PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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4 CHAPTER 1. INTRODUCTION for odd values of p. We take D-5 branes and D-1 branes and wrap them on K3 × S 1 and S 1 respectively. By T-dualising one can also think of this as D0-D4 system on K3 in type IIA string theory [5]. In the following we shall describe in detail the low energy field theory of this system. Type IIB string theory compactified on K3 × S 1 has 16 unbroken supersymmetries. If we wrap D1-D5 brane system on K3 × S 1 [6] in the manner described above the number of unbroken spacetime supersymmetries are 8. Now we go to a limit where the size of the K3 is much smaller than the size of the S 1 . In this limit the low energy theory describing the system is expected to be a (1 + 1) dimensional field theory based on S 1 . It can be shown that this theory is actually a (4, 4) superconformal sigma model with target space the moduli space of instantons on K3. If we want to calculate the degeneracy of some configuration which preserves only 4 out of 16 spacetime supersymmetries, which is precisely the amount of supersymmetries preserved by D1-D5 black hole, we can look for half-BPS states in this two dimensional theory. Since we are looking for supersymmetric states we can just look at the Ramond-Ramond sector of the field theory. In this sector the half-BPS states are precisely the ones in which say the right moving excitations are in their ground state and the left moving excitation is arbitrary. 1 In a unitary conformal field theory the number of states with large conformal weight can be calculated using Cardy formula, without knowing the details of the theory except the central charges of the left movers and right movers. Typically black holes with large charges correspond to states in the conformal field theory with large conformal dimensions. For them the degeneracy can be calculated using Cardy formula and the leading answer matches perfectly with the Bekenstein-Hawking entropy 1.3 Brief Introduction to E 8 ×E 8 Heterotic String Theory In the following we will briefly review the basic facts about heterotic string theory which will be useful for our purpose. Heterotic string has sixteen supersymmetries and E 8 × E 8 gauge symmetry in ten dimensions. Compactification on T 6 does not lead to any supersymmetry breaking. In any compactified string theory there are massless scalar fields whose asymptotic values determine the vacuum. They are called moduli fields. Heterotic string theory also has a set of moduli fields parametrized by 28 × 28 real symmetric matrix M which takes O(6,22;R) O(6;R)×O(22;R) value in and a complex scalar moduli field, τ valued in the upper half plane. Spacetime variation of M represents the fluctuations of the internal geometry and various nongeometric fields on the compactification manifold and τ represents the axion-dilaton of the four dimensional theory. At generic points on the moduli space, there are 28 U(1) gauge fiels. Sixteen of them come from that part of the E 8 × E 8 gauge field which is valued in the Cartan 1 In a supersymmetric field theory Witten index counts the difference between the number of Bosonic and Fermionic vacua. In this case there is a similar index called elliptic genus which is essentially the Witten index for say the right movers and the usual partition function for the left movers.It does not count the absolute number of states and being an index it is robust against deformation of the asymptotic moduli.
1.3. BRIEF INTRODUCTION TO E 8 × E 8 HETEROTIC STRING THEORY 5 sub-algebra. Six of them come from the rank two antisymmetric tensor field and the rest from the metric. There exist special points on the moduli space where the gauge symmetry is enhanced to a non-abelian group. The four dimensional theory has black hole solutions which preserve 1/4 -th of the original supersymmetry.They are called quarter BPS dyonic black holes. They are both electrically and magnetically charged under the U(1) gauge fields present in the theory. The electric and magnetic charge vectors, (Q, P ), take value in the 28 dimensional Narain lattice which is an even selfdual lattice of signature (6, 22) (a lattice is called even if the length squared of every vector belonging to it is an even integer and self-dual if the dual lattice is isomorphic to the original lattice). They are both 28 dimensional vectors. An explicit expression can be written down for the matrix M in terms of the internal components of metric, antisymmetric tensor field and gauge fields denoted by G ab , B ab and A I a, respectively [80,81]. Here 4 ≤ a, b ≤ 9 stand for the internal spacetime indices along the torus direction and 1 ≤ I ≤ 16 denote the Lie algebra indices along the cartan direction. We can combine the scalar fields G ab , B ab , and A I a into an O(6, 22; R) matrix. For this we regard G ab , B ab and A I a as 6 × 6, 6 × 6, and 6 × 16 matrices respectively, C ab = 1 2 AI a A I b as a 6 × 6 matrix, and define M to be the 28 × 28 dimensional matrix ⎛ ⎞ G −1 G −1 (B + C) G −1 A M = ⎝ (−B + C) G −1 (G − B + C) G −1 (G + B + C) (G − B + C) G −1 A ⎠ . (1.3.1) A T G −1 A G −1 (G + B + C) I 16 + A T G −1 A It can be checked that M satisfies MLM T = L, M T = M, L = ⎛ ⎝ 0 I ⎞ 6 0 I 6 0 0 ⎠ , (1.3.2) 0 0 −I 16 where I n denotes the n×n identity matrix. The first equation tells us that M is an O(6, 22; R) matrix. The matrix L has 6 positive eigenvalues +1 and 22 negative eigenvalues −1. This is also the metric on the Narain lattice [26, 27] and can be used to define the inner product between two charge vectors. 1.3.1 Duality Symmetries In four dimensions the heterotic string has two types of duality symmetries, T-duality and S-duality [81]. T duality group is O(6, 22; Z) and it acts on the charges and the moduli in the following way, and Q → ΩQ, P → ΩP (1.3.1) M → ΩMΩ T , τ → τ, (1.3.2)
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: 1.2. D BRANES 3 specific class of s
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 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
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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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