By J. L. Dupont, I. H. Madsen

ISBN-10: 354009721X

ISBN-13: 9783540097211

Collage of Aarhus, 50. Anniversary, eleven September 1978

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**Example text**

The possible choices are found just by analyzing how the 4 of SU(4)I splits in terms of representations of the rotation group K. There are just three possibilities which will give a topological symmetry for a given choice of the SU(2) component of K: (i) 4 → (2, 1)⊕(2, 1), (ii) 4 → (2, 1)⊕(1, 1)⊕(1, 1) and (iii) 4 → (2, 1)⊕(1, 2), each of which leads to a different topological quantum field theory. Choosing the other SU(2) component of K one would obtain the equivalent twists: 4 → (1, 2) ⊕ (1, 2), 4 → (1, 2) ⊕ (1, 1) ⊕ (1, 1) and 4 → (1, 2) ⊕ (2, 1).

Let us now focus on the expression within the parenthesis. 20) µν + (using again Bαβ ≡ σαβ Bµν ). 21) with Cµντ λ the Weyl tensor, we finally obtain, 1 1 √ d4 x g Tr − Dβ α˙ B βα Dγα˙ B γα + F +αβ [Bγα , Bβγ ] 2 2 X 1 √ + D µ B +νλ − B +µν R (gµτ gνλ − gντ gµλ ) − Cµντ λ B +τ λ . 6) be supersymmetric. 48 The Vafa-Witten theory The associated fermionic symmetry splits up as well into BRST (Q+ ≡ Q1 ) and antiBRST (Q− ≡ iQ2 ) parts. 23) satisfying the algebra, {Q+ , Q+ } = δg (φ), ¯ {Q− , Q− } = δg (−φ), {Q+ , Q− } = δg (C).

And finally, associated to the gauge symmetry, we have a commuting scalar field φ ∈ Ω0 (X, adP ) with ghost number +2 [107], and a multiplet of scalar fields φ¯ (commuting and with ghost number −2) and η (anticommuting and with ghost number −1), both also in Ω0 (X, adP ) and which enforce the horizontal projection M → M/G [20]. The BRST symmetry of the model is given by: [Q, Aµ ] = ψµ , [Q, C] = ζ, [Q, B + ] = ψ˜+ , µν µν [Q, φ] = 0, ˜ µ, {Q, χ ˜µ } = H + {Q, χ+ µν } = Hµν , ¯ = η, [Q, φ] {Q, ψµ } = Dµ φ, {Q, ζ } = i [C, φ], {Q, ψ˜+ } = i [B + , φ], µν µν ˜ µ ] = i [χ˜µ , φ], [Q, H + [Q, Hµν ] = i [χ+ µν , φ], ¯ φ].

### Algebraic Topology, Aarhus 1978: Proceedings of a Symposium held at Aarhus, Denmark, August 7-12, 1978 by J. L. Dupont, I. H. Madsen

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