Koszul duality in quantum field theory
Confluentes Mathematici, Volume 14 (2022) no. 2, pp. 87-138.

In this article, we introduce basic aspects of the algebraic notion of Koszul duality for a physics audience. We then review its appearance in the physical problem of coupling QFTs to topological line defects, and illustrate the concept with some examples drawn from twists of various simple supersymmetric theories. Though much of the content of this article is well-known to experts, the presentation and examples have not, to our knowledge, appeared in the literature before. Our aim is to provide an elementary introduction for those interested in the appearance of Koszul duality in supersymmetric gauge theories with line defects and, ultimately, its generalizations to higher-dimensional defects and twisted holography.

Published online:
DOI: 10.5802/cml.88
Classification: 81T35, 18M70, 81T60, 81T30
Keywords: Koszul duality, chiral algebra, defects and boundaries in QFT
Natalie M. Paquette 1, 2; Brian R. Williams 3

1 Institute for Advanced Study, School of Natural Sciences, Princeton NJ, 08540, USA
2 Department of Physics, University of Washington, Seattle, WA, 98195-1560, USA
3 School of Mathematics, University of Edinburgh, Edinburgh, UK
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Natalie M. Paquette and Brian R. Williams},
     title = {Koszul duality in quantum field theory},
     journal = {Confluentes Mathematici},
     pages = {87--138},
     publisher = {Institut Camille Jordan},
     volume = {14},
     number = {2},
     year = {2022},
     doi = {10.5802/cml.88},
     language = {en},
     url = {https://cml.centre-mersenne.org/articles/10.5802/cml.88/}
AU  - Natalie M. Paquette
AU  - Brian R. Williams
TI  - Koszul duality in quantum field theory
JO  - Confluentes Mathematici
PY  - 2022
SP  - 87
EP  - 138
VL  - 14
IS  - 2
PB  - Institut Camille Jordan
UR  - https://cml.centre-mersenne.org/articles/10.5802/cml.88/
DO  - 10.5802/cml.88
LA  - en
ID  - CML_2022__14_2_87_0
ER  - 
%0 Journal Article
%A Natalie M. Paquette
%A Brian R. Williams
%T Koszul duality in quantum field theory
%J Confluentes Mathematici
%D 2022
%P 87-138
%V 14
%N 2
%I Institut Camille Jordan
%U https://cml.centre-mersenne.org/articles/10.5802/cml.88/
%R 10.5802/cml.88
%G en
%F CML_2022__14_2_87_0
Natalie M. Paquette; Brian R. Williams. Koszul duality in quantum field theory. Confluentes Mathematici, Volume 14 (2022) no. 2, pp. 87-138. doi : 10.5802/cml.88. https://cml.centre-mersenne.org/articles/10.5802/cml.88/

[1] Nanna Havn Aamand. Chern-Simons Theory and the R-Matrix. May 2019. arXiv: [hep-th]. | arXiv

[2] Mina Aganagic. Knot Categorification from Mirror Symmetry, Part II: Lagrangians. May 2021. arXiv: [hep-th]. | arXiv

[3] Mina Aganagic, Kevin Costello, Jacob McNamara, and Cumrun Vafa. Topological Chern-Simons/Matter Theories. June 2017. arXiv: [hep-th]. | arXiv

[4] Paul S. Aspinwall, Tom Bridgeland, Alastair Craw, Michael R. Douglas, Anton Kapustin, Gregory W. Moore, Mark Gross, Graeme Segal, Balázs Szendröi, and P. M. H. Wilson. Dirichlet branes and mirror symmetry, Clay Math. Monogr., vol. 4. AMS, Providence, RI, 2009. | Zbl

[5] David Ayala and John Francis. Factorization homology of topological manifolds. J. Topol., 8(4):1045–1084, 2015. | DOI | Zbl

[6] David Ayala and John Francis. Poincaré/Koszul duality. Comm. Math. Phys., 365(3):847–933, 2019. | DOI | Zbl

[7] David Ayala, John Francis, and Hiro Lee Tanaka. Factorization homology of stratified spaces. Sel. Math. (N.S.), 23(1):293–362, 2017. | DOI | Zbl

[8] Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel. Koszul duality patterns in representation theory. J. Amer. Math. Soc., 9(2):473–527, 1996. | DOI | MR | Zbl

[9] Alexander A Beilinson, Victor A Ginsburg, and Vadim V Schechtman. Koszul duality. J. Geom. Phys., 5(3):317–350, 1988. | DOI | MR

[10] Dmitriy Belov and Gregory W. Moore. Classification of Abelian spin Chern-Simons theories. May 2005. arXiv: . | arXiv

[11] Joseph Bernstein, Izrail Moiseevich Gel’fand, and Sergei Izrail’evich Gel’fand. Algebraic bundles over P n and problems of linear algebra. Funktsional’nyi Analiz i ego Prilozheniya, 12(3):66–67, 1978.

[12] Mathew Bullimore, Tudor Dimofte, Davide Gaiotto, and Justin Hilburn. Boundaries, Mirror Symmetry, and Symplectic Duality in 3d 𝒩=4 Gauge Theory. JHEP, 10:108, 2016. arXiv: [hep-th]. | arXiv | DOI | MR | Zbl

[13] Damien Calaque, Giovanni Felder, Andrea Ferrario, and Carlo A. Rossi. Bimodules and branes in deformation quantization. Compos. Math., 147(1):105–160, 2011. | DOI | Zbl

[14] Alberto S. Cattaneo, Pavel Mnev, and Konstantin Wernli. Quantum Chern-Simons theories on cylinders: BV-BFV partition functions. December 2020. arXiv: [hep-th]. | arXiv

[15] Chi-Ming Chang, Ying-Hsuan Lin, Shu-Heng Shao, Yifan Wang, and Xi Yin. Topological Defect Lines and Renormalization Group Flows in Two Dimensions. JHEP, 01:026, 2019. arXiv: [hep-th]. | arXiv | DOI | Zbl

[16] Clay Córdova, Thomas T. Dumitrescu, and Kenneth Intriligator. Exploring 2-Group Global Symmetries. JHEP, 02:184, 2019. arXiv: [hep-th]. | arXiv | DOI | MR | Zbl

[17] Clay Córdova, Thomas T. Dumitrescu, and Kenneth Intriligator. 2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories. JHEP, 04:252, 2021. arXiv: [hep-th]. | arXiv | DOI | MR | Zbl

[18] Kevin Costello. Renormalization and effective field theory. Number 170. Amer. Math. Soc., 2011. | DOI | Zbl

[19] Kevin Costello. Supersymmetric gauge theory and the Yangian. March 2013. arXiv: [hep-th]. | arXiv

[20] Kevin Costello. M-theory in the Omega-background and 5-dimensional non-commutative gauge theory. October 2016. arXiv: [hep-th]. | arXiv

[21] Kevin Costello. Holography and Koszul duality: the example of the M2 brane. May 2017. arXiv: [hep-th]. | arXiv | DOI

[22] Kevin Costello, Tudor Dimofte, and Davide Gaiotto. Boundary Chiral Algebras and Holomorphic Twists. April 2020. arXiv: [hep-th]. | arXiv

[23] Kevin Costello and Davide Gaiotto. Twisted Holography. December 2018. arXiv: [hep-th]. | arXiv

[24] Kevin Costello, Davide Gaiotto, and Junya Yagi. Q-operators are ’t Hooft lines. March 2021. arXiv: [hep-th]. | arXiv

[25] Kevin Costello and Owen Gwilliam. Factorization Algebras in Quantum Field Theory: Volume 1. Cambridge University Press, 2016. | DOI | Zbl

[26] Kevin Costello and Owen Gwilliam. Factorization Algebras in Quantum Field Theory: Volume 2. Cambridge University Press, 2021. | DOI | Zbl

[27] Kevin Costello and Si Li. Quantization of open-closed BCOV theory, I. May 2015. arXiv: [hep-th]. | arXiv

[28] Kevin Costello and Si Li. Twisted supergravity and its quantization. June 2016. arXiv: [hep-th]. | arXiv

[29] Kevin Costello and Si Li. Anomaly cancellation in the topological string. Adv. Theor. Math. Phys., 24(7):1723–1771, 2020. arXiv: [hep-th]. | arXiv | DOI | MR | Zbl

[30] Kevin Costello and Natalie M. Paquette. Celestial holography & twisted holography (in preparation). 2021.

[31] Kevin Costello and Natalie M. Paquette. Twisted Supergravity and Koszul Duality: A case study in AdS 3 . Commun. Math. Phys., 384(1):279–339, 2021. arXiv: [hep-th]. | arXiv | DOI | Zbl

[32] Kevin Costello, Edward Witten, and Masahito Yamazaki. Gauge Theory and Integrability, I. ICCM Not., 06(1):46–119, 2018. arXiv: [hep-th]. | arXiv | DOI | MR | Zbl

[33] Tudor Dimofte. Koszul duality patterns in physics. String-Math 2017: https://stringmath2017.desy.de/sites/sites_conferences/site_stringmath2017/content/e45470/e56510/e56560/SM-Dimofte.pdf, 2017.

[34] Tudor Dimofte, Davide Gaiotto, and Natalie M. Paquette. Dual boundary conditions in 3d SCFT’s. JHEP, 05:060, 2018. arXiv: [hep-th]. | arXiv | DOI | MR | Zbl

[35] Chris Elliott and Owen Gwilliam. Spontaneous symmetry breaking: A view from derived geometry. J. Geom. Phys., 162:104096, 2021. arXiv: [math-ph]. | arXiv | DOI | MR | Zbl

[36] Chris Elliott, Pavel Safronov, and Brian R. Williams. A Taxonomy of Twists of Supersymmetric Yang–Mills Theory. February 2020. arXiv: [math-ph]. | arXiv

[37] Chris Elliott and Philsang Yoo. A physical origin for singular support conditions in geometric Langlands theory, 2019. arXiv: [math-ph]. | arXiv | DOI | MR | Zbl

[38] Davide Gaiotto and Jihwan Oh. Aspects of Ω-deformed M-theory. July 2019. arXiv: [hep-th]. | arXiv

[39] Davide Gaiotto and Miroslav Rapcak. Miura operators, degenerate fields and the M2-M5 intersection. December 2020. arXiv: [hep-th]. | arXiv

[40] Davide Gaiotto and Edward Witten. Probing Quantization Via Branes. July 2021. arXiv: [hep-th]. | arXiv

[41] Rajesh Gopakumar. Open-closed-open string duality. 2nd Johannesburg Workshop on String Theory : http://presentations.wits.ac.za/index.php?module=webpresent&action=view&id=gen13Srv42Nme14_57977_1273143257, 2010.

[42] Sergei Gukov and Edward Witten. Branes and quantization. Adv. Theor. Math. Phys., 13(5):1445–1518, 2009. | DOI | Zbl

[43] Owen Gwilliam, Eugene Rabinovich, and Brian R. Williams. Quantization of topological-holomorphic field theories: local aspects, 2021. arXiv: [math-ph]. | arXiv

[44] Owen Gwilliam and Brian R. Williams. Higher Kac-Moody algebras and symmetries of holomorphic field theories. October 2018. arXiv: [math.QA]. | arXiv

[45] Owen Gwilliam and Brian R. Williams. A one-loop exact quantization of Chern-Simons theory. October 2019. arXiv: [math-ph]. | arXiv

[46] Joseph Hirsh and Joan Millès. Curved Koszul duality theory, 2010. arXiv: [math.KT]. | arXiv

[47] Justin Kaidi, Zohar Komargodski, Kantaro Ohmori, Sahand Seifnashri, and Shu-Heng Shao. Higher central charges and topological boundaries in 2+1-dimensional TQFTs. July 2021. arXiv: [hep-th]. | arXiv

[48] Anton Kapustin and Natalia Saulina. Topological boundary conditions in abelian Chern-Simons theory. Nucl. Phys. B, 845:393–435, 2011. arXiv: [hep-th]. | arXiv | DOI | MR | Zbl

[49] Si Li. Vertex algebras and quantum master equation. December 2016. arXiv: [math.QA] | arXiv

[50] Jean-Louis Loday and Bruno Vallette. Algebraic operads, volume 346. Springer Science & Business Media, 2012. | DOI | Zbl

[51] Jacob Lurie. Derived algebraic geometry x: Formal moduli problems. Preprint accessible at author’s home page:http://www.math.harvard.edu/~ lurie, 2011.

[52] Jacob Lurie. Higher algebra. Prepublication book draft. http://www.math.harvard.edu/~ lurie/papers/higheralgebra.pdf, 2011.

[53] Gregory W. Moore and Edward Witten. Integration over the u plane in Donaldson theory. Adv. Theor. Math. Phys., 1:298–387, 1997. arXiv: . | arXiv | DOI | MR | Zbl

[54] Jihwan Oh and Junya Yagi. Poisson vertex algebras in supersymmetric field theories. August 2019. arXiv: [hep-th]. | arXiv

[55] Natalie M. Paquette and Ingmar Saberi. In progress. 2021.

[56] Leonid Positselski. Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence. Amer. Math. Soc., 2011. | DOI | MR | Zbl

[57] Carlo Antonio Rossi. The Chevalley-Eilenberg complex and deformation quantization in presence of two branes. Algebr. Represent. Theory, 16(3):819–841, 2013. | DOI | Zbl

[58] Pavel Safronov. Shifted geometric quantization, 2020. arXiv: [math.SG]. | arXiv

[59] Claudia Scheimbauer. On fully extended topological field theories. Ph.D. thesis at ETH Zurich, 2014.

[60] Boris Shoikhet. Koszul duality in deformation quantization and Tamarkin’s approach to Kontsevich formality. Adv. Math., 224(3):731–771, 2010. | DOI | Zbl

[61] Charles A Weibel. An introduction to homological algebra. Number 38. Cambridge University Press, 1995. | Zbl

[62] Edward Witten. Topological Quantum Field Theory. Commun. Math. Phys., 117:353, 1988. | DOI | MR | Zbl

[63] Edward Witten and Barton Zwiebach. Algebraic structures and differential geometry in 2-D string theory. Nucl. Phys. B, 377:55–112, 1992. arXiv: . | arXiv | DOI | MR

Cited by Sources: