Infinite Dimensional Groups and Manifolds by Tilman Wurzbacher

By Tilman Wurzbacher

The amount is a set of refereed learn papers on limitless dimensional teams and manifolds in arithmetic and quantum physics. Directed at examine mathematicians and theoretical physicists in addition to graduate scholars, the quantity offers an summary of significant issues of analysis on the leading edge of arithmetic and theoretical physics.

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74 6 Large N transitions and toric geometry 80 7 Conclusions 86 1 Introduction Enumerative geometry and knot theory have benefitted considerably from the insights and results in string theory and topological field theory. The theory of Gromov–Witten invariants has emerged mostly from the consideration of topological sigma models and topological strings, and mirror symmetry has provided a surprising point of view with powerful techniques and deep implications for the theory of enumerative invariants.

The result, which depends on the choice of complex structure of the Riemann surface, is then integrated over the moduli space M g . Fg can be evaluated again, like in the topological sigma model, as a sum over instanton sectors. It turns out [15] that Fg is a generating functional for the Gromov–Witten invariants Ng,β , or more precisely, Ng,β q β . 15) β It is also useful to introduce a generating functional for the all-genus free energy: ∞ F (gs , t) = 2g−2 Fg (t)gs . 16) g=0 The parameter gs can be regarded as a formal variable, but in the context of type II strings it is nothing but the string coupling constant.

The coefficients I0,3,β (φ1 , φ2 , φ3 ) “count” in some appropriate way the number of holomorphic maps from the sphere to the Calabi–Yau, in the topological sector specified by β, and in such a way that the point of insertion of Oφi gets mapped to the divisor Di . This is an example of a Gromov–Witten invariant, although to get the general picture we have to couple the model to gravity, as we will see very soon. 7): the trivial sector gives just the classical intersection number of the cohomology ring, and then there are quantum corrections associated to the worldsheet instantons.

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