Abstract
One of the open problems in understanding (0,2) mirror symmetry concerns the construction of Toda-like Landau-Ginzburg mirrors to (0,2) theories on Fano spaces. In this paper, we begin to fill this gap by making an ansatz for (0,2) Toda-like theories mirror to (0,2) supersymmetric nonlinear sigma models on products of projective spaces, with deformations of the tangent bundle, generalizing a special case previously worked out for \( {\mathrm{\mathbb{P}}}^1\times {\mathrm{\mathbb{P}}}^1 \). We check this ansatz by matching correlation functions of the B/2-twisted Toda-like theories to correlation functions of corresponding A/2-twisted nonlinear sigma models, computed primarily using localization techniques. These (0,2) Landau-Ginzburg models admit redundancies, which can lend themselves to multiple distinct-looking representatives of the same physics, which we discuss.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Kontsevich, Homological algebra of mirror symmetry, alg-geom/9411018 [INSPIRE].
R. Blumenhagen, R. Schimmrigk and A. Wisskirchen, (0, 2) mirror symmetry, Nucl. Phys. B 486 (1997) 598 [hep-th/9609167] [INSPIRE].
R. Blumenhagen and S. Sethi, On orbifolds of (0, 2) models, Nucl. Phys. B 491 (1997) 263 [hep-th/9611172] [INSPIRE].
B.R. Greene and M.R. Plesser, Duality in Calabi-Yau moduli space, Nucl. Phys. B 338 (1990) 15 [INSPIRE].
A. Adams, A. Basu and S. Sethi, (0, 2) duality, Adv. Theor. Math. Phys. 7 (2003) 865 [hep-th/0309226] [INSPIRE].
I.V. Melnikov and M.R. Plesser, A (0, 2) mirror map, JHEP 02 (2011) 001 [arXiv:1003.1303] [INSPIRE].
S.H. Katz and E. Sharpe, Notes on certain (0, 2) correlation functions, Commun. Math. Phys. 262 (2006) 611 [hep-th/0406226] [INSPIRE].
R. Donagi, J. Guffin, S. Katz and E. Sharpe, A mathematical theory of quantum sheaf cohomology, Asian J. Math. 18 (2014) 387 [arXiv:1110.3751] [INSPIRE].
R. Donagi, J. Guffin, S. Katz and E. Sharpe, Physical aspects of quantum sheaf cohomology for deformations of tangent bundles of toric varieties, Adv. Theor. Math. Phys. 17 (2013) 1255 [arXiv:1110.3752] [INSPIRE].
J. Guffin, Quantum sheaf cohomology, a precis, Mat. Contemp. 41 (2012) 17 [arXiv:1101.1305 ] [INSPIRE].
J. Guffin and S. Katz, Deformed quantum cohomology and (0, 2) mirror symmetry, JHEP 08 (2010) 109 [arXiv:0710.2354] [INSPIRE].
M. Kreuzer, J. McOrist, I.V. Melnikov and M.R. Plesser, (0, 2) deformations of linear σ-models, JHEP 07 (2011) 044 [arXiv:1001.2104] [INSPIRE].
J. McOrist, The revival of (0, 2) linear σ-models, Int. J. Mod. Phys. A 26 (2011) 1 [arXiv:1010.4667] [INSPIRE].
J. McOrist and I.V. Melnikov, Half-twisted correlators from the Coulomb branch, JHEP 04 (2008) 071 [arXiv:0712.3272] [INSPIRE].
J. McOrist and I.V. Melnikov, Old issues and linear σ-models, Adv. Theor. Math. Phys. 16 (2012) 251 [arXiv:1103.1322] [INSPIRE].
I.V. Melnikov, (0, 2) Landau-Ginzburg models and residues, JHEP 09 (2009) 118 [arXiv:0902.3908] [INSPIRE].
E. Sharpe, Notes on correlation functions in (0, 2) theories, hep-th/0502064 [INSPIRE].
E. Sharpe, Notes on certain other (0, 2) correlation functions, Adv. Theor. Math. Phys. 13 (2009) 33 [hep-th/0605005] [INSPIRE].
M.-C. Tan, Two-dimensional twisted σ-models and the theory of chiral differential operators, Adv. Theor. Math. Phys. 10 (2006) 759 [hep-th/0604179] [INSPIRE].
M.-C. Tan, Two-dimensional twisted σ-models, the mirror chiral de Rham complex and twisted generalised mirror symmetry, JHEP 07 (2007) 013 [arXiv:0705.0790] [INSPIRE].
J. Guffin and E. Sharpe, A-twisted heterotic Landau-Ginzburg models, J. Geom. Phys. 59 (2009) 1581 [arXiv:0801.3955] [INSPIRE].
C. Closset, W. Gu, B. Jia and E. Sharpe, Localization of twisted \( \mathcal{N} \) = (0, 2) gauged linear σ-models in two dimensions, JHEP 03 (2016) 070 [arXiv:1512.08058] [INSPIRE].
J. Guo, Z. Lu and E. Sharpe, Quantum sheaf cohomology on Grassmannians, arXiv:1512.08586 [INSPIRE].
I.V. Melnikov and S. Sethi, Half-twisted (0, 2) Landau-Ginzburg models, JHEP 03 (2008) 040 [arXiv:0712.1058] [INSPIRE].
J. McOrist and I.V. Melnikov, Summing the instantons in half-twisted linear σ-models, JHEP 02 (2009) 026 [arXiv:0810.0012] [INSPIRE].
E. Sharpe, An introduction to quantum sheaf cohomology, PoS(ICMP 2012)026.
R. Donagi, Z. Lu and I.V. Melnikov, Global aspects of (0, 2) moduli space: toric varieties and tangent bundles, Commun. Math. Phys. 338 (2015) 1197 [arXiv:1409.4353] [INSPIRE].
C. Vafa, Topological Landau-Ginzburg models, Mod. Phys. Lett. A 6 (1991) 337 [INSPIRE].
J. Guffin and E. Sharpe, A-twisted Landau-Ginzburg models, J. Geom. Phys. 59 (2009) 1547 [arXiv:0801.3836] [INSPIRE].
T. Pantev and E. Sharpe, GLSM’s for gerbes (and other toric stacks), Adv. Theor. Math. Phys. 10 (2006) 77 [hep-th/0502053] [INSPIRE].
S. Katz, private communication.
H. Iritani, A mirror construction for the big equivariant quantum cohomology of toric manifolds, arXiv:1503.02919 [INSPIRE].
C. Okonek, M. Schneider and H. Spindler, Vector bundles on complex projective spaces, Birkhäuser, Boston, U.S.A. (1980).
R. Friedman, Algebraic surfaces and holomorphic vector bundles, Springer-Verlag, New York U.S.A. (1998).
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: 1603.09634
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
Chen, Z., Sharpe, E. & Wu, R. Toda-like (0,2) mirrors to products of projective spaces. J. High Energ. Phys. 2016, 93 (2016). https://doi.org/10.1007/JHEP08(2016)093
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2016)093