PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

More documents

Recommendations

Info

60 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION where (ˇρ, ˇσ, ˇv) and (ρ, σ, v) are related via (4.4.7). ̂Φ(ˇρ, ˇσ, ˇv) given in (4.5.2.6) can be shown to satisfy this property. In fact the relevent pole of ̂Φ −1 comes from the first term on the right hand side of (4.5.2.6). The location of the zeroes of Φ 10 are given in (4.6.8), and it follows from this that the second term does not have a pole at v = 0. 4.5.3 Dyons of torsion 2 in heterotic string theory on T 6 We consider again heterotic string theory on T 6 and take the set A to consist of charge vectors of the form ⎛ ⎞ ⎛ ⎞ 1 2K + 1 Q = ⎜ 2m + 1 ⎟ ⎝ 1 ⎠ , P = ⎜ 2J + 1 ⎟ ⎝ 1 ⎠ , m, K, J ∈ Z . (4.5.3.1) 1 −1 This has Q 2 = 4(m + 1), P 2 = 4(K − J), Q · P = 2(K + J − m + 1) . (4.5.3.2) Furthermore gcd{Q i P j − Q j P i }=2. Thus we have a family of charge vectors with torsion 2. It was shown in [48,63] that for r = 2 there are three T-duality orbits for given (Q 2 , P 2 , Q·P ) – in the first Q is twice a primitive lattice vector, in the second P is twice a primitive lattice vector and in the third both Q and P are primitive but Q ± P are twice primitive lattice vectors. The dyon charges given in (4.5.3.1) are clearly of the third kind. In the notation of [48] the discrete T-duality invariants of these charges are (r 1 = 1, r 2 = 1, r 3 = 2, u 1 = 1). Note that as we vary m, J and K, Q 2 /2 and P 2 /2 take all possible even values and Q · P takes all possible values subject to the restriction that Q ± P are twice primitive lattice vectors. The latter condition requires Q · P to be even and Q · P − 1 2 Q2 − 1 2 P 2 to be a multiple of four. It now follows from the result of [48,63] that the T-duality orbit B of the set A consists of all the pairs (Q, P ) with (r 1 = 1, r 2 = 1, r 3 = 2, u 1 = 1) and even values of Q 2 /2, P 2 /2. Since Q 2 /2, P 2 /2 and Q · P are all even and Q 2 + P 2 + 2Q · P is a multiple of 8, it is natural to introduce new charge vectors and variables so that we have Q s ≡ Q/2, P s ≡ P/2, ˇρ s ≡ 4ˇρ, ˇσ s ≡ 4ˇσ, ˇv s ≡ 4ˇv , (4.5.3.3) 1 2 Q2 s = 1 2 (m + 1), 1 2 P s 2 = 1 2 (K − J), Q s · P s = 1 (K + J − m + 1) , (4.5.3.4) 2 quantized in half integer units subject to the constraint that 1 2 Q2 s + 1 2 P 2 s + Q s · P s = K + 1 , (4.5.3.5)
4.5. EXAMPLES 61 is an integer. Since (ˇρ s , ˇσ s , ˇv s ) are conjugate to (P 2 s /2, Q 2 s/2, Q s · P s ), the partition function (5.0.4) will be periodic under (ˇρ s , ˇσ s , ˇv s ) → (ˇρ s + 2, ˇσ s , ˇv s ), (ˇρ s , ˇσ s + 2, ˇv s ), (ˇρ s , ˇσ s , ˇv s + 2), (ˇρ s + 1, ˇσ s + 1, ˇv s + 1) . (4.5.3.6) The group generated by these transformations can be collectively represented by symplectic matrices of the form ⎛ ⎞ 1 0 ã 1 ã 3 ⎜ 0 1 ã 3 ã 2 ⎟ ⎝ 0 0 1 0 ⎠ , ã 1, ã 2 , ã 3 ∈ Z, ã 1 + ã 2 , ã 2 + ã 3 , ã 1 + ã 3 ∈ 2 Z , (4.5.3.7) 0 0 0 1 acting on the variables (ˇρ s , ˇσ s , ˇv s ). For future reference we note that the change of variables from (ˇρ, ˇσ, ˇv) to (ˇρ s , ˇσ s , ˇv s ) can be regarded as a symplectic transformation of the form ⎛ ⎞ 2 0 0 0 ⎜ 0 2 0 0 ⎟ ⎝ 0 0 1/2 0 ⎠ . (4.5.3.8) 0 0 0 1/2 We now need to determine the subgroup of the S-duality group that leaves the set B invariant. If we did not have the restriction that Q( 2 /2 and ) P 2 /2 are even, then this subgroup a b would consist of SL(2, Z) matrices of the form subject to the restriction a + b ∈ c d 2 Z + 1 and c + d ∈ 2 Z + 1 [48], – these conditions guarantee that the new charge vectors (Q ′′ , P ′′ ) are each primitive and hence have the same set of discrete T-duality invariants (r 1 = 1, r 2 = 1, r 3 = 2, u 1 = 1). We shall now argue that the same subgroup also leaves the set B invariant. For this we need to note that if we begin with a (Q, P ) for which Q 2 /2, P 2 /2 and Q · P are all even then their S-duality transforms given in (4.2.2) will automatically have the same properties. Thus requiring the transformed pair (Q ′′ , P ′′ ) to have even Q ′′2 /2 and P ′′2 /2, as is required for (Q ′′ , P ′′ ) to belong to the set B, does not put any additional restriction on the S-duality transformations. Since both Q and P are scaled by the same amount to get the rescaled charges Q s and P s , the S-duality group action on (Q s , P s ) is identical to that on (Q, P ) and hence its action on (ˇρ s , ˇσ s , ˇv s ) is identical to that on (ˇρ, ˇσ, ˇv). Using (4.2.14) we see that the representations of these symmetries as symplectic matrices are given by ⎛ ⎞ d b 0 0 c a 0 0 ⎟ ⎜ ⎝ 0 0 a −c 0 0 −b d ⎟ ⎠ , a, b, c, d ∈ Z, ad − bc = 1, a + c ∈ 2 Z + 1, b + d ∈ 2 Z + 1 , acting on the variables (ˇρ, ˇσ, ˇv) and also on (ˇρ s , ˇσ s , ˇv s ). (4.5.3.9)
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 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 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: 4.5. EXAMPLES 59 uncorrelated and c
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
show all

PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?