07.01.2014 Views

PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

SHOW MORE
SHOW LESS

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

YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.

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

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!