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PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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42 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION<br />

Comparing (4.2.10) and (4.2.7) we get<br />

̂Φ(ˇρ, ˇσ, ˇv) = ̂Φ(ˇρ ′′ , ˇσ ′′ , ˇv ′′ ) . (4.2.11)<br />

For future reference we shall rewrite the transformation laws (4.2.5) in a suggestive form.<br />

We define<br />

( )<br />

ˇρ ˇv ˇΩ = . (4.2.12)<br />

ˇv ˇσ<br />

Then the transformations (4.2.5) may be written as<br />

where A, B, C and D are 2 × 2 matrices, given by<br />

⎛<br />

( )<br />

A B<br />

= ⎜<br />

C D ⎝<br />

ˇΩ = (AˇΩ ′′ + B)(C ˇΩ ′′ + D) −1 , (4.2.13)<br />

d b 0 0<br />

c a 0 0<br />

0 0 a −c<br />

0 0 −b d<br />

⎞<br />

⎟<br />

⎠ . (4.2.14)<br />

Eq.(4.2.11) now gives (after replacing the dummy variable ˇΩ ′′ by ˇΩ on both sides),<br />

̂Φ((AˇΩ + B)(C ˇΩ + D) −1 ) = det(C ˇΩ + D) k ̂Φ(ˇΩ) , (4.2.15)<br />

for A, B, C, D given in (4.2.14). Here k is an arbitrary number. Since det(CΩ + D) = 1, we<br />

cannot yet ascertain the value of k.<br />

To this we can also append the translational symmetries of ̂Φ:<br />

̂Φ(ˇρ, ˇσ, ˇv) = ̂Φ(ˇρ + a 1 , ˇσ + a 2 , ˇv + a 3 ) , (4.2.16)<br />

where a i ’s are integer multiples of the T i ’s. It is convenient, although not necessary, to work<br />

with appropriately rescaled Q and/or P so that the T i ’s and hence the a i ’s are integers. This<br />

symmetry can also be rewritten as (4.2.15) with the choice<br />

⎛<br />

⎞<br />

( )<br />

1 0 a 1 a 3<br />

A B<br />

= ⎜ 0 1 a 3 a 2<br />

⎟<br />

C D ⎝ 0 0 1 0 ⎠ . (4.2.17)<br />

0 0 0 1<br />

Again since det(C ˇΩ + D) = 1 the choice of k is arbitrary.<br />

The alert reader would have noticed that although we have expressed the consequences of<br />

S-duality invariance and charge quantization conditions as symmetries of the function ̂Φ under<br />

a symplectic transformation, the symplectic transformations arising this way are trivial, – for<br />

all the transformations arising this way the matrix C vanishes and hence the transformations

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