s(et)m(w)m(een)-9 1599 y(v)m(ectors)j(on)e |
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(Narain)g(lattice)f(and)i(junctions:)1308 1850 y Ft(W)1424 |
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1865 y Fr(8)1491 1850 y Fm(\030)28 b Fs(z)t Ft(\006)p |
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Fs(;)115 b Ft(W)1984 1865 y Fr(9)2051 1850 y Fm(\030)28 |
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b Fs(y)t Ft(\012)2289 1809 y Fr(0)2350 1850 y FB(+)22 |
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b Fs(x\016)n(:)1019 b FB(\(3)3677 1829 y(.)3704 1850 |
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y(3\))-9 2102 y(The)39 b(problem)d(is)h(to)h(determine)f(ho)m(w)i(w)m |
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(e)f(should)g(\014x)g(the)h(co)s(e\016cien)m(ts)g Fs(x)p |
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FB(,)g Fs(y)i FB(and)d Fs(z)t FB(.)60 b(F)-8 b(or)37 |
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b Fs(z)t FB(,)i(w)m(e)g(can)-9 2252 y(pro)s(ceed)h(as)f(follo)m(ws.)61 |
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b(As)39 b(is)g(sho)m(wn)h(in)e(Fig.1,)h(the)g(junction)g |
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Ft(\006)g FB(con)m(tains)g(a)f(string)g(lo)s(op)g(with)g(R-R)-9 |
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2401 y(c)m(harge)31 b Fs(Q)372 2416 y Fr(9)439 2401 y |
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FB(=)d(1)p Fs(=)p FB(\(8)16 b Fm(\000)i Fs(n)p FB(\).)43 |
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b(By)31 b(the)f(relation)f(\(1)1764 2380 y(.)1790 2401 |
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y(2\),)i(w)m(e)g(kno)m(w)h(that)e(the)g(con\014guration)g |
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Ft(\006)g FB(on)g(the)h(t)m(yp)s(e)-9 2551 y(I)s(IB)c(side)g(corresp)s |
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(onds)h(to)f(a)g(heterotic)g(string)f(with)h(winding)f(n)m(um)m(b)s(er) |
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h Fs(n)2728 2566 y Fr(8)2795 2551 y FB(=)h(2)p Fs(=)p |
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FB(\(8)11 b Fm(\000)g Fs(n)p FB(\).)41 b(On)27 b(the)g(other)-9 |
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2700 y(hand,)33 b(the)g Fs(n)488 2715 y Fr(8)560 2700 |
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y FB(comp)s(onen)m(t)g(of)f(the)h(v)m(ector)g Ft(W)1746 |
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2715 y Fr(8)1818 2700 y FB(is)f Fs(l)1945 2715 y Fk(s)1982 |
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2700 y Fs(=)p FB(\()2069 2618 y Fm(p)p 2152 2618 49 4 |
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v 82 x FB(2)p Fs(R)2275 2715 y Fr(8)2314 2700 y FB(\))c(=)f(2)p |
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Fs(=)2581 2622 y Fm(p)p 2664 2622 229 4 v 78 x FB(8)22 |
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b Fm(\000)g Fs(n)q FB(.)43 b(Therefore,)34 b(w)m(e)g(\014nd)1576 |
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2967 y Ft(\006)27 b Fm(\030)1931 2900 y FB(1)p 1800 2944 |
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312 4 v 1800 2962 a Fm(p)p 1883 2962 229 4 v 78 x FB(8)21 |
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b Fm(\000)i Fs(n)2121 2967 y Ft(W)2237 2982 y Fr(8)2276 |
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2967 y Fs(:)1287 b FB(\(3)3677 2946 y(.)3704 2967 y(4\))-9 |
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3237 y(If)43 b(w)m(e)h(b)s(eliev)m(e)f(this)f(corresp)s(ondence,)48 |
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b(w)m(e)c(can)f(repro)s(duce)h(the)f(self-in)m(tersection)g(n)m(um)m(b) |
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s(er)g(of)f Ft(\006)h FB(in)-9 3387 y(T)-8 b(able)32 |
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b(I)s(I)h(as)g(an)f(inner)g(pro)s(duct)h(of)f(the)h(v)m(ector:)1161 |
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3649 y(\()p Ft(\006)22 b Fm(\001)g Ft(\006)p FB(\))28 |
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b(=)1702 3581 y(1)p 1612 3625 V 1612 3717 a(8)22 b Fm(\000)h |
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Fs(n)1851 3649 y Ft(W)1967 3664 y Fr(8)2028 3649 y Fm(\001)f |
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Ft(W)2194 3664 y Fr(8)2261 3649 y FB(=)28 b Fm(\000)2542 |
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3581 y FB(1)p 2452 3625 V 2452 3717 a(8)22 b Fm(\000)h |
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Fs(n)2691 3649 y(:)872 b FB(\(3)3677 3627 y(.)3704 3649 |
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y(5\))137 3901 y(In)34 b(this)e(w)m(a)m(y)-8 b(,)35 b(can)e(w)m(e)h |
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(establish)f(a)g(complete)f(corresp)s(ondence)j(b)s(et)m(w)m(een)h(the) |
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d(quan)m(tum)h(n)m(um)m(b)s(ers)-9 4051 y(of)24 b(heterotic)h(strings)g |
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(and)g(in)m(v)-5 b(arian)m(t)24 b(c)m(harges)i(of)e(junctions?)41 |
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b(Probably)-8 b(,)26 b(this)f(is)f(p)s(ossible.)41 b(Ho)m(w)m(ev)m(er,) |
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28 b(it)-9 4200 y(is)g(di\016cult.)42 b(The)30 b(origin)c(of)j(the)g |
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(di\016cult)m(y)g(is)f(the)i(existence)g(of)f(con\014gurations)f(lik)m |
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(e)g Ft(\006)p FB(.)42 b(The)30 b(p)s(osition)-9 4350 |
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y(of)k(the)h(D-string)d(lo)s(op)h(can)i(b)s(e)f(freely)h(mo)m(v)m(ed,)g |
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(op)-9 4499 y(dep)s(ends)40 b(on)f(the)h(p)s(osition.)61 |
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b(Ho)m(w)m(ev)m(er,)43 b(information)35 b(regarding)j(the)i(p)s |
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(osition)d(is)h(not)h(con)m(tained)g(in)-9 4649 y(the)c(quan)m(tum)g(n) |
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m(um)m(b)s(ers)h(on)e(the)h(heterotic)g(string)f(side.)50 |
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b(Therefore)36 b(it)e(is)g(not)g(clear)g(ho)m(w)i(w)m(e)g(should)-9 |
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4798 y(de\014ne)e(the)f(winding)e(n)m(um)m(b)s(er)i(on)g(the)g(t)m(yp)s |
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(e)g(I)s(IB)g(side)f(using)h(heterotic)f(quan)m(tum)h(n)m(um)m(b)s |
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(ers.)137 4948 y(There)40 b(is)e(only)h(one)g(case)g(in)f(whic)m(h)i |
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(suc)m(h)g(subtlet)m(y)f(do)s(es)h(not)e(exist.)63 b(The)39 |
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b(reason)g(w)m(e)h(need)g(the)-9 5098 y(string)29 b(attac)m(hed)i(on)e |
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(the)i(string)e(lo)s(op)f(of)i(the)g(junction)f Ft(\006)h |
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FB(is)f(that)h(the)g(mono)s(drom)m(y)f(around)h(the)g(lo)s(op)-9 |
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5247 y(c)m(hanges)43 b(the)f(c)m(harge)h(of)e(the)h(string)f(lo)s(op.) |
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70 b(Therefore,)45 b(in)c(the)h(\015at)g(bac)m(kground,)j(where)e(the)f |
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(R-R)-9 5397 y(c)m(harge)29 b(of)g(eac)m(h)g(O7-plane)f(is)g(canceled)i |
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(b)m(y)f(four)g(D7-branes)f(and)h(the)g(mono)s(drom)m(y)f(around)h(the) |
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g(cycle)1915 5687 y(7)p eop |
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8 7 bop -127 158 a FB(is)31 b(1,)h(con\014gurations)f(lik)m(e)g |
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Ft(\006)h FB(are)f(not)h(allo)m(w)m(ed,)f(and)h(the)g(de\014nition)f |
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(of)g(the)h(winding)f(n)m(um)m(b)s(er)h(is)f(v)m(ery)-127 |
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307 y(clear.)43 b(On)33 b(the)g(heterotic)f(side,)h(the)g(bac)m |
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(kground)g(is)g(sp)s(eci\014ed)g(b)m(y)g(the)g(follo)m(wing)d(Wilson)i |
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(line:)798 592 y Ft(w)879 607 y Fr(8)946 592 y FB(=)c(\(0)1137 |
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551 y Fr(8)1176 592 y Fs(;)17 b FB(\(1)p Fs(=)p FB(2\))1443 |
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551 y Fr(8)1481 592 y FB(\))p Fs(;)114 b Ft(w)1741 607 |
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y Fr(9)1808 592 y FB(=)28 b(\(0)1999 551 y Fr(4)2038 |
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592 y Fs(;)17 b FB(\(1)p Fs(=)p FB(2\))2305 551 y Fr(4)2343 |
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592 y Fs(;)g FB(0)2436 551 y Fr(4)2475 592 y Fs(;)g FB(\(1)p |
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Fs(=)p FB(2\))2742 551 y Fr(4)2781 592 y FB(\))p Fs(:)627 |
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b FB(\(3)3560 571 y(.)3586 592 y(6\))-127 877 y(On)30 |
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b(this)h(bac)m(kground,)h(there)f(is)f(no)g(subtlet)m(y)i(concerning)e |
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(the)h(de\014nition)f(of)f(c)m(harges,)k(and)d(b)m(y)i(means)-127 |
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1027 y(of)24 b(Eq.\(1)207 1005 y(.)235 1027 y(2\),)i(w)m(e)f(can)g |
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(represen)m(t)i(the)e(winding)e(n)m(um)m(b)s(ers)j(of)e(strings)g(on)h |
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(the)g(t)m(yp)s(e)g(I)s(IB)g(side)g(b)m(y)g(quan)m(tum)-127 |
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1176 y(n)m(um)m(b)s(ers)39 b(on)f(the)g(heterotic)g(side.)60 |
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b(If)38 b(all)e(the)j(comp)s(onen)m(ts)f(of)g(the)g(v)m(ector)2801 |
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1167 y Fj(e)2797 1176 y Ft(a)2851 1191 y Fr(0)2928 1176 |
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y FB(v)-5 b(anish,)40 b(w)m(e)f(obtain)e(a)-127 1326 |
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y(con\014guration)32 b(with)h(t)m(w)m(o)h(string)e(lo)s(ops)g(with)h(c) |
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m(harges)h(\()p Fs(P)2071 1341 y Fk(\026)2118 1326 y |
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Fs(;)17 b(Q)2239 1341 y Fk(\026)2285 1326 y FB(\))33 |
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b(\(Fig.2\).)43 b(If)33 b(the)h(comp)s(onen)m(ts)g(of)e(the)-127 |
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1475 y(v)m(ector)174 1466 y Fj(e)170 1475 y Ft(a)224 |
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1490 y Fr(0)298 1475 y FB(=)j Ft(k)25 b Fm(\000)h Fs(n)654 |
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1490 y Fr(8)693 1475 y Ft(w)774 1490 y Fr(8)839 1475 |
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y Fm(\000)f Fs(n)999 1490 y Fr(9)1039 1475 y Ft(w)1120 |
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1490 y Fr(9)1196 1475 y FB(tak)m(e)37 b(nonzero)h(v)-5 |
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b(alues,)37 b(eac)m(h)h(comp)s(onen)m(t)f(should)f(b)s(e)h(regarded)g |
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(as)-127 1625 y(the)f(n)m(um)m(b)s(er)g(of)f(strings)h(attac)m(hed)g |
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(to)f(eac)m(h)i(D7-brane.)51 b(As)37 b(a)e(result,)h(w)m(e)h(obtain)d |
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(the)i(con\014guration)-127 1774 y(in)e(Fig.3.)50 b(In)35 |
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b(this)g(case,)h(the)g(c)m(harges)g(of)e(the)i(string)e(lo)s(ops)g(are) |
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h(not)g(constan)m(t)h(on)f(them,)h(due)f(to)g(the)-127 |
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1924 y(fundamen)m(tal)28 b(strings)h(lying)e(b)s(et)m(w)m(een)k(the)e |
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(lo)s(ops)f(and)h(D7-branes.)42 b(Nev)m(ertheless,)32 |
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b(w)m(e)e(can)f(de\014ne)h(the)-127 2073 y(c)m(harge)c(\()p |
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Fs(P)273 2088 y Fk(\026)319 2073 y Fs(;)17 b(Q)440 2088 |
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y Fk(\026)487 2073 y FB(\))24 b(of)h(the)g(junction)g(uniquely)g(as)g |
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(one)h(half)d(of)i(the)g(sum)g(of)g(the)g(c)m(harges)h(that)f(go)g |
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(through)-127 2222 y(cycles)35 b(wrapp)s(ed)h(around)e(the)h |
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Fs(x)1112 2186 y Fr(8)1186 2222 y FB(and)f Fs(x)1432 |
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2186 y Fr(9)1506 2222 y FB(directions,)h(whic)m(h)g(are)f(represen)m |
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(ted)j(b)m(y)e(the)g(dashed)g(lines)-127 2372 y(in)44 |
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b(Fig.3.)78 b(The)46 b(c)m(harge)f(of)f(an)m(y)i(segmen)m(t)f(in)f(the) |
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h(con\014guration)f(in)g(Fig.3)f(is)h(determined)h(b)m(y)g(the)-127 |
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2521 y(information)34 b(w)m(e)k(ha)m(v)m(e)h(already)d(giv)m(en.)58 |
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b(In)37 b(Fig.3,)g(w)m(e)h(assume)g(that)e(the)i(branc)m(h)g(cuts)g(go) |
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f(out)m(w)m(ard.)-127 2671 y(By)d(means)g(of)g(con)m(tin)m(uous)h |
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(deformation,)d(w)m(e)j(can)f(collect)f(all)f(16)i(D7-branes)f(at)h |
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(the)g(same)g(p)s(osition.)-127 2820 y(As)28 b(w)m(e)g(mo)m(v)m(e)g |
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(D7-branes,)g(the)g(branc)m(h)g(cuts)h(attac)m(hed)f(on)f(them)g(are)g |
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(also)g(mo)m(v)m(ed.)42 b(If)28 b(the)f(branc)m(h)i(cuts)-127 |
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2970 y(go)39 b(across)h(strings,)h(the)f(c)m(harges)h(of)e(the)h |
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(strings)f(c)m(hange,)j(and)e(if)e(a)h(D7-brane)g(go)s(es)g(across)i |
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(strings)-127 3119 y(with)26 b(D-string)f(c)m(harge,)k(the)e(n)m(um)m |
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(b)s(er)g(of)f(strings)g(attac)m(hed)i(to)e(the)h(D7-brane)f(is)g(c)m |
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(hanged)i(through)e(the)-127 3268 y(Hanan)m(y-Witten)36 |
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b(e\013ect.)53 b(T)-8 b(aking)35 b(accoun)m(t)h(of)f(these)i(facts,)g |
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(w)m(e)f(obtain)f(the)h(con\014guration)e(in)h(Fig.4.)-127 |
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3418 y(In)d(this)f(con\014guration,)g(w)m(e)h(ha)m(v)m(e)h(separated)f |
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(eac)m(h)h(O7-plane)d(in)m(to)h(a)g(B-)g(and)g(a)g(C-brane,)i(and)e |
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(the)h(24)-127 3567 y(7-branes)g(w)m(e)i(ha)m(v)m(e)f(obtained)f(are)h |
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(lab)s(eled)e Fs(A)1601 3582 y Fr(1)1640 3567 y FB(,)i |
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Fm(\001)17 b(\001)g(\001)n FB(,)32 b Fs(A)1948 3582 y |
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Fr(16)2023 3567 y FB(,)g Fs(B)2156 3582 y Fr(1)2196 3567 |
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y FB(,)g Fs(C)2325 3582 y Fr(1)2365 3567 y FB(,)g Fm(\001)17 |
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b(\001)g(\001)n FB(,)33 b Fs(B)2674 3582 y Fr(4)2713 |
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3567 y FB(,)g Fs(C)2843 3582 y Fr(4)2914 3567 y FB(in)f(coun)m(ter-clo) |
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s(c)m(kwise)-127 3717 y(order.)44 b(As)33 b(a)f(result,)h(in)m(v)-5 |
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b(arian)m(t)31 b(c)m(harges)i(of)g(the)g(junctions)f(are)h(giv)m(en)f |
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(as)h(follo)m(ws:)1264 4002 y Fs(Q)1341 4017 y Fe(a)1381 |
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4027 y Ff(i)1439 4002 y FB(=)28 b Fs(k)1594 4017 y Fk(i)1621 |
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4002 y Fs(;)1825 b FB(\(3)3560 3980 y(.)3586 4002 y(7\))1250 |
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4176 y Fs(Q)1327 4191 y Fe(b)1372 4200 y Fd(1)1439 4176 |
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y FB(=)28 b(0)p Fs(;)1854 b FB(\(3)3560 4155 y(.)3586 |
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4176 y(8\))1259 4351 y Fs(Q)1336 4366 y Fe(c)1372 4375 |
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y Fd(1)1439 4351 y FB(=)28 b(0)p Fs(;)1854 b FB(\(3)3560 |
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4329 y(.)3586 4351 y(9\))1250 4525 y Fs(Q)1327 4540 y |
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Fe(b)1372 4549 y Fd(2)1439 4525 y FB(=)28 b Fm(\000)p |
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Fs(m)1705 4540 y Fr(8)1767 4525 y FB(+)22 b Fs(n)1923 |
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4540 y Fr(8)1985 4525 y Fm(\000)g Fs(n)2142 4540 y Fr(9)2204 |
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4525 y Fm(\000)h Fs(s;)1047 b FB(\(3)3511 4504 y(.)3538 |
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4525 y(10\))1259 4699 y Fs(Q)1336 4714 y Fe(c)1372 4723 |
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y Fd(2)1439 4699 y FB(=)28 b Fm(\000)p Fs(m)1705 4714 |
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y Fr(8)1767 4699 y FB(+)22 b Fs(n)1923 4714 y Fr(8)1985 |
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4699 y Fm(\000)g Fs(s;)1267 b FB(\(3)3511 4678 y(.)3538 |
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4699 y(11\))1250 4874 y Fs(Q)1327 4889 y Fe(b)1372 4898 |
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y Fd(3)1439 4874 y FB(=)28 b Fs(m)1628 4889 y Fr(8)1689 |
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4874 y FB(+)22 b Fs(m)1872 4889 y Fr(9)1934 4874 y Fm(\000)h |
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Fs(n)2092 4889 y Fr(8)2154 4874 y FB(+)f Fs(s;)1099 b |
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FB(\(3)3511 4852 y(.)3538 4874 y(12\))1259 5048 y Fs(Q)1336 |
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5063 y Fe(c)1372 5072 y Fd(3)1439 5048 y FB(=)28 b Fs(m)1628 |
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5063 y Fr(8)1689 5048 y FB(+)22 b Fs(m)1872 5063 y Fr(9)1934 |
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5048 y Fm(\000)h Fs(n)2092 5063 y Fr(9)2154 5048 y FB(+)f |
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Fs(s;)1099 b FB(\(3)3511 5027 y(.)3538 5048 y(13\))1250 |
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5222 y Fs(Q)1327 5237 y Fe(b)1372 5246 y Fd(4)1439 5222 |
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y FB(=)28 b Fm(\000)p Fs(m)1705 5237 y Fr(9)1767 5222 |
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y FB(+)22 b Fs(n)1923 5237 y Fr(9)1985 5222 y Fm(\000)g |
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Fs(s;)1267 b FB(\(3)3511 5201 y(.)3538 5222 y(14\))1259 |
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5397 y Fs(Q)1336 5412 y Fe(c)1372 5421 y Fd(4)1439 5397 |
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y FB(=)28 b Fm(\000)p Fs(m)1705 5412 y Fr(9)1767 5397 |
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y Fm(\000)23 b Fs(n)1925 5412 y Fr(8)1986 5397 y FB(+)f |
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Fs(n)2142 5412 y Fr(9)2204 5397 y Fm(\000)h Fs(s;)1047 |
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b FB(\(3)3511 5375 y(.)3538 5397 y(15\))1797 5687 y(8)p |
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9 8 bop -9 158 a FB(where)34 b Fs(s)e FB(is)g(de\014ned)i(b)m(y)1687 |
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344 y Fs(s)28 b FB(=)1874 277 y(1)p 1874 321 49 4 v 1874 |
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413 a(2)1974 237 y Fr(16)1950 261 y Fj(X)1953 444 y Fk(i)p |
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Fr(=1)2086 344 y Fs(k)2137 359 y Fk(i)2165 344 y Fs(:)1350 |
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b FB(\(3)3629 323 y(.)3655 344 y(16\))-9 582 y(Because)34 |
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b(the)f(v)m(ector)h Ft(k)e FB(tak)m(es)i(v)-5 b(alues)33 |
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b(on)f(the)h(lattice)e Fs(\000)14 b FB(,)33 b Fs(s)f |
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FB(is)g(alw)m(a)m(ys)h(an)g(in)m(teger.)137 731 y(Junctions,)c(whic)m |
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(h)g(are)f(dual)f(to)g(heterotic)h(string)f(states,)j(should)d(b)s(e)i |
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(prop)s(er;)g(i.e.,)g(their)e(in)m(v)-5 b(arian)m(t)-9 |
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881 y(c)m(harges)26 b(should)e(b)s(e)h(in)m(tegers.)41 |
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b(A)m(t)25 b(\014rst)g(sigh)m(t,)h(ho)m(w)m(ev)m(er,)j(Eq.\(3)2359 |
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859 y(.)2387 881 y(7\))24 b(giv)m(es)h(fractional)e(c)m(harges)i |
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Fs(Q)3571 896 y Fe(a)3611 906 y Ff(i)3666 881 y FB(when)-9 |
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1030 y Ft(k)39 b FB(con)m(tains)g(an)h(o)s(dd)f(n)m(um)m(b)s(er)g(of)g |
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(spinor)g(ro)s(ots)g Fs(\013)1915 1045 y Fk(s)1991 1030 |
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y FB(in)f(\(2)2198 1009 y(.)2225 1030 y(4\).)63 b(W)-8 |
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b(e)39 b(can)h(alw)m(a)m(ys)g(mak)m(e)f(this)g(junction)-9 |
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1180 y(prop)s(er)29 b(b)m(y)h(adding)e(n)m(ull)g(junctions.)43 |
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b(W)-8 b(e)29 b(ha)m(v)m(e)i(t)m(w)m(o)e(indep)s(enden)m(t)i(n)m(ull)d |
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(junctions)h Ft(N)3204 1195 y Fr(1)3272 1180 y FB(and)g |
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Ft(N)3546 1195 y Fr(2)3585 1180 y FB(,)h(whic)m(h)-9 |
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1329 y(go)e(around)h(all)e(24)i(junctions.)42 b(Because)31 |
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b(the)e(compacti\014cation)e(manifold)f(is)j Fs(S)2983 |
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1293 y Fr(2)3022 1329 y FB(,)h(these)g(junctions)f(can)-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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(follo)m(wing)d(in)m(v)-5 b(arian)m(t)31 b(c)m(harges:)150 |
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1724 y Ft(N)238 1739 y Fr(1)305 1724 y FB(:)d(\(0)447 |
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1683 y Fr(16)521 1724 y Fs(;)17 b FB(1)p Fs(;)g FB(1)p |
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Fs(;)g Fm(\000)p FB(1)p Fs(;)g Fm(\000)p FB(1)p Fs(;)g |
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FB(1)p Fs(;)g FB(1)p Fs(;)g Fm(\000)p FB(1)p Fs(;)g Fm(\000)p |
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FB(1\))p Fs(;)112 b Ft(N)1838 1739 y Fr(2)1905 1724 y |
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FB(:)28 b(\(1)2047 1683 y Fr(16)2121 1724 y Fs(;)17 b |
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Fm(\000)p FB(15)p Fs(;)g Fm(\000)p FB(13)p Fs(;)g FB(11)p |
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Fs(;)g FB(9)p Fs(;)g Fm(\000)p FB(7)p Fs(;)g Fm(\000)p |
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FB(5)p Fs(;)g FB(3)p Fs(;)g FB(1\))p Fs(:)157 b FB(\(3)3629 |
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1702 y(.)3655 1724 y(17\))-9 1969 y(If)28 b(all)f Fs(Q)293 |
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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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(half)g(o)s(dd)g(in)m(teger,)h(b)m(y)h(adding)e(the)h(n)m(ull)e |
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(junction)h(\()p Ft(N)3140 1984 y Fr(1)3194 1969 y FB(+)14 |
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b Ft(N)3372 1984 y Fr(2)3411 1969 y FB(\))p Fs(=)p FB(2,)29 |
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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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b(F)-8 b(urthermore,)30 b(using)g(n)m(ull)f(junctions,)h(w)m(e)h(can)g |
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(\014x)f(an)m(y)h(t)m(w)m(o)g(of)e Fs(Q)3625 2134 y Fe(b)3670 |
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2144 y Ff(i)3731 2119 y FB(and)-9 2268 y Fs(Q)68 2283 |
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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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b(alues,)26 b(and)e(in)f(the)i(relations)e(\(3)1678 2247 |
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y(.)1704 2268 y(7\)-\(3)1911 2247 y(.)1937 2268 y(15\))h(w)m(e)h |
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(adopted)g(the)f(con)m(v)m(en)m(tion)i Fs(Q)3314 2283 |
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y Fe(b)3359 2292 y Fd(1)3426 2268 y FB(=)h Fs(Q)3606 |
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2283 y Fe(c)3642 2292 y Fd(1)3709 2268 y FB(=)h(0.)137 |
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2418 y(Finally)-8 b(,)35 b(let)g(us)i(con\014rm)e(that)h(the)h(in)m |
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(tersection)f(n)m(um)m(b)s(er)h(de\014ned)g(in)e(Ref.)3037 |
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2381 y Fr(2\))3140 2418 y FB(is)h(consisten)m(t)h(with)-9 |
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2567 y(the)k(inner)f(pro)s(duct)h(of)f(the)h(v)m(ector)h |
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Ft(K)p FB(.)67 b(According)40 b(to)h(Ref.)2362 2531 y |
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Fr(2\))2428 2567 y FB(,)i(the)e(in)m(tersection)f(n)m(um)m(b)s(ers)i |
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(among)-9 2717 y(unit)32 b(junctions)g(are)267 2972 y(\()p |
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Ft(a)359 2987 y Fk(i)410 2972 y Fm(\001)21 b Ft(a)513 |
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2987 y Fk(j)550 2972 y FB(\))28 b(=)g(\()p Ft(b)820 2987 |
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y Fk(i)870 2972 y Fm(\001)22 b Ft(b)982 2987 y Fk(j)1019 |
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2972 y FB(\))27 b(=)h(\()p Ft(c)1276 2987 y Fk(i)1326 |
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2972 y Fm(\001)22 b Ft(c)1426 2987 y Fk(j)1462 2972 y |
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FB(\))28 b(=)f Fs(\016)1674 2987 y Fk(ij)1735 2972 y |
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Fs(;)114 b FB(\()p Ft(a)1968 2987 y Fk(i)2019 2972 y |
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Fm(\001)22 b Ft(b)2131 2987 y Fk(j)2168 2972 y FB(\))27 |
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b(=)h Fm(\000)2424 2904 y FB(1)p 2424 2949 V 2424 3040 |
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a(2)2483 2972 y Fs(;)114 b FB(\()p Ft(a)2716 2987 y Fk(i)2767 |
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2972 y Fm(\001)22 b Ft(c)2867 2987 y Fk(j)2903 2972 y |
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FB(\))28 b(=)3082 2904 y(1)p 3082 2949 V 3082 3040 a(2)3141 |
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2972 y Fs(;)374 b FB(\(3)3629 2950 y(.)3655 2972 y(18\))1350 |
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3319 y(\()p Ft(b)1450 3334 y Fk(i)1501 3319 y Fm(\001)22 |
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b Ft(c)1601 3334 y Fk(j)1637 3319 y FB(\))28 b(=)1806 |
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3144 y Fj(8)1806 3219 y(<)1806 3369 y(:)1930 3249 y Fm(\000)p |
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FB(1)112 b(\()p Fs(i)28 b Fm(\024)h Fs(j)6 b FB(\))1969 |
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3398 y(1)138 b(\()p Fs(i)28 b(>)f(j)6 b FB(\))p Fs(:)3542 |
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3319 y FB(\(3)3629 3297 y(.)3655 3319 y(19\))-9 3590 |
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y(\(In)28 b(our)f(con)m(v)m(en)m(tion,)j(in)m(tersection)d(n)m(um)m(b)s |
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(ers)i(ha)m(v)m(e)f(signs)g(opp)s(osite)f(to)g(those)h(in)f(Ref.)3229 |
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3554 y Fr(2\))3295 3590 y FB(.\))42 b(Using)27 b(these)-9 |
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3739 y(equations,)33 b(w)m(e)g(obtain)1288 3889 y(\()p |
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Ft(J)22 b Fm(\001)g Ft(J)p FB(\))27 b(=)h Ft(k)1742 3848 |
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y Fr(2)1804 3889 y FB(+)22 b(2)p Fs(m)2036 3904 y Fr(8)2075 |
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3889 y Fs(n)2133 3904 y Fr(8)2195 3889 y FB(+)g(2)p Fs(m)2427 |
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3904 y Fr(9)2467 3889 y Fs(n)2525 3904 y Fr(9)2564 3889 |
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y Fs(:)951 b FB(\(3)3629 3867 y(.)3655 3889 y(20\))-9 |
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4091 y(This)33 b(is)f(iden)m(tical)f(to)h(the)h(inner)f(pro)s(duct)h |
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Ft(K)1695 4054 y Fr(2)1767 4091 y FB(on)f(the)h(Narain)f(lattice.)1541 |
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4427 y Fl(x)p Fy(4.)93 b(Conclusion)137 4649 y FB(W)-8 |
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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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(w)m(een)g(the)f(quan)m(tum)f(n)m(um)m(b)s(ers)h(of)f(heterotic)-9 |
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4799 y(strings)30 b(compacti\014ed)f(on)h Fs(T)1074 4763 |
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y Fr(2)1144 4799 y FB(and)g(in)m(v)-5 b(arian)m(t)28 |
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b(c)m(harges)k(of)d(junctions)h(on)g Fs(S)2805 4763 y |
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Fr(2)2875 4799 y FB(with)f(24)h(7-branes.)43 b(In)30 |
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b(or-)-9 4948 y(der)e(to)f(\014nd)i(the)f(necessary)i(relations,)d(w)m |
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(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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@endspecial -127 2972 a Fh(Fig.)32 b(2.)74 b(A)31 b(dual)g |
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(momen)m(tum)i Fg(m)3201 2939 y Fi(0)3201 2994 y Fk(\026)3279 |
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2972 y Fh(and)f(winding)32 3096 y(n)m(um)m(b)s(er)e Fg(n)418 |
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3063 y Fi(0)418 3119 y Fk(\026)464 3096 y Fh(.)41 b(The)30 |
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b(string)f(c)m(harges)j(\()p Fg(P)1390 3110 y Fk(\026)1437 |
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3096 y Fg(;)15 b(Q)1549 3110 y Fk(\026)1596 3096 y Fh(\))30 |
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b(are)h(obtained)f(from)g(Eq.\(1)2617 3077 y(.)2643 3096 |
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y(2\).)1603 3374 y FB(\(2)p Fs(P)1753 3389 y Fr(8)1792 |
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3374 y Fs(;)17 b FB(2)p Fs(Q)1962 3389 y Fr(8)2001 3374 |
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y FB(\))2641 4245 y(\(2)p Fs(P)2791 4260 y Fr(9)2830 |
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4245 y Fs(;)g FB(2)p Fs(Q)3000 4260 y Fr(9)3039 4245 |
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y FB(\))1811 3872 y Fs(k)1862 3887 y Fk(i)1912 3872 y |
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4331 y Fk(n)1525 4340 y Fd(8)p 1482 4347 V 1503 4404 |
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a Fr(2)1188 4121 y Fs(k)1239 4136 y Fk(i)1289 4121 y |
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