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%!PS-Adobe-2.0
%%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software
%%Title: arXiv:hep-th/9901001v3 10 May 1999
%%Pages: 13
%%PageOrder: Ascend
%%BoundingBox: 0 0 612 792
%%EndComments
%DVIPSWebPage: (www.radicaleye.com)
%DVIPSCommandLine: dvips -R -P pk -K1 -z imamura2.dvi -o
%DVIPSParameters: dpi=600, compressed, comments removed
%DVIPSSource: TeX output 2024.04.27:1735
%%BeginProcSet: texc.pro
/TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S
N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72
mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0
0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{
landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize
mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[
matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round
exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{
statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0]
N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin
/FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array
/BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2
array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N
df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A
definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get
}B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub}
B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr
1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3
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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
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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
(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
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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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3305 y(is)f(unique)h(18)24 b(+)f(2-dimensional)32 b(Loren)m(tzian)i
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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
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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
(24)e(7-)-127 4201 y(branes.)73 b(Although)41 b(there)i(are)f(man)m(y)h
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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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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
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(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
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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)
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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)
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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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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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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
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(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
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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
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
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905 y(carrying)26 b(these)j(c)m(harges)f(are)f(transformed)f(b)m(y)i
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(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
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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
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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)
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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
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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
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y Fr(osc)3031 4711 y Fm(\000)h FB(1\))p Fs(;)345 b FB(\(2)3677
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5098 y(the)h(NS)g(sector)h(and)e(0)h(for)f(the)h(R)g(sector.)57
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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
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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
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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
(torus)-127 457 y Fs(T)-56 420 y Fk(d)17 457 y FB(is)c(rectangular.)43
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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
843 y FB(+)g Ft(w)1730 858 y Fk(\026)1798 843 y Fm(\001)g
Ft(k)g Fm(\000)2039 804 y Fr(1)p 2039 820 36 4 v 2039
877 a(2)2084 843 y Ft(w)2165 858 y Fk(\026)2234 843 y
Fm(\001)g Ft(w)2365 858 y Fk(\027)2408 843 y Fs(n)2466
858 y Fk(\027)2509 747 y Fj(\021)2581 843 y FB(+)2716
795 y Fk(R)2769 803 y Ff(\026)p 2689 820 149 4 v 2689
829 a Fi(p)p 2748 829 36 3 v 55 x Fr(2)p Fk(l)2804 892
y Ff(s)2847 843 y Fs(n)2905 858 y Fk(\026)1830 1007 y
Ft(k)h Fm(\000)f Fs(n)2069 1022 y Fk(\026)2116 1007 y
Ft(w)2197 1022 y Fk(\026)3003 592 y Fj(1)3003 738 y(C)3003
788 y(C)3003 838 y(C)3003 891 y(A)3092 838 y Fs(;)354
b FB(\(2)3560 816 y(.)3586 838 y(2\))-127 1219 y(where)33
b Fs(R)228 1234 y Fk(\026)307 1219 y FB(are)f(compacti\014cation)e
(radii,)g Ft(k)i FB(is)f(a)h(v)m(ector)g(on)g(a)g(16-dimensional)c
(self-dual)j(ev)m(en)i(lattice)-127 1368 y Fs(\000)14
b FB(,)35 b(and)f Ft(w)281 1383 y Fk(\026)362 1368 y
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eop
%%Page: 5 5
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%%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
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4201 y(yond.")68 4351 y FB(6\))49 b(Y.Imam)m(ura,)31
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FB(,)204 4500 y Fc(\\)p Fs(E)326 4515 y Fr(8)400 4500
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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
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%%Page: 11 11
11 10 bop 186 158 a FB(9\))48 b(A.Mikhailo)m(v,)29 b(N.Nekraso)m(v)i
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%%Page: 12 12
12 11 bop 2434 465 a Ft(\012)2515 429 y Fr(0)2442 590
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181 y Fr(1)1679 272 y(0)1729 112 y Fj(\023)-20 b(\022)1896
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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
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@endspecial -127 5204 a Fh(Fig.)32 b(3.)74 b(When)471
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%%Page: 13 13
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@endspecial -9 3294 a Fh(Fig.)32 b(4.)73 b(As)33 b(a)h(result)e(of)h
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