Abstract
We present free field realizations for the associated vertex operator algebras of a number of four-dimensional \( \mathcal{N} \) = 2 superconformal field theories. Our constructions utilize an exceptionally small set of chiral bosons whose number matches the complex dimensionality of the Higgs branch of the superconformal field theory. In the case of theories whose Higgs branches support additional degrees of freedom (free vector multiplets or decoupled interacting SCFTs), the corresponding “free field realizations” include additional ingredients: symplectic fermions in the case of vector multiplets and a C2 co-finite VOA in the case of a residual interacting SCFT. The resulting picture is that the associated VOA can be constructed from the Higgs branch effective theory via free field realization. Our constructions also provide a natural realization of the R-filtration of the associated VOA.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli and B.C. van Rees, Infinite chiral symmetry in four dimensions, Commun. Math. Phys.336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
C. Beem, W. Peelaers, L. Rastelli and B.C. van Rees, Chiral algebras of class S, JHEP05 (2015) 020 [arXiv:1408.6522] [INSPIRE].
T. Arakawa, Chiral algebras of class S and Moore-Tachikawa symplectic varieties, arXiv:1811.01577 [INSPIRE].
P. Liendo, I. Ramirez and J. Seo, Stress-tensor OPE in N = 2 superconformal theories, JHEP02 (2016) 019 [arXiv:1509.00033] [INSPIRE].
M. Lemos and P. Liendo, N = 2 central charge bounds from 2d chiral algebras, JHEP04 (2016) 004 [arXiv:1511.07449] [INSPIRE].
C. Beem, Flavor symmetries and unitarity bounds in N = 2 superconformal field theories, Phys. Rev. Lett.122 (2019) 241603 [arXiv:1812.06099] [INSPIRE].
C. Beem and L. Rastelli, Vertex operator algebras, Higgs branches and modular differential equations, JHEP08 (2018) 114 [arXiv:1707.07679] [INSPIRE].
T. Arakawa, Associated varieties of modules over Kac-Moody algebras and C 2-cofiniteness of W-algebras, Int. Math. Res. Notices2015 (2015) 11605 [arXiv:1004.1554] [INSPIRE].
C. Beem, C. Meneghelli and L. Rastelli, work in progress.
D. Adamovic, Realizations of simple affine vertex algebras and their modules: the cases \( \hat{sl(2)} \)and \( \hat{osp\Big(1,2}\Big) \), arXiv:1711.11342.
J. Song, Macdonald index and chiral algebra, JHEP08 (2017) 044 [arXiv:1612.08956] [INSPIRE].
F. Bonetti, C. Meneghelli and L. Rastelli, VOAs labelled by complex reflection groups and 4d SCFTs, JHEP05 (2019) 155 [arXiv:1810.03612] [INSPIRE].
T. Arakawa and K. Kawasetsu, Quasi-lisse vertex algebras and modular linear differential equations, arXiv:1610.05865 [INSPIRE].
T. Arakawa and A. Moreau, Joseph ideals and lisse minimal W-algebras, arXiv:1506.00710 [INSPIRE].
M. Dedushenko and S. Gukov, IR duality in 2D N = (0, 2) gauge theory with noncompact dynamics, Phys. Rev.D 99 (2019) 066005 [arXiv:1712.07659] [INSPIRE].
R. Eager, G. Lockhart and E. Sharpe, Hidden exceptional symmetry in the pure spinor superstring, arXiv:1902.09504 [INSPIRE].
H. Shimizu, Y. Tachikawa and G. Zafrir, Anomaly matching on the Higgs branch, JHEP12 (2017) 127 [arXiv:1703.01013] [INSPIRE].
T. Arakawa and A. Moreau, Arc spaces and chiral symplectic cores, arXiv:1802.06533.
D. Adamovic, A construction of admissible \( {A}_1^{(1)} \)-modules of level −4/3, math.QA/0401023.
A.M. Semikhatov, The MFF singular vectors in topological conformal theories, Mod. Phys. Lett.A 9 (1994) 1867 [hep-th/9311180] [INSPIRE].
D. Adamovic, A realization of certain modules for the N = 4 superconformal algebra and the affine Lie algebra \( {A}_2^{(1)} \), arXiv:1407.1527 [INSPIRE].
P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP12 (2007) 088 [arXiv:0711.0054] [INSPIRE].
A. Joseph, The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. Sci. École Norm. Sup.9 (1976) 1.
V.G. Kac and M. Wakimoto, Modular invariant representations of infinite dimensional Lie algebras and superalgebras, Proc. Nat. Acad. Sci.85 (1988) 4956 [INSPIRE].
T. Arakawa and A. Moreau, Sheets and associated varieties of affine vertex algebras, Adv. Math.320 (2017) 157.
M. Günaydin and O. Pavlyk, Minimal unitary realizations of exceptional U-duality groups and their subgroups as quasiconformal groups, JHEP01 (2005) 019 [hep-th/0409272] [INSPIRE].
A. Joseph, Minimal realizations and spectrum generating algebras, Commun. Math. Phys.36 (1974) 325 [INSPIRE].
C. Beem and L. Rastelli, Infinite chiral symmetry in four and six dimensions, seminar by L. Rastelli at Harvard University, U.S.A., November 2014.
C. Córdova and S.-H. Shao, Schur indices, BPS particles and Argyres-Douglas theories, JHEP01 (2016) 040 [arXiv:1506.00265] [INSPIRE].
M. Buican, Z. Laczko and T. Nishinaka, N = 2 S-duality revisited, JHEP09 (2017) 087 [arXiv:1706.03797] [INSPIRE].
C. Beem, C. Meneghelli, W. Peelaers and L. Rastelli, VOAs and rank-two instanton SCFTs, arXiv:1907.08629 [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge theories and Macdonald polynomials, Commun. Math. Phys.319 (2013) 147 [arXiv:1110.3740] [INSPIRE].
L. Rastelli and S.S. Razamat, The superconformal index of theories of class S, in New dualities of supersymmetric gauge theories, J. Teschner ed., Springer, Cham, Switzerland (2016), pg. 261 [arXiv:1412.7131] [INSPIRE].
M. Buican and T. Nishinaka, Argyres-Douglas theories, the Macdonald index and an RG inequality, JHEP02 (2016) 159 [arXiv:1509.05402] [INSPIRE].
Y. Tachikawa, On ‘categories’ of quantum field theories, in Proceedings, International Congress of Mathematicians (ICM 2018), Rio de Janeiro, Brazil, 1-9 August 2018, pg. 2695 [arXiv:1712.09456] [INSPIRE].
R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction, Ann. Math.134 (1991) 375.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1903.07624
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Beem, C., Meneghelli, C. & Rastelli, L. Free field realizations from the Higgs branch. J. High Energ. Phys. 2019, 58 (2019). https://doi.org/10.1007/JHEP09(2019)058
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2019)058