Abstract
We relate Liouville/Toda CFT correlators on Riemann surfaces with boundaries and cross-cap states to supersymmetric observables in four-dimensional \( \mathcal{N}=2 \) gauge theories. Our construction naturally involves four-dimensional theories with fields defined on different ℤ2 quotients of the sphere (hemisphere and projective space) but nevertheless interacting with each other. The six-dimensional origin is a ℤ2 quotient of the setup giving rise to the usual AGT correspondence. To test the correspondence, we work out the ℝℙ4 partition function of four-dimensional \( \mathcal{N}=2 \) theories by combining a 3d lens space and a 4d hemisphere partition functions. The same technique reproduces known ℝℙ2 partition functions in a form that lets us easily check two-dimensional Seiberg-like dualities on this nonorientable space. As a bonus we work out boundary and cross-cap wavefunctions in Toda CFT.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L.F. Alday, M. Fluder and J. Sparks, The large-N limit of M2-branes on Lens spaces, JHEP 10 (2012) 057 [arXiv:1204.1280] [INSPIRE].
L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [INSPIRE].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
L.F. Alday and Y. Tachikawa, Affine SL(2) conformal blocks from 4d gauge theories, Lett. Math. Phys. 94 (2010) 87 [arXiv:1005.4469] [INSPIRE].
A. Belavin, V. Belavin and M. Bershtein, Instantons and 2d superconformal field theory, JHEP 09 (2011) 117 [arXiv:1106.4001] [INSPIRE].
A. Bawane, S. Benvenuti, G. Bonelli, N. Muteeb and A. Tanzini, \( \mathcal{N}=2 \) gauge theories on unoriented/open four-manifolds and their AGT counterparts, arXiv:1710.06283.
A.A. Belavin, M.A. Bershtein, B.L. Feigin, A.V. Litvinov and G.M. Tarnopolsky, Instanton moduli spaces and bases in coset conformal field theory, Commun. Math. Phys. 319 (2013) 269 [arXiv:1111.2803] [INSPIRE].
G. Bonelli, K. Maruyoshi, A. Tanzini and F. Yagi, \( \mathcal{N}=2 \) gauge theories on toric singularities, blow-up formulae and W-algebrae, JHEP 01 (2013) 014 [arXiv:1208.0790].
F. Benini and S. Cremonesi, Partition functions of \( \mathcal{N}=\left(2,2\right) \) gauge theories on S 2 and vortices, Commun. Math. Phys. 334 (2015) 1483 [arXiv:1206.2356] [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].
V. Belavin and B. Feigin, Super Liouville conformal blocks from N = 2 SU(2) quiver gauge theories, JHEP 07 (2011) 079 [arXiv:1105.5800] [INSPIRE].
F. Benini and B. Le Floch, Supersymmetric localization in two dimensions, J. Phys. A 50 (2017) 443003 [arXiv:1608.02955] [INSPIRE].
F. Benini, T. Nishioka and M. Yamazaki, 4D index to 3D index and 2D TQFT, Phys. Rev. D 86 (2012) 065015 [arXiv:1109.0283] [INSPIRE].
F. Benini and W. Peelaers, Higgs branch localization in three dimensions, JHEP 05 (2014) 030 [arXiv:1312.6078] [INSPIRE].
R.E. Behrend, P.A. Pearce, V.B. Petkova and J.-B. Zuber, Boundary conditions in rational conformal field theories, Nucl. Phys. B 570 (2000) 525 [hep-th/9908036] [INSPIRE].
F. Benini, D.S. Park and P. Zhao, Cluster algebras from dualities of 2D \( \mathcal{N}=\left(2,2\right) \) quiver gauge theories, Commun. Math. Phys. 340 (2015) 47 [arXiv:1406.2699] [INSPIRE].
V. Belavin and N. Wyllard, N = 2 superconformal blocks and instanton partition functions, JHEP 06 (2012) 173 [arXiv:1205.3091] [INSPIRE].
J.L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].
C. Cordova and D.L. Jafferis, Toda theory from six dimensions, arXiv:1605.03997 [INSPIRE].
C. Cordova and S.-H. Shao, An index formula for supersymmetric quantum mechanics, arXiv:1406.7853 [INSPIRE].
H.-Y. Chen and T.-H. Tsai, On Higgs branch localization of Seiberg-Witten theories on an ellipsoid, PTEP 2016 (2016) 013B09 [arXiv:1506.04390] [INSPIRE].
M. Dedushenko, work in progress.
M. Dedushenko, Y. Fan, S. Pufu and R. Yacoby, Correlators of Coulomb branch operators in three dimensions, work in progress.
N. Drukker, D. Gaiotto and J. Gomis, The virtue of defects in 4D gauge theories and 2D CFTs, JHEP 06 (2011) 025 [arXiv:1003.1112] [INSPIRE].
N. Doroud, J. Gomis, B. Le Floch and S. Lee, Exact results in D = 2 supersymmetric gauge theories, JHEP 05 (2013) 093 [arXiv:1206.2606] [INSPIRE].
J. Erdmenger, Z. Guralnik and I. Kirsch, Four-dimensional superconformal theories with interacting boundaries or defects, Phys. Rev. D 66 (2002) 025020 [hep-th/0203020] [INSPIRE].
V.A. Fateev, Normalization factors, reflection amplitudes and integrable systems, hep-th/0103014 [INSPIRE].
G. Felder, J. Fröhlich, J. Fuchs and C. Schweigert, Conformal boundary conditions and three-dimensional topological field theory, Phys. Rev. Lett. 84 (2000) 1659 [hep-th/9909140] [INSPIRE].
G. Felder, J. Fröhlich, J. Fuchs and C. Schweigert, Correlation functions and boundary conditions in RCFT and three-dimensional topology, Compos. Math. 131 (2002) 189 [hep-th/9912239] [INSPIRE].
M. Fujitsuka, M. Honda and Y. Yoshida, Higgs branch localization of 3D \( \mathcal{N}=2 \) theories, PTEP 2014 (2014) 123B02 [arXiv:1312.3627] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro conformal blocks and thermality from classical background fields, JHEP 11 (2015) 200 [arXiv:1501.05315] [INSPIRE].
S. Fredenhagen, Boundary conditions in Toda theories and minimal models, JHEP 02 (2011) 052 [arXiv:1012.0485] [INSPIRE].
V. Fateev and S. Ribault, Conformal Toda theory with a boundary, JHEP 12 (2010) 089 [arXiv:1007.1293] [INSPIRE].
V. Fateev, A.B. Zamolodchikov and A.B. Zamolodchikov, Boundary Liouville field theory. 1. Boundary state and boundary two point function, hep-th/0001012 [INSPIRE].
D. Gaiotto, Boundary F-maximization, arXiv:1403.8052 [INSPIRE].
D. Gaiotto, Open Verlinde line operators, arXiv:1404.0332 [INSPIRE].
D. Gang, Chern-Simons theory on L(p, q) lens spaces and localization, arXiv:0912.4664 [INSPIRE].
A. Gadde and S. Gukov, 2D index and surface operators, JHEP 03 (2014) 080 [arXiv:1305.0266] [INSPIRE].
J. Gomis and B. Le Floch, ’t Hooft operators in gauge theory from Toda CFT, JHEP 11 (2011) 114 [arXiv:1008.4139] [INSPIRE].
J. Gomis and S. Lee, Exact Kähler potential from gauge theory and mirror symmetry, JHEP 04 (2013) 019 [arXiv:1210.6022] [INSPIRE].
J. Gomis, B. Le Floch, Y. Pan and W. Peelaers, Intersecting surface defects and two-dimensional CFT, Phys. Rev. D 96 (2017) 045003 [arXiv:1610.03501] [INSPIRE].
E. Gava, K.S. Narain, M.N. Muteeb and V.I. Giraldo-Rivera, N = 2 gauge theories on the hemisphere HS 4, Nucl. Phys. B 920 (2017) 256 [arXiv:1611.04804] [INSPIRE].
J. Gomis, T. Okuda and V. Pestun, Exact results for ’t Hooft loops in gauge theories on S 4, JHEP 05 (2012) 141 [arXiv:1105.2568] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The 4D superconformal index from q-deformed 2D Yang-Mills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [INSPIRE].
D. Gaiotto and J. Teschner, Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories, I, JHEP 12 (2012) 050 [arXiv:1203.1052] [INSPIRE].
S. Gukov, Surface operators, arXiv:1412.7127.
D. Gaiotto and E. Witten, Supersymmetric boundary conditions in N = 4 super Yang-Mills theory, J. Statist. Phys. 135 (2009) 789 [arXiv:0804.2902] [INSPIRE].
D. Gaiotto and E. Witten, S-duality of boundary conditions in N = 4 super Yang-Mills theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
N. Hama and K. Hosomichi, Seiberg-Witten theories on ellipsoids, JHEP 09 (2012) 033 [arXiv:1206.6359] [INSPIRE].
C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5D SCFT, JHEP 07 (2015) 063 [arXiv:1406.6793] [INSPIRE].
K. Hori, H. Kim and P. Yi, Witten index and wall crossing, JHEP 01 (2015) 124 [arXiv:1407.2567] [INSPIRE].
K. Hosomichi, S. Lee and J. Park, AGT on the S-duality wall, JHEP 12 (2010) 079 [arXiv:1009.0340] [INSPIRE].
D. Honda and T. Okuda, Exact results for boundaries and domain walls in 2d supersymmetric theories, JHEP 09 (2015) 140 [arXiv:1308.2217] [INSPIRE].
P. Hořava, Open strings from three-dimensions: Chern-Simons-Witten theory on orbifolds, J. Geom. Phys. 21 (1996) 1 [hep-th/9404101] [INSPIRE].
K. Hori, C.Y. Park and Y. Tachikawa, 2D SCFTs from M2-branes, JHEP 11 (2013) 147 [arXiv:1309.3036] [INSPIRE].
K. Hori and M. Romo, Exact results in two-dimensional (2, 2) supersymmetric gauge theories with boundary, arXiv:1308.2438 [INSPIRE].
N. Ishibashi, The boundary and crosscap states in conformal field theories, Mod. Phys. Lett. A 4 (1989) 251 [INSPIRE].
Y. Imamura, H. Matsuno and D. Yokoyama, Factorization of the S 3/ℤ n partition function, Phys. Rev. D 89 (2014) 085003 [arXiv:1311.2371] [INSPIRE].
Y. Imamura and D. Yokoyama, S 3/ℤ n partition function and dualities, JHEP 11 (2012) 122 [arXiv:1208.1404] [INSPIRE].
S. Jackson, L. McGough and H. Verlinde, Conformal bootstrap, universality and gravitational scattering, Nucl. Phys. B 901 (2015) 382 [arXiv:1412.5205] [INSPIRE].
C. Jego and J. Troost, Notes on the Verlinde formula in non-rational conformal field theories, Phys. Rev. D 74 (2006) 106002 [hep-th/0601085] [INSPIRE].
H. Kim, S. Lee and P. Yi, Exact partition functions on ℝℙ2 and orientifolds, JHEP 02 (2014) 103 [arXiv:1310.4505] [INSPIRE].
H. Kanno and Y. Tachikawa, Instanton counting with a surface operator and the chain-saw quiver, JHEP 06 (2011) 119 [arXiv:1105.0357] [INSPIRE].
B. Le Floch, Quiver description of M5 brane surface operators, work in progress.
D.C. Lewellen, Sewing constraints for conformal field theories on surfaces with boundaries, Nucl. Phys. B 372 (1992) 654 [INSPIRE].
A. Lewkowycz, G.J. Turiaci and H. Verlinde, A CFT perspective on gravitational dressing and bulk locality, JHEP 01 (2017) 004 [arXiv:1608.08977] [INSPIRE].
M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Continuous multiscale entanglement renormalization ansatz as holographic surface-state correspondence, Phys. Rev. Lett. 115 (2015) 171602 [arXiv:1506.01353] [INSPIRE].
A. Maloney and S.F. Ross, Holography on non-orientable surfaces, Class. Quant. Grav. 33 (2016) 185006 [arXiv:1603.04426] [INSPIRE].
T.G. Mertens, G.J. Turiaci and H.L. Verlinde, Solving the Schwarzian via the conformal bootstrap, JHEP 08 (2017) 136 [arXiv:1705.08408] [INSPIRE].
Y. Nakayama and H. Ooguri, Bulk locality and boundary creating operators, JHEP 10 (2015) 114 [arXiv:1507.04130] [INSPIRE].
F. Nieri and S. Pasquetti, Factorisation and holomorphic blocks in 4D, JHEP 11 (2015) 155 [arXiv:1507.00261] [INSPIRE].
N. Nekrasov and E. Witten, The Omega deformation, branes, integrability and Liouville theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [INSPIRE].
K. Ohta and Y. Sasai, Exact results in quiver quantum mechanics and BPS bound state counting, JHEP 11 (2014) 123 [arXiv:1408.0582] [INSPIRE].
K. Ohta and Y. Sasai, Coulomb branch localization in quiver quantum mechanics, JHEP 02 (2016) 106 [arXiv:1512.00594] [INSPIRE].
Y. Pan, 5D Higgs branch localization, Seiberg-Witten equations and contact geometry, JHEP 01 (2015) 145 [arXiv:1406.5236] [INSPIRE].
W. Peelaers, Higgs branch localization of \( \mathcal{N}=1 \) theories on S 3 × S 1, JHEP 08 (2014) 060 [arXiv:1403.2711] [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].
B. Ponsot, Liouville theory on the pseudosphere: bulk boundary structure constant, Phys. Lett. B 588 (2004) 105 [hep-th/0309211] [INSPIRE].
Y. Pan and W. Peelaers, Ellipsoid partition function from Seiberg-Witten monopoles, JHEP 10 (2015) 183 [arXiv:1508.07329] [INSPIRE].
B. Ponsot and J. Teschner, Boundary Liouville field theory: boundary three point function, Nucl. Phys. B 622 (2002) 309 [hep-th/0110244] [INSPIRE].
I. Runkel, Boundary structure constants for the A series Virasoro minimal models, Nucl. Phys. B 549 (1999) 563 [hep-th/9811178] [INSPIRE].
G. Sarkissian, Some remarks on D-branes and defects in Liouville and Toda field theories, Int. J. Mod. Phys. A 27 (2012) 1250181 [arXiv:1108.0242] [INSPIRE].
S. Sugishita and S. Terashima, Exact results in supersymmetric field theories on manifolds with boundaries, JHEP 11 (2013) 021 [arXiv:1308.1973] [INSPIRE].
J. Teschner, Remarks on Liouville theory with boundary, hep-th/0009138 [INSPIRE].
J. Teschner and G. Vartanov, 6j symbols for the modular double, quantum hyperbolic geometry and supersymmetric gauge theories, Lett. Math. Phys. 104 (2014) 527 [arXiv:1202.4698] [INSPIRE].
G. Turiaci and H. Verlinde, On CFT and quantum chaos, JHEP 12 (2016) 110 [arXiv:1603.03020] [INSPIRE].
Y. Tachikawa and K. Yonekura, Gauge interactions and topological phases of matter, PTEP 2016 (2016) 093B07 [arXiv:1604.06184] [INSPIRE].
E.P. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
H. Verlinde, Poking holes in AdS/CFT: bulk fields from boundary states, arXiv:1505.05069 [INSPIRE].
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
M. Yamazaki, Quantum trilogy: discrete Toda, Y-system and chaos, arXiv:1610.06925 [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Liouville field theory on a pseudosphere, hep-th/0101152 [INSPIRE].
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: 1708.04631
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Le Floch, B., Turiaci, G.J. AGT/ℤ2. J. High Energ. Phys. 2017, 99 (2017). https://doi.org/10.1007/JHEP12(2017)099
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2017)099