nLab
Rozansky-Witten theory

Contents

Context

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Knot theory

Contents

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 Σ 3\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 4n\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 4n\mathcal{M}^{4n} and depend only on the underlying holomorphic symplectic manifold (Kapranov 99). Generally, they are defined for 4n\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 4n\mathcal{M}^{4n} and composed with an integration over 4n\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 A^\hat A-genus

Proposition

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

For 4n\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 A^( 4n)\sqrt{{\widehat A}(\mathcal{M}^{4n})} of the A-hat genus of 4n\mathcal{M}^{4n}.

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

References

Original articles:

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:

Review:

Further relation to Chern-Simons theory:

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:

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)

Last revised on November 12, 2020 at 02:42:41. See the history of this page for a list of all contributions to it.