Abstract
We propose a way to define and compute invariants of general smooth 4-manifolds based on topological twists of non-Lagrangian 4d \( \mathcal{N}=2\kern0.5em \mathrm{and}\kern0.5em \mathcal{N}=3 \) theories in which the problem is reduced to a fairly standard computation in topological A-model, albeit with rather unusual targets, such as compact and non-compact Gepner models, asymmetric orbifolds, \( \mathcal{N}=\left(2,2\right) \) linear dilaton theories, “self-mirror” geometries, varieties with complex multiplication, etc.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Witten, Topological quantum field theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].
M. Dedushenko, S. Gukov and P. Putrov, Vertex algebras and 4-manifold invariants, arXiv:1705.01645 [INSPIRE].
P.C. Argyres and M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B 448 (1995) 93 [hep-th/9505062] [INSPIRE].
D. Xie, General Argyres-Douglas theory, JHEP 01 (2013) 100 [arXiv:1204.2270] [INSPIRE].
O. Aharony and M. Evtikhiev, On four dimensional N = 3 superconformal theories, JHEP 04 (2016) 040 [arXiv:1512.03524] [INSPIRE].
I. García-Etxebarria and D. Regalado, N = 3 four dimensional field theories, JHEP 03 (2016) 083 [arXiv:1512.06434] [INSPIRE].
O. Aharony and Y. Tachikawa, S-folds and 4d N = 3 superconformal field theories, JHEP 06 (2016) 044 [arXiv:1602.08638] [INSPIRE].
P.C. Argyres, Y. Lü and M. Martone, Seiberg-Witten geometries for Coulomb branch chiral rings which are not freely generated, JHEP 06 (2017) 144 [arXiv:1704.05110] [INSPIRE].
F. Benini, Y. Tachikawa and D. Xie, Mirrors of 3d Sicilian theories, JHEP 09 (2010) 063 [arXiv:1007.0992] [INSPIRE].
M. Bershadsky, A. Johansen, V. Sadov and C. Vafa, Topological reduction of 4D SYM to 2D σ-models, Nucl. Phys. B 448 (1995) 166 [hep-th/9501096] [INSPIRE].
J.A. Harvey, G.W. Moore and A. Strominger, Reducing S duality to T duality, Phys. Rev. D 52 (1995) 7161 [hep-th/9501022] [INSPIRE].
C. Lozano and M. Mariño, Donaldson invariants of product ruled surfaces and two-dimensional gauge theories, Commun. Math. Phys. 220 (2001) 231 [hep-th/9907165] [INSPIRE].
A. Losev, N. Nekrasov and S.L. Shatashvili, Freckled instantons in two-dimensions and four-dimensions, Class. Quant. Grav. 17 (2000) 1181 [hep-th/9911099] [INSPIRE].
J.M. Maldacena and C. Núñez, Supergravity description of field theories on curved manifolds and a no go theorem, Int. J. Mod. Phys. A 16 (2001) 822 [hep-th/0007018] [INSPIRE].
A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].
D. Gay and R. Kirby, Trisecting 4-manifolds, Geom. Topol. 20 (2016) 3097 [arXiv:1205.1565].
P. Ozsváth and Z. Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006) 326.
S. Gukov, Gauge theory and knot homologies, Fortsch. Phys. 55 (2007) 473 [arXiv:0706.2369] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, in Arbeitstagung Bonn 2013, Progr. Math. 319, Birkhäuser/Springer, Cham Switzerland, 2016, pg. 155 [arXiv:1306.4320] [INSPIRE].
D. Baraglia and L.P. Schaposnik, Higgs bundles and (A, B, A)-branes, Commun. Math. Phys. 331 (2014) 1271 [arXiv:1305.4638] [INSPIRE].
S. Gukov, Three-dimensional quantum gravity, Chern-Simons theory and the A polynomial, Commun. Math. Phys. 255 (2005) 577 [hep-th/0306165] [INSPIRE].
M. Kontsevich, Homological algebra of mirror symmetry, in Proceedings of the International Congress of Mathematicians, Birkhäuser, Basel Switzerland, (1995), pg. 120 [alg-geom/9411018] [INSPIRE].
A. Polishchuk and E. Zaslow, Categorical mirror symmetry: the elliptic curve, Adv. Theor. Math. Phys. 2 (1998) 443 [math/9801119] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Walls, lines and spectral dualities in 3d gauge theories, JHEP 05 (2014) 047 [arXiv:1302.0015] [INSPIRE].
H.-J. Chung, T. Dimofte, S. Gukov and P. Sulkowski, 3d-3d correspondence revisited, JHEP 04 (2016) 140 [arXiv:1405.3663] [INSPIRE].
S. Gukov, M. Mariño and P. Putrov, Resurgence in complex Chern-Simons theory, arXiv:1605.07615 [INSPIRE].
T. Dimofte and S. Gukov, Chern-Simons theory and S-duality, JHEP 05 (2013) 109 [arXiv:1106.4550] [INSPIRE].
P. Putrov, J. Song and W. Yan, (0, 4) dualities, JHEP 03 (2016) 185 [arXiv:1505.07110] [INSPIRE].
N.J. Hitchin, The selfduality equations on a Riemann surface, Proc. Lond. Math. Soc. 55 (1987) 59 [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Bethe/gauge correspondence on curved spaces, JHEP 01 (2015) 100 [arXiv:1405.6046] [INSPIRE].
E. Witten, On S duality in Abelian gauge theory, Selecta Math. 1 (1995) 383 [hep-th/9505186] [INSPIRE].
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of N = 2 SCFTs III: enhanced Coulomb branches and central charges, arXiv:1609.04404 [INSPIRE].
W. Lerche, D. Lüst and N.P. Warner, Duality symmetries in N = 2 Landau-Ginzburg models, Phys. Lett. B 231 (1989) 417 [INSPIRE].
E.J. Chun, J. Lauer and H.P. Nilles, Equivalence of ZN orbifolds and Landau-Ginzburg models, Int. J. Mod. Phys. A 7 (1992) 2175 [INSPIRE].
K.S. Narain, M.H. Sarmadi and C. Vafa, Asymmetric orbifolds, Nucl. Phys. B 288 (1987) 551 [INSPIRE].
I. Brunner, M. Herbst, W. Lerche and J. Walcher, Matrix factorizations and mirror symmetry: the cubic curve, JHEP 11 (2006) 006 [hep-th/0408243] [INSPIRE].
M. Herbst, W. Lerche and D. Nemeschansky, Instanton geometry and quantum A-infinity structure on the elliptic curve, hep-th/0603085 [INSPIRE].
E.P. Verlinde, Global aspects of electric-magnetic duality, Nucl. Phys. B 455 (1995) 211 [hep-th/9506011] [INSPIRE].
A. Gadde and S. Gukov, 2d index and surface operators, JHEP 03 (2014) 080 [arXiv:1305.0266] [INSPIRE].
A.D. Shapere and Y. Tachikawa, Central charges of N = 2 superconformal field theories in four dimensions, JHEP 09 (2008) 109 [arXiv:0804.1957] [INSPIRE].
D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
S.K. Donaldson, Polynomial invariants for smooth manifolds, Topology 29 (1990) 257 [INSPIRE].
M. Mariño, G.W. Moore and G. Peradze, Superconformal invariance and the geography of four manifolds, Commun. Math. Phys. 205 (1999) 691 [hep-th/9812055] [INSPIRE].
L.D. Faddeev and S.L. Shatashvili, Algebraic and Hamiltonian methods in the theory of non-Abelian anomalies, Theor. Math. Phys. 60 (1985) 770 [Teor. Mat. Fiz. 60 (1984) 206] [INSPIRE].
L. Álvarez-Gaumé and E. Witten, Gravitational anomalies, Nucl. Phys. B 234 (1984) 269 [INSPIRE].
K. Hori et al., Mirror symmetry, in Clay mathematics monographs, volume 1, AMS, Providence U.S.A., (2003).
P.C. Argyres and J.R. Wittig, Infinite coupling duals of N = 2 gauge theories and new rank 1 superconformal field theories, JHEP 01 (2008) 074 [arXiv:0712.2028] [INSPIRE].
E. Witten, Supersymmetric Yang-Mills theory on a four manifold, J. Math. Phys. 35 (1994) 5101 [hep-th/9403195] [INSPIRE].
A. Johansen, Twisting of N = 1 SUSY gauge theories and heterotic topological theories, Int. J. Mod. Phys. A 10 (1995) 4325 [hep-th/9403017] [INSPIRE].
A. Losev, N. Nekrasov and S.L. Shatashvili, Issues in topological gauge theory, Nucl. Phys. B 534 (1998) 549 [hep-th/9711108] [INSPIRE].
C. Vafa, Topological Landau-Ginzburg models, Mod. Phys. Lett. A 6 (1991) 337 [INSPIRE].
I.V. Melnikov and M.R. Plesser, A-model correlators from the Coulomb branch, hep-th/0507187 [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE].
D. Tong, Superconformal vortex strings, JHEP 12 (2006) 051 [hep-th/0610214] [INSPIRE].
D. Gaiotto, S. Gukov and N. Seiberg, Surface defects and resolvents, JHEP 09 (2013) 070 [arXiv:1307.2578] [INSPIRE].
S. Gukov, P. Putrov and C. Vafa, Fivebranes and 3-manifold homology, JHEP 07 (2017) 071 [arXiv:1602.05302] [INSPIRE].
T. Nishinaka and Y. Tachikawa, On 4d rank-one N = 3 superconformal field theories, JHEP 09 (2016) 116 [arXiv:1602.01503] [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].
M. Honda and Y. Yoshida, Supersymmetric index on T 2 × S 2 and elliptic genus, arXiv:1504.04355 [INSPIRE].
F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfaces, Proc. Symp. Pure Math. 96 (2017) 13 [arXiv:1605.06120] [INSPIRE].
A. Gadde, S.S. Razamat and B. Willett, “Lagrangian” for a non-Lagrangian field theory with N = 2 supersymmetry,Phys. Rev. Lett. 115 (2015) 171604 [arXiv:1505.05834] [INSPIRE].
K. Maruyoshi and J. Song, Enhancement of supersymmetry via renormalization group flow and the superconformal index, Phys. Rev. Lett. 118 (2017) 151602 [arXiv:1606.05632] [INSPIRE].
K. Maruyoshi and J. Song, N = 1 deformations and RG flows of N = 2 SCFTs, JHEP 02 (2017) 075 [arXiv:1607.04281] [INSPIRE].
A. Gadde, S.S. Razamat and B. Willett, On the reduction of 4d N = 1 theories on S 2, JHEP 11 (2015) 163 [arXiv:1506.08795] [INSPIRE].
A. Amariti, L. Cassia and S. Penati, Surveying 4d SCFTs twisted on Riemann surfaces, JHEP 06 (2017) 056 [arXiv:1703.08201] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, The long flow to freedom, JHEP 02 (2017) 056 [arXiv:1611.02763] [INSPIRE].
L. Fredrickson, D. Pei, W. Yan and K. Ye, Argyres-Douglas theories, chiral algebras and wild Hitchin characters, arXiv:1701.08782 [INSPIRE].
S. Gukov, Quantization via mirror symmetry, Jpn. J. Math. 6 (2011) 65 [arXiv:1011.2218] [INSPIRE].
O.J. Ganor, N.P. Moore, H.-Y. Sun and N.R. Torres-Chicon, Janus configurations with SL(2, Z)-duality twists, strings on mapping tori and a tridiagonal determinant formula, JHEP 07 (2014) 010 [arXiv:1403.2365] [INSPIRE].
E. Frenkel and E. Witten, Geometric endoscopy and mirror symmetry, Commun. Num. Theor. Phys. 2 (2008) 113 [arXiv:0710.5939] [INSPIRE].
S. Gukov and D. Pei, Equivariant Verlinde formula from fivebranes and vortices, Commun. Math. Phys. 355 (2017) 1 [arXiv:1501.01310] [INSPIRE].
S. Mukhi and C. Vafa, Two-dimensional black hole as a topological coset model of c = 1 string theory, Nucl. Phys. B 407 (1993) 667 [hep-th/9301083] [INSPIRE].
D. Ghoshal and C. Vafa, c = 1 string as the topological theory of the conifold, Nucl. Phys. B 453 (1995) 121 [hep-th/9506122] [INSPIRE].
H. Ooguri and C. Vafa, Two-dimensional black hole and singularities of CY manifolds, Nucl. Phys. B 463 (1996) 55 [hep-th/9511164] [INSPIRE].
A. Giveon, D. Kutasov and O. Pelc, Holography for noncritical superstrings, JHEP 10 (1999) 035 [hep-th/9907178] [INSPIRE].
A. Giveon and D. Kutasov, Little string theory in a double scaling limit, JHEP 10 (1999) 034 [hep-th/9909110] [INSPIRE].
T. Eguchi and Y. Sugawara, Modular invariance in superstring on Calabi-Yau n-fold with ADE singularity, Nucl. Phys. B 577 (2000) 3 [hep-th/0002100] [INSPIRE].
W. Lerche, On a boundary CFT description of nonperturbative N = 2 Yang-Mills theory, hep-th/0006100 [INSPIRE].
T. Eguchi and Y. Sugawara, SL(2, R)/U(1) supercoset and elliptic genera of noncompact Calabi-Yau manifolds, JHEP 05 (2004) 014 [hep-th/0403193] [INSPIRE].
T. Eguchi and Y. Sugawara, Conifold type singularities, N = 2 Liouville and SL(2 : R)/U(1) theories, JHEP 01 (2005) 027 [hep-th/0411041] [INSPIRE].
S.K. Ashok, R. Benichou and J. Troost, Non-compact Gepner models, Landau-Ginzburg orbifolds and mirror symmetry, JHEP 01 (2008) 050 [arXiv:0710.1990] [INSPIRE].
S.K. Ashok and J. Troost, Elliptic genera of non-compact Gepner models and mirror symmetry, JHEP 07 (2012) 005 [arXiv:1204.3802] [INSPIRE].
K. Hori and A. Kapustin, Duality of the fermionic 2D black hole and N = 2 Liouville theory as mirror symmetry, JHEP 08 (2001) 045 [hep-th/0104202] [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: 1707.01515
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
Gukov, S. Trisecting non-Lagrangian theories. J. High Energ. Phys. 2017, 178 (2017). https://doi.org/10.1007/JHEP11(2017)178
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2017)178