PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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82 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION the quarter BPS partition function near the pole that controls the jump in the index at this particular wall. As an example we can consider the Z 2 CHL model of §4.5.4. The decay (Q, P ) → (Q, 0) + (0, P ) is controlled by the behaviour of ̂Φ near ˇv = 0. Thus if we did not know the spectrum of magnetically charged half BPS states in this theory, we could study the behaviour of ̂Φ near ˇv = 0 to get this information. In this case since all the other walls are related to this one by S-duality transformation, this is the only independent information we can get. However for more complicated models there can be more information. To illustrate this we shall consider the example of the Z 6 CHL model [69,70] mentioned in footnote 9. Our set A consists of charge vectors of the form ⎛ ⎞ ⎛ ⎞ 0 K Q = ⎜ m/6 ⎟ ⎝ 0 ⎠ , P = ⎜ J ⎟ ⎝ 1 ⎠ , m, K, J ∈ Z , (4.7.1) −1 0 as in (4.5.4.1). We now consider the decay associated with the matrix ( ) a0 b 0 = c 0 d 0 From (4.3.1) we see that this corresponds to the decay ( ) 1 1 . (4.7.2) 2 3 (Q, P ) → (M 0 , 2M 0 ) + (N 0 , 3N 0 ) , M 0 ≡ 3Q − P, N 0 ≡ −2Q + P . (4.7.3) The charge vectors M 0 and N 0 are not related to Q or P by a T-duality transformation since they correspond to charges that are triple and double twisted respectively. Furthermore the dyon charges (M 0 , 2M 0 ) and (N 0 , 3N 0 ) cannot be related by S-duality group Γ 1 (6) to either a purely electric or a purely magnetic state whose index is known. On the other hand the partition function of quarter BPS states of the type given in (4.7.3) is known [17, 24]. Thus the latter can be used to extract information about the partition function of these half BPS states. From (4.3.2) it follows that the relevant zero of ̂Φ we need to examine is at 6ˇρ + ˇσ + 5ˇv = 0 . (4.7.4) The zeroes of ̂Φ have been classified in [15, 24]. For a generic Z N model ̂Φ has double zeroes at n 2 (ˇσˇρ − ˇv 2 ) + jˇv + n 1ˇσ − ˇρm 1 + m 2 = 0 m 1 ∈ N Z, n 1 , m 2 , n 2 ∈ Z, j ∈ 2 Z + 1, m 1 n 1 + m 2 n 2 + j2 4 = 1 4 . (4.7.5)
4.7. REVERSE APPLICATIONS 83 For the N = 6 model, taking m 1 = −6, n 1 = 1, m 2 = n 2 = 0, j = 5 , (4.7.6) we see that ̂Φ indeed has a zero at (4.7.4). Thus by examining the known expression for ̂Φ near this zero we can determine the half-BPS partition functions of interest. This can be done in a straightforward manner following the general procedure described in [15, 24]. We note in passing that in an arbitrary Z N model with the set A chosen as ⎛ ⎞ ⎛ ⎞ 0 K Q = ⎜ m/N ⎟ ⎝ 0 ⎠ , P = ⎜ J ⎟ ⎝ 1 ⎠ , m, K, J ∈ Z , (4.7.7) −1 0 the walls of marginal stability are controlled by matrices [44] ( ) a0 b 0 subject to the conditions c 0 d 0 a 0 d 0 − b 0 c 0 = 1, a 0 , b 0 , c 0 , d 0 ∈ Z, c 0 d 0 ∈ N Z . (4.7.8) According to our hypothesis this decay will be controlled by a double zero of ̂Φ at This corresponds to the choice ˇρc 0 d 0 + ˇσa 0 b 0 + ˇv(a 0 d 0 + b 0 c 0 ) = 0 . (4.7.9) m 1 = −c 0 d 0 , n 1 = a 0 b 0 , m 2 = n 2 = 0, j = a 0 d 0 + b 0 c 0 , (4.7.10) in eq.(4.7.5). We now see that the m i ’s, n i ’s and j given in (4.7.10) satisfies all the constraints mentioned in (4.7.5) ( as a consequence ) of (4.7.8). Thus our proposal that the decay associated a0 b with the matrix 0 is always controlled by the zero at (4.7.9) is at least consistent c 0 d 0 with the locations of the zeroes of ̂Φ for Z N orbifold models.
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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
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This proves the desired result. Nex
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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 103: 4.7. REVERSE APPLICATIONS 81 Thus i
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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