PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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50 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION respectively, acting on the original variables (ˇρ, ˇσ, ˇv). Again we could hope that a part of this symmetry is a symmetry of ̂Φ. We shall illustrate these by several examples in §4.5. 4.4 Black hole entropy Another set of constraints may be derived by requiring that the formula for the index of quarter BPS states match the entropy of the black hole carrying the same charges in the limit when the charges are large. The consequences of this constraint have been analyzed in detail in the past [7, 8, 12, 15] and reviewed in [24]. Hence our discussion will be limited to a review of the salient features. In the approximation where we keep the supergravity part of the action containing only the two derivative terms, the black hole entropy is given by π √ Q 2 P 2 − (Q · P ) 2 . (4.4.1) In all known cases this result is reproduced by the asymptotic behaviour of (5.0.14) for large charges. Furthermore the leading asymptotic behaviour comes from the residue of the partition function at the pole at [7, 8, 12, 15, 24] ˇρˇσ − ˇv 2 + ˇv = 0 , (4.4.2) up to translations of ˇρ, ˇσ and ˇv by their periods. We shall assume that this result continues to hold in the general case. Thus ̂Φ(ˇρ, ˇσ, ˇv) must have a zero at (4.4.2). In order to find the behaviour of ̂Φ near this zero one needs to know the first non-leading correction to the leading formula (4.4.1) for the black hole entropy. A priori these corrections depend on the complete set of four derivative terms in the quantum effective action of the theory and are difficult to calculate. However in all known examples one finds that the entropy calculated just by including the Gauss-Bonnet term in the effective action reproduces correctly the first non-leading correction to the statistical entropy. If we assume that this result continues to hold for a general theory then we can use this to determine the behaviour of ̂Φ near (4.4.2) in terms of the coefficient of the Gauss-Bonnet term in the effective action. Since this procedure has been extensively studied in [7,8,12,15] and reviewed in [24], we shall only quote the result. Typically the Gauss Bonnet term in the Lagrangian has the form ∫ d 4 x √ − det g φ(a, S) { R µνρσ R µνρσ − 4R µν R µν + R 2} , (4.4.3) where τ = a + iS is the axion-dilaton modulus and the function φ(a, S) has the form φ(a, S) = − 1 ((k + 2) ln S + ln g(a + iS) + ln g(−a + iS)) + constant . (4.4.4) 64π2
4.5. EXAMPLES 51 Here k is the same integer that appeared in (4.2.15) and g(τ) transforms as a modular form of weight k + 2 under the S-duality group. In a given theory g(τ) can be calculated in string perturbation theory [61,62]. To the first non-leading order in the inverse power of charges, the effect of this term is to change the black hole entropy to [24] S BH = π √ ( √ ) Q 2 P 2 − (Q · P ) 2 + 64 π 2 Q · P φ P , Q2 P 2 − (Q · P ) 2 + · · · (4.4.5) 2 P 2 The analysis of [7,8,12,15,24] shows that this behaviour can be reproduced if we assume that near the zero at (4.4.2) ̂Φ(ˇρ, ˇσ, ˇv) ∝ (2v − ρ − σ) k {v 2 g(ρ) g(σ) + O(v 4 )} , (4.4.6) where ˇρˇσ − ˇv2 ˇρˇσ − (ˇv − 1)2 ρ = , σ = , v = ˇρˇσ − ˇv2 + ˇv . (4.4.7) ˇσ ˇσ ˇσ If we assume that eq.(4.4.6) holds in general, then it gives us information about the behaviour of ̂Φ(ˇρ, ˇσ, ˇv) near the zero at (4.4.2). On the other hand if we can determine ̂Φ from other considerations then the validity of (4.4.6) would provide further evidence for the postulate that in N = 4 supersymmetric string theories the Gauss-Bonnet term gives the complete correction to black hole entropy to first non-leading order. 4.5 Examples In this section we shall describe several applications of the general procedure described in §4.1. Some of them will involve known cases and will provide a test for our procedure, while others will be new examples where we shall derive a set of constraints on certain dyon partition functions which have not yet been computed from first principles. 4.5.1 Dyons with unit torsion in heterotic string theory on T 6 We consider a dyon of charge (Q, P ) in the heterotic string theory on T 6 . Q and P take values in the Narain lattice Λ [26, 27]. Let S 1 and ˜S 1 be two circles of T 6 , each labelled by a coordinate with period 2π and let us denote by n ′ , ñ the momenta along S 1 and ˜S 1 , by −w ′ , − ˜w the fundamental string winding numbers along S 1 and ˜S 1 , by N ′ , Ñ the Kaluza-Klein monopole charges associated with S 1 and ˜S 1 , and by −W ′ , −˜W the H-monopole charges associated with S 1 and ˜S 1 [24]. Then in the four dimensional subspace consisting of charge vectors ⎛ ⎞ ⎛ ⎞ ñ ˜W Q = ⎜ n ′ ⎟ ⎝ ˜w ⎠ , P = ⎜ W ′ ⎟ ⎝ Ñ ⎠ , (4.5.1.1) w ′ N ′
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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
Page 13 and 14:
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
Page 23 and 24: Chapter 1 Introduction Einstein pro
Page 25 and 26: 1.2. D BRANES 3 specific class of s
Page 27 and 28: 1.3. BRIEF INTRODUCTION TO E 8 × E
Page 29 and 30: 1.4. DEGENERACY OF QUARTER-BPS DYON
Page 31 and 32: 1.5. MOTIVATION AND A BRIEF REVIEW
Page 33 and 34: Chapter 2 Duality orbits, dyon spec
Page 35 and 36: 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 71: 4.3. CONSTRAINTS FROM WALL CROSSING
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 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
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BIBLIOGRAPHY 101 [30] M. Stern and
Page 125 and 126:
BIBLIOGRAPHY 103 [63] S. Banerjee a
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