By Barletta E., Dragomir S., Duggal K.L.

ISBN-10: 0821843044

ISBN-13: 9780821843048

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**Extra info for Foliations in Cauchy-Riemann geometry**

**Example text**

The creation operators, we obtain a new basis for ∆ ∼ = Λ∗ ( 5 ): ˆ α¯ (1), Γ ˆ α¯ 1 α¯ 2 (1), Γ ˆ α¯ 1 α¯ 2 α¯ 3 (1), Γ ˆ α¯ 1 α¯ 2 α¯ 3 α¯ 4 (1), Γ ˆ α¯ 1 α¯ 2 α¯ 3 α¯ 4 α¯ 5 (1)} . 19) This basis will greatly simplify the calculations involved in solving the Killing spinor equations for certain spinor configurations. 3 Ns = 1 B ACKGROUNDS In this section, backgrounds with one Killing spinor will be investigated with the major aim of solving the Killing spinor equations, and the strategy will be as follows.

By counting dimensions, this can be seen to correspond to the decomposition ¯ ⊕ 45 ⊕ 45 ¯ ⊕5+5 ¯. 57) Similarly, the component FIJKL respects the decomposition (3,1)+(1,3) Λ4 (Ê10 ) ∼ = Λ(4,0)+(0,4) ⊕ Λ0 (2,2) ⊕ Λ0 (1,1) ⊕ Λ(2,0)+(0,2) ⊕ Λ0 ⊕Ê. 58) This makes explicit the branching§ of the 210 of SO(10) under SU (5), as ¯ ⊕ 40 ⊕ 40 ¯ ⊕ 75 ⊕ 10 ⊕ 10 ¯ ⊕ 24 ⊕ 1 . 59) From the table, we see that the Ns = 1 Killing spinor equations determine all components of the field strength except for the traceless part of the (2, 2)-component of the magnetic flux, denoted F02,2 .

E. ˚ ¯vα . 4) is a gauge-fixing condition for allowable diffeomorphisms of Xm . Later on, we will see further indications that this is a justifiable gauge choice. 6). 2), this equation can be re-cast in the form θ(˚ g ) = 2d˚ Φ, from which we see that (Xm ,˚ g , J) being conformally balanced is equivalent to the zeroth-order dilatino equation being satisfied. 6) is satisfied to first-order in α′ , we must demonstrate that Xm remains conformally balanced after the field deformations are taken into account.

### Foliations in Cauchy-Riemann geometry by Barletta E., Dragomir S., Duggal K.L.

by Ronald

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