# Theory X on lens spaces

A class of novel 3d topological field theories

This project is joint with Monica Kang.

One important source of physical dualities in recent years is the six dimensional \(\mathcal{N}=(2,0)\) superconformal field theory affectionately known as “Theory \(\mathfrak{X}\)”. Theory \(\mathfrak{X}\) in particular provides a 6d origin for the geometric Langlands conjecture (Witten, 2009), the AGT correspondence (Alday et al., 2010), and (importantly for this project) the 3d-3d correspondence (Dimofte et al., 2013; Dimofte et al., 2014). Roughly, for two 3-manifolds \(M_1\) and \(M_2\) the 3d-3d correspondence identifies the partition function of the compactified theory \(\mathfrak{X}[M_1]\) on \(M_2\) with the partition function of \(\mathfrak{X}[M_2]\) on \(M_1\) by relating both of them to the partition function of \(\mathfrak{X}\) on \(M_1\times M_2\).

In this project we are investigating the situation where one of the 3-manifolds is a lens space \(L(p,q)\). More precisely, we are interested in compactifying a 4d-topological/2d-holomorphic twist \(\mathfrak{X}^{twist}\) of Theory \(\mathfrak{X}\) on a lens space equipped with a transversely holomorphic foliation (Closset et al., 2014). This structure is precisely what is needed for the compactification to “eat up” the two holomorphic directions and one of the topological directions, in principle leaving us with a three dimensional topological quantum field theory (TQFT).

Previous work in the literature has shown that for \(L(p,1)\) type lens spaces the remaining 3d theory is a complex Chern-Simons theory with level determined by \(p\) and the transversely holomorphic foliation on \(L(p,1)\) (Córdova & Jafferis, 2017; Mikhaylov, 2018). Compactifying on \(L(p,q)\) with \(q\neq1\) coprime to \(p\) will yield a family of previously unknown 3d TQFTs - a naive but ultimately incorrect attempt at compactification suggests that these theories might be deserving of the moniker *complex Chern-Simons at fractional level* - and consequently a family of new topological invariants for 3-manifolds \(M\) given by the resulting partition functions \(\mathfrak{X}^{twist}[L(p,q)](M)\). Via the 3d-3d correspondence, these 3-manifold invariants can alternately be interpreted physically as the lens space partition functions of a different theory: namely the 1d-topological/2d-holomorphic theory \(\mathfrak{X}^{twist}[M]\).

Interestingly, the compactification on general \(L(p,q)\) turns out to be significantly harder than the \(L(p,1)\) case already treated in the literature. The (comparative!) simplicity of the \(L(p,1)\) case ultimately boils down to the fact that in this case the resulting theory has a weakly coupled description:

- In the approach of Córdova and Jafferis (Cordova & Jafferis, 2013) the description of \(L(p,1)\) as a circle bundle over \(S^2\) is used to reduce the problem to 5d \(\mathcal{N}=2\) supersymmetric Yang-Mills (SYM) in a particular supergravity background.
- Alternately, in the approach of Mikhaylov (Mikhaylov, 2018) \(L(p,1)\) is represented as a degenerate torus fibration over a closed interval \(I\), and so via torus reduction the problem is factored through a Janus configuration (missing reference) of 4d \(\mathcal{N}=4\) SYM on the product manifold \(I\times M\). It is shown that the left hand boundary condition is of type \((\pm1,0)\) and the right hand boundary condition is of type \((\pm 1,\pm p)\), which correspond to inserting terms proportional to the Chern-Simons action functional supported on the boundaries into the path integral.

Neither of these approaches generalises to the other \(L(p,q)\) lens spaces, and we in fact expect that the 3d topological theory \(\mathfrak{X}^{twist}[L(p,q)]\) will *not* have a weakly coupled description. This novelty makes the theory interesting, however it also makes the problem of describing it (e.g. in a functorial manner) extremely difficult. As an intermediate step, we are currently investigating the circle compactification of the this theory, i.e. the 2d TQFT \(\mathfrak{X}^{twist}[L(p,q)\times S^1]\): as this theory factors though a twist of 5d \(\mathcal{N}=2\) SYM it has a Lagrangian description, and we plan to study the properties of this 2d theory in order to determine potential properties and candidates for the full 3d TQFT.

#### References:

- Geometric Langlands From Six DimensionsPreprint, arXiv: 0905.2720, 2009
- Liouville correlation functions from four-dimensional gauge theories
*Letters in Mathematical Physics*, 2010 - 3-manifolds and 3d indices
*Advances in Theoretical and Mathematical Physics*, 2013 - Gauge theories labelled by three-manifolds
*Communications in Mathematical Physics*, 2014 - From rigid supersymmetry to twisted holomorphic theories, 2014
- Complex Chern-Simons from M5-branes on the squashed three-sphere
*Journal of High Energy Physics*, 2017 - Teichmüller TQFT vs. Chern-Simons theory, 2018
- Five-Dimensional Maximally Supersymmetric Yang-Mills in Supergravity BackgroundsPreprint, arXiv: 1305.2886, 2013