nLab Rozansky-Witten theory

Idea

What is called Rozansky-Witten theory (Rozansky-Witten 96) is a topological twist of D=3 N=4 super Yang-Mills theory, or else the assignment of topological invariant quantum observables which such a perturbative quantum field theory assigns to a given 3-manifold $\Sigma^3$, or more generally to a Wilson loop knot inside a 3-manifold – the Rozansky-Witten invariants.

The key result of Rozansky-Witten 96 is that, after gauge fixing and some subtle field identifications, the Feynman rules of RW-twisted D=3 N=4 super Yang-Mills theory are those of perturbative Chern-Simons theory, in that the only relevant propagator is the Chern-Simons propagator and the only relevant Feynman diagrams are trivalent, except that the Lie algebra weights assigned by Chern-Simons theory to a Feynman-Jacobi diagram, are replaced by other weight systems, now called Rozansky-Witten weight systems.

Precisely: Recalling that perturbative Chern-Simons Wilson loop observables are given by evaluating Lie algebra weight systems on the Jacobi diagram-valued universal Vassiliev invariant

we have that Wilson loop Rozansky-Witten invariants (Rozansky-Witten 96 (2.26)) are given by evaluating Rozansky-Witten invariants on the Jacobi diagram-valued universal Vassiliev invariant

graphics from Sati-Schreiber 19c

These Rozansky-Witten weight systems depend only on the hyperkähler manifold $\mathcal{M}^{4n}$ which is the (classical) Coulomb branch of the RW-twisted D=3 N=4 super Yang-Mills theory, and in fact they are independent of the Riemannian geometry of $\mathcal{M}^{4n}$ and depend only on the underlying holomorphic symplectic manifold (Kapranov 99). Generally, they are defined for $\mathcal{M}^{4n}$ any hyperkähler manifold which is either asymptotically flat (ALE spaces) or compact topological space (compact hyperkähler manifolds).

Via the equivalent reformulation by Kapranov 99 one finds (Roberts-Willerton 10) that the Rozansky-Witten invariants are structurally Lie algebra weight systems themselves, but internal to the derived category of coherent sheaves of $\mathcal{M}^{4n}$ and composed with an integration over $\mathcal{M}^{4n}$ which makes the resulting Dolbeault cohomology-valued weights become ground field-valued.

Konsevich 97 gives a formulation of Rozansky-Witten invariants via characteristic classes of foliations and Gelfand-Fuks cohomology. He devised a formal construction, again depending on a trivalent graph of a cohomology class of the Lie algebra of formal (in the sense of formal power series) Hamiltonian vector fields on any arbitrary finite-dimensional symplectic vector space. Characteristic classes of foliations may induce examples where this construction applies; one of the examples yields Rozansky-Witten .

Properties

As KK-compactification of M5-brane worldvolume

Rozansky-Witten theory may be identified with topologically twisted KK-compactification of the D=6 N=(2,0) SCFT on the M5-brane (Gukov-Putrov-Vafa 17, Sections 3.2 and 4.2)

Examples

Rozanzky-Witten Wilson loop of unknot is $\hat A$-genus

Proposition

(Rozansky-Witten Wilson loop of unknot is square root of A-hat genus)

For $\mathcal{M}^{4n}$ a hyperkähler manifold (or just a holomorphic symplectic manifold) the Rozansky-Witten invariant Wilson loop observable associated with the unknot in the 3-sphere is the square root $\sqrt{{\widehat A}(\mathcal{M}^{4n})}$ of the A-hat genus of $\mathcal{M}^{4n}$.

This is Roberts-Willerton 10, Lemma 8.6, using the Wheels theorem and the Hitchin-Sawon theorem.

Related concepts

• Casson invariant

• Seiberg-Witten invariant

• super Chern-Simons theory

References

Original articles:

• Lev Rozansky, Edward Witten, Hyper-Kähler geometry and invariants of 3-manifolds, Selecta Math., New Ser. 3 (1997), 401–458 (arXiv:hep-th/9612216, doi:10.1007/s000290050016, MR98m:57041)

• Maxim Kontsevich, Rozansky–Witten invariants via formal geometry, Compositio Mathematica 115: 115–127, 1999, doi, arXiv:dg-ga/9704009, MR2000h:57057

• Mikhail Kapranov, Rozansky–Witten invariants via Atiyah classes, Compositio Math. 115 (1999), no. 1, 71–113 (MR2000h:57056, doi, alg-geom/9704009)

• Justin Roberts, Justin Sawon, Generalisations of Rozansky-Witten invariants, Geom. Topol. Monogr. 4 (2002) 263-279 (arXiv:math/0112210)

Unified description of Rozansky-Witten weight systems as Lie algebra weight systems for Lie algebra objects in the derived category of quasi-coherent sheaves, and unified Wheels theorem:

• Justin Roberts, Simon Willerton, On the Rozansky-Witten weight systems, Algebr. Geom. Topol. 10 (2010) 1455-1519 (arXiv:math/0602653)

Review:

• Justin Sawon, Rozansky-Witten invariants of hyperkähler manifold, Cambridge 2000 (arXiv:math/0404360)

• Justin Roberts, Rozansky-Witten theory (arXiv:math/0112209)

Further relation to Chern-Simons theory:

• Anton Kapustin, Natalia Saulina, Chern-Simons-Rozansky-Witten topological field theory, Nucl. Phys. B823 (2009) 403-427 (arXiv:0904.1447, spire:817599/)

• Jian Qiu, Maxim Zabzine, Odd Chern-Simons theory, Lie algebra cohomology and characteristic classes, Commun. Math. Phys. 300:789-833, 2010 (arxiv/0912.1243)

As topologically twisted KK-compactification of the D=6 N=(2,0) SCFT on the M5-brane (see D=3 N=4 super Yang-Mills theory and 3d-3d correspondence):

• Sergei Gukov, Pavel Putrov, Cumrun Vafa, Sections 3.2 and 3.4 of: Fivebranes and 3-manifold homology, J. High Energ. Phys. (2017) 2017: 71 (arXiv:1602.05302)

• Cyril Closset, Heeyeon Kim, Section 6.1 of: Comments on twisted indices in 3d supersymmetric gauge theories, JHEP 08 (2016) 059 (arXiv:1605.06531)

Discussion of Rozansky-Witten theory as a boundary field theory:

• Anton Kapustin, Lev Rozansky, Natalia Saulina, Three-dimensional topological field theory and symplectic algebraic geometry I (arXiv:0810.5415)

Relation to equivariant cohomology:

• Sergei Gukov, Po-Shen Hsin, Hiraku Nakajima, Sunghyuk Park, Du Pei, Nikita Sopenko, Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants (arXiv:2005.05347)

• Jian Qiu, Rozansky-Witten theory, Localised then Tilted (arXiv:2011.05375)

Discussion of Rozansky-Witten defect QFT as an extended TQFT via the cobordism hypothesis:

• Ilka Brunner, Nils Carqueville, Daniel Roggenkamp, Truncated affine Rozansky–Witten models as extended TQFTs, Commun. Math. Phys. (2023) [arXiv:2201.03284, doi:10.1007/s00220-022-04614-4]

reviewed in:

• Nils Carqueville, Extended defect TQFTs, talk at QFT and Cobordism, CQTS (Mar 2023) [web, pdf, pdf, video:YT]

Relation to super Chern-Simons theory:

• Nathan Geer, Matthew B. Young, Three dimensional topological quantum field theory from $U_q(\mathfrak{gl}(1\vert1))$ and $U(1\vert 1)$ Chern–Simons theory [arXiv:2210.04286]

Review:

• Matthew B. Young, Non-semisimple TFT and $U(1\vert 1)$ Chern-Simons theory, talk at QFT and Cobordism, CQTS (Mar 2023)[web]

On Rozansky-Witten theory as extended functorial field theory:

• Lorenzo Riva, Higher categories of push-pull spans, I: Construction and applications [arXiv:2404.14597]

On line defects in Rozansky-Witten theory and their braiding:

• Sergei Gukov, Babak Haghighat, Nicolai Reshetikhin: Foams and KZ-equations in Rozansky-Witten theories [arXiv:2407.19757]