Abstract
We compute the exact all-orders perturbative expansion for the partition function of 2d SU(2) Yang-Mills theory on closed surfaces around higher critical points of the classical action. We demonstrate that the expansion can be derived from the lattice partition function for all genera using a distributional generalization of the Poisson summation formula. We then recompute the expansion directly, using a stationary phase version of supersymmetric localization. The result of localization is a novel effective action which is itself a distribution rather than a function of the supersymmetric moduli. We comment on possible applications to A-twisted models and their analogs in higher dimensions.
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14 August 2024
An Erratum to this paper has been published: https://doi.org/10.1007/JHEP08(2024)101
References
A.A. Migdal, Recursion Equations in Gauge Theories, Sov. Phys. JETP 42 (1975) 413 [INSPIRE].
E. Witten, On quantum gauge theories in two-dimensions, Commun. Math. Phys. 141 (1991) 153 [INSPIRE].
E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303 [hep-th/9204083] [INSPIRE].
M. Blau and G. Thompson, Lectures on 2-d gauge theories: Topological aspects and path integral techniques, in the proceedings of the Summer School in High-energy Physics and Cosmology (Includes Workshop on Strings, Gravity, and Related Topics, Jul 29–30 (1993) [hep-th/9310144] [INSPIRE].
D.J. Gross, Two-dimensional QCD as a string theory, Nucl. Phys. B 400 (1993) 161 [hep-th/9212149] [INSPIRE].
D.J. Gross and W. Taylor, Two-dimensional QCD is a string theory, Nucl. Phys. B 400 (1993) 181 [hep-th/9301068] [INSPIRE].
M.R. Douglas and V.A. Kazakov, Large N phase transition in continuum QCD in two-dimensions, Phys. Lett. B 319 (1993) 219 [hep-th/9305047] [INSPIRE].
D.J. Gross and A. Matytsin, Instanton induced large N phase transitions in two-dimensional and four-dimensional QCD, Nucl. Phys. B 429 (1994) 50 [hep-th/9404004] [INSPIRE].
A. Bassetto, F. De Biasio and L. Griguolo, Lightlike Wilson loops and gauge invariance of Yang-Mills theory in (1 + 1)-dimensions, Phys. Rev. Lett. 72 (1994) 3141 [hep-th/9402029] [INSPIRE].
A. Bassetto and L. Griguolo, Two-dimensional QCD, instanton contributions and the perturbative Wu-Mandelstam-Leibbrandt prescription, Phys. Lett. B 443 (1998) 325 [hep-th/9806037] [INSPIRE].
N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, Supersymmetric Wilson loops on S3, JHEP 05 (2008) 017 [arXiv:0711.3226] [INSPIRE].
V. Pestun, Localization of the four-dimensional N = 4 SYM to a two-sphere and 1/8 BPS Wilson loops, JHEP 12 (2012) 067 [arXiv:0906.0638] [INSPIRE].
S. Giombi and V. Pestun, Correlators of local operators and 1/8 BPS Wilson loops on S2 from 2d YM and matrix models, JHEP 10 (2010) 033 [arXiv:0906.1572] [INSPIRE].
A. Bassetto et al., Correlators of supersymmetric Wilson-loops, protected operators and matrix models in N = 4 SYM, JHEP 08 (2009) 061 [arXiv:0905.1943] [INSPIRE].
S. Giombi and S. Komatsu, Exact Correlators on the Wilson Loop in \( \mathcal{N} \) = 4 SYM: Localization, Defect CFT, and Integrability, JHEP 05 (2018) 109 [Erratum ibid. 11 (2018) 123] [arXiv:1802.05201] [INSPIRE].
D.J. Gross and A. Matytsin, Some properties of large N two-dimensional Yang-Mills theory, Nucl. Phys. B 437 (1995) 541 [hep-th/9410054] [INSPIRE].
S. Giombi and V. Pestun, The 1/2 BPS ’t Hooft loops in N = 4 SYM as instantons in 2d Yang-Mills, J. Phys. A 46 (2013) 095402 [arXiv:0909.4272] [INSPIRE].
B.E. Rusakov, Loop averages and partition functions in U(N) gauge theory on two-dimensional manifolds, Mod. Phys. Lett. A 5 (1990) 693 [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
I. Aniceto, G. Basar and R. Schiappa, A Primer on Resurgent Transseries and Their Asymptotics, Phys. Rept. 809 (2019) 1 [arXiv:1802.10441] [INSPIRE].
D. Dorigoni and P. Glass, The grin of Cheshire cat resurgence from supersymmetric localization, SciPost Phys. 4 (2018) 012 [arXiv:1711.04802] [INSPIRE].
D. Dorigoni and P. Glass, Picard-Lefschetz decomposition and Cheshire Cat resurgence in 3D \( \mathcal{N} \) = 2 field theories, JHEP 12 (2019) 085 [arXiv:1909.05262] [INSPIRE].
T. Fujimori and P. Glass, Resurgence in 2-dimensional Yang-Mills and a genus-altering deformation, PTEP 2023 (2023) 053B03 [arXiv:2212.11988] [INSPIRE].
I. Aniceto, J.G. Russo and R. Schiappa, Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories, JHEP 03 (2015) 172 [arXiv:1410.5834] [INSPIRE].
M. Honda, Borel Summability of Perturbative Series in 4D N = 2 and 5D N = 1 Supersymmetric Theories, Phys. Rev. Lett. 116 (2016) 211601 [arXiv:1603.06207] [INSPIRE].
M. Honda and D. Yokoyama, Resumming perturbative series in the presence of monopole bubbling effects, Phys. Rev. D 100 (2019) 025012 [arXiv:1711.10799] [INSPIRE].
M. Honda, How to resum perturbative series in 3d N = 2 Chern-Simons matter theories, Phys. Rev. D 94 (2016) 025039 [arXiv:1604.08653] [INSPIRE].
M.F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984) 1 [INSPIRE].
M.F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. Lond. A 308 (1982) 523 [INSPIRE].
K.G. Wilson, Confinement of Quarks, Phys. Rev. D 10 (1974) 2445 [INSPIRE].
P. Goddard, J. Nuyts and D.I. Olive, Gauge Theories and Magnetic Charge, Nucl. Phys. B 125 (1977) 1 [INSPIRE].
A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].
D. Ghoshal, C. Imbimbo and D. Kumar, Weak Coupling Expansion of Yang-Mills Theory on Recursive Infinite Genus Surfaces, JHEP 10 (2014) 181 [arXiv:1407.6380] [INSPIRE].
A.L. Durán, R. Estrada and R.P. Kanwal, Extensions of the Poisson Summation Formula, J. Math. Anal. Appl. 218 (1998) 581.
F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfaces, Proc. Symp. Pure Math. 96 (2017) 13 [arXiv:1605.06120] [INSPIRE].
A.S. Schwarz and O. Zaboronsky, Supersymmetry and localization, Commun. Math. Phys. 183 (1997) 463 [hep-th/9511112] [INSPIRE].
E.H. Leeb-Lundberg, Unstable Instantons in A-model Localization, arXiv:2312.11347 [INSPIRE].
F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].
C. Closset, S. Cremonesi and D.S. Park, The equivariant A-twist and gauged linear sigma models on the two-sphere, JHEP 06 (2015) 076 [arXiv:1504.06308] [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of two-dimensional N = 2 gauge theories with rank-one gauge groups, Lett. Math. Phys. 104 (2014) 465 [arXiv:1305.0533] [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic Genera of 2d \( \mathcal{N} \) = 2 Gauge Theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].
L.C. Jeffrey and F.C. Kirwan, Localization for nonabelian group actions, alg-geom/9307001 [INSPIRE].
M. Bershtein, G. Bonelli, M. Ronzani and A. Tanzini, Exact results for \( \mathcal{N} \) = 2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants, JHEP 07 (2016) 023 [arXiv:1509.00267] [INSPIRE].
S.M. Hosseini, I. Yaakov and A. Zaffaroni, Topologically twisted indices in five dimensions and holography, JHEP 11 (2018) 119 [arXiv:1808.06626] [INSPIRE].
Acknowledgments
LG and DS are supported, in part, by Istituto Nazionale di Fisica Nucleare (INFN) through the “Gauge and String Theory” (GAST) research project. The research of RP is funded, in part, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer 277101999 – TRR 183 (project A03 and B01). JP acknowledges financial support from the European Research Council (grant BHHQG-101040024). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. IY is supported, in part, by UKRI consolidated grants ST/P000711/1 and ST/T000775/1.
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Griguolo, L., Panerai, R., Papalini, J. et al. Localization and resummationof unstable instantons in 2d Yang-Mills. J. High Energ. Phys. 2024, 188 (2024). https://doi.org/10.1007/JHEP06(2024)188
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DOI: https://doi.org/10.1007/JHEP06(2024)188