nLab topologically twisted D=4 super Yang-Mills theory

Idea

A deformation of super Yang-Mills theory that yields a topological field theory in 4 dimensions. This is in higher dimensional analogy to how the topological string-twisting of the superstring yields the topological A-model and B-model 2d topological field theories.

For $N = 4$ supersymmetry

Given N=4 D=4 super Yang-Mills theory, the twisting is induced by a choice of subgroup inclusion of the special orthogonal group $SO(4)$ into the R-symmetry group $SO(6)$. Then choose a supersymmetry $Q$ which is invariant under the resulting combined action of $SO(4)$ on spacetime and via R-symmetry and consider the subspace of quantum states/quantum observables which are in the kernel of $Q$. This subspace defines a topological field theory which is called the corresponding twisted topological super Yang-Mills theory.

Before the twisting super Yang-Mills theory depends on the complex coupling constant

$$\tau = \frac{\theta}{2\pi} + \frac{4 \pi i }{g_{YM}^2}$$

(with $\theta$ the theta angle), after the twisting there is an additional complex parameter $t$ encoding the choice of topological supercharge. The twisted theory however only depends on the combination

$$\Psi \coloneqq \frac{\theta}{2 \pi} + \frac{4 \pi i}{g_{YM}^2} \frac{t - t^{-1}}{t + t^{-1}} \,.$$

(Witten 11, p. 29)

The resulting 4d TQFT is also called the Kapustin-Witten TQFT.

For $N = 2$ supersymmetry

For N=2 D=4 super Yang-Mills theory the twisting follows the same idea, but is a little but more intricate (Witten 11, section 5.1.1)

Table of relations via holographic, compactifications and twists

gauge theory induced via AdS-CFT correspondence

M-theory perspective via AdS7-CFT6	F-theory perspective
11d supergravity/M-theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$	compactificationon elliptic fibration followed by T-duality
7-dimensional supergravity
$\;\;\;\;\downarrow$ topological sector
7-dimensional Chern-Simons theory
$\;\;\;\;\downarrow$ AdS7-CFT6 holographic duality
6d (2,0)-superconformal QFT on the M5-brane with conformal invariance	M5-brane worldvolume theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface	double dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondence	D3-brane worldvolume theory with type IIB S-duality
$\;\;\;\;\; \downarrow$ topological twist
topologically twisted N=2 D=4 super Yang-Mills theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
A-model on $Bun_G$, Donaldson theory

$\,$

gauge theory induced via AdS5-CFT4
type II string theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^5$
$\;\;\;\; \downarrow$ topological sector
5-dimensional Chern-Simons theory
$\;\;\;\;\downarrow$ AdS5-CFT4 holographic duality
N=4 D=4 super Yang-Mills theory
$\;\;\;\;\; \downarrow$ topological twist
topologically twisted N=4 D=4 super Yang-Mills theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
A-model on $Bun_G$ and B-model on $Loc_G$, geometric Langlands correspondence

Formalization

A formalization of the topological twisting in the framework of perturbative BV-quantization of field theory via factorization algebras of local quantum observables is proposed in (Costello 11, section 15, 16, …).

The definition there essentially amounts to saying that a choice of topological twisting is a choice of action of the semidirect product supergroup

$$\mathbb{G}_m \ltimes \Pi \mathbb{G}_{ad}$$

of the multiplicative group acting on the odd-shifted additive group via the given super Poincare Lie algebra.

We notice that this group is the automorphism group of the odd line

$$\mathbf{Aut}(\mathbb{R}^{0|1}) \simeq \mathbb{G}_m \ltimes \Pi \mathbb{G}_{ad}$$

for which it is well known that an action of it is equivalent to a choice of differential $Q$ and corresponding grading.

This chosen differential $Q$ among the supersymmetry generators in the super Poincare Lie algebra is the choice of what in the physics literature is called the twisting “BRST operator”.

The twisted theory itself is then defined to be given by the factorization algebra of observables which is essentially the homotopy fixed points of this $\mathbf{Aut}(\mathbb{R}^{0|1})$-infinity-action.

Related concepts

• topological Yang-Mills theory

• topological twist

• D=4 Yang-Mills theory

• N=1 D=4 super Yang-Mills theory

• N=2 D=4 super Yang-Mills theory

• N=4 D=3 super Yang-Mills theory

• N=4 D=4 super Yang-Mills theory

• D=5 super Yang-Mills theory

• S-duality, geometric Langlands duality

• chiral ring, quantum cohomology

References

The idea of topological twisting of supersymmetric quantum field theory goes back to

• Edward Witten, Topological quantum field theory, Comm. Math. Phys. Volume 117, Number 3 (1988), 353-386 (Euclid)

often referred to as “cohomological field theory”

• Edward Witten, Introduction to cohomological field theory, InternationalJournal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 (pdf)

where it is N=2 D=4 super Yang-Mills theory that is twisted and related to Donaldson theory.

A superspace formulation that makes the form of the Lagrangian more transparent is in:

• James Horne. Superspace versions of topological theories. Nuclear Physics B 318, no. 1 (1989): 22-52. (doi90046-1))

A geometric interpretation and derivation of the Lagrangian using equivariant de Rham cohomology is in:

• Michael Atiyah, Lisa Jeffrey. Topological Lagrangians and cohomology. Journal of Geometry and Physics 7, no. 1 (1990): 119-136. (doi90023-V))

based on the results in

• Varghese Mathai, Daniel Quillen. Superconnections, Thom classes and equivariant differential forms. Topology 25, 85 (1986) (doi:10.1016/0040-9383(86)90007-8)

The analogous twisting of N=4 D=4 super Yang-Mills theory is due to

• Cumrun Vafa, Edward Witten, A Strong Coupling Test of S-Duality, Nucl. Phys. B 431 (1994) 3-77 [arXiv:hep-th/9408074]

This Vafa-Witten theory reviewed and further developed in:

• Zhi-Cong Ong, Meng-Chwan Tan, Vafa-Witten Theory: Invariants, Floer Homologies, Higgs Bundles, a Geometric Langlands Correspondence, and Categorification [arXiv:2203.17115]

The $N=4$-case and one of the possible $N=2$-twists yield instanton invariants captured by the Seiberg-Witten theory generalization of Donaldson theory. Another variant of the $N=2$ twist was described in

• Neil Marcus, The other topological twisting of N=4 Yang-Mills, Nucl.Phys. B452 (1995) 331-345 (arXiv:hep-th/9506002)

and yields a geometric interpretation of the geometric Langlands correspondence, as found in

• Anton Kapustin, Edward Witten, Electric-Magnetic Duality And The Geometric Langlands Program (arXiv:hep-th/0604151)

A detailed analysis of the three twists of the $N=4$ theory is in

• Carlos Lozano, Duality in Topological Quantum Field Theories, PhD thesis (arXiv:hep-th/9907123)

Discussion of generalization of the twisting to quantum field theory on curved spacetime is in

• Jeong-Hyuck Park, Dimitrios Tsimpis, Topological twisting of conformal supercharges, Nucl. Phys. B776:405-430, 2007 (arXiv:hep-th/0610159)

Section 2.2.1 of

• Edward Witten, Fivebranes and knots (arXiv:1101.3216)

briefly recalls the topological twisting of N=4 D=4 super Yang-Mills theory. Section 5.1.1 there discusses the twisting of N=2 D=4 super Yang-Mills theory (induced from the 6d (2,0)-superconformal QFT on the M5-brane), which was introduced in section 3.1.2 of

• Davide Gaiotto, Gregory Moore and Andrew Neitzke, Wall-crossing, Hitchin Systems, and the WKB Approximation, (arXiv:0907.3987)

For more on this see the references listed at N=2 D=4 super Yang-Mills theory – References – Construction from 5-branes.

More mathematically formalized discussion of topologically twisted supersymmetric theories in the framework of BV-BRST formalism perturbation theory (and with an eye towards the factorization algebra formulation) is in

• Kevin Costello, Notes on supersymmetric and holomorphic field theories in dimension 2 and 4, talk at Geometric and Algebraic Structures in Mathematics, Stony Brook (2011) [arXiv:1111.4234]

More in

• Chris Elliott, Pavel Safronov, Topological twists of supersymmetric observables (arXiv:1805.10806)

In

• Kevin Costello, Supersymmetric gauge theory and the Yangian (arXiv:1303.2632)

is discussed that the holomorphically twisted $N=1$ theory is controled by the Yangian in analogy to how Chern-Simons theory is controled by a quantum group.