Quantum Kostka and the rank one problem for π°π©2m
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Natalie L. F. Hobson
Abstract
We give necessary and sufficient conditions to specify vector bundles of conformal blocks for π°π©2m with rectangular weights which have ranks 0, 1, and larger than 1. We show that the first Chern classes of such rank one bundles determine a finitely generated subcone of the nef cone.
Acknowledgements
The author would like to thank Angela Gibney and Dave Swinarski for many useful discussions and feedback. She especially thanks Swinarski for his contributions in writing several Macaulay2 programs to implement the lemmas and propositions in this paper. These programs can be found on the authorβs website. She thanks Prakash Belkale, Swarnava Mukhopadhyay, and Anna Kazanova for their comments. She thanks the referee for providing an example to motivate Definition 3.2, suggestions for Lemma 5.3, and many helpful comments. Many computations for this paper were performed using the ConformalBlocks package [12] in Macaulay2.
Funding The author gratefully acknowledges support from the RTG in Algebra, Algebraic Geometry, and Number Theory, at the University of Georgia (NSF grant DMS-1344994).
References
[1] V. Alexeev, A. Gibney, D. Swinarski, Higher-level sl2 conformal blocks divisors on M0,nProc. Edinb. Math. Soc. (2) 57 (2014), 7β30. MR3165010 Zbl 1285.1401210.1017/S0013091513000941Search in Google Scholar
[2] P. Belkale, Quantum generalization of the Horn conjecture. J. Amer. Math. Soc. 21 (2008), 365β408. MR2373354 Zbl 1134.1402910.1090/S0894-0347-07-00584-XSearch in Google Scholar
[3] P. Belkale, A. Gibney, A. Kazanova, Scaling of conformal blocks and generalized theta functions over πgnMath. Z. 284 (2016), 961β987. MR3563262 Zbl 0665703110.1007/s00209-016-1682-1Search in Google Scholar
[4] P. Belkale, A. Gibney, S. Mukhopadhyay, Nonvanishing of conformal blocks divisors on M0,nTransform. Groups 21 (2016), 329β353. MR3492039 Zbl 0659211210.1007/s00031-015-9357-2Search in Google Scholar
[5] A. Bertram, I. Ciocan-Fontanine, W. Fulton, Quantum multiplication of Schur polynomials. J. Algebra 219 (1999), 728β746. MR1706853 Zbl 0936.0508610.1006/jabr.1999.7960Search in Google Scholar
[6] N. Bourbaki, Lie groups and Lie algebras. Chapters 1β3. Springer 1989. MR979493 Zbl 0672.22001Search in Google Scholar
[7] P. Di Francesco, P. Mathieu, D. SΓ©nΓ©chal, Conformal field theory Springer 1997. MR1424041 Zbl 0869.5305210.1007/978-1-4612-2256-9Search in Google Scholar
[8] N. Fakhruddin, Chern classes of conformal blocks. In: Compact moduli spaces and vector bundles volume 564 of Contemp. Math. 145β176, Amer. Math. Soc. 2012. MR2894632 Zbl 1244.1400710.1090/conm/564/11148Search in Google Scholar
[9] N. Giansiracusa, A. Gibney, The cone of type A level 1, conformal blocks divisors. Adv. Math. 231 (2012), 798β814. MR2955192 Zbl 1316.1405110.1016/j.aim.2012.05.017Search in Google Scholar
[10] A. Marian, D. Oprea, R. Pandharipande, The first Chern class of the Verlinde bundles. In: String-Math 2012, volume 90 of Proc. Sympos. Pure Math. 87β111, Amer. Math. Soc. 2015. MR3409789 Zbl 0668964310.1090/pspum/090/01521Search in Google Scholar
[11] S. Mukhopadhyay, Rank-Level duality and Conformal Block divisors. arXiv:1308.0854 [math.AG]Search in Google Scholar
[12] D. Swinarski, Conformalblocks: a macaulay2 package for computing conformal blocks divisors. http://www.math.uiuc.edu/Macaulay2/Search in Google Scholar
[13] A. Tsuchiya, K. Ueno, Y. Yamada, Conformal field theory on universal family of stable curves with gauge symmetries. In: Integrable systems in quantum field theory and statistical mechanics volume 19 of Adv. Stud. Pure Math. 459β566, Academic Press 1989. MR1048605 Zbl 0696.1701010.2969/aspm/01910000Search in Google Scholar
[14] E. Witten, The Verlinde algebra and the cohomology of the Grassmannian. In: Geometry, topology, & physics 357β422, Int. Press, Cambridge, MA 1995. MR1358625 Zbl 0863.53054Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Minimal hypersurfaces in βn Γ Sm
- Higher order Dehn functions for horospheres in products of Hadamard spaces
- Unitals with many Baer secants through a fixed point
- Is a complete, reduced set necessarily of constant width?
- Pseudo-embeddings of the (point, k-spaces)-geometry of PG(n, 2) and projective embeddings of DW(2n β 1, 2)
- Classification of 8-dimensional rank two commutative semifields
- On non-KΓ€hler degrees of complex manifolds
- Quantum Kostka and the rank one problem for π°π©2m
- The diffeomorphism type of small hyperplane arrangements is combinatorially determined
- Superforms, tropical cohomology, and PoincarΓ© duality
- Eigenvalues of the weighted Laplacian under the extended Ricci flow
Articles in the same Issue
- Frontmatter
- Minimal hypersurfaces in βn Γ Sm
- Higher order Dehn functions for horospheres in products of Hadamard spaces
- Unitals with many Baer secants through a fixed point
- Is a complete, reduced set necessarily of constant width?
- Pseudo-embeddings of the (point, k-spaces)-geometry of PG(n, 2) and projective embeddings of DW(2n β 1, 2)
- Classification of 8-dimensional rank two commutative semifields
- On non-KΓ€hler degrees of complex manifolds
- Quantum Kostka and the rank one problem for π°π©2m
- The diffeomorphism type of small hyperplane arrangements is combinatorially determined
- Superforms, tropical cohomology, and PoincarΓ© duality
- Eigenvalues of the weighted Laplacian under the extended Ricci flow
