a leading order potential [287]: V = V$_{0}$ + ∑ i Λ 4 i cos( φ$_{i}$/f$_{i}$ ) + ... (473) where φ$_{i}$ correspond to the partners of K¨ ahler moduli. The ellipses contain higher order contributions. The advantage of this potential is that there are no H $^{2}$φ2 i type contributions at the lowest order provided there is a shift symmetry, therefore the SUGRA η problem can be evaded. The axions have shift symmetry, which are only broken at non-perturbative level. In string theory case, the assumption behind the potential is following. In a compactified framework, it is assumed that all the moduli are heavy and thus stabilized by prior dynamics, including that of the volume modulus. Only the axions of T$_{i}$ = φ$_{i}$/f$_{i}$ + iM 2 $_{s}$R 2 i are light [287]. The assumption of decoupling the dynamics of K¨ ahler modulus from the axions is still a debatable issue, see the discussion in [48]. These axions can then obtain an exponential superpotential, W ∼ ∑ $_{i}$W$_{i}$ e 2 $^{πT$_{i}$}$, correction similar to the KKLT setup, in addition to a constant superpotential piece, W$_{0}$ . The shift symmetry is now protected by the choice of K¨ ahler potential [48] K = − ln [ i C$_{ijk}$ 3! ( T$_{i}$ − T$_{i}$ )( T$_{j}$ − T$_{j}$ )( T$_{k}$ − T$_{k}$ ) ] , (474) where C$_{ijk}$ is the Calab-Yau intersection number. After rearranging the potential for the axions, V ≈ ∑ V$_{i}$ ≈ ∑ i | ∂$_{i}$W | $^{2}$, expanding them around their minima and for a canonical choice of the kinetic terms, the Lagrangian simplifies to the lowest order in expansion: L = 1 2 ∂$_{µ}$φ$_{i}$∂ $^{µ}$φ$_{j}$ − 1 2 m 2 $_{i}$φ 2 i + · · · . (475) The exact calculation of m$_{i}$ is hard, assuming all of the mass terms to be the same m$_{i}$ ∼ 10 13 GeV, and N > ( M$_{P}$/f ) $^{2}$, it is possible to match the current observations with a tilt in the spectrum, n ∼ 0 . 97, and large tensor to scalar ratio: r ∼ 8 /N$_{Q}$ ∼ 0 . 13 for N$_{Q}$ ∼ 60. 4. Inflation due to K¨ ahler modulus Various realizations of modular inflation have been studied in the context of volume modulus, out of which the racetrack models [902–904], and the K¨ ahler moduli inflation are 228