Fivebranes and 4-Manifolds

Note, this cannot be deduced from the Rokhlin’s theorem as in the case of the E ₈ manifold.

Sometimes, to avoid clutter, we suppress the choice of the gauge group, G, which in most of our applications will be either G = U(N) or G = SU(N) for some N ≥ 1. It would be interesting to see if generalization to G of Cartan type D or E leads to new phenomena. We will not aim to do this analysis here.

Recall, that a free Fermi multiplet contributes to the central charge (c _L , c _R ) = (1, 0).

Another nice property of such 4-manifolds is that they admit an achiral Lefschetz fibration over the disk [Har79].

But not all! See Fig. 3 for an instructive (counter)example.

Depending on the context, sometimes M ₃ will refer to a single component of the boundary.

While this problem has been successfully solved for a large class of 3-manifolds [DGG1, CCV, DGG2], unfortunately it will not be enough for our purposes here and we need to resort to matching M ₃ with T[M ₃] based on identification of vacua, as was originally proposed in [DGH11]. One reason is that the methods of loc. cit. work best for 3-manifolds with sufficiently large boundary and/or fundamental group, whereas in our present context M ₃ is itself a boundary and, in many cases, is a rational homology sphere. As we shall see below, 3d \(\mathcal{N} = 2\) theories T[M ₃] seem to be qualitatively different in these two cases; typically, they are (deformations of) superconformal theories in the former case and massive 3d \(\mathcal{N} = 2\) theories in the latter. Another, more serious issue is that 3d theories T[M ₃] constructed in [DGG1] do not account for all flat connections on M ₃, which will be crucial in our applications below. This second issue can be avoided by considering larger 3d theories T ^(ref)[M ₃] that have to do with refinement/categorification and mix all branches of flat connections [FGSA, FGP13]. Pursuing this approach should lead to new relations with rich algebraic structure and functoriality of knot homologies.

The converse is not true since some line defects in 2d come from line operators in 3d.

Explaining how to do this is precisely the goal of the present section.

Note, in [VW94] the symmetry group U(1) _U is enhanced to the global symmetry group SU(2) _U due to larger R-symmetry of the starting point.

When M ₄ is non-compact χ(M ₄) should be replaced by the regularized Euler characteristic, and when G = U(N) one needs to remove by hand the zero-mode to ensure that the partition function does not vanish identically.

Here and in what follows the instanton number is not necessarily an integer.

Let us note that H ₂(M ₄ ⁺) ≠ H ₂(B) ⊕ H ₂(M ₄ ⁻). However, the lattice H ₂(M ₄ ⁺) can be obtained from the lattice H ₂(B) ⊕ H ₂(M ₄ ⁻) by the so-called gluing procedure that will be described in detail shortly.

Such lift exists because the manifold is Spin ^c .