PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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48 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION The integration over τ run parallel to the real axis over unit period with the imaginary part fixed at some large positive value M. Substituting (4.3.22) into (4.3.21) and picking up the residue from the pole at ˇv ′ = 0 we get the change in the index to be in agreement with (4.3.20). (−1) Q′·P ′ +1 Q ′ · P ′ d h (a 0 Q ′ , c 0 Q ′ )d h (b 0 P ′ , d 0 P ′ ) , (4.3.24) To summarize, (4.3.2) gives us the locations of the zeroes of ̂Φ, whereas eq.(4.3.9) and more generally (4.3.22) give us information about the behaviour of ̂Φ near this zero. We shall now show that these results suggest additional symmetries of ̂Φ of the type described in (4.2.15). Typically in any theory the partition functions of half BPS states have modular properties. Let us for definiteness consider the decay (Q, P ) → (Q, 0) + (0, P ). In this case the functions φ m (ˇρ) and φ e (ˇσ) transform as modular forms of a subgroup of SL(2, Z) since they arise from quantization of a fundamental string or a dual magnetic string. These relations take the form φ m ((αˇρ + β)(γ ˇρ + δ) −1 ) = (γ ˇρ + δ) k+2 φ m (ˇρ), φ e ((pˇσ + q)(rˇσ + s) −1 ) = (rˇσ + s) k+2 φ e (ˇσ) , ( ) ( ) (4.3.25) α β p q where k is an integer specific to the theory under study, and and belong to γ δ r s appropriate subgroups of SL(2, Z). Given that φ m and φ e have these symmetries, we conclude from (4.3.9) that near ˇv = 0, ̂Φ also has some additional symmetries. Even though there is no guarantee that these will be symmetries of the full quarter BPS partition function, one could hope that some part of these do lift to symmetries of the partition function and hence of ̂Φ. Those which do can be represented by symplectic transformations of the type (4.2.15) with ⎛ ⎞ ⎛ ⎞ ( ) α 0 β 0 ( ) 1 0 0 0 A B = ⎜ 0 1 0 0 ⎟ A B C D ⎝ γ 0 δ 0 ⎠ and = ⎜ 0 p 0 q ⎟ C D ⎝ 0 0 1 0 ⎠ . (4.3.26) 0 0 0 1 0 r 0 s The first transformation generates while the second transformation generates ˇρ → αˇρ + β γˇv2 ˇv , ˇσ → ˇσ − , ˇv → γ ˇρ + δ γ ˇρ + δ γ ˇρ + δ , (4.3.27) ˇρ → ˇρ − rˇv2 rˇσ + s , pˇσ + q ˇv ˇσ → , ˇv → rˇσ + s rˇσ + s . (4.3.28) Both transformations leave the ˇv = 0 surface invariant. Furthermore applying these transformations on (4.2.15) and using (4.3.9) near ˇv = 0 we generate the transformation laws (4.3.25).
4.3. CONSTRAINTS FROM WALL CROSSING 49 The symplectic transformations given in (4.3.26), if present, give us the additional symmetries required to have ̂Φ transform as a modular form under a non-trivial subgroup of Sp(2, Z). We can use this to determine the subgroup of Sp(2, Z) under which we expect ̂Φ to transform as a modular form and also the weight k of the modular form. However since we do not know a priori which part of the symmetry groups of φ e and φ m will lift to the symmetries of ̂Φ, this is not a fool proof method. Nevertheless these can serve as guidelines for making an educated guess. The behaviour of ̂Φ near the other zeroes given in (4.3.2) could provide us with additional information. If the zero of ̂Φ at (4.3.2) is related to the one at ˇv = 0 by an S-duality transformation then( this information ) is not new. Since S-duality transformation acts by multiplying ( ) a0 b the matrix 0 a0 b associated with a wall from the left [44], this means that if 0 c 0 d 0 c 0 d 0 itself is an S-duality transformation then we ( do not) get a new information. ( To this ) we must a0 b also add the information that multiplying 0 λ 0 from the right by c 0 d 0 0 λ −1 for any λ ( ) 0 1 or by does not change the wall [44]. However in many cases even after imposing −1 0 these equivalence relations one finds inequivalent walls. 9 In such cases the associated zero of ̂Φ cannot be related ( to the) zero at ˇv = 0 by an S-duality transformation, and we get new information. 10 a0 b Let 0 be the matrix associated with such a decay. If the corresponding c 0 d 0 partition functions ( ) φ m (τ; b( 0 , d 0 ) and) φ e (τ; a 0 , c 0 ) have modular groups containing matrices of α1 β the form 1 p1 q and 1 respectively, then they may be regarded as symplectic γ 1 δ 1 r 1 s 1 transformations generated by the matrices and ⎛ ⎞ d 0 b 0 0 0 ⎜ c 0 a 0 0 0 ⎟ ⎝ 0 0 a 0 −c 0 ⎠ 0 0 −b 0 d 0 ⎛ ⎞ d 0 b 0 0 0 ⎜ c 0 a 0 0 0 ⎟ ⎝ 0 0 a 0 −c 0 ⎠ 0 0 −b 0 d 0 ⎞ ⎛ ⎞ α 1 0 β 1 0 d 0 b 0 0 0 ⎜ 0 1 0 0 ⎟ ⎜ c 0 a 0 0 0 ⎟ ⎝ γ 1 0 δ 1 0 ⎠ ⎝ 0 0 a 0 −c 0 ⎠ (4.3.29) 0 0 0 1 0 0 −b 0 d 0 −1 ⎛ ⎞ ⎛ ⎞ 1 0 0 0 d 0 b 0 0 0 ⎜ 0 p 1 0 q 1 ⎟ ⎜ c 0 a 0 0 0 ⎟ ⎝ 0 0 1 0 ⎠ ⎝ 0 0 a 0 −c 0 ⎠ (4.3.30) 0 r 1 0 s 1 0 0 −b 0 d 0 −1 ⎛ ( ) 1 1 9 For example in Z 6 CHL model with S-duality group Γ 1 (6) the wall corresponding to the matrix ( ) 2 3 1 0 is not equivalent to the wall corresponding to . We shall discuss this example in some detail in §4.7. 0 1 10 Typically the number of such additional zeroes is a finite number, providing us with a finite set of additional information.
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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
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
Page 65 and 66: 4.3. CONSTRAINTS FROM WALL CROSSING
Page 67 and 68: 4.3. CONSTRAINTS FROM WALL CROSSING
Page 69: 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 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
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Bibliography [1] J.Polchinski ”Di
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BIBLIOGRAPHY 101 [30] M. Stern and
Page 125 and 126:
BIBLIOGRAPHY 103 [63] S. Banerjee a
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