We provide a physical definition of new homological invariants H a (M 3 ) of 3manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on M 3 times a 2-disk, D 2 , whose Hilbert space of BPS states plays the role of a basic building block in categorification of various partition functions of 3d N = 2 theory T [M 3 ]: D 2 × S 1 half-index, S 2 × S 1 superconformal index, and S 2 × S 1 topologically twisted index. The first partition function is labeled by a choice of boundary condition and provides a refinement of Chern-Simons (WRT) invariant. A linear combination of them in the unrefined limit gives the analytically continued WRT invariant of M 3 . The last two can be factorized into the product of half-indices. We show how this works explicitly for many examples, including Lens spaces, circle fibrations over Riemann surfaces, and plumbed 3-manifolds.C. Categorification of the Turaev-Viro invariants 733 It is in fact (Tor H1(M3, Z)) * /Z2 that is canonically identified with components of abelian flat connections.However, as the distinction between Tor H1(M3, Z) and its dual is only important in section 2.2, we will use the same set of labels {a, b, . . .} for elements in both groups. 4 A related conjecture was made in [28]. However it did not include the S-transform, which is crucial for restoring integrality and categorification. 5 The constant positive integer c depends only on M3 and in a certain sense measures its "complexity". In many simple examples c = 0, and the reader is welcome to ignore 2 −c factor which arises from some technical subtleties. Its physical origin will be explained later in the paper. 6 Later in the text we will sometimes use slightly redefined quantities, Za(q) → q ∆ Za(q), where ∆ is a common, a independent rational number.7 Recall that Tor H1(M3, Z), as a finitely generated abelian group, can be decomposed into Tor H1(M3, Z) = i Zp i . We ask for a fairly weak condition that Z2 doesn't appear in this decomposition. In other words, M3 is a Z2-homology sphere. Equivalently, there is a unique Spin structure on M3, so that there is no ambiguity in specifying Nahm-pole boundary condition for N = 4 SU (2) SYM on M3 × R+ [18]. The general case, in principle, could also be worked out. We leave it as an exercise to an interested reader.8 The presence of 1/2 factors that produce 2 −c overall factor in (2.7) can be interpreted as presence of factors ∼ = C[x] with deg q x = 0 in Ha[M3]. The q-graded Euler charecteristic of C[x] is naively divergent: 1+1+1+. . ., but its zeta-regularization gives 1/2.

We study complex Chern-Simons theory on a Seifert manifold M 3 by embedding it into string theory. We show that complex Chern-Simons theory on M 3 is equivalent to a topologically twisted supersymmetric theory and its partition function can be naturally regularized by turning on a mass parameter. We find that the dimensional reduction of this theory to 2d gives the low energy dynamics of vortices in four-dimensional gauge theory, the fact apparently overlooked in the vortex literature. We also generalize the relations between 1) the Verlinde algebra, 2) quantum cohomology of the Grassmannian, 3) Chern-Simons theory on Σ × S 1 and 4) index of a spin c Dirac operator on the moduli space of flat connections to a new set of relations between 1) the "equivariant Verlinde algebra" for a complex group, 2) the equivariant quantum K-theory of the vortex moduli space, 3) complex Chern-Simons theory on Σ × S 1 and 4) the equivariant index of a spin c Dirac operator on the moduli space of Higgs bundles.

We study reductions of 6d theories on a d-dimensional manifold Md, focusing on the interplay between symmetries, anomalies, and dynamics of the resulting (6 −d)-dimensional theory T[Md]. We refine and generalize the notion of “polarization” to polarization on Md, which serves to fix the spectrum of local and extended operators in T[Md]. Another important feature of theories T[Md] is that they often possess higher-group symmetries, such as 2-group and 3-group symmetries. We study the origin of such symmetries as well as physical implications including symmetry breaking and symmetry enhancement in the renormalization group flow. To better probe the IR physics, we also investigate the ’t Hooft anomaly of 5d Chern-Simons matter theories. The present paper focuses on developing the general framework as well as the special case of d = 0 and 1, while an upcoming paper will discuss the case of d = 2, 3 and 4.