Gauge Theories Labelled by Three-Manifolds

By Tudor Dimofte, D. Gaiotto, Sergei Gukov. 2011

489 citations recorded by the source corpus.

We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional $${\mathcal{N} = 2}$$N=2 gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independence of the SL(2) Chern-Simons partition function on the choice of triangulation translates to a statement that $${S^{3}_{b}}$$Sb3 partition functions of two mirror 3d $${\mathcal{N} = 2}$$N=2 gauge theories are equal. Three-dimensional $${\mathcal{N} = 2}$$N=2 field theories associated to 3-manifolds can be thought of as theories that describe boundary conditions and duality walls in four-dimensional $${\mathcal{N} = 2}$$N=2 SCFTs, thus making the whole construction functorial with respect to cobordisms and gluing.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/7de9b2e1822fa1485e57c343ba58417b502a09d4

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Citations in the corpus

Partition Functions of $${\mathcal{N}=(2,2)}$$N=(2,2) Gauge Theories on S2 and Vortices (2012, background): “A similar role is being played by the S3 and S2 × S1 partition functions [57,58], hence one could expect our result on S2 to fit into the pattern.”
The gravity dual of supersymmetric gauge theories on a squashed three-sphere (2011, background): “For example, the double sine function and the U(1)(2)-squashed sphere (which is a three-dimensional ellipsoid) are important ingredients in the AGT correspondence [15] and its 3d/3d version [16, 17].”
The (2, 0) superconformal bootstrap (2015, unknown): “tensor multiplet is D[0;0;0;2;0], which we denote simply by D[2;0]. A[c 1,c 2,c 3;d 1,d 2] Q 14 = 0 [ 0, 0,-1;0,1] A[c 1,c 2,0;d 1,d 2] Q 13Q 14 = 0 [ 0,-1, 0;0,2] A[c 1,0,0;d 1,d 2] Q 12Q 13Q 14 = 0 [-1, 0, 0;0,3] A[0,0,0;d 1,d 2] Q 11Q 12Q 13Q 14 = 0 [ 0, 0, 0;0,4] B[c 1,c 2,0;d 1,d 2] Q 13 = 0 [ ”
BPS spectra and 3-manifold invariants (2017, result): “However, as it was argued in [75] it coincides with the result given by triangulation decomposition, as in [76,77], and therefore, if interpreted as the partition function of the complex Chern-Simons theory, it misses contributions of the reducible flat connections [67].”
Comments on twisted indices in 3d supersymmetric gauge theories (2016, background): “It is dual to a single free chiral multiplet T+ of U(1)T charge 1, corresponding to the lowest gauge-invariant monopole operator for the ‘half’ Coulomb branch of the gauge theory [28, 72].”
Supersymmetric partition functions on Riemann surfaces (2016, background): “2 U(1)1/2 with one chiral multiplet We consider the following duality [33, 34].”
Gluing equations for PGL(n, ℂ)–representations of 3–manifolds (2012, background): “(b) The loop invariants of [7] (c) The 3D index of [6, 5, 10, 11] The input of [4] is an ideal triangulation T of a 3-manifold with torus boundary components, and the output is a polynomial in q-commuting variables (one per meridian and longitude of each torus boundary component).”
Fivebranes and 3-manifold homology (2016, background): “rcumvent the problem of compactifying 6d (0;2) theory on M 3 by guessing the basic building blocks of T[M 3] associated to ideal tetrahedra in a way consistent with gluing and Pachner moves, see e.g. [48,71]. Usually, this approach leads to a 3d N= 2 theory that does not account for all reducible at”
Fivebranes and 4-manifolds (2013, methodology): “ses; typically, they are (deformations of) superconformal theories in the former case and massive 3d N = 2 theories in the latter. Another, more serious issue is that 3d theories T[M3] constructed in [6] do not account for allﬂat connections on M3, which will be crucial in our applications below. Th”
The Coulomb Branch of 3d N = 4 Theories (2015, background): “related to the 3d-3d correspondence [26] and to integrable systems [27, 28].”
A topologically twisted index for three-dimensional supersymmetric theories (2015, background): “It is well known that the theory above is dual to the theory of a free chiral multiplet, which can be identified with the monopole T [37, 38].”
4d Index to 3d Index and 2d TQFT (2011, methodology): “Similarly, in the limit S(1) T → 0 (γ → 0 in the notation of section 4), we expect to recover a Chern-Simons theory with a non-compact gauge group [42, 43, 44, 45, 46, 47].”
Bootstrapping the superconformal index with surface defects (2012, methodology): “al indices" if the theory is conformal). These observables have been indispensable tools in the exploration of the new classes of 4dand 3dtheories, dubbed respectively class S[15, 16] and class R[17]2, which arise by twisted compactications of the 6d(2,0) theory on Riemann surfaces and on hyper”
3-Manifolds and 3d Indices (2011, background): “equence of 2{3 moves.2 Thus for every such manifold Mwe get a specic IR xed point T M in the class R! The invariants L[T M], Z b[T M], I[T M] must coincide with geometric invariants of M. Indeed, in [8,19] it was proven that L[T M] is the space of hyperbolic metrics, or equivalently at SL(2;C) conne”
Comments on Chern-Simons contact terms in three dimensions (2012, background): “Many other dualities have been shown to require relative Chern-Simons counterterms [33-43].”
1-efficient triangulations and the index of a cusped hyperbolic 3-manifold (2013, methodology): “Finally in the appendix, we give a detailed and self-contained proof of the invariance of the 3D index of 1–efficient triangulations under 2–3 moves, following [16] and Dimofte, Gaiotto and Gukov [10].”