PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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78 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION The behaviour of ̂Φ near ˇv s ′ = 0 can be found by the usual procedure and result is { ̂Φ(ˇρ, ˇσ, ˇv) −1 = − 1 ( ) ˇρ ′ −24 ( ) } η s ˇρ ′ −24 + η s + 2 4π2ˇv ′2 s 4 4 { ( ) ˇσ ′ −24 ( ) × η s ˇσ ′ −24 ( ) + η s + 1 ˇσ ′ −24 ( ) } + η s + 2 ˇσ ′ −24 + η s + 3 4 4 4 4 { − 1 ( ) ˇρ ′ −24 ( ) } η s ˇρ ′ −24 + η s + 2 ) η(ˇσ π2ˇv ′ ′2 s 4 4 s) −24 ′0 + O (ˇv s (4.6.28) Note that the first set of terms represent correctly the factorization behaviour given in (4.5.3.21), but the second set of terms are extra. Thus the wall crossing formula gets modified for the decay into non-primitive states. Using (4.6.28) we can compute the jump in the index across the wall ∆d(Q, P ) = 1 ∫ { iM ′ (−1)Q·P +1 Q ′ · P ′ 1 +1 ( ) ˇρ dˇρ ′ ′ −24 ( ) } 16 iM 1 ′ −1 s η s ˇρ ′ −24 + η s + 2 e −iπ ˇρ′ sP ′2 /4 4 4 ∫ { iM ′ 2 +1/2 ( ) ˇσ dˇσ ′ ′ −24 ( ) iM 2 ′ −1/2 s η s ˇσ ′ −24 + η s + 1 4 4 ( ) ˇσ ′ −24 ( ) } +η s + 2 ˇσ ′ −24 + η s + 3 e −iπˇσ′ sQ ′2 /4 Defining 4 + 1 ∫ { iM ′ (−1)Q·P +1 Q ′ · P ′ 1 +1 dˇρ ′ s η 4 × ∫ iM ′ 2 +1/2 we can express (4.6.29) as iM ′ 1 −1 4 ( ) ˇρ ′ −24 s + η 4 ( ) } ˇρ ′ −24 s + 2 4 e −iπ ˇρ′ sP ′2 /4 iM ′ 2 −1/2 dˇσ ′ s η(ˇσ ′ s) −24 e −iπˇσ′ sQ ′2 /4 . (4.6.29) ∆d(Q, P ) = (−1) Q 1·P 2 −Q 2·P 1 +1 (Q 1 · P 2 − Q 2 · P 1 ) (Q 1 , P 1 ) = (Q − P, 0), (Q 2 , P 2 ) = (P, P ) , (4.6.30) { ( 1 d h (Q 1 , P 1 ) + d h 2 Q 1, 1 )} 2 P 1 d h (Q 2 , P 2 ) . (4.6.31) The second term is extra compared to (4.3.7); it represents the effect of non-primitivity of the final state dyons. Finally let us turn to the analysis of the black hole entropy. For this we need to identify the zeroes of ̂Φ at ˇρˇσ − ˇv 2 + ˇv = 0 and show that ̂Φ has the behaviour given in (4.5.3.25) near this pole. This is easily done using (4.6.1) and the locations of the zeroes of Φ 10 given in
4.6. A PROPOSAL FOR THE PARTITION FUNCTION OF DYONS OF TORSION TWO79 (4.6.8). One finds that the only term that has a zero at the desired location is the first term inside the first square bracket in (4.6.1). Furthermore this term is proportional to the dyon partition function 1/Φ 10 (ˇρ, ˇσ, ˇv) of the unit torsion states discussed in §4.5.1. Thus this term clearly will have the desired factorization property given in (4.5.3.25). Our proposal for the dyon partition function can be easily generalized to the torsion 2, primitive Q, P and odd Q 2 /2, P 2 /2 dyons discussed at the end of §4.5.3. This requires changing the signs of appropriate terms in (4.6.1) so that the partition function is odd under ˇρ → ˇρ + 1 and also under ˇσ → ˇσ + 1 . The result is 2 2 1 ̂Φ(ˇρ, ˇσ, ˇv) [ = 1 1 16 Φ 10 (ˇρ, ˇσ, ˇv) − 1 Φ 10 (ˇρ, ˇσ + 1, ˇv) − 1 Φ 2 10 (ˇρ + 1 , ˇσ, ˇv) 2 1 + Φ 10 (ˇρ + 1, ˇσ + 1, ˇv) + 1 Φ 2 2 10 (ˇρ + 1, ˇσ + 1, ˇv + 1) 4 4 4 1 − Φ 10 (ˇρ + 1, ˇσ + 3, ˇv + 1) − 1 Φ 4 4 4 10 (ˇρ + 3, ˇσ + 1, ˇv + 1) 4 4 4 1 + Φ 10 (ˇρ + 3, ˇσ + 3, ˇv + 1) + 1 Φ 4 4 4 10 (ˇρ + 1, ˇσ + 1, ˇv + 1) 2 2 2 1 − Φ 10 (ˇρ + 1, ˇσ, ˇv + 1) − 1 Φ 2 2 10 (ˇρ, ˇσ + 1, ˇv + 1) + 1 Φ 2 2 10 (ˇρ, ˇσ, ˇv + 1) 2 1 + Φ 10 (ˇρ + 3, ˇσ + 3, ˇv + 3) − 1 Φ 4 4 4 10 (ˇρ + 3, ˇσ + 1, ˇv + 3) 4 4 4 ] 1 − Φ 10 (ˇρ + 1, ˇσ + 3, ˇv + 3) + 1 Φ 4 4 4 10 (ˇρ + 1, ˇσ + 1, ˇv + 3) 4 4 4 [ 1 + Φ 10 (ˇρ + ˇσ + 2ˇv, ˇρ + ˇσ − 2ˇv, ˇσ − ˇρ) ] 1 − Φ 10 (ˇρ + ˇσ + 2ˇv + 1, ˇρ + ˇσ − 2ˇv + 1, ˇσ − ˇρ + 1) . (4.6.32) 2 2 2 This together with (4.6.1) exhausts all the dyons of torsion two with Q, P primitive since there are no dyons of this type with Q 2 /2 even, P 2 /2 odd or vice versa. To see this we note that since (Q ± P ) are 2× primitive vectors, (Q ± P ) 2 /2 must be multiples of four. Taking the sum and difference we find that (Q 2 + P 2 )/2 and Q · P must be even. Since (4.6.1) and (4.6.32) contains information about all the torsion two dyons with primitive (Q, P ), the full partition function for such dyons in obtained by taking the sum of these two functions. This gives the result quoted in (4.0.23). Given this result on torsion two dyons in string theory we can go to appropriate gauge theory limit to extract information about torsion two dyons in gauge theories as in [63,68]. For simplicity we shall consider SU(3) gauge theories. If we denote by α 1 and α 2 the two simple roots of SU(3), then, since the metric L reduces to the negative of the Cartan metric of the
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Counting Microscopic Degeneracy of
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Declaration This thesis is a presen
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Acknowledgments Firstly I would lik
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Synopsis Black Holes are classical
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proposed formula reproduces the kno
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Contents 1 Introduction 1 1.1 Broad
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Chapter 1 Introduction Einstein pro
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1.2. D BRANES 3 specific class of s
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1.3. BRIEF INTRODUCTION TO E 8 × E
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1.4. DEGENERACY OF QUARTER-BPS DYON
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1.5. MOTIVATION AND A BRIEF REVIEW
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Chapter 2 Duality orbits, dyon spec
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2.2. T-DUALITY ORBITS OF DYON CHARG
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2.3. AN ALTERNATIVE PROOF 19 Q and
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2.4. PREDICTIONS FOR GAUGE THEORY 2
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2.4. PREDICTIONS FOR GAUGE THEORY 2
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Chapter 3 S-duality Action on Discr
Page 49 and 50: This proves the desired result. Nex
Page 51 and 52: For appropriate choice of (α, β,
Page 53 and 54: Chapter 4 Generalities of Quarter B
Page 55 and 56: may not be convergent in another re
Page 57 and 58: 35 where Λ is a large positive num
Page 59 and 60: equiring that we reproduce the blac
Page 61 and 62: 4.1. THE DYON PARTITION FUNCTION 39
Page 63 and 64: 4.2. CONSEQUENCES OF S-DUALITY SYMM
Page 65 and 66: 4.3. CONSTRAINTS FROM WALL CROSSING
Page 73 and 74: 4.5. EXAMPLES 51 Here k is the same
Page 75 and 76: 4.5. EXAMPLES 53 Next we turn to th
Page 77 and 78: 4.5. EXAMPLES 55 Let us determine t
Page 79 and 80: 4.5. EXAMPLES 57 We shall now set a
Page 81 and 82: 4.5. EXAMPLES 59 uncorrelated and c
Page 83 and 84: 4.5. EXAMPLES 61 is an integer. Sin
Page 85 and 86: 4.5. EXAMPLES 63 1 ((Q−P 2 )/2)2
Page 87 and 88: 4.5. EXAMPLES 65 Finally we turn to
Page 89 and 90: 4.5. EXAMPLES 67 IIB description of
Page 91 and 92: 4.5. EXAMPLES 69 Note that Q and P
Page 93 and 94: 4.5. EXAMPLES 71 On the other hand
Page 95 and 96: 4.6. A PROPOSAL FOR THE PARTITION F
Page 97 and 98: 4.6. A PROPOSAL FOR THE PARTITION F
Page 99: 4.6. A PROPOSAL FOR THE PARTITION F
Page 103 and 104: 4.7. REVERSE APPLICATIONS 81 Thus i
Page 105 and 106: 4.7. REVERSE APPLICATIONS 83 For th
Page 107 and 108: Chapter 5 Partition Functions of To
Page 109 and 110: crossing formula for decay into a p
Page 111 and 112: 89 As a consequence of (5.0.19) and
Page 113 and 114: 91 In the primed variables the desi
Page 115 and 116: Furthermore, in order to reproduce
Page 117 and 118: The spectrum in gauge theory contai
Page 119 and 120: Chapter 6 Concluding Remarks Comple
Page 121 and 122: Bibliography [1] J.Polchinski ”Di
Page 123 and 124: BIBLIOGRAPHY 101 [30] M. Stern and
Page 125 and 126: BIBLIOGRAPHY 103 [63] S. Banerjee a
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