Multiple mirrors and the JKLMR conjecture. (English. Russian original) Zbl 1516.81164
Theor. Math. Phys. 213, No. 1, 1441-1452 (2022); translation from Teor. Mat. Fiz. 213, No. 1, 149-162 (2022).
Summary: We address the problem of the fulfillment of the conjecture proposed by H. Jockers et al. [Commun. Math. Phys. 325, No. 3, 1139–1170 (2014; Zbl 1301.81253)]
(JKLMR conjecture) on the equality of the partition function of a supersymmetric gauged linear sigma model on the sphere \(S^2\) and the exponential of the Kähler potential on the moduli space of Calabi-Yau manifolds. The problem is considered for a specific class of Calabi-Yau manifolds that does not belong to the Fermat type class. We show that the JKLMR conjecture holds when a Calabi-Yau manifold \(X(1)\) of such type has a mirror partner \(Y(1)\) in a weighted projective space that also admits a Calabi-Yau manifold of Fermat type \(Y(2)\). Moreover, the mirror \(X(2)\) for \(Y(2)\) has the same special geometry on the moduli space of complex structures as the original \(X(1)\).
MSC:
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 81T60 | Supersymmetric field theories in quantum mechanics |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 81T15 | Perturbative methods of renormalization applied to problems in quantum field theory |
| 81T16 | Nonperturbative methods of renormalization applied to problems in quantum field theory |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 32Q15 | Kähler manifolds |
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 14J33 | Mirror symmetry (algebro-geometric aspects) |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 32Q25 | Calabi-Yau theory (complex-analytic aspects) |
Citations:
Zbl 1301.81253References:
| [1] | Candelas, P.; Ossa, X. C. de la, Moduli space of Calabi-Yau manifolds, Nucl. Phys. B, 355, 455-481 (1991) · Zbl 0732.53056 · doi:10.1016/0550-3213(91)90122-E |
| [2] | Aleshkin, K.; Belavin, A., A new approach for computing the geometry of the moduli spaces for a Calabi-Yau manifold, J. Phys. A, 51 (2018) · Zbl 1387.81280 · doi:10.1088/1751-8121/aa9e7a |
| [3] | Jockers, H.; Kumar, V.; Lapan, J. M.; Morrison, D. R.; Romo, M., Two-sphere partition functions and Gromov-Witten invariants, Commun. Math. Phys., 325, 1139-1170 (2014) · Zbl 1301.81253 · doi:10.1007/s00220-013-1874-z |
| [4] | Benini, F.; Cremonesi, S., Partition functions of \({\mathcal{N}=(2,2)}\) gauge theories on \(S^2\) and vortices, Commun. Math. Phys., 334, 1483-1527 (2015) · Zbl 1308.81131 · doi:10.1007/s00220-014-2112-z |
| [5] | Doroud, N.; Gomis, J.; Floch, B. Le; Lee, S., Exact results in \(D=2\) supersymmetric gauge theories, JHEP, 05 (2013) · Zbl 1342.81573 · doi:10.1007/JHEP05(2013)093 |
| [6] | Aleshkin, K.; Belavin, A.; Litvinov, A., JKLMR conjecture and Batyrev construction, J. Stat. Mech., 2019 (0000) · Zbl 1539.81089 · doi:10.1088/1742-5468/ab081a |
| [7] | Aleshkin, K.; Belavin, A., GLSM for Berglund-Hübsch type Calabi-Yau manifolds, JETP Lett., 110, 711-714 (2019) · doi:10.1134/S0021364019230012 |
| [8] | Artem’ev, A. A.; Kochergin, I. V., On the calculation of the special geometry for a Calabi-Yau loop manifold and two constructions of the mirror manifold, JETP Lett., 112, 263-268 (2020) · doi:10.1134/S0021364020170051 |
| [9] | Kochergin, I. V., Calabi-Yau manifolds in weighted projective spaces and their mirror gauged linear sigma models, Phys. Rev. D, 105 (2022) · doi:10.1103/PhysRevD.105.066008 |
| [10] | Krawitz, M., FJRW rings and Landau-Ginzburg mirror symmetry (0000) |
| [11] | Kelly, T. L., Berglund-Hubsch-Krawitz mirrors via Shioda maps, Adv. Theor. Math. Phys., 17, 1425-1449 (2013) · Zbl 1316.14076 · doi:10.4310/ATMP.2013.v17.n6.a8 |
| [12] | Belavin, A.; Belavin, V.; Koshevoy, G., Periods of the multiple Berglund-Hübsch-Krawitz mirrors, Lett. Math. Phys., 111 (2021) · Zbl 1467.14098 · doi:10.1007/s11005-021-01439-5 |
| [13] | Kreuzer, M.; Skarke, H., On the classification of quasihomogeneous functions, Commun. Math. Phys., 150 (1992) · Zbl 0767.57019 · doi:10.1007/BF02096569 |
| [14] | Belavin, A.; Eremin, B., On the equivalence of Batyrev and BHK mirror symmetry constructions, Nucl. Phys. B, 961 (2020) · Zbl 1473.14075 · doi:10.1016/j.nuclphysb.2020.115271 |
| [15] | Witten, E., Phases of \(N=2\) theories in two-dimensions, Mirror symmetry II, 143-211 (1996), Providence, RI: AMS, Providence, RI · Zbl 0910.14019 · doi:10.1090/amsip/001/09 |
| [16] | Berglund, P.; Hubsch, T., A generalized construction of Calabi-Yau models and mirror symmetry, SciPost Phys., 4 (2018) · doi:10.21468/SciPostPhys.4.2.009 |
| [17] | Candelas, P.; Ossa, X. C. de la; Green, P. S.; Parkes, L.; Yau, S.-T., A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Mirror Symmetry I, 31-95 (1998), Providence, RI: AMS, Providence, RI · Zbl 0904.32019 · doi:10.1090/amsip/009/02 |
| [18] | Belavin, A. A.; Eremin, B. A., Mirror pairs of quintic orbifolds, JETP Lett., 112, 370-375 (2020) · doi:10.1134/S002136402018006X |
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