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3d Superconformal Theories from Three-Manifolds

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New Dualities of Supersymmetric Gauge Theories

Part of the book series: Mathematical Physics Studies ((MPST))

Abstract

We review here some aspects of the 3d \(\mathcal {N}=2\) SCFT’s that arise from the compactification of M5 branes on 3-manifolds.

A citation of the form [V:x] refers to article number x in this volume.

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Notes

  1. 1.

    Here \(S^3_b\) denotes a “squashed” 3-sphere with ellipsoidal metric. It is also useful to note that complex \(SL(K,{\mathbb C})\) Chern-Simons theory has two coupling constants or levels \((k,\sigma )\), one quantized and the other continuous, cf. Sects. 2.1 and 2.2. It is only the quantized level that is being fixed in (2). The general pattern following from work of [11] is that \(T_K[C]\) on a squashed Lens space \(L(k,1)_b\) is equivalent to \(SL(K,{\mathbb C})\) Chern-Simons at level k.

  2. 2.

    To be precise: after compactification on \(S^1\), the subsectors only contain SUSY vacua corresponding to irreducible \(SL(K,{\mathbb C})\) flat connections on M, with given boundary conditions, rather than all flat connections as prescribed by (2). The relation between these subsectors and the “full” \(T_K[M]\) began to be analyzed in [27].

  3. 3.

    In a few examples, nonabelian duals are already known: the basic tetrahedron theory has an SU(2) dual discussed in [31]; and the theory for the basic S-duality wall in 4d \(\mathcal {N}=2\) SU(2) theory with \(N_f=4\) (associated to the manifold in Fig. 4b) has an SU(2) dual found in [24]. Some basic ideas about smooth gluing were also discussed in [5].

  4. 4.

    The counting goes as follows. First, the cotangent bundle \(T^*M\) is a noncompact Calabi-Yau manifold. M-theory on a generic Calabi-Yau background preserves eight supercharges (cf. [34, 35]). An M5 brane wrapping a special Lagrangian cycle in the Calabi-Yau (such as the zero-section M in \(T^*M\)) is half-BPS, and preserves four of the eight supercharges.

  5. 5.

    See, e.g., the supergravity solutions of [36] involving special Lagrangian 3-cycles. For the analogous compactifications on 2d surfaces, the flow of the metric to constant curvature was analyzed in [37].

  6. 6.

    Notable exceptions include spheres, tori, lens spaces, and more general Seifert-fibered manifolds, which have the structure of an \(S^1\) fibration over a surface. The 3d theories resulting from compactification on such manifolds are qualitatively different from the hyperbolic case. For example, compactification on a 3-torus yields \(\mathcal {N}=8\) SYM in 3d, while compactification on the 3-sphere yields a gapped theory that breaks SUSY.

  7. 7.

    Also described in Sect. 3.1–3.2 of Families of \(\mathcal {N}=2\) field theories by D. Gaiotto.

  8. 8.

    This “shrinking” procedure turns parts of M that look like \(S^1\times {\mathbb R}\times {\mathbb R}_+\) (i.e. the neighborhoods of tubes) into defects. An identical setup was used to create defects in [32].

  9. 9.

    It is also possible to arrive at a theory where the center of SU(K), or subgroups of the center, are not gauged. Then instead of getting a relation to \(SL(K,{\mathbb C})\) connections, we find a relation to \(PSL(K,{\mathbb C})\) connections, or similar. The details are subtle (see [8]), but the correct relation can ultimately be derived by examining the charges of fundamental line operators in \(T_K[M]\).

  10. 10.

    The structure of Hitchin equations in two dimensions and their relation to 4d \(\mathcal {N}=2\) theory on a circle is reviewed in [V:3].

  11. 11.

    One can attempt to use algebraic Mostow rigidity [39] to analyze the problem. This requires knowing that the representation \(\rho :\pi _1(M)\rightarrow SL(K,{\mathbb C})\) defined by the holonomies of a flat connection \(\mathcal {A}\) is a lattice. That is, \(\rho (\pi _1(M))\subset SL(K,{\mathbb C})\) is a discrete subgroup, with no accumulation points, such that \(SL(K,{\mathbb C})/\rho (\pi _1(M))\) has finite volume. This is true if M is hyperbolic and \(\mathcal {A}\) is the flat connection related to the hyperbolic metric; but is unknown in general.

  12. 12.

    See Sect. 2 of Hitchin systems in \(\mathcal {N}=2\) field theory by A. Neitzke.

  13. 13.

    Explicitly, if we re-introduce the radius \(\beta \) of the compactification circle, these dimensionless coordinates arise as \(x = \exp \big (\beta m_{3d}+i \oint _{S^1}A\big )\), where A is the background gauge field for a 3d flavor symmetry, and \(m_{3d}\) is its real mass. A factor of \(\beta \) also enters (8) to keep \(\widetilde{W}\) dimensionless.

  14. 14.

    This Lagrangian and its quantization also plays a role in the study of surface operators in 4d \(\mathcal {N}=2\) theories, and their lifts to 3d defects in 5d theories—see Sect. 2.4 of [V:8].

  15. 15.

    See also A review on SUSY gauge theories on \(S^3\) by K. Hosomichi.

  16. 16.

    Quantization of \(SL(2,{\mathbb R})\) flat connections on a surface is reviewed in this volume in Supersymmetric gauge theories, quantization of \(\mathcal {M}_\mathrm{flat}\) , and conformal field theory by J. Teschner.

  17. 17.

    Here we mean low energy from the point of view of M-theory dynamics, which is still UV for 3d field theories on \({\mathbb R}^3\). See related comments below about being able to choose arbitrary metric for M.

  18. 18.

    More generally, we have \(Z_{3d}=\mathrm{Re}[\zeta ^{-1}Z_{4d}]\), where the phase \(\zeta \) characterizes the \(4d\rightarrow 3d\) supersymmetry breaking. The 4d R-symmetry group \(SU(2)_R\times U(1)_r\) is broken to \(U(1)_R\) (a Cartan of \(SU(2)_R\)), and this \(\zeta \) is rotated by the broken \(U(1)_r\). This same phase also happens to select the complex structure that one should use for the hyperkähler moduli spaces of flat connections [29, 48], as discussed in Sect. 2.

  19. 19.

    Similar half-BPS configurations in 3d \(\mathcal {N}=2\) theories were discussed in [79].

  20. 20.

    The correction requires solving the potential problem \(\nabla ^2\sigma = \partial _3f\). Then \(\lambda =\mathrm{Re}\,\lambda ^{SW}-f\,dx^3+d\sigma \).

  21. 21.

    It may seem like \(\Pi =\Pi '\) in this example. This is not the case, due to the relative orientation on the two halves. The setup corresponding to \(\Pi =\Pi '\) involves X getting Dirichlet b.c. on one side and Y getting Dirichlet b.c. on the other, with the remaining (Neumann) halves coupled directly by a superpotential \(W=Y^-X^+\) at \(x^3=0\). This flows immediately to \(T_K[C]\) on all of \({\mathbb R}^4\).

  22. 22.

    Such manifolds were called “admissible” in [7].

  23. 23.

    These coordinates generalize Thurston’s classic shear coordinates in Teichmüller theory, later studied by Penner, Fock, and others.

  24. 24.

    Recall that lines in \({\mathbb C}^2\) are just points in \({\mathbb {CP}}^1\), so an SL(2)-invariant cross-ratio can be formed.

  25. 25.

    Strictly speaking, this is true only for a sufficiently generic or refined triangulation of M. In particular, one must make sure that the \(({\mathbb C}^*)^{N-d}\) action in the quotient is transverse to the product Lagrangian \(\prod _i \mathcal {L}_{\Delta _i}\).

  26. 26.

    Note that the half-integer background Chern-Simons term is corrected by the standard parity anomaly of a 3d \(\mathcal {N}=2\) theory (cf. [74]) to be an integer in the IR, given any nonzero real mass for \(\phi _z\).

  27. 27.

    The signs, and indeed the full lift to logarithms of the edge-coordinates, becomes relevant when keeping track of a choice of \(U(1)_R\) symmetry for a theory. Then symplectic \(Sp(2N,{\mathbb Z})\) actions are promoted to affine-symplectic \(ISp(2N,{\mathbb Z})\) actions.

  28. 28.

    In Sect. 1, we also talked about isolating 3d theories \(T_K[M,\mathbf p]\) based on a pants decomposition \(\mathbf p\) of the topological boundary of M. This was meant to correspond to decoupling a nonabelian 4d gauge theory in some duality frame. Such a choice is already built in to the definition of a framed manifold M: a pants decomposition for a boundary component \(\mathcal {C}\) corresponds to a splitting of that boundary into a network of small annuli connected by big 3-punctured spheres when selecting a framing.

  29. 29.

    Just like in the gluing of classical Lagrangian submanifolds, some extra regularity conditions need to be imposed on a 3d triangulation to truly guarantee the existence of the gluing operators \(\mathcal {O}_I\). See Sect. 4.1 of [1] or the Appendix A of [7].

  30. 30.

    The precise \(Sp(4,{\mathbb Z})\) action first removes the background Chern-Simons coupling for the anti-diagonal subgroup of \(U(1)_r\times U(1)_s\), and then gauges it. It is a nice exercise to demonstrate this.

  31. 31.

    Two technical clarifications here: first, the choice of eigenvalue \(\lambda \) versus \(\lambda ^{-1}\) depends on the choice of framing for the flat connection at the small annulus; second, to get a well defined sign for \(\lambda \) one actually needs to lift to SL(2) rather than PSL(2) holonomies around the small annulus.

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Acknowledgments

It is a pleasure to thank Christopher Beem, Clay Córdova, Davide Gaiotto, and Sergei Gukov for discussions and advice during the writing of this review, and especially Andrew Neitzke and Jeorg Teschner for careful readings and comments.

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    Tudor Dimofte

  2. Trinity College, CB2 1TQ, Cambridge, UK

    Tudor Dimofte

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Dimofte, T. (2016). 3d Superconformal Theories from Three-Manifolds. In: Teschner, J. (eds) New Dualities of Supersymmetric Gauge Theories. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-18769-3_11

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