Communications in Mathematical Physics - Volume 253 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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44 M. Mari˜no There are also various interesting physical contexts in which the results of this paper might be relevant. Let us end by mentioning a few of them: 1) The computation of Rozansky-Witten invariants [38] involves the universal perturbative invariants of three-manifolds that are extracted from Chern-Simons theory, but the weight system is now associated to a hyperK¨ahler manifold (see [39] for a very nice review). Therefore, the universal perturbative invariants of Chern-Simons theory are relevant in Rozansky-Witten theory.

Group theory factors. We first present the group theory factors associated to the connected graphs that give a basis of A(∅)conn up to order 5. The evaluation of these factors is straightforward by using the graphical techniques of Cvitanovi´c [12], and rather immediate for all of them (except for rω (G), that gives the quartic Casimir in the adjoint and has been computed for all gauge groups in the second reference of [12]). Our conventions are as in [12]: the Lie algebra in the defining representation has Hermitian generators Ti , i = 1, .

27) in the orthogonal ensemble. As a simple example, let us consider again k = (0, 1, 0, · · · ). 18). 3. Universal perturbative invariants up to order 5. Using the above ingredients, it is easy to find the universal perturbative invariants of Seifert spaces up to order 5. 10) (so for example E1 = −n + ni=1 pi−2 ). One finds, 1 P φ − (2 + E1 ) , 48 H P 2 1 = (1 + E1 + E2 ), 1152 H 1 P 3 = E3 , 13824 H 1 P 4 = 11059200 H Iθ = − Iθ2 Iθ3 Iθ4 × 82E4 − 46E3 − 18(1 + E2 + E3 )E1 − 9E22 − 18E2 − 9E12 − 9 , 1 P 4 2E4 − 6E3 + 2E1 (1 + E2 − E3 ) + E22 + 2E2 + E12 + 1 , 1382400 H 1 P 5 Iθ5 = 55E5 + E4 (27E1 − 56) − E3 (9E2 + 36E1 + 8) , 66355200 H P 5 1 Iωθ = 5E5 − E4 (3E1 + 16) + E3 (E2 + 4E1 + 12) .

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