# Kähler Potential of Calabi-Yau volume

At tree level, the Kähler potential is given by (neglecting complex structure)

$K = -\ln(-\mathrm{i}(\tau - \bar{\tau})) - 2\ln(V_{CY})$

where $V_{CY} = \frac{1}{6} \kappa_{abc}t^at^bt^c$ ia the the two cycle volume.

In some literature this is written in form of Kähler moduli variables as $V_{CY}=-i(\rho_a - \bar{\rho_a})t^a$ where $\rho = b + i\tau$. $\tau$ here is 4 cycle modulus.

In some other literature this is given as $V_{CY}=-3i(\rho - \bar{\rho})$ where $\rho = b + ie^{4u}$. $u$ fixes the volume of Calabi-Yau.

So my question is are these two equivalent? Would the $\rho\bar\rho$ component of the Kähler metric be the same?

• Which literature? Which pages? – Qmechanic Apr 19 '15 at 20:03
• the latter appears in Becker Becker book or Gidding Kachru Polchenski paper(arxiv.org/abs/hep-th/0105097) and the former appeared in Large Volume scenario paper (hepth - 0502058) – sol0invictus Apr 19 '15 at 20:30
• BBS - pg 498 and BBCQ (hepth-0502058)pg 6 above equation 12 they have given deifnition of $\rho$. – sol0invictus Apr 19 '15 at 21:19