PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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94CHAPTER 5. PARTITION FUNCTIONS OF TORSION > 1 DYONS IN HETEROTIC STRIN term in Φ 10 (ˇρ, s2ˇσ + k¯s 2 , sˇv + l¯s) −1 which has powers of e 2πiˇρ , e 2πiˇσ and e 2πiˇv compatible with the requirement (5.0.56) is e −2πiˇρ−2πi(s2ˇσ+ k¯s 2 )−2πi(sˇv+ l¯s ) . (5.0.58) This is in fact the leading term in the expansion in the limit (5.0.57). Substituting this into (5.0.4) and performing the sum over k and l we see that only the s = r term in the sum survives and the result is r e −2πi(ˇρ+r2ˇσ+rˇv) . (5.0.59) This corresponds to dyons with with an index of (−1) r+1 r. Q 2 /2 = −r 2 , P 2 /2 = −1, Q · P = −r , (5.0.60) We can also determine the walls of marginal stability which border the domain in which these dyons exist. This requires determining the region in the (I(ˇρ), I(ˇσ), I(ˇv)) space in which the expansion (5.0.59) of Φ 10 (ˇρ, r 2ˇσ, rˇv) −1 is valid since then we can determine the associated walls of marginal stability using (5.0.24). For this we shall utilize the known results for r = 1; in this case the walls of marginal stability bordering the domain in which (5.0.59) is valid correspond to the decays into (Q, 0) + (0, P ), (Q − P, 0) + (P, P ) and (0, P − Q) + (Q, Q) respectively [68]. Using (5.0.24) we now see that validity of the expansion of Φ 10 (ˇρ, ˇσ, ˇv) −1 given by (5.0.59) with r = 1 is bounded by the following surfaces in the (I(ˇρ), I(ˇσ), I(ˇv)) space: I(ˇv) = 0 , I(ˇv + ˇσ) = 0, I(ˇv + ˇρ) = 0 . (5.0.61) We can now simply scale ˇσ by r 2 and ˇv by r to determine the region of validity of the expansion (5.0.59) for Φ 10 (ˇρ, r 2ˇσ, rˇv): I(ˇv) = 0 , I(ˇv + rˇσ) = 0, I(rˇv + ˇρ) = 0 . (5.0.62) Comparing these with (5.0.24) we now see that the corresponding walls of marginal stability are associated with the decays into (Q, 0) + (0, P ), (Q − rP, 0) + (rP, P ), (0, P − 1 ) r Q + (Q, 1 ) r Q . (5.0.63) Let us now compare these results with dyons in N = 4 supersymmetric SU(3) gauge theory. If we denote by α 1 and α 2 a pair of simple roots of SU(3) with α 2 1 = α 2 2 = −2 and α 1 · α 2 = 1, then the analysis of [57, 58, 75, 77, 78] shows that the gauge theory contains dyons of charge (Q, P ) = (rα 1 , −α 2 ) , (5.0.64) with index (−1) r+1 r. These are precisely the dyons of the type given in (5.0.60). Furthermore using string junction picture [29, 32], ref. [77] also determined the walls of marginal stability bordering the domain in which these dyons exist. These also coincide with (5.0.63).
The spectrum in gauge theory contains other dyons of torsion r related to the ones described above by S-duality transformation. Since our construction is manifestly S-duality invariant, the results for these dyons can also be reproduced from the general formula given in (5.0.4). Gauge theory also contains other dyons which are not related to the ones described here by S-duality [57, 58]. These typically require higher gauge groups and has additional fermionic zero modes besides the ones required by broken supersymmetry. Quantization of these additional zero modes gives rise to additional bose-fermi degeneracy, and as a result the index being computed here vanishes for these dyons. This is also apparent from the fact that these dyons typically exist only in a subspace of the full moduli space; as a result when we move away from this subspace the various states combine and become non-BPS. Some aspects of these dyons have been discussed recently in [75, 76]. 95
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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
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may not be convergent in another re
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35 where Λ is a large positive num
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equiring that we reproduce the blac
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4.1. THE DYON PARTITION FUNCTION 39
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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 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: Furthermore, in order to reproduce
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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