By C. Lozano

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**Additional info for Duality in Topological Quantum Field Theories [thesis]**

**Example text**

Likewise, as there are two independent twisted supersymmetries, and the D-brane generically preserves 1/2 of the supersymmetries left unbroken by the compactification, we need a 7 manifold preserving 1/8-th of the supersymmetry. This is a 7-manifold of G2 holonomy, which indeed has supersymmetric 4-cycles whose normal bundle is precisely Ω2,+ (X) (the bundle of self-dual two-forms on X), as expected. A similar analysis [9] shows that the second twist arises in compactifications on an 8-dimensional Spin(7)-holonomy manifold, while the amphicheiral theory is realized on supersymmetric cycles of SU(4)-holonomy Calabi-Yau 4-folds.

Instead of writing the full expression for the Mathai-Quillen form, we define the action to be {Q, Ψ} for some appropriate gauge invariant gauge fermion Ψ [20]. The use of gauge fermions was introduced in the context of topological quantum field theory in [68] (see [10] for a review). 31) in T M and Ω0 (X, adP ). 41) ˜ ∈ T(A,C,B+ ) M. 40) the expression: where (ψ, ζ, ψ) √ Ψproj = X i + g Tr φ¯ −Dµ ψ µ + [ψ˜µν , B +µν ] + i[ζ, C] 2 . 42). However, as in the case of the Mathai-Quillen formulation of Donaldson-Witten theory [6], one must add another piece to the gauge fermion to make full contact with the corresponding twisted supersymmetric theory.

Now, the Montonen-Olive duality conjecture [83] follows simply as the statement that the electric and magnetic factors are exchanged under an inversion of the coupling constant e0 → 1/e0 . Let us consider the example H = SU(N) in detail. Since SU(N) is simply laced, SU(N) and SU(N)v have the same Lie algebra su(N). Also, we can identify the coweight SU (N ) SU (N ) SU (N )v = lattice of SU(N) with the root lattice of su(N). Thus, Λweight = Λcoweight = Λroot v v v SU (N ) SU (N ) SU (N ) H v H , SU(N) has no center (Λweight /Λroot ≃ Center(H)), .