PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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84 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION
Chapter 5 Partition Functions of Torsion > 1 Dyons in Heterotic String Theory on T 6 Since the original proposal of Dijkgraaf, Verlinde and Verlinde [7] for quarter BPS dyon spectrum in heterotic string theory compactified on T 6 , there has been extensive study of dyon spectrum in a variety of N = 4 supersymmetric string theories [8–18,20,21,24,44–46] and also in N = 8 and N = 2 supersymmetric string theories [47, 79]. However it has been realised for some time that even in heterotic string theory on T 6 the proposal of [7] gives the correct dyon spectrum only for a subset of dyons, – those with unit torsion, ı.e. for which the electric and magnetic charge vectors Q and P satisfy gcd(Q ∧ P )=1 [20,48,63]. In the previous chapter we wrote down a general set of constraints which must be satisfied by the partition function of quarter BPS dyons in any N = 4 supersymmetric string theory and used these constraints to propose a candidate for the dyon partition function for torsion two dyons in heterotic string theory on T 6 [71]. In the present chapter we extend our analysis to dyons of arbitrary torsion and propose a form of the partition function of such dyons. Proposal for the partition function: We consider the set B of all dyons of charge vectors (Q, P ) in heterotic string theory on T 6 , with Q being r times a primitive vector, P a primitive vector and Q/r and P admitting a primitive embedding in the Narain lattice [26, 27], ı.e. all lattice vectors lying in the plane of Q and P can be expressed as integer linear combinations of Q/r and P . These dyons have torsion r, ı.e. gcd(Q ∧ P ) = r. It was shown in [48, 63] that given any pair (Q, P ) of this type with the same values of Q 2 , P 2 and Q·P , they are related by T-duality transformation. We denote by d(Q, P ) the index measuring the number of bosonic supermultiplets minus the number of fermionic supermultiplets of quarter BPS dyons carrying charges (Q, P ) – up to a normalization this can be identified with the helicity supertrace B 6 introduced in [62]. T-duality invariance of the theory tells us that d(Q, P ) must be a function of the T-duality invariants, and hence has the form f(Q 2 , P 2 , Q · P ). Then the dyon partition 85
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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,
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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 and 104: 4.7. REVERSE APPLICATIONS 81 Thus i
Page 105: 4.7. REVERSE APPLICATIONS 83 For th
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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