PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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66 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION subspace of charges given in (4.5.1.1), now the momentum n ′ along S 1 is quantized in units of 1/2 whereas the Kaluza-Klein monopole charge N ′ along S 1 is quantized in units of 2 [15]. We shall take the set A to be consisting of charge vectors of the form For this state we have ⎛ ⎞ 0 Q = ⎜ m/2 ⎟ ⎝ 0 ⎠ , −1 ⎛ P = ⎜ ⎝ K J 1 0 ⎞ ⎟ ⎠ , m, K, J ∈ Z . (4.5.4.1) Q 2 = −m, P 2 = 2K, Q · P = −J . (4.5.4.2) As usual we denote by B the set of all (Q, P ) which are related to the ones given in (4.5.4.2) by a T-duality transformation. Since Q 2 /2, P 2 /2 and Q · P are quantized in units of 1/2, 1 and 1 respectively, ̂Φ satisfies the periodicity conditions (4.2.16) with a 1 ∈ Z, a 2 ∈ 2 Z, a 3 ∈ Z . (4.5.4.3) Comparison of (4.5.4.1) and (4.5.1.1) shows that the winding charge −w ′ along S 1 is 1 for this state. Thus it represents a twisted sector state. Our next task is to determine the subgroup of the S-duality group that leaves the set ( B invariant. ) In this case the full S-duality group is Γ 0 (2), generated by matrices of the form a b with a, b, d ∈ Z, c ∈ 2 Z, ad − bc = 1. It was shown in [24] that the set B is c d closed under the full S-duality group. Thus the full S-duality group must be a symmetry of the partition function. We now turn to the constraints from the wall crossing formula. Consider first the wall associated with the decay (Q, P ) → (Q, 0) + (0, P ), – this in fact is the only case we need to analyze since all the walls are related to this one by S-duality transformation [44]. First of all note from (4.5.4.1), (4.5.4.2) that for a given Q 2 = −m the charge vector Q ∈ A is fixed uniquely. Thus the index of half-BPS states with charge (Q, 0) can be regarded as a function of Q 2 . On the other hand for a given P 2 = 2K there is a family of P ∈ A labelled by J, but these can be transformed ⎛ to the⎞vector corresponding to J = 0 by the T-duality 1 0 0 J transformation matrix [24] ⎜ 0 1 −J 0 ⎟ ⎝ 0 0 1 0 ⎠ . Thus the index of the charge vector (0, P ) can 0 0 0 1 also be expressed as a function of P 2 . Finally we see from (4.5.4.2) that the allowed values of Q 2 and P 2 are uncorrelated. Thus we can use eqs.(4.3.9), (4.3.10) to extract the behaviour of ̂Φ near ˇv = 0. The electric partition function can be calculated by examining the spectrum of twisted sector states in the heterotic string theory [64–67]. On the other hand the magnetic partition function can be calculated by examing the spectrum of D1-D5 system in a dual type
4.5. EXAMPLES 67 IIB description of the theory [24]. The results are Eq.(4.3.9) then gives, near ˇv = 0, φ e (ˇσ) = η(ˇσ) 8 η(ˇσ/2) 8 , φ m (ˇρ) = η(ˇρ) 8 η(2ˇρ) 8 . (4.5.4.4) ̂Φ(ˇρ, ˇσ, ˇv) ∝ {ˇv 2 η(ˇσ) 8 η(ˇσ/2) 8 η(ˇρ) 8 η(2ˇρ) 8 + O(ˇv 4 ) } . (4.5.4.5) φ e (ˇσ) and φ m (ˇρ) transform as modular forms of weight 8 under and ˇσ → pˇσ + q , p, r, s ∈ Z, q ∈ 2 Z, ps − qr = 1 , (4.5.4.6) rˇσ + s ˇρ → αˇρ + β , α, β, δ ∈ Z, γ ∈ 2 Z, αδ − βγ = 1 . (4.5.4.7) γ ˇρ + δ The corresponding groups are Γ 0 (2) and Γ 0 (2) respectively. Thus from (??), (4.2.17), (4.3.26) we see that if (4.5.4.6) and (4.5.4.7) lift to symmetries of the full partition function then the partition function transforms as a modular form of weight 6 under the Sp(2, Z) transformations of the form ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ d b 0 0 1 0 a 1 a 3 α 0 β 0 1 0 0 0 ⎜ c a 0 0 ⎟ ⎝ 0 0 a −c ⎠ , ⎜ 0 1 a 3 a 2 ⎟ ⎝ 0 0 1 0 ⎠ , ⎜ 0 1 0 0 ⎟ ⎝ γ 0 δ 0 ⎠ and ⎜ 0 p 0 q ⎟ ⎝ 0 0 1 0 ⎠ , 0 0 −b d 0 0 0 1 0 0 0 1 0 r 0 s (4.5.4.8) with ( ) ( ) ( ) a b α β p q ∈ Γ c d 0 (2), ∈ Γ γ δ 0 (2), ∈ Γ 0 (2), r s a 1 , a 3 ∈ Z, a 2 ∈ 2 Z . (4.5.4.9) All the Sp(2, Z) matrices in (4.5.4.8) subject to the constraints (4.5.4.9) have the form ⎛ ⎞ 1 ∗ ∗ ∗ ⎜ 0 1 ∗ 0 ⎟ ⎝ 0 0 1 0 ⎠ mod 2 . (4.5.4.10) 0 ∗ ∗ 1 Furthermore the set of matrices (4.5.4.10) are closed under matrix multiplication. Thus the group generated by the set of Sp(2, Z) matrices (4.5.4.8) subject to the condition (4.5.4.9) is contained in the group Ǧ of Sp(2, Z) matrices (4.5.4.10). All the symmetries listed in (4.5.4.8) are indeed symmetries of the dyon partition function of this model proposed in [12] and proved in [15]. Furthermore near ˇv = 0 the partition function is known to have the factorization property given in (4.5.4.5) [44–46]. One question that one
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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
Page 37 and 38: 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: 4.5. EXAMPLES 65 Finally we turn to
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 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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