PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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52 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION the metric L takes the form L = ( ) 0 I2 , (4.5.1.2) I 2 0 where I 2 denotes 2 × 2 identity matrix. In this subspace we consider a three parameter family of charge vectors (Q, P ) with ⎛ ⎞ ⎛ ⎞ 0 K Q = ⎜ m ⎟ ⎝ 0 ⎠ , P = ⎜ J ⎟ ⎝ 1 ⎠ , m, K, J ∈ Z . (4.5.1.3) −1 0 This has Q 2 = −2m, P 2 = 2K, Q · P = −J . (4.5.1.4) We shall identify this set of charge vectors as the set A. As required, Q 2 , P 2 and Q · P are independent linear functions of m, K and J so that for a pair of distinct values of (m, K, J) we get a pair of distinct values of (Q 2 , P 2 , Q · P ). All the charge vectors in this family have unit torsion, ı.e. if we express the charges as linear combinations ∑ Q i e i and ∑ P i e i of primitive basis elements e i of the lattice Λ, then the torsion r(Q, P ) ≡ gcd{Q i P j − Q j P i } , (4.5.1.5) is equal to 1. In this case it is known that Q 2 , P 2 and Q · P are the complete set of T-duality invariants [63], ı.e. beginning with a pair (Q, P ) with unit torsion we can reach any other pair with unit torsion and same values of Q 2 , P 2 and Q · P via a T-duality transformation. Since the set A contains all integer triplets (Q 2 /2, P 2 /2, Q · P ) we conclude that the set B is the set of all (Q, P ) with unit torsion. The corresponding partition function is known [7] – it is the inverse of the weight ten Igusa cusp form Φ 10 of the full Sp(2, Z) group. We shall now examine how Φ 10 satisfies the various constraints derived in the previous sections. First of all note that since S-duality transformation does not change the torsion r, the full SL(2, Z) group is a symmetry of this set. Furthermore in this set Q 2 /2, P 2 /2 and Q · P are all quantized in integer units. Thus the partition function is invariant under translation of ˇρ, ˇσ and ˇv by arbitrary integer units. These correspond to symplectic transformations of the form ⎛ ⎞ ⎛ ⎞ d b 0 0 1 0 a 1 a 3 ⎜ c a 0 0 ⎟ ⎝ 0 0 a −c ⎠ , and ⎜ 0 1 a 3 a 2 ⎟ ⎝ 0 0 1 0 ⎠ 0 0 −b d 0 0 0 1 ( ) a b ∈ SL(2, Z), a c d 1 , a 2 , a 3 ∈ Z . (4.5.1.6) Clearly each of these transformations belong to Sp(2, Z) and is a symmetry of Φ 10 .
4.5. EXAMPLES 53 Next we turn to the constraints from the wall crossing formula. In this case all the walls are related by S-duality transformation to the wall corresponding to the decay (Q, P ) → (Q, 0) + (0, P ). So it is sufficient to study the consequences of the wall crossing formula at this wall. Clearly Q 2 and P 2 given in (4.5.1.4) are uncorrelated. Furthermore in heterotic string theory on T 6 all Q’s with a given Q 2 are related by T -duality transformation [41]. The same is true for P . Thus the subtleties mentioned below eq.(4.3.11) are absent, and the behaviour of ̂Φ(ˇρ, ˇσ, ˇv) near ˇv = 0 is expected to be given by (4.3.9). In this case both the electric and the magnetic half-BPS partition functions are given by η(τ) −24 where η denotes the Dedekind function. Thus we have, as a consequence of the wall crossing formula, ̂Φ(ˇρ, ˇσ, ˇv) ∝ {ˇv 2 (η(ˇρ)) 24 (η(ˇσ)) 24 + O(ˇv 4 )} . (4.5.1.7) η(τ) 24 transforms as a modular form of weight 12 under an SL(2, Z) transformation. From eqs.(4.3.26) it follows that these SL(2, Z) transformations may be regarded as the following symplectic transformations of ˇρ, ˇσ, ˇv Furthermore ̂Φ should have weight 12 − 2 = 10. ⎛ ⎞ ⎛ ⎞ α 0 β 0 1 0 0 0 ⎜ 0 1 0 0 ⎟ ⎝ γ 0 δ 0 ⎠ and ⎜ 0 p 0 q ⎟ ⎝ 0 0 1 0 ⎠ 0 0 0 1 0 r 0 s ( ) ( ) α β p q ∈ SL(2, Z), ∈ SL(2, Z) . (4.5.1.8) γ δ r s Let us now compare these with the known properties of Φ 10 . Φ 10 (ˇρ, ˇσ, ˇv) is indeed known to have the factorization property (4.5.1.7). Furthermore since Φ 10 transforms as a modular form of weight 10 under the full Sp(2, Z) group, and since (4.5.1.8) are Sp(2, Z) matrices, they represent symmetries of Φ 10 . Thus we see that in this case the full set of symmetries of φ m and φ e lift to symmetries of ̂Φ. It is worth noting that the matrices given in (4.5.1.6) and (4.5.1.8) generate the full Sp(2, Z) group. Thus in this case by assuming that the full modular groups of φ e and φ m lift to symmetries of the partition function we could determine the symmetries of the partition function. Finally let us consider the constraints coming from the knowledge of black hole entropy. In this case the function g(τ) appearing in (4.4.4) is given by η(τ) 24 . Thus (4.4.6) takes the fom ̂Φ(ˇρ, ˇσ, ˇv) ∝ (2v − ρ − σ) 10 {v 2 η(ρ) 24 η(σ) 24 + O(v 4 )} , (4.5.1.9) where (ˇρ, ˇσ, ˇv) and (ρ, σ, v) are related via eq.(4.4.7). The Siegel modular form Φ 10 (ˇρ, ˇσ, ˇv) satisfies these properties. In fact since (4.4.7) represents an Sp(2, Z) transformation, the property (4.5.1.9) of ̂Φ(ˇρ, ˇσ, ˇv) follows from the factorization property (4.5.1.7). This however will not be the case in a more generic situation.
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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
Page 13 and 14:
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
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
Page 65 and 66: 4.3. CONSTRAINTS FROM WALL CROSSING
Page 73: 4.5. EXAMPLES 51 Here k is the same
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 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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