By Marcos Marino
Lately, the previous concept that gauge theories and string theories are an identical has been carried out and built in numerous methods, and there are by means of now a number of versions the place the string concept / gauge conception correspondence is at paintings. essentially the most vital examples of this correspondence relates Chern-Simons thought, a topological gauge thought in 3 dimensions which describes knot and three-manifold invariants, to topological string idea, that's deeply on the topic of Gromov-Witten invariants. This has ended in a few extraordinary kinfolk among three-manifold geometry and enumerative geometry. This booklet provides the 1st coherent presentation of this and different comparable themes. After an advent to matrix versions and Chern-Simons conception, the publication describes intimately the topological string theories that correspond to those gauge theories and develops the mathematical implications of this duality for the enumerative geometry of Calabi-Yau manifolds and knot thought. it really is written in a pedagogical sort and may be valuable analyzing for graduate scholars and researchers in either arithmetic and physics keen to benefit approximately those advancements.
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Extra info for Chern-Simons Theory, Matrix Models, and Topological Strings
Wilson loops was started by Guadagnini et al. (1990), and a nice review of its development can be found in Labastida (1999). The resulting structure can be formalized in a beautiful way by introducing an algebra of diagrams, as we did for the partition functions, and the corresponding universal perturbative invariants are Vassiliev invariants of knots; see Bar-Natan (1995) and Ohtsuki (2002) for further information. 2 Group factors The computation of S involves the evaluation of group factors of Feynman diagrams, which we have denoted by rG (Γ) above.
Each of the Z (c) (M ) will be an asympotic series in 1/k of the form ∞ (c) Z (c) (M ) = Z1−loop (M ) exp S (c) x . , 1990). The one-loop correction (c) Z1−loop (M ) was ﬁrst analyzed by Witten (1989), and has been studied in great detail since then (Freed and Gompf, 1991; Jeﬀrey, 1992; Rozansky, 1995). 18) where Hc0,1 are the cohomology groups with values in the Lie algebra of G as(c) sociated to the ﬂat connection A(c) , τR is the Reidemeister–Ray–Singer torsion of A(c) , Hc is the isotropy group of A(c) , and ϕ is a certain phase.
3 CHERN–SIMONS THEORY AND KNOT INVARIANTS Canonical quantization and surgery As was shown by Witten (1989), Chern–Simons theory is exactly solvable by using non-perturbative methods and the relation to the Wess–Zumino–Witten (WZW) model. In order to present this solution, it is convenient to recall some basic facts about the canonical quantization of the model. Let M be a three-manifold with boundary given by a Riemann surface Σ. We can insert a general operator O in M , which will, in general, be a product of Wilson loops along diﬀerent knots and in arbitrary representations of the gauge group.