In Chern-Simons theory with a compact gauge group, perturbation theory is often developed in the background of a trivial (or reducible) flat connection A(ρ). One advantage of Chern-Simons theory with complex gauge group is that the classical phase space Mflat(GC,Σ) is a hyper-K¨ahler manifold, a fact that greatly simplifies the quantization problem in any of the existing frameworks (such as geometric quantization ) [45], distortion quantization [46, 47], or “brane quantization” [48]).
Such a transition is the only type expected to occur in moduli space MH (see [21]), so the index Ω(t∞) is independent of the hypermultiplet moduli. The refined index encodes much of the information present in the Hilbert space HBP S.
Physical wall-crossing formulas
Let the wall be attms, and let+ andt− be moduli on the stable and unstable sides of the wall, respectively. Hilbert space on. the stable side of the wall is a product of Hilbert spaces for the components times an electromagnetic angular momentum multiple.
BPS states of D-branes
The modulus space M(γ) of D-branes can often be described as a modulus space of quiver representations. The classical module space of this gauge theory, MQM, is equivalent to the classical module space of D-branes.
Invisible walls
In the refined case, we can similarly repeat the refined primitive wall intersection formula (2.2.7) to find it. Since the refined index does not change in the 3-center case, it cannot change in the multi-center case either.
Wall crossing for the conifold
Subsequently, D2 +N D0 particles begin to bind until, after passing through a D2 +D0 wall, one ends up in the Szendr¨oi region of moduli space. In the Szendrøoi region itself, this eventually becomes Szendrói's “non-commutative Donaldson-Thomas” partition function [64].
Crystals and quivers
In the Szendr¨oi chamber C1, the generator function (3.1.22) improves the Szendr¨oi partition function for the noncommutative Donaldson-Thomas theory on the conifold. In [64] it is claimed that every fixed point in the moduli space of cyclic representations is uniquely identified by such a melting configuration.
Pyramid crystals and wall crossing
As shown in Figure 3.4, this crystal has two different types of atoms, corresponding to the two vertices of the Klebanov-Witten quiver. We let the weight wn be a function acting linearly on formal sums of partitions, and define Θ(n) as the formal sum of all possible partitions of a dimer lattice with asymptotic boundary conditions corresponding to the length-n crystal. 3.3.7) The operation that we claim is the combinatorial equivalent of wall crossing is described in [76] as dimer shift.
Refined crystals
To see this, note that as n→ ∞ the neighborhoods of the top and bottom vertices of the dimer lattice continue to reduce to effective brick-like lattices, now shown in Figure 3.14. Kontsevich and Soibelman also define a "q-distorted" version of the generalized Donaldson-Thomas invariants: the motivational Donaldson-Thomas invariants.
Classical and motivic KS wall crossing
The motif DT invariants are functions of L1/2, the square root of the motif of the affine line. Apart from the potentially naive fact that both motivic and refined invariants are quite natural distortions of the classical BPS indices/invariants, we will argue in this chapter that motivic and refined invariants have identical wall-crossing behavior.
Refined = Motivic
However, this time consider bound states of all charges γ1+N γ2 with N ≥1.5 Also assume, for simplicity, that no states of charge γ1+N γ2 exist in the spectrum on the unstable side of the wall. Or, more correctly, it would be true if we added γ1+N γ2. states on the unstable side of the semi-primitive formula.).
Examples: SU (2) Seiberg-Witten theory
The low energy theory of the D branes and the resulting BPS states in the gauge theory can be described by representations of a. In a certain limit of the complex structure (corresponding to the decoupling of supergravity from the gauge theory) this becomes the Seiberg-Witten curve [82] (see also [63]).
Basics
As discussed in the introduction, one can then consider perturbation theory around a given flat connection A(ρ) ∈ Mflat(GC;M) corresponding to the homomorphism ρ. By standard methods in quantum field theory—essentially a stationary phase approximation to the path integral—each component Z(ρ)(M;t) can be expanded in inverse powers of t.
Coefficients and Feynman diagrams
As we pointed out in the Introduction, such 3-manifolds provide some of the most interesting examples for Chern-Simons theory with complex gauge groups. To describe the flat connection A(geom) explicitly, recall that H3 can be defined as the upper half-space with the standard hyperbolic metric. 5.2.7) The components of the vielbein and spin connection corresponding to this metric can be written as.
Arithmeticity
This makes, for example, the arithmetical nature of the perturbative coefficients Sn(ρ) as clear as possible. Z(S3;~) = Z(0)(S3;~).) This normalization does not affect the arithmetical nature of the perturbative coefficients Sn(geom), because for M =S3 all the Sn(0)' e rational numbers.
Examples
Actually, in this case one can rigorously prove the correctness of the expansion: the two formulas in (5.4.10) were proved in [105] and the rationality of the numbers defined by (5.4.11) in and [108] , and can therefore check that the numerically determined values are the true ones (see also [101] for a generalization of this analysis). We note that in both the examples treated here, the first statement of the conjecture (5.4.7) can be strengthened.
We also include a brief discussion of "brane quantization" for Chern-Simons theory with a complex gauge group from [3], as it is closely related to geometric quantization. However, quantization of Chern-Simons theory with complex gauge group has a number of good properties.
Analytic continuation and the Volume Conjecture
It is labeled with a representation R=Rλ of the gauge group G, which we assume to be an irreducible representation with highest weight λ∈ t∨. If one takes the limit with ~−1 ∈ iπ1Z, which corresponds to the allowed values of the coupling constant in Chern-Simon's theory with compact gauge group G, then one can never see the exponential asymptotics (5.1.9) with Im(S0( ρ) )>0.
Hierarchy of differential equations
A more detailed explanation of the limit (6.2.3) will be presented in Chapter 7, where we discuss Wilson loops for complex and compact groups. The semiclassical expression (6.3.11) gives the value of the classical Chern-Simons functional (5.1.1), evaluated on the flat gauge connection A(ρ), denoted by the homomorphismρ.
Classical and quantum symmetries
This action itself cannot be a symmetry of the theory because it does not preserve the symptomatic form (6.1.14). However, we can try to combine it with the transformation ~→ −~ to obtain a symmetry of the complex phase space (Mflat(GC,Σ), ω).
Brane quantization
This choice must be consistent with the involution action τ, which means that τ :Y → Y raises to an action on L such that τ|N = id. 6 Note that while elsewhere we consider only the "holomorphic" sector of the theory (which is sufficient in the perturbative approach), here we write the full complex form in Mflat(GC,Σ) derived from the classical Chern-Simons action (5.1. 1), including the contributions of both A and ¯A fields.
Examples
As for the geometric branch, it is then replaced in the rest of the hierarchy of equations. In the second part of the chapter, we incorporate these results into Wilson loops and Chern-Simons theory.
Some representation theory
It operates on the space of functions on G valued in the vector space of the representation σ that satisfies. This is generally true for maximal parabolic subgroups: principal series representations are induced from representations of the maximal torus that trivially act on the off-diagonal part of the parabolic subgroup.
Representations as coadjoint orbits
In the case of complex groupGC, principal series representations are obtained by considering real polarizations on the line bundle overGC/Hλ, whereλ is now a non-complex linear element of dualg∗. The set of all sections of this line bundle is the representation space of a representation induced from Hλ by eiλ. It is easy to see that the generators of Lie algebrasu(2) act on local functions f(z) exactly as in (7.1.4).
Quantum mechanics for Wilson loops
Therefore, we immediately conclude that the Hilbert space of this quantum mechanical system is the space of the representation with the highest weight λ. Somewhat less trivially, we claim that the quantization of the HamiltonianiH=−iTr(λg−1Atg) is precisely the matrix At that acts on the Hilbert space in the representation with the highest weightλ.
Wilson loops vs. boundary conditions
From Section 7.3, we know that we can write the path integral of the theory by introducing the Wilson cycle asR. In the structure of the Chern-Simons unitarity relevant for Euclidean quantum gravity, met,¯t∈R, mew∈R representations are also allowed.
Hyperbolic geometry
The region of large values of x3 in F corresponds to the region near the top of the complement of the figure-of-eight knot M. The shape parameters z and w are related to the eigenvalue of the holonomy SL(2,C), l, along the length8 of the knot as follows.
Hikami’s invariant
To build a quantum invariant, however, only half of the variables pj really "belong" to one tetrahedron. In the classical limit, this integral can be naively evaluated in the saddle approximation, and Hikami's claim is that the saddle ratios exactly coincide with the boundary conditions for triangulation on M .
Quantum dilogarithm
Here and hereafter we use the abbreviation e(x) =e2πix.) This is the third of these functions in the normalization. The asymptotic formula (8.3.2) can be improved to the asymptotic expansion Li2 x;e2~. here Bn is the nth Bernoulli number) with the polylogarithm of the negative index Li2−n(x) ∈ Q 1.
Finally, we also note that we use only those critical points of the full integrand (8.4.12) which correspond to the critical points of V in the limit ~→0. The resulting analytic continuations become integrals of products of quantum dilogarithms just like partition functions of the state integral model.
Figure-eight knot 4 1
We claim that this corresponds to the geometric and conjugate branches of the A-polynomial described in Section 6.6, which can be verified by computing 1 s(x, m) = exp ∂u∂ V(p, u). We now calculate the perturbative invariants Sn(geom)(u) and Sn(conj)(u) by doing a full saddle point approximation of the integral (9.1.2) on (dummy) contours passing through the two critical points.
Three-twist knot 5 2
Since the A-polynomial has three non-abelian branches, one expects to find three saddle points in the integrand of (9.2.1). It is easy to solve for y = m4(−x)−1m4x , but the subsequent equation for x is (as expected) a third degree irreducible polynomial:. 9.2.4) The solutions to this polynomial have the same arithmetic structure as the solutions to the A-polynomial.
Direct analytic continuation
Bar-Natan, Perturbative Aspects of Chern-Simons Topological Quantum Field Theory, Dissertation (Ph.D.)–Princeton University (1991). Murakami, SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial,math/0608324v2.
