PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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68 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION can ask is: do the matrices given in (4.5.4.8) generate the full symmetry group of the partition function (which is known in this case)? It turns out that the answer is no. This group does not include the matrix ⎛ ⎞ 1 0 0 0 ⎜ 0 1 0 0 ⎟ ⎝ 0 −1 1 0 ⎠ , (4.5.4.11) −1 0 0 1 since this is not of the form given in (4.5.4.10). This generates the transformation ˇρ → ˇρ (1 − ˇv) 2 − ˇσˇρ , ˇσ → ˇσ ˇρˇσ + ˇv(1 − ˇv) , ˇv → (1 − ˇv) 2 − ˇσˇρ (1 − ˇv) 2 − ˇσˇρ , (4.5.4.12) and is known to be a symmetry of the partition function. 12 Finally we turn to the constraints from black hole entropy. In this case the function g(τ) is given by [62, 64]: g(τ) = η(τ) 8 η(2τ) 8 . (4.5.4.13) Thus (4.4.6) takes the fom ̂Φ(ˇρ, ˇσ, ˇv) ∝ (2v − ρ − σ) 6 {v 2 η(ρ) 8 η(2ρ) 8 η(σ) 8 η(2σ) 8 + O(v 4 )} , (4.5.4.14) where (ρ, σ, v) and (ˇρ, ˇσ, ˇv) are related via (4.4.7). The dyon partition function of the Z 2 CHL model is known to satisfy this property. In fact historically this is the property that was used to guess the form of the partition function [12]. This analysis can be easily generalized to the dyons of Z N CHL orbifolds carrying twisted sector electric charges. 4.5.5 Dyons in ZZ 2 CHL model with untwisted sector electric charge We again consider the Z 2 CHL model introduced in §4.5.4, but now take the set A to consist of dyons with charge vectors ⎛ ⎞ ⎛ ⎞ 0 2K + 1 Q = ⎜ (2m + 1)/2 ⎟ ⎝ 0 ⎠ , P = ⎜ J ⎟ ⎝ 1 ⎠ , m, K, J ∈ Z . (4.5.5.1) −2 0 Since w ′ = −2 for this state, it represents an untwisted sector state. For this state we have Q 2 = −2(2m + 1), P 2 = 2(2K + 1), Q · P = −2J . (4.5.5.2) 12 This is the symmetry refered to as g 3 (1, 0) in [12] in a different representation.
4.5. EXAMPLES 69 Note that Q and P are both primitive. Since Q · P is quantized in units of 2, we shall define Q s = Q √ 2 , P s = P √ 2 , ˇρ s = 2ˇρ, ˇσ s = 2ˇσ, ˇv s = 2ˇv . (4.5.5.3) Thus we have Q 2 s = − (2m + 1) , P 2 s = 2K + 1, Q s · P s = −J . (4.5.5.4) Since Q 2 s/2 is quantized in units of 1/2, we expect the partition function to have ˇσ s period 2. However except for an overall additive factor of 1/2, Q 2 s/2 is actually quantized in integer units. Thus the partition function has the additional property that it is odd under ˇσ s → ˇσ s +1. Similarly since Ps 2 is an odd integer, the partition function picks up a minus sign under ˇρ s → ˇρ s + 1. We shall call these symmetries of ̂Φ. Finally since Q s · P s is quantized in integer units, the period in the ˇv s direction is also unity. The corresponding symplectic transformations acting on (ˇρ s , ˇσ s , ˇv s ) are of the form ⎛ ⎞ 1 0 ã 1 ã 3 ⎜ 0 1 ã 3 ã 2 ⎟ ⎝ 0 0 1 0 ⎠ , ã 1, ã 2 , ã 3 ∈ Z . (4.5.5.5) 0 0 0 1 Under this transformation the partition function picks up a multiplier factor of (−1) ea 1+ea 2 . Our next task is to determine the subgroup of the S-duality group Γ 0 (2) that leaves the set B – defined( as the) T-duality orbit of A – invariant. For this let us apply the S-duality a b transformation ∈ Γ c d 0 (2) on the charge vector (4.5.5.1). This gives ⎛ Q ′ = aQ + bP = ⎜ ⎝ b(2K + 1) (2m + 1)a/2 + bJ b −2a ⎞ ⎟ ⎠ , P ′ = cQ + dP = ⎛ ⎜ ⎝ d(2K + 1) (2m + 1)c/2 + dJ d −2c (4.5.5.6) We need to choose a, b, c, d such that (4.5.5.6) is inside the set B, ı.e. it can be brought to the form (4.5.5.1) after a T-duality transformation. The T-duality transformations acting within this four dimensional subspace are generated by matrices of the form [24]: ⎛ ⎞ ⎛ n 1 −m 1 ⎜ −l 1 k 1 ⎟ ⎝ k 1 l 1 ⎠ and ⎜ ⎝ m 1 n 1 −m 2 k 2 −l 2 k 2 l 2 m 2 n 2 n 2 ⎞ ⎟ ⎠ , ( ki l i ⎞ ⎟ ⎠ . ) ∈ Γ m i n 0 (2) . i (4.5.5.7)
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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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2.2. T-DUALITY ORBITS OF DYON CHARG
Page 39 and 40: 2.2. T-DUALITY ORBITS OF DYON CHARG
Page 41 and 42: 2.3. AN ALTERNATIVE PROOF 19 Q and
Page 43 and 44: 2.4. PREDICTIONS FOR GAUGE THEORY 2
Page 45 and 46: 2.4. PREDICTIONS FOR GAUGE THEORY 2
Page 47 and 48: 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: 4.5. EXAMPLES 67 IIB description of
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 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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PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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