PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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xiv Superstring theories live in ten dimensions. To reduce the number of large dimensions to four one compactifies the six extra dimensions and obtain in this way a theory, which at low energies, reduces to a field theory containing gravity. Black holes arise as solutions of these low energy field theories. In string theory it is possible to count micro states of many supersymmetric black holes. The entropy obtained in this way matches exactly with the Bekenstein-Hawking entropy. In this thesis I shall focus my attention on a particular N = 4 string theory in four dimensions which is obtained by compactifying Heterotic string theory on T 6 . This theory has black hole solutions in four dimensions which preserve 1/4 -th of the original supersymmetry (they are usually called quarter BPS dyonic black holes). They carry both electric and magnetic charges- (Q, P )- under 28U(1) gauge fields which are present in the theory. The charge vectors take value in the 28 dimensional Narain lattice which is an even selfdual lattice of signature (6, 22). Q and P are both 28 dimensional vectors.This theory also has a set of massless scalar fields which transform in a specific way under the T and S duality symmetries of the four dimensional theory, along with the charge vectors (Q, P ). The microscopic degeneracy of these dyonic black holes was known only for a subset of charge vectors which satisfy a specific irreducibility criterion. Our work extends these results to all charge vectors. The details are as follows: In our first work, we derived the complete set of T-duality invariants which characterize a pair of charge vectors (Q, P ). Using this we could identify the complete set of dyonic black holes to which the previously derived degeneracy formula can be extended. By going near special points in the moduli space of the theory we derived the spectrum of quarter BPS dyons in N = 4 supersymmetric gauge theory with simply laced gauge groups. The results are in agreement with those derived from field theory analysis. In our next work we studied the action of S-duality group on the discrete T-duality invariants and studied its consequence for the dyon degeneracy formula. In particular we found that for dyons with torsion r, the degeneracy formula, expressed as a function of Q 2 , P 2 and Q.P , is required to be manifestly invariant under only a subgroup of the S-duality group. This subgroup is isomorphic to Γ 0 (r). Our analysis also showed that for a given torsion r, all other discrete T-duality invariants are characterized by the elements of the coset SL(2,Z) . Γ 0 (r) In the third work we proposed a general set of constraints on the partition function of quarter BPS dyons in any N = 4 supersymmetric string theory by drawing insight from known examples, and studied the consequences of this proposal. The main ingredients of our analysis are duality symmetries, wall crossing formula and black hole entropy. We used our analysis to constrain the dyon partition function for two hitherto unknown cases - the partition function of dyons of torsion two (i.e. gcd(Q ∧ P ) = 2) in heterotic string theory on T 6 and the partition function of dyons carrying untwisted sector electric charge in Z 2 CHL model. With the help of these constraints we proposed a candidate for the partition function of dyons of torsion two in heterotic string theory on T 6 . This leads to a novel wall crossing formula for decay of quarter BPS dyons into half BPS dyons with non-primitive charge vectors. In an appropriate limit the
proposed formula reproduces the known result for the spectrum of torsion two dyons in gauge theory. The original proposal of Dijkgraaf, Verlinde and Verlinde for the quarter BPS dyon partition function in heterotic string theory on T 6 is known to correctly produce the degeneracy of dyons of torsion 1, i.e. dyons for which gcd(Q ∧ P ) = 1. In our last work we proposed a generalization of this formula for dyons of arbitrary torsion. Our proposal satisfies the constraints coming from S-duality invariance, wall crossing formula, black hole entropy and the gauge theory limit. Furthermore using our proposal we derive a general wall crossing formula that is valid even when both the decay products are nonprimitive half-BPS dyons. xv
Page 1: Counting Microscopic Degeneracy of
Page 5: Declaration This thesis is a presen
Page 9: Acknowledgments Firstly I would lik
Page 13: Synopsis Black Holes are classical
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 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
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4.3. CONSTRAINTS FROM WALL CROSSING
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4.5. EXAMPLES 51 Here k is the same
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4.5. EXAMPLES 53 Next we turn to th
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4.5. EXAMPLES 55 Let us determine t
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4.5. EXAMPLES 57 We shall now set a
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4.5. EXAMPLES 59 uncorrelated and c
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4.5. EXAMPLES 61 is an integer. Sin
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4.5. EXAMPLES 63 1 ((Q−P 2 )/2)2
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4.5. EXAMPLES 65 Finally we turn to
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4.5. EXAMPLES 67 IIB description of
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4.5. EXAMPLES 69 Note that Q and P
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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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