when 10$_{H}$ and 10$_{H}$ develops a VEV. The relevant superpotential is given by [78] W = κS [ 10$_{H}$10$_{H}$ − M 2 ] + λ$_{1}$ 10$_{H}$10$_{H}$5$_{h}$ + λ$_{2}$ 10$_{H}$10$_{H}$5$_{h}$ (296) + y ( d ) ij 10$_{i}$10$_{j}$5$_{h}$ + y ( u,ν ) ij 10$_{i}$5$_{j}$5$_{h}$ + y ( e ) ij 1$_{i}$5$_{j}$5$_{h}$ , where κ, λ$_{1}$, λ$_{2}$ are constants and the appropriate Yukawas are given by Y$_{ij}$ , i.e. Y ( µ,ν ) ij , provide masses to up-type quarks and neutrino Dirac masses. The U (1)$_{R}$ symme- try eliminates terms such as S 2 and S 3 from the superpotential. Higher dimensional baryon number violating operators such as 10$_{i}$10$_{j}$10$_{k}$5$_{l}$ 〈 S 〉 /M 2 $_{P}$, 10$_{i}$5$_{j}$5$_{k}$1$_{l}$ 〈 S 〉 /M 2 $_{P}$, etc. are suppressed as a consequence of U (1)$_{R}$. The proton decay proceeds via dimen- sion six operators mediated by the superheavy gauge bosons. The dominant decay mode is p → e $^{+}$/µ$^{+}$, π0 and the estimated lifetime is of order 10 36 yrs. [694, 695]. This Z$_{2}$ ensures that the LSP is absolutely stable. Inflation happens for κ ≤ 10 − 2 and matches the standard predictions, i.e. n$_{s}$ ≈ 1 with negligible gravity waves and running of the spectral tilt [75, 78] for M ∼ 10 16 GeV. By introducing soft-SUSY breaking mass terms with minimal K¨ ahler potential it is possible to bring down the spectral tilt, n$_{s}$ ∼ 0 . 96 − 0 . 97 for M$_{s}$ ∼ 8 × 10 15 − 2 × 10 16 GeV [693]. • 5D SO(10): Inflationary models from 5D SO (10) were also constructed in Refs. [77, 696]. There are certain advantages of orbifold constructions of five dimensional (5D) SUSY GUTs, in which SO (10) can be readily broken to its maximal subgroup H [697–699], with the doublet-triplet splitting problem addressed due to construction where SO (10) is compactified on S $^{1}$/ ( Z$_{2}$ × Z ′ $_{2}$) (where Z$_{2}$ reflects, y → − y , and Z ′ 2 reflects y ′ → − y ′ with y ′ = y + y$_{c}$/ 2. The two orbifold points are y = 0 and y = y$_{c}$/ 2). In order to realize inflation in 4 D , the N = 2 SUSY in the 5D bulk is broken down to N = 1 SUSY on the orbifold fixed points, below the compactification scale π/y$_{c}$ , where the branes are located. The F -term inflation potential can be constructed on the branes, assuming that the inter brane separation is fixed. The two branes preserve different symmetries, on one the full SO (10) is preserved while in the other SU (4)$_{c}$ × SU (2)$_{L}$ × SU (2)$_{R}$ [700]. There also exists a bulk scalar field, S , which couples to the singlets of SO (10) on the branes. For instance, on the brane where SO (10) is preserved, the superpotential would be 139