Covariant constancy of quantum Steenrod operations. (English) Zbl 1489.53120
Summary: We prove a relationship between quantum Steenrod operations and the quantum connection. In particular, there are operations extending the quantum Steenrod power operations that, when viewed as endomorphisms of equivariant quantum cohomology, are covariantly constant. We demonstrate how this property is used in computations of examples.
MSC:
| 53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 55S10 | Steenrod algebra |
| 55N91 | Equivariant homology and cohomology in algebraic topology |
| 57R91 | Equivariant algebraic topology of manifolds |
References:
| [1] | Albers, P.: A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology. Int. Math. Res. Notes (IMRN) (2008) (art. ID rnm134) · Zbl 1158.53066 |
| [2] | Betz, M.; Cohen, R., Graph moduli spaces and cohomology operations, Turkish J. Math., 18, 23-41 (1994) · Zbl 0869.57035 |
| [3] | Cineli, E., Ginzburg, V., Gurel, B.: From pseudo-rotations to holomorphic curves via quantum Steenrod squares. Int. Math. Res. Notes (IMRN) 2274-2297 (2022) |
| [4] | Cohen, R.; Norbury, P., Morse field theory, Asian J. Math., 16, 661-711 (2012) · Zbl 1262.58009 · doi:10.4310/AJM.2012.v16.n4.a5 |
| [5] | Crauder, B., Miranda, R.: Quantum cohomology of rational surfaces. In: The moduli space of curves (Texel Island, 1994), pp. 33-80. (Birkhäuser, 1995) · Zbl 0843.14014 |
| [6] | Di Francesco, P., Itzykson, C.: Quantum intersection rings. In: The moduli space of curves (Texel Island, 1994), pp. 81-148. (Birkhäuser, 1995) · Zbl 0868.14029 |
| [7] | Donaldson, S.K.: Floer homology and algebraic geometry. In Vector bundles in algebraic geometry, pp. 119-138. (Cambridge University Press, Cambridge, 1993) · Zbl 0829.57011 |
| [8] | Fukaya, K.: Morse homotopy and its quantization. In: Geometric topology (Athens, GA, 1993), pp. 409-440. (American Mathematics Society, New York, 1997) · Zbl 0891.57035 |
| [9] | Hofer, H., Salamon, D.: Floer homology and Novikov rings. In: The Floer memorial, pp. 483-524. (Birkhauser, 1995) · Zbl 0842.58029 |
| [10] | Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., Zaslow, E.: Mirror symmetry. (American Mathematics Society/Clay Mathematics Institute, 2003) · Zbl 1044.14018 |
| [11] | Katić, J., Compactification of mixed moduli spaces in Morse-Floer theory, Rocky Mt. J. Math., 38, 3, 923-939 (2008) · Zbl 1160.53375 · doi:10.1216/RMJ-2008-38-3-923 |
| [12] | Katić, J.; Milinković, D., Piunikhin-Salamon-Schwarz isomorphisms for Lagrangian intersections, Differ. Geom. Appl., 22, 2, 215-227 (2005) · Zbl 1064.37041 · doi:10.1016/j.difgeo.2004.10.008 |
| [13] | Lu, G., An explicit isomorphism between Floer homology and quantum homology, Pac. J. Math., 213, 2, 319-363 (2004) · Zbl 1063.53088 · doi:10.2140/pjm.2004.213.319 |
| [14] | McDuff, D., Salamon, D.: \(J\)-holomorphic curves and symplectic topology. (American Mathematics Society, New York, 2004) · Zbl 1064.53051 |
| [15] | Pandharipande, R.: Rational curves on hypersurfaces (after A. Givental). In: Séminaire Bourbaki 1997/98, Astérisque 252, pp. 307-340 (1998) · Zbl 0932.14029 |
| [16] | Schwarz, M., A quantum cup-length estimate for symplectic fixed points, Invent. Math., 133, 2, 353-397 (1998) · Zbl 0951.53053 · doi:10.1007/s002220050248 |
| [17] | Seidel, P., Connections on equivariant Hamiltonian Floer cohomology, Comment. Math. Helv., 93, 587-644 (2018) · Zbl 1407.53097 · doi:10.4171/CMH/445 |
| [18] | Seidel, P.: Formal groups and quantum cohomology. Geom. Topol. (to appear) · Zbl 1539.53099 |
| [19] | http://math.mit.edu/ seidel/code.html. Accessed 20 May 2022 |
| [20] | Shelukhin, E., Pseudorotations and Steenrod squares, J. Mod. Dyn., 16, 289-304 (2020) · Zbl 1466.53094 · doi:10.3934/jmd.2020010 |
| [21] | Shelukhin, E., Pseudorotations and Steenrod squares revisited, Math. Res. Lett., 28, 1255-1261 (2021) · Zbl 1483.53102 · doi:10.4310/MRL.2021.v28.n4.a13 |
| [22] | Wilkins, N., A construction of the quantum Steenrod squares and their algebraic relations, Geom. Topol., 24, 885-970 (2020) · Zbl 1467.53095 · doi:10.2140/gt.2020.24.885 |
| [23] | Zinger, A.: Pseudocycles and integral homology. Preprint arxiv:math/0605535 · Zbl 1213.57031 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
