By Sen Hu

This useful monograph has arisen partly from E Witten's lectures on topological quantum box thought within the spring of 1989 at Princeton college. at the moment Witten unified a number of vital mathematical works by way of quantum box idea, such a lot particularly the Donaldson polynomial, the Gromov-Floer homology and the Jones polynomials. In his lectures, between different issues, Witten defined his intrinsic 3-dimensional development of Jones polynomials through Chern-Simons gauge concept. He supplied either a rigorous facts of the geometric quantization of the Chern-Simons motion and a truly illuminating view as to how the quantization arises from quantization of the distance of connections. He built a projective flat connection for the Hilbert area package deal over the gap of complicated buildings, which turns into the Knizhik-Zamolodchikov equations in a distinct case. His building ends up in many attractive purposes, comparable to the derivation of the skein relation and the surgical procedure formulation for knot invariant, an explanation of Verlinde's formulation, and the institution of a reference to conformal box theory.In this publication, Sen Hu has further fabric to supply the various info omitted of Witten's lectures and to replace a few new advancements. In bankruptcy four he provides a development of knot invariant through illustration of mapping classification teams in accordance with the paintings of Moore-Seiberg and Kohno. In bankruptcy 6 he deals an method of developing knot invariant from string idea and topological sigma versions proposed by means of Witten and Vafa. The localization precept is a strong instrument to construct mathematical foundations for such cohomological quantum box theories.In addition, a few hugely correct fabric via S S Chern and E Witten has been integrated as appendices for the benefit of readers: (1) complicated Manifold with out capability idea by way of S S Chern, pp148-154. (2) "Geometric quantization of Chern-Simons gauge idea" through S Axelrod, S D Pietra and E Witten.

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**Example text**

It is a vector bundle over the space of complex structures on A. A complex structure is also an almost complex structure J : TM + T M ,J 2 = - I d . Let 65 be a small deformation of J , then we have ( J 6J)2 = - I d . This is J6J 6 d d = 0. Let 6J be a deformation of complex structure J . The following is a connection on the Hilbert s p x c vector bundle. + + [Vi! Vj] = lCwi3: [V;, Vj] = 0, [V;, V,] = 0. It is importmit to note thal: 1) 6% preserves holornorphicity. This can be verified by showing that it.

We know that it is a vector bundle of complex structures over the punctured sphere. , zn)}. So we have a trivial vector bundle + Vj, 8 vj, @I ... @ Vj, + Cn - A. The projective flat connection can be explicitly described in this case. Let { I p } be an orthonomal basis of s l ( 2 , C ) with respect t o the CartanKilling form. Let is the projection from I$,@ Vj, @ ... @ Vjv, to The Knizhik-Zamolodchikov equations are: where l3@ :=O,i= 8 zi 5,. , n , whcre @ is a section of the Hilbert space vector bundle.

For each flat connection A we have a covariant derivst'ive d~ = d A . Thc tangent space of M is the space of first cohomology H i A ( C ,E @g ) . M is also a symplectic variety with respect to the syrnplectic form w above and one can easily check t,hat w only depend on dA-cohornology classes. This way we push the symplectic form down t o a syrnplectic form on the syrnplectic quotieat. As we have seen in Chapter One it is pretty easy t o quantize an affine symplectic space. W e use holomorphic quantization here.