Abstract
Five-dimensional Sp(N) supersymmetric Yang-Mills admits a ℤ2 version of a theta angle θ. In this note, we derive a double quantization of the Seiberg-Witten geometry of \( \mathcal{N} \) = 1 Sp(1) gauge theory at θ = π, on the manifold S1 × ℝ4. Crucially, ℝ4 is placed on the Ω-background, which provides the two parameters to quantize the geometry. Physically, we are counting instantons in the presence of a 1/2-BPS fundamental Wilson loop, both of which are wrapping S1. Mathematically, this amounts to proving the regularity of a qq-character for the spin-1/2 representation of the quantum affine algebra \( {U}_q\left(\hat{A_1}\right) \), with a certain twist due to the θ-angle. We motivate these results from two distinct string theory pictures. First, in a (p, q)-web setup in type IIB, where the loop is characterized by a D3 brane. Second, in a type I′ string setup, where the loop is characterized by a D4 brane subject to an orientifold projection. We comment on the generalizations to the higher rank case Sp(N) when N > 1, and the SU(N) theory at Chern-Simons level κ when N > 2.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
H.-C. Kim, S.-S. Kim and K. Lee, 5-dim Superconformal Index with Enhanced En Global Symmetry, JHEP 10 (2012) 142 [arXiv:1206.6781] [INSPIRE].
C.-M. Chang, O. Ganor and J. Oh, An index for ray operators in 5d En SCFTs, JHEP 02 (2017) 018 [arXiv:1608.06284] [INSPIRE].
D.R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys. B 483 (1997) 229 [hep-th/9609070] [INSPIRE].
M.R. Douglas, S.H. Katz and C. Vafa, Small instantons, del Pezzo surfaces and type-I′ theory, Nucl. Phys. B 497 (1997) 155 [hep-th/9609071] [INSPIRE].
E. Witten, An SU(2) Anomaly, Phys. Lett. B 117 (1982) 324 [INSPIRE].
N. Nekrasov, BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters, JHEP 03 (2016) 181 [arXiv:1512.05388] [INSPIRE].
M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Y. Manin, Construction of Instantons, Phys. Lett. A 65 (1978) 185 [INSPIRE].
D. Tong and K. Wong, Instantons, Wilson lines, and D-branes, Phys. Rev. D 91 (2015) 026007 [arXiv:1410.8523] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, in The Unity of Mathematics, Progress in Mathematics 244, Springer (2006), pp. 525–596 [hep-th/0306238] [INSPIRE].
M. Jimbo, A q-Analog of U(\( \mathfrak{gl} \)(N + 1)), Hecke Algebra and the Yang-Baxter Equation, Lett. Math. Phys. 11 (1986) 247 [INSPIRE].
V.G. Drinfeld, A New realization of Yangians and quantized affine algebras, Sov. Math. Dokl. 36 (1988) 212 [INSPIRE].
V. Chari and A. Pressley, Quantum affine algebras and their representations, hep-th/9411145 [INSPIRE].
V. Chari, Minimal affinizations of representations of quantum groups: The Uq(\( \mathfrak{g} \)) module structure, hep-th/9411144 [INSPIRE].
E. Frenkel and N. Reshetikhin, The q-characters of representations of quantum affine algebras and deformations of W-algebras, in Recent Developments in Quantum Affine Algebras and Related Topics, Contemporary Mathematics 248, American Mathematical Society, Providence Rhode Island U.S.A. (1999) [math.QA/9810055].
J. Shiraishi, H. Kubo, H. Awata and S. Odake, A Quantum deformation of the Virasoro algebra and the Macdonald symmetric functions, Lett. Math. Phys. 38 (1996) 33 [q-alg/9507034] [INSPIRE].
H. Awata, H. Kubo, S. Odake and J. Shiraishi, Quantum WN algebras and Macdonald polynomials, Commun. Math. Phys. 179 (1996) 401 [q-alg/9508011] [INSPIRE].
E. Frenkel and N. Reshetikhin, Deformations of \( \mathcal{W} \)-algebras associated to simple Lie algebras, Commun. Math. Phys. 197 (1998) 1 [q-alg/9708006].
H. Nakajima, t-analogue of the q-characters of finite dimensional representations of quantum affine algebras, in Physics and Combinatorics , proceedings of the Nagoya 2000 International Workshop, Nagoya University, Nagoya, Japan, 21–26 August 2000, World Scientific (2000), pp. 196–219 [math.QA/0009231].
O. Aharony, A. Hanany and B. Kol, Webs of (p, q) five-branes, five-dimensional field theories and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [INSPIRE].
B. Assel and A. Sciarappa, Wilson loops in 5d \( \mathcal{N} \) = 1 theories and S-duality, JHEP 10 (2018) 082 [arXiv:1806.09636] [INSPIRE].
H.-C. Kim, Line defects and 5d instanton partition functions, JHEP 03 (2016) 199 [arXiv:1601.06841] [INSPIRE].
T. Kimura and V. Pestun, Quiver W-algebras, Lett. Math. Phys. 108 (2018) 1351 [arXiv:1512.08533] [INSPIRE].
A. Mironov, A. Morozov and Y. Zenkevich, Ding-Iohara-Miki symmetry of network matrix models, Phys. Lett. B 762 (2016) 196 [arXiv:1603.05467] [INSPIRE].
N. Haouzi and C. Kozçaz, Supersymmetric Wilson Loops, Instantons, and Deformed \( \mathcal{W} \)-Algebras, arXiv:1907.03838 [INSPIRE].
D. Tong, The holographic dual of AdS3 × S3 × S3 × S1, JHEP 04 (2014) 193 [arXiv:1402.5135] [INSPIRE].
N. Nekrasov and N.S. Prabhakar, Spiked Instantons from Intersecting D-branes, Nucl. Phys. B 914 (2017) 257 [arXiv:1611.03478] [INSPIRE].
J.-E. Bourgine, M. Fukuda, K. Harada, Y. Matsuo and R.-D. Zhu, (p, q)-webs of DIM representations, 5d \( \mathcal{N} \) = 1 instanton partition functions and qq-characters, JHEP 11 (2017) 034 [arXiv:1703.10759] [INSPIRE].
D. Gaiotto and H.-C. Kim, Duality walls and defects in 5d \( \mathcal{N} \) = 1 theories, JHEP 01 (2017) 019 [arXiv:1506.03871] [INSPIRE].
N. Haouzi and J. Oh, On the Quantization of Seiberg-Witten Geometry, arXiv:2004.00654 [INSPIRE].
O. Bergman, D. Rodríguez-Gómez and G. Zafrir, Discrete θ and the 5d superconformal index, JHEP 01 (2014) 079 [arXiv:1310.2150] [INSPIRE].
A. Iqbal and C. Vafa, BPS Degeneracies and Superconformal Index in Diverse Dimensions, Phys. Rev. D 90 (2014) 105031 [arXiv:1210.3605] [INSPIRE].
S.-S. Kim and F. Yagi, 5d En Seiberg-Witten curve via toric-like diagram, JHEP 06 (2015) 082 [arXiv:1411.7903] [INSPIRE].
G. Zafrir, Brane webs and O5-planes, JHEP 03 (2016) 109 [arXiv:1512.08114] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Discrete theta angle from an O5-plane, JHEP 11 (2017) 041 [arXiv:1707.07181] [INSPIRE].
O. Bergman and G. Zafrir, 5d fixed points from brane webs and O7-planes, JHEP 12 (2015) 163 [arXiv:1507.03860] [INSPIRE].
large N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].
S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity, Eur. Phys. J. C 22 (2001) 379 [hep-th/9803001] [INSPIRE].
N. Drukker and B. Fiol, All-genus calculation of Wilson loops using D-branes, JHEP 02 (2005) 010 [hep-th/0501109] [INSPIRE].
J. Gomis and F. Passerini, Holographic Wilson Loops, JHEP 08 (2006) 074 [hep-th/0604007] [INSPIRE].
S. Yamaguchi, Wilson loops of anti-symmetric representation and D5-branes, JHEP 05 (2006) 037 [hep-th/0603208] [INSPIRE].
P. Putrov, J. Song and W. Yan, (0, 4) dualities, JHEP 03 (2016) 185 [arXiv:1505.07110] [INSPIRE].
B. Assel and J. Gomis, Mirror Symmetry And Loop Operators, JHEP 11 (2015) 055 [arXiv:1506.01718] [INSPIRE].
U. Danielsson, G. Ferretti and I.R. Klebanov, Creation of fundamental strings by crossing D-branes, Phys. Rev. Lett. 79 (1997) 1984 [hep-th/9705084] [INSPIRE].
C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5d SCFT, JHEP 07 (2015) 063 [Addendum JHEP 04 (2016) 094] [arXiv:1406.6793] [INSPIRE].
B. Collie and D. Tong, Instantons, Fermions and Chern-Simons Terms, JHEP 07 (2008) 015 [arXiv:0804.1772] [INSPIRE].
S. Kim, K.-M. Lee and S. Lee, Dyonic Instantons in 5-dim Yang-Mills Chern-Simons Theories, JHEP 08 (2008) 064 [arXiv:0804.1207] [INSPIRE].
L. Jeffrey and F. Kirwan, Localization for nonabelian group actions, alg-geom/9307001.
C. Cordova and S.-H. Shao, An Index Formula for Supersymmetric Quantum Mechanics, arXiv:1406.7853 [INSPIRE].
K. Hori, H. Kim and P. Yi, Witten Index and Wall Crossing, JHEP 01 (2015) 124 [arXiv:1407.2567] [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].
T. Banks, N. Seiberg and E. Silverstein, Zero and one-dimensional probes with N = 8 supersymmetry, Phys. Lett. B 401 (1997) 30 [hep-th/9703052] [INSPIRE].
T.D. Brennan, A. Dey and G.W. Moore, ’t Hooft defects and wall crossing in SQM, JHEP 10 (2019) 173 [arXiv:1810.07191] [INSPIRE].
T.D. Brennan, Monopole Bubbling via String Theory, JHEP 11 (2018) 126 [arXiv:1806.00024] [INSPIRE].
N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional N = 2 quiver gauge theories, arXiv:1211.2240 [INSPIRE].
A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].
I.V. Cherednik, Factorizing Particles on a Half Line and Root Systems, Theor. Math. Phys. 61 (1984) 977 [Teor. Mat. Fiz. 61 (1984) 35] [INSPIRE].
E.K. Sklyanin, Boundary Conditions for Integrable Quantum Systems, J. Phys. A 21 (1988) 2375 [INSPIRE].
K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [INSPIRE].
S. Shadchin, Saddle point equations in Seiberg-Witten theory, JHEP 10 (2004) 033 [hep-th/0408066] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2005.13565
Rights and permissions
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.
About this article
Cite this article
Haouzi, N. Quantum geometry and θ-angle in five-dimensional super Yang-Mills. J. High Energ. Phys. 2020, 35 (2020). https://doi.org/10.1007/JHEP09(2020)035
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2020)035