Fivebranes and 4-Manifolds

Abstract

We describe rules for building 2d theories labeled by 4-manifolds. Using the proposed dictionary between building blocks of 4-manifolds and 2d $\mathcal{N} = (0,2)$ theories, we obtain a number of results, which include new 3d $\mathcal{N} = 2$ theories T[M ₃] associated with rational homology spheres and new results for Vafa–Witten partition functions on 4-manifolds. In particular, we point out that the gluing measure for the latter is precisely the superconformal index of 2d (0, 2) vector multiplet and relate the basic building blocks with coset branching functions. We also offer a new look at the fusion of defect lines/walls, and a physical interpretation of the 4d and 3d Kirby calculus as dualities of 2d $\mathcal{N} = (0,2)$ theories and 3d $\mathcal{N} = 2$ theories, respectively.

Appendices

Appendix 1: M5-Branes on Calibrated Submanifolds and Topological Twists

We study the twisted compactification of 6d (2, 0) theory on a four-manifold M ₄. In each of the cases listed in Table 5, such compactification produces a superconformal theory T[M ₄] in the two non-compact dimensions. Via the computation of the T ² partition function explained in the main text, the cases (a)–(c) correspond to previously studied topological twists of $\mathcal{N} = 4$ super-Yang-Mills which, in turn, are summarized in Table 6.

Table 5 Supersymmetric M5 brane compactifications on a negatively curved 4-manifold M ₄

Table 6 Topological twists of $\mathcal{N} = 4$ super-Yang-Mills

Specifically, in the first case (a) the $\mathcal{N} = 4$ SYM is thought of as an $\mathcal{N} = 2$ gauge theory with an extra adjoint multiplet and the Donaldson–Witten twist [Wit88]. Its path integral localizes on solutions to the non-abelian monopole equations. The untwisted rotation group of the DW theory is then twisted by the remaining SU(2) symmetry to obtain the case (b). This twist (a.k.a. GL twist) was first considered by Marcus [Mar95] and related to the geometric Langlands program in [KW07]. The last case (c) is of most interest to us as it corresponds to (0, 2) SCFT in 2d. On a 4-manifold M ₄, this twist is the standard Vafa–Witten twist [VW94].

Appendix 2: Orthogonality of Affine Characters

The Weyl–Kac formula for affine characters of $\widehat{\mathfrak{s}\mathfrak{u}}(2)_{k}$ is

$$\displaystyle{ \chi _{\lambda }^{\widehat{\mathfrak{s}\mathfrak{u}}(2)_{k} }(q,a) = \frac{\Theta _{\lambda +1}^{(k+2)}(a;q) - \Theta _{-\lambda -1}^{(k+2)}(a;q)} {\Theta _{1}^{(2)}(a;q) - \Theta _{-1}^{(2)}(a;q)} }$$

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where

$$\displaystyle{ \Theta _{\lambda }^{(k)}(a;q):= e^{-2\pi ikt}\sum _{ n\in \mathbb{Z}+\lambda /2k}q^{kn^{2} }a^{kn} = e^{-2\pi ikt}q^{ \frac{\lambda ^{2}} {4k} }\sum _{n}q^{kn^{2}+\lambda n }a^{kn+\lambda } }$$

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Using the Weyl–Kac denominator formula the character can be rewritten as

$$\displaystyle{ \chi _{\lambda }^{\widehat{\mathfrak{s}\mathfrak{u}}(2)_{k} }(q,a) = \frac{e^{-2\pi i(k+2)t}q^{ \frac{(\lambda +1)^{2}} {4(k+2)} }\sum _{n}q^{(k+2)n^{2} }a^{(k+2)n}(q^{(\lambda +1)n}a^{(\lambda +1)} - q^{-(\lambda +1)n}a^{-(\lambda +1)})} {a^{-1}(q;q)\theta (a^{2};q)}. }$$

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Consider the integral

$$\displaystyle\begin{array}{rcl} & & \oint \frac{da} {2\pi ia}(q;q)_{\infty }^{2}\theta (a^{2};q)\theta (a^{-2};q)\chi _{\lambda }^{\widehat{\mathfrak{s}\mathfrak{u}}(2)_{k} }(q,a)\chi _{\lambda '}^{\widehat{\mathfrak{s}\mathfrak{u}}(2)_{k} }(q,a) \\ & & \quad = e^{-2\pi i(k+2)t}q^{ \frac{(\lambda +1)^{2}} {4(k+2)} + \frac{(\lambda '+1)^{2}} {4(k+2)} } \\ & & \qquad \times \sum _{n,m}\left [q^{(k+2)(n^{2}+m^{2})+(\lambda +1)n+(\lambda '+1)m }\oint \frac{da} {2\pi ia}a^{(k+2)(n+m)+(\lambda +1)+(\lambda '+1)}\right. \\ & & \qquad - q^{(k+2)(n^{2}+m^{2})+(\lambda +1)n-(\lambda '+1)m }\oint \frac{da} {2\pi ia}a^{(k+2)(n-m)+(\lambda +1)-(\lambda '+1)} \\ & & \qquad - q^{(k+2)(n^{2}+m^{2})-(\lambda +1)n+(\lambda '+1)m }\oint \frac{da} {2\pi ia}a^{(k+2)(-n+m)-(\lambda +1)+(\lambda '+1)} \\ & & \qquad + q^{(k+2)(n^{2}+m^{2})-(\lambda +1)n-(\lambda '+1)m }\oint \frac{da} {2\pi ia}a^{(k+2)(-n-m)-(\lambda +1)-(\lambda '+1)} \propto \delta _{\lambda,\lambda '}{}\end{array}$$

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This shows that $\widehat{\mathfrak{s}\mathfrak{u}}(2)_{k}$ characters are orthogonal with respect to the measure

$$\displaystyle{ (q;q)_{\infty }^{2}\theta (a^{2};q)\theta (a^{-2};q) }$$

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but this measure is exactly the index of SU(2) (0, 2) vector multiplet. The orthogonality of $\hat{\mathfrak{u}}(1)_{k}$ characters can be verified in a similar way. We conjecture that $\widehat{\mathfrak{s}\mathfrak{u}}(N)_{k}$ ($\hat{\mathfrak{u}}(N)_{k}$) characters are orthogonal with respect to SU(N) (U(N)) vector multiplet measure as well.