Flux compactifications and the scalar potential

Question

Does the scalar potential:

$$V=e^K(K^{I \bar{J}})D_IW D_{\bar{J}}\bar{W}-3|W|^2$$

arise **because** of the presence of fluxes? 

If the fluxes are "turned off", does this mean $F_3=0$ and $H_3=0$, or that the integral of these field strengths over a particular cycle is zero (_i.e._ there are no non-trivial sources available in the theory)? 

I usually see the $F_3$ and $H_3$ referred to as fluxes but I always thought these were field strengths.

To be specific this whole confusion arises from studying [*The Effective Action of $\mathcal{N}= 1$ Calabi-Yau orientifolds*][1]. Footnote $9$ says not having fluxes would result in not having the $V$ potential in the $4$D action; wouldn't also the kinetic terms for the field strength vanish?

  [1]: https://arxiv.org/abs/hep-th/0403067

Answer 1

$F_3$ and $H_3$ are indeed the field strengths of 2-form gauge fields (respectively in the RR and in the NS-NS sectors). According to Maxwell equations for abelian p-form gauge fields, a vacuum configuration is obtained by taking the field strength to be an harmonic form. On a compact manifold, there is a unique harmonic representative in each cohomology class and so one can identify a vacuum configuration of the gauge field with a cohomology class for the field strength. This cohomology class is determined (up to some subtleties) by the integrals of the field strength through non-trivial cycles. These integrals are classically called fluxes and a non-trivial vacuum configurations of gauge fields is generally called "with fluxes".

Fluxes "turned off" means $F_3=H_3=0$, which is equivalent to the vanishing of all the fluxes, i.e. integrals over all the 3-cycles. If we consider $IIB$ on a Calabi-Yau 3-fold, without fluxes and without branes, the 4d effective theory in the non-compact dimensions has $\mathcal{N}=2$ SUSY and so there is no superpotential: $V=0$.

Comments

So, also the kinetic terms for the gauge fields will vanish right? Is it ok for this to happen?

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I don't really understand this question. The condition fluxes "turned off" is a condition on the vacuum/background configuration of IIB superstring theory on $\mathbb{R}^{1,3} \times X$, where $X$ is the compact Calabi-Yau 3-fold. The dynamics of the string theory will involve excitations above this background and in particular, it is possible to have kinetic terms describing excitations of the gauge field even if the background configuration has zero field strength (which by the way is what happens in usual gauge theory on flat space without non-trivial topology to have fluxes). Moreover, in the 4d effective action, the gauge fields come by dimensional reduction from the 4-form gauge field of IIB in 10 dimensions and not from the 2-form gauge fields related to $F_3$ and $H_3$ (in fact, the (self-dual) field strength of the 4-form field strength has automatically zero flux because there is no non-trivial 5-cycles in general in a Calabi-Yau 3-fold).