By Susanne Reffert

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Extra resources for Toroidal orbifolds : resolutions, orientifolds and applications in string phenomenology

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27) lead to 5 3 R1 = 3 D1,3 + E2,µ,γ . 35) µ=3 γ=1 The linear relations for D2,β are the same as those for D1,α : 3 3 E1,γ + 2 R2 = 6 D2,1 + E3,ν , E2,µ,γ + 3 ν=1,3 µ=1,3 γ=1 γ=1 E3,ν , R2 = 2 D2,2 + ν=2,4,5,6 3 R2 = 3 D2,3 + E2,µ,γ . 19): 5 R3 = 3 D3,γ + 2 E1,γ + E2,µ,γ γ = 1, . . , 3. 8 The intersection ring There is a purely combinatorial way to determine the intersection ring of the resolved torus orbifold. 4 for the local patches. Recall that first, the intersection numbers between three distinct divisors were determined, and then the linear relations were used to compute all the remaining intersection numbers.

2 Example C: T 6 /Z6 × Z6 This, being the point group of largest order, is the most tedious of all examples. It is presented here to show that the procedure is not so tedious after all. 6 The big picture Group el. θ1 (θ1 )2 (θ1 )3 θ2 (θ2 )2 (θ2 )3 θ1 θ2 1 2 2 θ (θ ) θ1 (θ2 )3 θ1 (θ2 )4 θ1 (θ2 )5 (θ1 )2 θ2 (θ1 )3 θ2 (θ1 )4 θ2 (θ1 )2 (θ2 )2 (θ1 )2 (θ2 )3 (θ1 )2 (θ2 )4 (θ1 )3 (θ2 )2 (θ1 )3 (θ2 )3 Order 6 3 2 6 3 2 6×6 6×3 6×2 6×6 6 3×6 2×6 6×6 3×3 3×2 3 2×3 2 Fixed Set Conj. 2: Fixed point set for Z6 × Z6 .

The orbifold has db untwisted K¨ahler moduli and 12 (dg − db ) untwisted complex structure moduli. 7). 13) √ with y = − 13 (3R1 R3 cos θ13 − 3R1 R3 cos θ23 ). There are five real continuous deformation parameters for the metric, R12 , R32 , R52 , θ13 and θ23 . One can immediately identify the three 2 × 2 blocks corresponding to the two G2 factors and the SU(3) factor. The angles between the two G2 factors, θ13 and θ23 are not constrained by the orbifold twist. For the choice θ13 = θ23 = π/2, the metric of the rigid root lattice G22 × SU(3) is recovered.

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Toroidal orbifolds : resolutions, orientifolds and applications in string phenomenology by Susanne Reffert

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