Supersymmetric twists in twistor space and holography

Problem and Motivation
Supersymmetric field theories contain BPS sectors, characterized as cohomology classes of supercharges, which are often simpler to study than the full theory. These sectors are typically accessed via a "twist," a procedure where part of the R-symmetry is promoted to a spacetime symmetry, and a supercharge is promoted to a gauge redundancy (BRST operator). While the holomorphic twist of supersymmetric theories on spacetime is well-understood (localizing theories to holomorphic structures on $\mathbb{C}^2$), the behavior of these twists within twistor space requires further investigation. Twistor space naturally encodes self-dual sectors of Yang-Mills and gravity, and while the minimal twist of $N=1$ super Yang-Mills (sYM) in twistor space was previously described abstractly, a concrete construction using BV-BRST formalism for extended supersymmetry ($N=2, 4$) and supergravity was lacking. Furthermore, the holographic duals of these twisted theories in the framework of "chiral holography" needed explicit computation to verify conjectures regarding their localization properties.

Methodology
The authors employ the Batalin-Vilkovisky (BV) formalism to construct supersymmetric twists of holomorphic field theories defined on supertwistor space ($PT^{3|4}$). The methodology proceeds in several stages:

1. Twistor Space Setup: The paper utilizes the supertwistor space $PT^{3|4}$ as the total space of a holomorphic fibration over $\mathbb{CP}^1$. The theories considered are self-dual supersymmetric Yang-Mills (sYM) and self-dual supergravity, formulated as holomorphic Chern-Simons or holomorphic BF theories on this space.
2. BV-BRST Construction: The authors adapt the BV-BRST formalism to the specific gauge redundancies present in twistor space. A key technical challenge identified is that twistor space possesses extra gauge redundancies (Dolbeault cohomology classes) compared to spacetime. Consequently, the supersymmetry transformations used for the twist do not commute with all gauge transformations, requiring the addition of specific interaction terms to the BV action to account for the non-trivial structure of the twisted gauge algebra.
3. Twist Implementation:
   • Minimal Twist: The authors perform the minimal twist by selecting a nilpotent supercharge $Q$ (parametrized by a projective spinor $\epsilon_\alpha$) and adding it to the BRST operator. This is done for $N=1$, $N=2$, and $N=4$ vector multiplets, as well as for the chiral algebra twist (involving a combination of $Q$ and a superconformal charge $\tilde{S}$).
   • Localization Analysis: By analyzing the ghost numbers and gauge transformations in the twisted BV action, the authors demonstrate that the action becomes BRST-exact (topological) everywhere except at specific loci on the twistor sphere defined by the vanishing of the spinor contractions (e.g., $\langle \epsilon \lambda \rangle = 0$).
   • Reduction to Spacetime: Using the Woodhouse gauge and expanding fields in weighted spherical harmonics on the $\mathbb{CP}^1$ fibers, the authors explicitly reduce the localized twistor space actions to spacetime.
4. Holographic Duals: In the context of chiral holography (based on the topological B-model), the authors compute the bulk duals. They identify the bulk $psl(4|4)$ algebra generators with specific polyvector fields in the bulk geometry. By turning on background superpotentials corresponding to the twisting supercharges, they induce a localization of the bulk theory, solving the deformed equations of motion for the BCOV (Bershadsky-Cecotti-Ooguri-Vafa) theory.

Key Contributions and Results

• Explicit BV Constructions: The paper provides concrete BV-BRST constructions for the minimal twists of $N=1$, $N=2$, and $N=4$ self-dual sYM, as well as the minimal twist of $N=1$ self-dual supergravity, on twistor space. This expands upon previous abstract descriptions.
• Localization to Spacetime: A primary result is the demonstration that the minimal twist of gauge theories in twistor space localizes the theory to a point on the twistor sphere (the fiber of the projective spinor bundle). Away from this point, the action is BRST-exact. At the localization point, the theory reduces to a holomorphic field theory on $\mathbb{C}^2$ with a complex structure determined by the choice of the supercharge. This reproduces the known minimal twist results on spacetime.
• Chiral Algebra Twist: The authors apply a further twist corresponding to the chiral algebra twist (using $Q + \tilde{S}$). They show that this localizes the theory to a plane contained within spacetime, reproducing the chiral algebra twist of $N=4$ sYM. This provides a geometric interpretation in twistor space for the plane where the chiral algebra lives.
• Holographic Duals: The paper computes the holographic duals for the minimal and chiral algebra twists of $N=4$ sYM within the chiral holography framework.
  • For the minimal twist, the bulk dual is shown to localize to a specific locus, reproducing the results of twisted holography conjectured in previous literature.
  • For the chiral algebra twist, the bulk dual is found to localize to a deformed conifold geometry, matching the results of [16].
• Comparison with Prior Work: The authors note that while their results for the chiral algebra twist match the geometric localization to a spacetime plane, they differ from the work of [17], which found localization to the twistor sphere. The authors attribute this discrepancy to the choice of twisting supercharges in [17], which are not standard from a spacetime perspective, though they acknowledge that the final holographic duality is recovered in both cases.

Significance and Claims
The paper claims that the minimal supersymmetric twist of self-dual Yang-Mills in twistor space coincides perturbatively with the analogous twist on spacetime. This establishes twistor space as a consistent framework for studying BPS sectors, where the choice of complex structure becomes manifest through the localization point on the twistor sphere.

The authors state that their construction of the holographic duals reproduces known results in the twisted holography literature, thereby validating the chiral holography framework for these specific twisted sectors. They highlight that the localization mechanism in twistor space provides a natural geometric interpretation for the reduction of the theory to spacetime (for minimal twists) and to a plane (for chiral algebra twists).

The paper concludes modestly, noting that while the twisted theories are topological-holomorphic, the resolution of short-distance singularities in operator products (a known issue in holomorphic BF theories) requires further analysis similar to that done for chiral algebras. Additionally, the authors point out a puzzle regarding the use of axial gauge in twistor space, which renders the theory effectively free at tree level, potentially obscuring known loop corrections in the BPS sector, and suggest this warrants revisiting.