Lecture 26 8 251 Spring 2007 8 251 Final Exam Review Practice Problems Problem 1 Assume standard space lling Virasoro brane a Calculate L 6 0 in terms of normal order oscillations I b Calculate L 6 1 0 Hint 0I 0 0 mom vacuum Problem 2 D2 z 0 D1 y 0 z D 26 a Find the M 2 formula in terms of N b Ground states Describe the lowest mass levels c Generating f x a r xr where a r is the number of states of M 2 r Problem 3 Consider the Ramond sector of a superstring in D 18 Naive M 2 1 2 n 0 I n nI ndI n dIn a Range of I Write the precise n o M 2 1 Lecture 26 8 251 Spring 2007 b Write the generating function fR x Solutions Problem 1 a L 6 1 6 p p 2 p Z p 0 1 6 0 7 1 8 2 5 1 4 2 3 3 2 4 1 5 0 6 1 7 2 Since n 0 s are annihilation operators and kill the vacuum state 1 3 3 2 4 1 5 0 6 7 1 8 2 2 1 L 6 0 3 3 2 4 1 5 0 2 b 1 I I I I L 6 1 0 3 3 2 4 1 5 1 0 7 1 1 0 L 6 0 7 0 2 Alternative method I I LI 6 1 0 L 6 1J 0 1 L 6 0 I 7 2 know Lecture 26 8 251 Spring 2007 Problem 2 a D2 D1 D2 D1 x x NN y stretched ND xx4 x25 localized localized DD z DD DD 1 ND coordinate all of the rest are DD M 2 Na T L 2 1 1 1 48 24 16 1 15 1 a 16 16 1 a 1 p q 16 2 2 T L 2 aN D M 2 N 15 4 11 N 16 16 16 b Ground state p p2 pq in book but here q 1 GS p No mo mentum in ND fractional oscillators or all the DDs The states live on the D1 brane c 3 GS has M 2 11 16 Suppose GS 0 and just oscillator 1 then could build 3 3 1 0 1 2 0 so f x x 11 16 11 1 x x2 x 16 3 1 1 x Lecture 26 8 251 Spring 2007 But we have more f x x 11 16 11 x 16 23 23 23 1 1 1 1 x 1 x2 1 x3 1 1 xn 23 n 1 Now have to account for the fractional oscillators 11 f x x 16 1 1 n 23 1 xn 1 2 1 x n 1 Plug into Mathematica to get speci c answers 4
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