Non-Abelian localization for Chern-Simons theory

This chapter presents various aspects of Chern-Simons theory. In a groundbreaking paper, Witten (1989) showed that Chern-Simons gauge theory, which is a quantum field theory in three dimensions, provides a physical description of a wide class of invariants of three-manifolds and of knots and links in three-manifolds. The partition function and correlation functions of Wilson loops in Chern-Simons theory can be computed in a variety of ways. In order to define the partition function of Chern-Simons theory at the quantum level, one has to specify a framing of the three-manifold. It turns out that the evaluation of correlation functions also involves a choice of framing of the knots. A good starting point for understanding framing is to take Chern-Simons theory with gauge group U. The relation between Chern-Simons theory and string theory involves the vacuum expectation values of Wilson loop operators for arbitrary irreducible representations of U(N). This means that N has to be bigger than the number of boxes of any representation under consideration.

We show that partition function of Chern-Simons theory on three-sphere with classical and exceptional groups (actually on the whole corresponding lines in Vogel's plane) can be represented as ratio of respectively triple and double sine functions (last function is essentially a modular quantum dilogarithm). The product representation of sine functions gives Gopakumar-Vafa structure form of partition function, which in turn gives a corresponding integer invariants of manifold after geometrical transition. In this way we suggest to extend gauge/string duality to exceptional groups, although one still have to resolve few problems. In both classical and exceptional cases an additional terms, non-perturbative w.r.t. the string coupling constant, appear. The full universal partition function of Chern-Simons theory on three-sphere is shown to be the ratio of quadruple sine functions. We also briefly discuss the matrix model for exceptional line.

In this paper we study the symplectic and Poisson geometry of moduli spaces of flat connections over quilted surfaces. These are surfaces where the structure group varies from region to region in the surface, and where a reduction (or relation) of structure occurs along the boundaries of the regions. Our main theoretical tool is a new form moment-map reduction in the context of Dirac geometry. This reduction framework allows us to extend the results of [30, 40] to allow more general relations of structure groups, and to investigate both the symplectic and Poisson geometry of the resulting moduli spaces from a unified perspective. The moduli spaces we construct in this way include a number of important examples, including Poisson Lie groups and their Homogeneous spaces, moduli spaces for meromorphic connections over Riemann surfaces (following the work of Philip Boalch), and various symplectic groupoids. Realizing these examples as moduli spaces for quilted surfaces provides new insights into their geometry.

Invariants for framed links in $S^3$ obtained from Chern-Simons gauge field theory based on an arbitrary gauge group (semi-simple) have been used to construct a three-manifold invariant. This is a generalization of a similar construction developed earlier for SU(2) Chern-Simons theory. The procedure exploits a theorem of Lickorish and Wallace and also those of Kirby, Fenn and Rourke which relate three-manifolds to surgeries on framed unoriented links. The invariant is an appropriate linear combination of framed link invariants which does not change under Kirby calculus. This combination does not see the relative orientation of the component knots. The invariant is related to the partition function of Chern-Simons theory. This thus provides an efficient method of evaluating the partition function for these field theories. As some examples, explicit computations of these manifold invariants for a few three-manifolds have been done.

Though not obvious at first glance, these ideas are all connected with each other. In the following sections we shall give an account of some of these ideas and indicate how everything ties together. The relation between the algebraic problem of counting solutions of equations in a group with two-dimensional Yang–Mills theory has been explored in several works by Mulase et al. [6, 7]. (We shall not describe relations with Chern–Simons theory.) Ideas from quantum Yang–Mills may be useful in understanding representation varieties of finitely generated groups. For the case of surface groups, it still remains to carry out a complete study of the topological and symplectic structure of the various strata of the moduli space of flat connections from the quantum Yang–Mills point of view, though much progress has been made (initiated by Witten’s works [11,12]).

The Poisson structure of the moduli space of flat connections on a two dimensional Riemann surface is described in terms of lattice gauge fields and Poisson-Lie groups.

In two dimensional conformal field theories the limit of large central charge plays the role of a semi-classical limit. Certain universal observables, such as conformal blocks involving the exchange of the identity operator, can be expanded around this classical limit in powers of the central charge c. This expansion is an asymptotic series, so — via the same resurgence analysis familiar from quantum mechanics — necessitates the existence of non-perturbative effects. In the case of identity conformal blocks, these new effects have a simple interpretation: the CFT must possess new primary operators with dimension of order the central charge. This constrains the data of CFTs with large central charge in a way that is similar to (but distinct from) the conformal bootstrap. We study this phenomenon in three ways: numerically, analytically using Zamolodchikov’s recursion relations, and by considering non-unitary minimal models with large (negative) central charge. In the holographic dual to a CFT2, the expansion in powers of c is the perturbative loop expansion in powers of ћ. So our results imply that the graviton loop expansion is an asymptotic series, whose cure requires the inclusion of new saddle points in the gravitational path integral. In certain cases these saddle points have a simple interpretation: they are conical excesses, particle-like states with negative mass which are not in the physical spectrum but nevertheless appear as non-manifold saddle points that control the asymptotic behaviour of the loop expansion. This phenomenon also has an interpretation in SL(2, ℝ) Chern-Simons theory, where the non-perturbative effects are associated with the non-Teichmüller component of the moduli space of flat connections.

A short overview is presented of: (i) the quantum field measure associated with pure Yang-Mills gauge theory on a compact surface, (ii) an approach to the symplectic structure on the [moduli] space of [flat] connections over compact oriented surfaces (with and without boundary), and (iii) the relation between the classical Yang-Mills measure and the volume measure on the moduli space of flat connections.

We study the quantization of the moduli space of flat connections on a surface of genus one, using the real polarization of this space described in [10]. The quantum wave functions in this formalism are exponential functions supported along the integral fibres of the polarization. The space of wave functions obtained in this way is isomorphic to a space of theta functions. We use our construction to construct part of what may be a topological field theory in genus one, and to compute the associated invariants of some three manifolds. These computations agree with those of Witten [12], but the invariants are expressed as sums of quantities computed at a discrete set of connections with curvature concentrated on a link in the three manifold. A similar prescription is used to produce knot invariants.

Previous path integral treatments of Yang–Mills on a Riemann surface automatically sum over principal fiber bundles of all possible topological types in computing quantum expectations. This paper extends the path integral formulation to treat separately each topological sector. The formulation is sufficiently explicit to calculate Wilson line expectations exactly. Further, it suggests two new measures on the moduli space of flat connections, one of which proves to agree with the small-volume limit of the Yang–Mills measure.

It is known that every nonorientable surface Σ has an orientable double cover Σ~. The covering map induces an involution on the moduli space ℳ~ of gauge equivalence classes of flat G-connections on Σ~. We identify the relation between the moduli space ℳ and the fixed point set of the moduli space ℳ~. In particular, ℳ is isomorphic to the fixed point set of ℳ~ if and only if the order of the center of G is odd. One important application is that we give a way to construct a minimal Lagrangian submanifold of the moduli space ℳ~.

Some computations of Donaldson's invariants via flat connections

We consider a semi-classical treatment, in the regime of weak gauge coupling, of supersymmetric Yang-Mills theory in a space-time of the form T^3xR with SU(n)/Z_n gauge group and a non-trivial gauge bundle. More specifically, we consider the theories obtained as power series expansions around a certain class of normalizable vacua of the classical theory, corresponding to isolated points in the moduli space of flat connections, and the perturbative corrections to the free energy eigenstates and eigenvalues in the weakly interacting theory. The perturbation theory construction of the interacting Hilbert space is complicated by the divergence of the norm of the interacting states. Consequently, the free and interacting Hilbert furnish unitarily inequivalent representation of the algebra of creation and annihilation operators of the quantum theory. We discuss a consistent redefinition of the Hilbert space norm to obtain the interacting Hilbert space and the properties of the interacting representation. In particular, we consider the lowest non-vanishing corrections to the free energy spectrum and discuss the crucial importance of supersymmetry for these corrections to be finite.

In this paper, we explain how generalized dynamical r-matrices can be obtained by (quasi-)Poisson reduction. New examples of Poisson structures and Poisson groupoid actions naturally appear in this setting. As an application, we use a generalized dynamical r-matrix induced by the gauge fixing procedure to give a new finite dimensional description of the Atiyah-Bott symplectic structure on the moduli space of flat connections on a surface.

According to Atiyah-Bott [ABA] the moduli space of flat connections on a compact oriented 2-manifold with prescribed holonomies around the boundary is a finite-dimensional symplectic manifold, possibly singular. A standard approach [W1W2] to computing invariants (symplectic volumes, Riemann-Roch numbers, etc.) of the moduli space is to study the “factorization” of invariants under gluing of 2-manifolds along boundary components. Given such a factorization result, any choice of a “pants decomposition” of the 2-manifold reduces the computation of invariants to the three-holed sphere.KeywordsModulus SpaceConjugacy ClassBoundary ComponentSymplectic FormMaximal TorusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.