Celebratio Mathematica

L. Baulieu and I. M. Singer: “Topological Yang–Mills symmetry,” pp. 12–19 in Conformal field theories and related topics (Annecy-le-Vieux, France, 14–16 March 1988), published as Nuclear Phys. B Proc. Suppl. 5B : 2. Issue edited by R. Stora, P. Binétruy, and P. Sorba. North-Holland (Amsterdam), December 1988. MR 1002954 Zbl 0958.58500 incollection

Paul Sorba	Related
Laurent Baulieu	Related
Pierre Binétruy	Related
Raymond Stora	Related

L. Baulieu and I. M. Singer: “The topological sigma model,” Comm. Math. Phys. 125 : 2 (June 1989), pp. 227–237. MR 1016870 Zbl 0684.57016 article

L. Baulieu and I. M. Singer: “Conformally invariant gauge fixed actions for 2-D topological gravity,” Comm. Math. Phys. 135 : 2 (1991), pp. 253–265. MR 1087384 Zbl 0725.53071 article

L. Baulieu, H. Kanno, and I. M. Singer: “Special quantum field theories in eight and other dimensions,” Comm. Math. Phys. 194 : 1 (May 1998), pp. 149–175. MR 1628298 Zbl 0910.53054 article

We build nearly topological quantum field theories in various dimensions. We give special attention to the case of eight dimensions for which we first consider theories depending only on Yang–Mills fields. Two classes of gauge functions exist which correspond to the choices of two different holonomy groups in \( SO(8) \), namely \( SU(4) \) and \( Spin(7) \). The choice of \( SU(4) \) gives a quantum field theory for a Calabi–Yau fourfold. The expectation values for the observables are formally holomorphic Donaldson invariants. The choice of \( Spin(7) \) defines another eight dimensional theory for a Joyce manifold which could be of relevance in \( M \)- and \( F \)-theories. Relations to the eight dimensional supersymmetric Yang–Mills theory are presented. Then, by dimensional reduction, we obtain other theories, in particular a four dimensional one whose gauge conditions are identical to the non-abelian Seiberg–Witten equations. The latter are thus related to pure Yang–Mills self-duality equations in 8 dimensions as well as to the \( N=1 \), \( D=10 \) super Yang–Mills theory. We also exhibit a theory that couples 3-form gauge fields to the second Chern class in eight dimensions, and interesting theories in other dimensions.

Hiroaki Kanno	Related
Laurent Baulieu	Related

L. Baulieu, H. Kanno, and I. M. Singer: “Cohomological Yang–Mills theory in eight dimensions,” pp. 365–373 in Dualities in gauge and string theories (Seoul and Sokcho, S. Korea, 17–28 February 1997). Edited by Y. M. Cho and S. Nam. World Scientific (River Edge, NJ), 1998. MR 1712267 Zbl 1052.53508 incollection

We construct nearly topological Yang–Mills theories on eight dimensional manifolds with a special holonomy group. These manifolds are the Joyce manifold with \( \operatorname{Spin}(7) \) holonomy and the Calabi–Yau manifold with \( \operatorname{SU}(4) \) holonomy. An invariant closed four form \( T_{\mu\nu\rho\sigma} \) on the manifold allows us to define an analogue of the instanton equation, which serves as a topological gauge fixing condition in BRST formalism. The model on the Joyce manifold is related to the eight dimensional supersymmetric Yang–Mills theory. Topological dimensional reduction to four dimensions gives non-abelian Seiberg–Witten equation.

Laurent Baulieu	Related
Hiroaki Kanno	Related
Y. M. Cho	Related
Soonkeon Nam	Related

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Isadore M. Singer

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