No es on (p, q)-s ings
Ryan J. Buchanan
Decembe , 2025
1 Wha a e (p, q)-s ings?
A (p, q)-s ing is a bound s a e consis ing o p F1-s ings and q D1-s ings ca ying wo ypes o
cha ge: a Ramond-Ramond (RR) cha ge, and a Kalb-Ramond (KR) cha ge. These a ise as wo-
o ms, and a e deno ed espec i ely by B(2) and C(2). The RR cha ges couple o he undamen al
s ings, also known as F1-s ings, whe eas he KR wo- o m couples o a D1-b ane. Fo Σ he
wo ldshee o a (p, q)-s ing, we ind ha he ac ion is
SWS ⊃ZΣ
pB(2) +qC(2).
S ing Tensions The below o mula summa izes he ela ionship be ween s ing ension and
coupling o he wo- o ms:
S ing Type Tension Fo mula Coupling
F-s ing TF1 B(2)
D-s ing TF1/gsC(2)
(p, q)-s ing TF1pp2+q2/g2
sp B(2) +q C(2)
Table 1: Summa y o S ing Types, Tension Fo mulas, and Couplings
SL(2,Z) Duali y Type IIB s ing heo y enjoys an SL(2,Z) (S-duali y) symme y, unde which
he F1 and D1 s ings a e exchanged, and gene al bound s a es, he (p, q)-s ings, a e ela ed by
duali y ans o ma ions. The axio-dila on, τ=C0+ie−ϕ, ans o ms ac ionally unde his duali y
and en e s in o he (p, q)-s ing ension o mula.
Wo ldshee Ac ion in Di e en ial Fo m Language The wo ldshee ac ion o a (p, q)-s ing
includes a coupling o he wo- o m gauge po en ials:
SWS ⊃ZΣpB(2) +qC(2),
whe e Σ is he s ing wo ldshee . He e, B(2) (NSNS) and C(2) (RR) a e bo h 2- o ms, and he
coe icien s p, q e lec he s ing’s cha ges unde each wo- o m.
1
Gene al Tension Fo mula The ension o a (p, q)-s ing is de e mined by i s cha ges and he
dila on:
T(p,q)=TF1sp2+q2
g2
s
whe e gs=eϕis he s ing coupling, TF1 =1
2πα′is he F-s ing ension, and p, q ∈Zlabel he
cha ges.
Physical Con ex (p, q)-s ings in e pola e be ween he undamen al F-s ing (p= 1, q = 0) and
he D-s ing (p= 0, q = 1). Thei exis ence is c ucial e idence o non-pe u ba i e duali ies in
ype IIB s ing heo y, and hey play an essen ial ole in s ing junc ions, b ane cons uc ions, and
he web o duali ies ha connec di e en s ing heo ies.
Addi ional No es on (p, q)-S ings
Beyond hei basic de ini ion and ension o mula, (p, q)-s ings exhibi se e al s uc u al ea u es
ha cla i y hei ole in ype IIB s ing heo y. Fi s , hese objec s a e 1/2-BPS s a es, p ese ing
six een supe cha ges, and hei ensions a e he e o e p o ec ed agains quan um co ec ions. This
p o ec ion unde lies he exac , duali y-co a ian exp ession
T(p,q)=TF1 |p+q τ|
√Im τ, τ =C0+i
gs
,
which makes he ull SL(2,Z) symme y mani es . The pai (p, q) ans o ms as an SL(2,Z)
double , and he allowed cha ges o m a wo-dimensional in eg al la ice in a ian unde modula
ans o ma ions.
A u he s uc u al poin conce ns s abili y: a (p, q)-s ing ep esen s an i educible BPS bound
s a e only when gcd(p, q) = 1. I (p, q) = m(p′, q′) wi h m > 1, he con igu a ion co esponds o m
pa allel copies o he p imi i e (p′, q′)-s ing a he han a single composi e objec .
(p, q)-s ings also pa icipa e in s ing junc ions. Cha ge conse a ion allows i alen e ices
sa is ying
(p1, q1)+(p2, q2)=(p3, q3),
and he ensions o he indi idual segmen s balance as ec o s in he plane. Such junc ions and he
esul ing (p, q)-webs play impo an oles in F- heo y, se en-b ane physics, and he cons uc ion o
i e-dimensional supe con o mal ield heo ies.
Finally, he supe g a i y desc ip ion o a (p, q)-s ing can be ob ained by ac ing wi h an SL(2,Z)
duali y ans o ma ion on he undamen al s ing solu ion, p o iding a use ul geome ic pic u e o
how hese objec s in e pola e be ween he F1 and D1 limi s.
Re e ences
[1] J. H. Schwa z, An SL(2,Z) Mul iple o Type IIB Supe s ings, Physics Le e s B 360 (1995)
13–18. doi:10.1016/0370-2693(95)01125-5.a Xi :hep- h/9508143.
[2] J. Polchinski, S ing Theo y, Volume 2: Supe s ing Theo y and Beyond, Camb idge Uni e si y
P ess, 1998.
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[3] A. Sen, S ing Theo y Dynamics in Va ious Dimensions, Nuclea Physics B 450 (1995) 103–114.
doi:10.1016/0550-3213(95)00398-3.a Xi :hep- h/9504027.
[4] J. H. Schwa z, Lec u es on Supe s ing and M-Theo y Duali ies, Nuclea Physics B (P oc.
Suppl.) 55 (1997) 1–32. doi:10.1016/S0920-5632(97)00070-4.a Xi :hep- h/9607201.
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