Flat connection contribution to topology changing amplitudes in an ensemble of seifert fibered homology spheres

The Fintushel-Stern pseudofree orbifolds are exploited to construct wave functions of universes created as a result of the interaction of cones on lens spaces. We also study the problem of the definition of topology changing amplitudes for tunneling topology changes, described by cobordisms with Seifert fibered homology sphere boundaries. It is demonstrated that such topology changes are accompanied by creation or annihilation of the lens spaces. The topology-changing amplitude calculations are carried out in the stationary phase approximation for Kodama wave functions. In this approximation the changing amplitudes factorize and they are expressed by means of Chern-Simons invariants of flat connections over Seifert fibered homology spheres and lens spaces.     

Siebenmann-type cobordisms with borders and topology changes by quantum tunneling

Siebenmann-type cobordisms are constructed to describe topology changes with the Seifert fibered homology spheres in in- and out-states. We study the problem of determining of topology-changing amplitudes for these quantum tunneling processes. The calculations are performed in the stationary phase approximation for Kodama wave functions. In this approximation the amplitudes are expressed in terms of Chern-Simons invariants of flatSU(2)-connections over the cobordism boundary components. The topology-change amplitudes found are factorized into the Kodama wave functions for the lens spaces. The results are compared with those for Fintushel-Stern-type cobordisms which have been previously investigated.     

Vassiliev Knot Invariants and Chern-Simons Perturbation Theory to All Orders

At any order, the perturbative expansion of the expectation values of Wilson lines in Chern-Simons theory gives certain integral expressions. We show that they all lead to knot invariants. Moreover these are finite type invariants whose order coincides with the order in the perturbative expansion. Together they combine to give a universal Vassiliev invariant. Received: 26 March 1996 / Accepted: 7 November 1996     

Witten's invariants of rational homology spheres at prime values ofK and trivial connection contribution

We establish a relation between the coefficients of asymptotic expansion of the trivial connection contribution to Witten's invariant of rational homology spheres and the invariants that T. Ohtsuki extracted from Witten's invariant at prime values ofK. We also rederive the properties of primeK invariants discovered by H. Murakami and T. Ohtsuki. We do this by using the bounds on Taylor series expansion of the Jones polynomial of algebraically split links, studied in our previous paper. These bounds are enough to prove that Ohtsuki's invariants are of finite type. The relation between Ohtsuki's invariants and trivial connection contribution is verified explicitly for lens spaces and Seifert manifolds.Work supported by the National Science Foundation under Grant No. PHY-92 009978.     

Computer calculation of Witten's 3-manifold invariant

Witten's 2+1 dimensional Chern-Simons theory is exactly solvable. We compute the partition function, a topological invariant of 3-manifolds, on generalized Seifert spaces. Thus we test the path integral using the theory of 3-manifolds. In particular, we compare the exact solution with the asymptotic formula predicted by perturbation theory. We conclude that this path integral works as advertised and gives an effective topological invariant.The first author is supported by NSF grant DMS-8805684, an Alfred P. Sloan Research Fellowship, and a Presidential Young Investigators award. The second author is supported by NSF grant DMS-8902153     

Odd Chern-Simons Theory, Lie Algebra Cohomology and Characteristic Classes

We investigate the generic 3D topological field theory within the AKSZ-BV framework. We use the Batalin-Vilkovisky (BV) formalism to construct explicitly cocycles of the Lie algebra of formal Hamiltonian vector fields and we argue that the perturbative partition function gives rise to secondary characteristic classes. We investigate a toy model which is an odd analogue of Chern-Simons theory, and we give some explicit computation of two point functions and show that its perturbation theory is identical to the Chern-Simons theory. We give a concrete example of the homomorphism taking Lie algebra cocycles to Q-characteristic classes, and we reinterpret the Rozansky-Witten model in this light.     

A contribution of the trivial connection to the Jones polynomial and Witten's invariant of 3d manifolds,I

We use a path integral formulation of the Chern-Simons quantum field theory in order to give a simple semi-rigorous proof of a recently conjectured limitation on the 1/K expansion of the Jones polynomial of a knot and its relation to the Alexander polynomial. A combination of this limitation with the finite version of the Poisson resummation allows us to derive a surgery formula for the contribution of the trivial connection to Witten's invariant of rational homology spheres. The 2-loop part of this formula coincides with Walker's surgery formula for the Casson-Walker invariant. This proves a conjecture that the Casson-Walker invariant is proportional to the 2-loop correction to the trivial connection contribution. A contribution of the trivial connection to Witten's invariant of a manifold with nontrivial rational homology is calculated for the case of Seifert manifolds.Work supported in part by the National Science Foundation under Grants No. PHY-92 09978 and 9009850 and by the R. A. Welch Foundation.     

Chern-Simons invariants of 3-manifolds decomposed along tori and the circle bundle over the representation space ofT 2

We describe a cut-and-paste method for computing Chern-Simons invariant of flatG-connections on 3-manifolds decomposed along tori, especially forG=SU(2) andSL(2,C). We use this method to make computations ofSU(2) Chern-Simons invariants of graph manifolds which generalize Fintushel and Stern's computations for Seifert-fibered spaces. We also use this technique to give a simple derivation of a formula of Yoshida relating the flatSL(2,C) Chern-Simons invariant of the holonomy representation to the volume and the metric Chern-Simons invariant for cusped hyperbolic 3-manifolds.     

Primitive Vassiliev Invariants and Factorization in Chern-Simons Perturbation Theory

The general structure of the perturbative expansion of the vacuum expectation value of a Wilson line operator in Chern-Simons gauge field theory is analyzed. The expansion is organized according to the independent group structures that appear at each order. It is shown that the analysis is greatly simplified if the group factors are chosen in a certain way that we call canonical. This enables us to show that the logarithm of a polynomial knot invariant can be written in terms of primitive Vassiliev invariants only. Received: 2 May 1996 / Accepted: 21 March 1997     

Topological phases of quantum theories. Chern-Simons theory

We analyze the vacuum structure (degeneracy, nodes and symmetries) of some quantum theories with special emphasis on the study of its dependence on the geometry and topology of the classical configuration space. The study of the topological limit shows that many low energy properties of those quantum theories can be inferred from the structure of their topological phases. After reviewing some simple pure quantum mechanical models (planar rotor, magnetic monopole and quantum Hall effect) we focus on the study of the rich relationship existing between topologically massive gauge theories and their topological phases, Chern-Simons theories. In particular we show that, although in a finite volume the degeneracy of the quantum vacuum of gauge theories depends on the topology of the underlying Riemann surface, in an infinite volume the vacuum is unique. Finally, the topological structure of Chern-Simons theory is analyzed in a covariant formalism within a geometric regularization scheme. We discuss in some detail the structure of the different metric dependent contributions to the Chern-Simons partition function and the associated topological invariants.     

Composite A,B,C,D-IRF-model invariants

In this work the introduction of generalized A,B,C,D interaction-round-a-face model invariants related to composite braid group representations will be proposed. The invariant polynomials are obtained in the framework of Witten's Chern-Simons theory summarizing recent works on link invariants. The primary intention is to present explicitly neglected results in the latter area and to outline in a pedagogical way the computation of a variety of known and new invariants. The close relationship of the topological interpretation of link invariants and the notion of generalized knot polynomials derived from integrable models in statistical mechanics is emphasized.     

Invariance of fermion determinant under large gauge transformations

We argue that in the perturbative framework the natural symmetry of the fermionic determinant is the perturbative gauge transformation (p.g.t.) which differs from the usual gauge transformation of the effective action through the absence of terms independent of the coupling constant. Calculated in a non-perturbative framework appropriate for large gauge function, the sum of these latter terms vanish. In three dimensions the invariance of the full fermion determinant under large gauge transformations is thus ensured due to the invariance under p.g.t. of the Chern-Simons term arising in some perturbative regularisations.     

Chern-Simons topological Lagrangians in odd dimensions and their Kaluza-Klein reduction

By clarifying the behavior of generic Chern-Simons secondary invariants under infinitesimal variation and finite gauge transformation, it is proved that they are eligible to be a candidate term in the Lagrangian in odd dimensions (2k ? 1 for gauge theories and 4k ? 1 for gravity). The coefficients in front of these terms may be quantized because of topological reasons. As a possible application, the dimensional reduction of such actions in Kaluza-Klein theory is discussed. The difficulty in defining the Chern-Simons action for topologically nontrivial field configurations is pointed out and resolved.     

Curve counting,instantons and McKay correspondences

We survey some features of equivariant instanton partition functions of topological gauge theories on four and six dimensional toric Kähler varieties, and their geometric and algebraic counterparts in the enumerative problem of counting holomorphic curves. We discuss the relations of instanton counting to representations of affine Lie algebras in the four-dimensional case, and to Donaldson–Thomas theory for ideal sheaves on Calabi–Yau threefolds. For resolutions of toric singularities, an algebraic structure induced by a quiver determines the instanton moduli space through the McKay correspondence and its generalizations. The correspondence elucidates the realization of gauge theory partition functions as quasi-modular forms, and reformulates the computation of noncommutative Donaldson–Thomas invariants in terms of the enumeration of generalized instantons. New results include a general presentation of the partition functions on ALE spaces as affine characters, a rigorous treatment of equivariant partition functions on Hirzebruch surfaces, and a putative connection between the special McKay correspondence and instanton counting on Hirzebruch–Jung spaces.     

Invariants of Spin Networks Embedded in Three-Manifolds

We study the invariants of spin networks embedded in a three-dimensional manifold which are based on the path integral for SU(2) BF-Theory. These invariants appear naturally in Loop Quantum Gravity, and have been defined as spin-foam state sums. By using the Chain-Mail technique, we give a more general definition of these invariants, and show that the state-sum definition is a special case. This provides a rigorous proof that the state-sum invariants of spin networks are topological invariants. We derive various results about the BF-Theory spin network invariants, and we find a relation with the corresponding invariants defined from Chern-Simons Theory, i.e. the Witten-Reshetikhin-Turaev invariants. We also prove that the BF-Theory spin network invariants coincide with V. Turaev’s definition of invariants of coloured graphs embedded in 3-manifolds and thick surfaces, constructed by using shadow-world evaluations. Our framework therefore provides a unified view of these invariants.     

Residue formulas for the largek asymptotics of Witten's invariants of Seifert manifolds. The case ofSU(2)

We derive the largek asymptotics of the surgery formula forSU(2) Witten's invariants of general Seifert manifolds. The contributions of connected components of the moduli space of flat connections are identified. The contributions of irreducible connections are presented in the residue form. This allows us to express them in terms of intersection numbers on their moduli spaces.Address after September 25: L. Rozansky, School of Mathematics, Institute for Advanced Study, Princeton, N.J. 08540, USA.     

Quantum invariants at the sixth root of unity

A general topological formula is given for theSU (2) quantum invariant of a 3-manifoldM at the sixth root of unity. It is expressed in terms of the homology, Witt invariants and signature defects of the various 2-fold covers ofM, and thus ties in with basic 4-dimensional invariants. A discussion of the range of values of these quantum invariants is included, and explicit evaluations are made for lens spaces.     

Chern-Simons-Witten invariants of lens spaces and torus bundles,and the semiclassical approximation

We derive explicit formulas for the Chern-Simons-Witten invariants of lens spaces and torus bundles overS ¹, for arbitrary values of the levelk. Most of our results are for the groupG=SU(2), though some are for more general compact groups. We explicitly exhibit agreement of the limiting values of these formulas ask with the semiclassical approximation predicted by the Chern-Simons path integral.Partially supported by an NSF Graduate FellowshipAddress as of September 1, 1991: School of Natural Science, Institute for Advanced Study, Princeton, NJ 08540; USA     

Semiclassical limit for Chern-Simons theory on compact hyperbolic spaces

The invariant integration method for Chern-Simons theory defined on compact hyperbolic spaces of the form Γℍ³ is verified in the semiclassical approximation. The semiclassical limit for the partition function is calculated. The contribution to the sum over topologies in three-dimensional quantum gravity is briefly discussed. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 2, 65–69 (25 July 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Seiberg--Witten Monopoles in Three Dimensions

Dimensional reduction of the Seiberg--Witten equations leads to the equations of motion of a U(1) Chern--Simons theory coupled to a massless spinorial field. A topological quantum field theory is constructed for the moduli space of gauge equivalence classes of solutions of these equations. The Euler characteristic of the moduli space is obtained as the partition function which yields an analogue of Casson's invariant.A mathematically rigorous definition of the invariant isdeveloped for homology spheres using the theory of spectral flow ofself-adjoint Fredholm operators.