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The $\mathcal{W}$-algebra bootstrap of 6d $\mathcal{N}=(2,0)$ theories
by Mitchell Woolley
Submission summary
| Authors (as registered SciPost users): | Mitchell Woolley |
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| Preprint Link: | https://arxiv.org/abs/2506.08094v3 (pdf) |
| Date accepted: | Aug. 25, 2025 |
| Date submitted: | Aug. 11, 2025, 1:06 p.m. |
| Submitted by: | Mitchell Woolley |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
6d (2,0) SCFTs of type $\mathfrak{g}$ have protected subsectors that were conjectured in arxiv:1404.1079 to be captured by $\mathcal{W}_\mathfrak{g}$ algebras. We write down the crossing equations for mixed four-point functions $\langle S_{k_1} S_{k_2}S_{k_3}S_{k_4}\rangle$ of 1/2-BPS operators $S_{k_i}$ in 6d (2,0) theories and detail how a certain twist reduces this system to a 2d meromorphic CFT multi-correlator bootstrap problem. We identify the relevant 6d (2,0) $\mathcal{W}$-algebras of type $\mathfrak{g} = \{A_{N-1},D_N\}$ as truncations of $\mathcal{W}_{1+\infty}$ and solve OPE associativity conditions for their structure constants, both using $\texttt{OPEdefs}$ and the holomorphic bootstrap of arxiv:1503.07111. With this, we solve the multi-correlator bootstrap for twisted 6d four-point correlators $\mathcal{F}_{k_1k_2k_3k_4}$ involving all $S_{k_i}$ up to $\{k_i\}=4$ and extract closed-form expressions for 6d OPE coefficients. We describe the implications of our CFT data on conformal Regge trajectories of the (2,0) theories and finally, demonstrate the consistency of our results with protected higher-derivative corrections to graviton scattering in M-theory on $AdS_7\times S^4/\mathbb{Z}_\mathfrak{o}$.
Author indications on fulfilling journal expectations
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- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
List of changes
1) Clarifying the filtration problem in 6d: We added three pieces of text that identify this problem and its practical effect on our calculation. In particular, when defining the chiral algebra twist in Section 3.2, we point out how the cohomological reduction coarsens our organization of 6d operators by effectively removing a quantum number, and how the map between 6d and 2d operators is ambiguous, given the numerous ways one might realize a 6d operator of fixed weight with different 2d (non-)composite quasiprimaries. We briefly comment on the analogous problem in 4d N=2 theories. In Section 5.2, we describe the practical effect this has on our expansion matching strategy for computing 6d data. We also describe an example (the one in our response to Referee 1) of how additional 2d and 6d principles can be used to partially lift this ambiguity. Finally, we attribute our inability to distinguish B[4,0]j and B[6,0]{j-2} in <4444> to the filtration problem and offer some preliminary comments on interpreting the resulting sum rule.
2) Clarifying our data extraction strategy: We reorganized Section 5.2, first by being more upfront in identifying the filtration problem, but also detailing our strategy to determine individual CFT data contributing to higher-rank correlators by first pinning down their contributions to lower-rank correlators. The main examples of this are the determination of B[2,0]j from <22pp> and B[p,0]j from <2p2p>. Furthermore, we partitioned the results of each correlator into sub(sub)sections and in each case wrote sentences on how our strategy fixes low-lying semi-short data using lower-rank correlators, allowing for an isolated expression for the highest semi-short datum.
3) Reference to AGT: In Section 4 we included a paragraph briefly describing another important appearance of W symmetry in the context of 6d (2,0) theories, namely the AGT correspondence. We include references to subsequent constructions that give evidence to the compatibility of the 6d/W-algebra correspondence with AGT.
4) Comparison with chiral algebras of 4d N=4 SCFTs: In Section 5.1 we included a brief discussion of the analogous chiral algebra construction in 4d N=4 SCFTs. We discuss the relative utility of these constructions by pointing out that certain 4d N=4 chiral algebra results are equally obtainable by a free-field computation, whereas W-algebras are indispensable for interpreting the protected correlator in 6d. We also discuss differences in what these chiral algebras can determine holographically.
5) Twisted holography: We improve our chiral algebra holography discussion in Section 7 by briefly describing the twisted holography work by Gaiotto and Costello using topological string theory.
6) Errors: --- We corrected our statement of the parameters that W_{1+infinity} depends on, and the way that these are fixed upon a truncation to W_g. --- We corrected our statement to say that W_g are obtainable as the quantum Drinfel'd-Sokolov reduction of an affine Kac-Moody algebra \hat{g}. --- We corrected numerous mispellings and grammatical and mathematical typos.
Published as SciPost Phys. 19, 074 (2025)