PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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22CHAPTER 2. DUALITY ORBITS, DYON SPECTRUM AND GAUGE THEORY LIMIT OF H first consider the domain bounded by a straight line passing through 0, a straight line passing through 1 and a circle passing through 0 and 1, – the domain called R in [24, 44]. This has vertices at 0, 1 and ∞. In this domain the only non-zero values of d(Q, P ) for Q 2 < 0, P 2 < 0 are obtained at Q 2 = P 2 = −2. For Q 2 = P 2 = −2 the result for d(Q, P ) is [15, 44, 68] { 0 for Q · P ≥ 0 d(Q, P ) = j(−1) j−1 for Q · P = −j, j > 0 . (2.4.4) The condition (2.4.2) on Q · P now shows that for (Q, P ) describing the elements of the root lattice, non-vanishing index exists only for Q 2 = P 2 = −2, Q · P = −1. Translated to a condition on the charge vectors in the gauge theory this gives 6 d gauge (q, p) = { 1 for q 2 = p 2 = 2, q ◦ p = 1, r gauge (q, p) = 1 0 for other (q, p) with r gauge (q, p) = 1 . (2.4.5) This condition in turn implies that q and −p can be regarded as the simple roots of an su(3) subalgebra of the full gauge algebra, with the Cartan metric of su(3) being equal to the restriction of the Cartan metric of the full algebra. Thus we learn that in the domain R the only dyons with r gauge (q, p) = 1 and non-vanishing index are the ones which can be regarded as SU(3) dyons for some level one su(3) subalgebra of the gauge algebra, with q and −p identified with the simple roots α and β of the su(3) algebra. The index in other domains can be found using the S-duality invariance of the theory. An S-duality transformation of the form τ → (aτ + b)/(cτ + d) maps the domain R to another domain with vertices a c , b d , a + b c + d . (2.4.6) Under the same S-duality transformation the charge vector (q, p) = (α, −β) gets mapped to (q, p) = (aα − bβ, cα − dβ) . (2.4.7) It can be easily seen that r(Q, P ) remains invariant under an SL(2, Z) transformation: r(Q, P ) = r(aQ + bP, cQ + dP ), a, b, c, d ∈ Z, ad − bc = 1 . (2.4.8) Thus we conclude that in the domain (2.4.6), the index of gauge theory dyons with r gauge (q, p) = 1 is given by { d gauge (q, p) = 1 for (q, p) = (aα − bβ, cα − dβ), , (2.4.9) 0 otherwise with (α, β) labelling the simple roots of some level one su(3) subalgebra of the full gauge algebra. 6 It has been shown in appendix ?? that for states with Q 2 = P 2 = −2, Q·P = ±1 the condition r(Q, P ) = 1 is satisfied automatically. Thus we do not need to state this as a separate condition.
2.4. PREDICTIONS FOR GAUGE THEORY 23 This general result agrees with the known results for quarter BPS dyons in gauge theories [29, 32, 57, 58, 78]. 7 In particular in the representation of SU(N) dyons as string network with ends on a set of parallel D3-branes, this is a reflection of the fact that networks with three external strings ending on three D3-barnes are the only quarter BPS configurations at a generic point in the moduli space [32]. 7 In codimension ≥ 1 subspaces of the moduli space the dyon spectrum, computed in some approximation, has a rich structure [32,57,58,78]. However the index associated with these dyons vanish and these results are not in contradiction with the spectrum of string theory.
Page 1: Counting Microscopic Degeneracy of
Page 5: Declaration This thesis is a presen
Page 9: Acknowledgments Firstly I would lik
Page 13 and 14: Synopsis Black Holes are classical
Page 15: proposed formula reproduces the kno
Page 19 and 20: 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: 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 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
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4.6. A PROPOSAL FOR THE PARTITION F
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4.7. REVERSE APPLICATIONS 81 Thus i
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4.7. REVERSE APPLICATIONS 83 For th
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Chapter 5 Partition Functions of To
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crossing formula for decay into a p
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89 As a consequence of (5.0.19) and
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91 In the primed variables the desi
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Furthermore, in order to reproduce
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The spectrum in gauge theory contai
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Chapter 6 Concluding Remarks Comple
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Bibliography [1] J.Polchinski ”Di
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BIBLIOGRAPHY 101 [30] M. Stern and
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BIBLIOGRAPHY 103 [63] S. Banerjee a
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PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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