# Fivebranes and 3-manifold homology

@article{Gukov2016FivebranesA3,
title={Fivebranes and 3-manifold homology},
author={Sergei Gukov and Pavel Putrov and Cumrun Vafa},
journal={Journal of High Energy Physics},
year={2016},
volume={2017},
pages={1-82}
}
• Published 17 February 2016
• Mathematics
• Journal of High Energy Physics
A bstractMotivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that fivebrane compactifications provide a universal description of various old and new homological invariants of 3-manifolds. In terms of 3d/3d correspondence, such invariants are given by the Q-cohomology of the Hilbert space of partially topologically twisted 3d N=2$$\mathcal{N}=2$$ theory T[M3] on a Riemann surface with defects. We demonstrate this by…
104 Citations

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## References

SHOWING 1-10 OF 120 REFERENCES
Fivebranes and Knots
We develop an approach to Khovanov homology of knots via gauge theory (previous physics-based approches involved other descriptions of the relevant spaces of BPS states). The starting point is a
Equivariant Verlinde Formula from Fivebranes and Vortices
• Mathematics
• 2015
We study complex Chern–Simons theory on a Seifert manifold M3 by embedding it into string theory. We show that complex Chern–Simons theory on M3 is equivalent to a topologically twisted
Hyper-Kähler geometry and invariants of three-manifolds
• Mathematics
• 1996
Abstract We study a 3-dimensional topological sigma-model, whose target space is a hyper-Kähler manifold X. A Feynman diagram calculation of its partition function demonstrates that it is a finite
3d-3d correspondence revisited
• Mathematics
• 2016
A bstractIn fivebrane compactifications on 3-manifolds, we point out the importance of all flat connections in the proper definition of the effective 3d N=2$$\mathcal{N}=2$$ theory. The Lagrangians
3d analogs of Argyres-Douglas theories and knot homologies
• Mathematics
• 2013
A bstractWe study singularities of algebraic curves associated with 3d $\mathcal{N}=2$ theories that have at least one global flavor symmetry. Of particular interest is a class of theories TK
Knot Homology from Refined Chern-Simons Theory
• Mathematics
• 2011
We formulate a refinement of SU(N) Chern-Simons theory on a three-manifold via the refined topological string and the (2,0) theory on N M5 branes. The refined Chern-Simons theory is defined on any
Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Mathematics
• 2008
We define new invariants of 3d Calabi-Yau categories endowed with a stability structure. Intuitively, they count the number of semistable objects with fixed class in the K-theory of the category
Knot Homology and Refined Chern–Simons Index
• Mathematics
• 2015
We formulate a refinement of SU(N) Chern–Simons theory on a three-manifold M via an index in the (2, 0) theory on N M5 branes. The refined Chern–Simons theory is defined on any M with a semi-free
Fivebranes and 4-manifolds
• Mathematics
• 2013
We describe rules for building 2d theories labeled by 4-manifolds. Using the proposed dictionary between building blocks of 4-manifolds and 2d N=(0,2) theories, we obtain a number of results, which