3-Manifolds and 3d Indices

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

382 citations recorded by the source corpus.

We identify a large class R of three-dimensional N = 2 superconformal field theories. This class includes the effective theories T_M of M5-branes wrapped on 3-manifolds M, discussed in previous work by the authors, and more generally comprises theories that admit a UV description as abelian Chern–Simons-matter theories with (possibly non-perturbative) superpotential. Mathematically, class R might be viewed as an extreme quantum generalization of the Bloch group; in particular, the equivalence relation among theories in class R is a quantum-field-theoretic “2 to 3 move.” We proceed to study the supersymmetric index of theories in class R, uncovering its physical and mathematical properties, including relations to algebras of line operators and to 4d indices. For 3-manifold theories T_M, the index is a new topological invariant, which turns out to be equivalent to non-holomorphic SL(2,ℂ) Chern–Simons theory on M with a previously unexplored “integration cycle.”

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

View Paper: https://www.semanticscholar.org/paper/b27b4b0b6f1a42dbe5e290d32d4ed740065ad730

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Sergei Gukov
2. Tudor Dimofte
3. Pavel Putrov
4. C. Vafa
5. Marcos Mariño
6. F. Benini
7. Sara Pasquetti
8. E. Witten

In Mathematical Physics

1. C. Vafa
2. Sergei Gukov
3. F. Benini
4. Pavel Putrov
5. Tudor Dimofte
6. Marcos Mariño
7. Sara Pasquetti
8. E. Witten

In Nuclear and High Energy Physics

1. F. Benini
2. Marcos Mariño
3. Cyril Closset
4. Sergei Gukov
5. N. Seiberg
6. Pavel Putrov
7. D. Gaiotto
8. T. Dumitrescu

Citations in the corpus

• The geometry of supersymmetric partition functions (2013, background): “As in four dimensions, a suitable trace over states on S(2) in Hamiltonian quantization defines a supersymmetric index I(y, u), where y and u are (generally complex) fugacities that couple to the Hamiltonian H, which generates translations along R, and an Abelian flavor charge Qf that commutes with ”
• Localization techniques in quantum field theories (2016, methodology): “Our present discussion can be summarized in the following table: dim multiplet #super special function references derivation S2 N = 2 vector 4 Υ1(x| ) [36,37] Chapter 3 S3 N = 2 vector 4 S2(x| ) [35] Chapter 6 S4 N = 2 vector 8 Υ2(x| ) [16] Chapter 10 S5 N = 1 vector 8 S3(x| ) [38–40] Chapter 14 S6 ”
• 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].”
• 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.”
• BPS spectra and 3-manifold invariants (2017, result): “he rst two of the Kirby moves in Figure 6 that correspond to relations in the SL(2;Z) group. 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,”