DBI Analysis of Open String Bound States on Non-compact D-branes

CHAPTER 7. A FIRST EXAMPLE 91with Γ the Euler gamma function, andu 3 = (−z) −a 2F 1(a, a + 1 − c; a + 1 − b; z−1 ) , (7.32)u 4 = (−z) −b 2F 1(b + 1 − c, b; b + 1 − a; z−1 ) . (7.33)The large r solution to our hypergeometric equation thus becomesu = C 1 (−1) 1 4(5+ √ ( )11+4kE 2 ) 1 + C21 + r 2(1 + √ 1 + 4kE 2 ), 1 4( 1× 2 F 14+ C 2 (−1) 4(3− 1 √ ( )11+4kE 2 ) 1 + C21 + r 2× 2 F 1( 14(5 − √ 1 + 4kE 2 ), 1 44(1+ √ 1+4kE 2 )(5 + √ )1 + 4kE 2 ; 1 + 1 √)1 + 4kE22 ; z −14(1− √ 1+4kE 2 )(1 − √ )1 + 4kE 2 ; 1 − 1 √)1 + 4kE22 ; z −1 ,(7.34)withC 1 =C 2 =( )Γ − 1 2√1 + 4kE 2( ) ( ), (7.35)Γ 34 − 1 4√1 + 4kE 2 Γ 54 − 1 4√1 + 4kE 2( )Γ 12√1 + 4kE 2( ) ( ). (7.36)Γ 34 + 1 4√1 + 4kE 2 Γ 54 + 1 4√1 + 4kE 2This looks nothing less than atrocious, but can be approximated for r −→ ∞ by using2F 1 (a, b; c; 0) = 1, resulting inu = C 1 (−1) 1 4(5+ √ 1+4kE 2 ) ( 1 + C 2)1 4(1+ √ 1+4kE 2 ) r12(1+ √ 1+4kE 2 )+ C 2 (−1) 1 4(3− √ 1+4kE 2 ) ( 1 + C 2)1 4(1− √ 1+4kE 2 ) r12(1− √ 1+4kE 2 ) . (7.37)7.2 D1-brane on the trumpetWe will redo the calculation carried out in the previous section for the trumpet geometry.The computation is very similar, and hence considerably less steps will be shown explicitely.Recall that this setup suffers from bad string coupling behaviour at the origin,and that hence we should be careful when interpreting the <strong>DBI</strong> analysis, as we shouldin fact consider extra perturbative terms. However, because this setup is T-dual to thewell-behaved D2-brane in the cigar, these results nevertheless give a first indication <strong>of</strong>what to expect.