Abstract
We record an observation about the Witten indices in two families of gauged linear sigma models: the U(2) model for linear sections of Grassmannians, and the U(1) model for quadric complete intersections. We describe how the Witten indices are related to the Euler characteristics of the singular skew-symmetric or symmetric determinantal varieties featuring in the analysis of their opposite phases, and we discuss the extent to which these relationships can be reconciled with standard Born-Oppenheimer arguments.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K. Hori and D. Tong, Aspects of non-Abelian gauge dynamics in two-dimensional N = (2, 2) theories, JHEP 05 (2007) 079 [hep-th/0609032] [INSPIRE].
A. Caldararu, J. Distler, S. Hellerman, T. Pantev and E. Sharpe, Non-birational twisted derived equivalences in Abelian GLSMs, Commun. Math. Phys. 294 (2010) 605 [arXiv:0709.3855] [INSPIRE].
E. Rodland, The Pfaffian Calabi-Yau, its mirror, and their link to the Grassmannian G(2, 7), Composit. Math. 122 (2000) 135 [math/9801092].
L. Borisov and A. Caldararu, The Pfaffian-Grassmannian derived equivalence, J. Alg. Geom. 18 (2009) 201 [math/0608404].
A. Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, Adv. Math. 218 (2008) 1340 [math/0510670].
N. Addington, W. Donovan and E. Segal, The Pfaffian-Grassmannian equivalence revisited, Alg. Geom. 2 (2015) 332 [arXiv:1401.3661] [INSPIRE].
A. Kuznetsov, Homological projective duality, Publ. Math. I.H.E.S. 105 (2007) 157 [math/0507292].
A. Kuznetsov, Homological projective duality for Grassmannians of lines, math/0610957.
N.M. Addington, E.P. Segal and E. Sharpe, D-brane probes, branched double covers and noncommutative resolutions, Adv. Theor. Math. Phys. 18 (2014) 1369 [arXiv:1211.2446] [INSPIRE].
V. den Bergh, Non-commutative crepant resolutions, in The legacy of Niels Henrik Abel, Springer, Berlin Germany, (2004), pg. 749 [math.RA/0211064].
L. Borisov and A. Libgober, Stringy E-functions of Pfaffian-Grassmannian double mirrors, arXiv:1502.03702 [INSPIRE].
S. Cappell, L. Maxim and J. Shaneson, Intersection cohomology invariants of complex algebraic varieties, in Singularities I, Contemp. Math. 474 (2008) 15.
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].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, The long flow to freedom, JHEP 02 (2017) 056 [arXiv:1611.02763] [INSPIRE].
A. Beauville and R. Donagi, La variete des droites d’une hypersurface cubique de dimension 4 (in French), C. R. Acad. Sci. Paris Ser. I Math. 3014 (1985) 703.
B. Hassett, Special cubic fourfolds, Composit. Math. 120 (2000) 1.
A. Kuznetsov, Derived categories of cubic fourfolds, Progr. Math. 282 (2010) 219, Birkhauser Inc., Boston U.S.A., (2010) [arXiv:0808.3351].
N. Addington and R. Thomas, Hodge theory and derived categories of cubic fourfolds, Duke Math. 163 (2014) 1885 [arXiv:1211.3758].
M.C.N. Cheng, F. Ferrari, S.M. Harrison and N.M. Paquette, Landau-Ginzburg orbifolds and symmetries of K3 CFTs, JHEP 01 (2017) 046 [arXiv:1512.04942] [INSPIRE].
M. Gross and S. Pavanelli, A Calabi-Yau threefold with Brauer group (Z/Z 8)2, Proc. Amer. Math. Soc. 136 (2008) 1 [math/0512182].
E. Sharpe, GLSM’s, gerbes and Kuznetsov’s homological projective duality, J. Phys. Conf. Ser. 462 (2013) 012047 [arXiv:1004.5388] [INSPIRE].
K. Hori, Duality in two-dimensional (2, 2) supersymmetric non-Abelian gauge theories, JHEP 10 (2013) 121 [arXiv:1104.2853] [INSPIRE].
A. Libgober and L. Maxim, Hodge polynomials of singular hypersurfaces, Michigan Math. J. 60 (2011) 661.
H. Jockers, V. Kumar, J.M. Lapan, D.R. Morrison and M. Romo, Non-Abelian 2D gauge theories for determinantal Calabi-Yau varieties, JHEP 11 (2012) 166 [arXiv:1205.3192] [INSPIRE].
E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, in Geometry, topology and physics, Conf. Proc. Lecture Notes Geom. Topology, IV, U.S.A., (1995), pg. 357 [hep-th/9312104] [INSPIRE].
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
R. Donagi and E. Sharpe, GLSM’s for partial flag manifolds, J. Geom. Phys. 58 (2008) 1662 [arXiv:0704.1761] [INSPIRE].
B. Jia, E. Sharpe and R. Wu, Notes on non-Abelian (0, 2) theories and dualities, JHEP 08 (2014) 017 [arXiv:1401.1511] [INSPIRE].
J. Guo, B. Jia and E. Sharpe, Chiral operators in two-dimensional (0,2) theories and a test of triality, JHEP 06 (2015) 201 [arXiv:1501.00987] [INSPIRE].
J. Halverson, V. Kumar and D.R. Morrison, New methods for characterizing phases of 2D supersymmetric gauge theories, JHEP 09 (2013) 143 [arXiv:1305.3278] [INSPIRE].
J. Harris and L. Tu, On symmetric and skew-symmetric determinantal varieties, Topology 23 (1984) 71.
P. Pragacz, Enumerative geometry of degeneracy loci, Ann. Sci. École Norm. Sup. 21 (1988) 413.
K. Hori and J. Knapp, Linear σ-models with strongly coupled phases — one parameter models, JHEP 11 (2013) 070 [arXiv:1308.6265] [INSPIRE].
K. Hori and J. Knapp, A pair of Calabi-Yau manifolds from a two parameter non-Abelian gauged linear σ-model, arXiv:1612.06214 [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: 1702.00730
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
Wong, K. Two-dimensional gauge dynamics and the topology of singular determinantal varieties. J. High Energ. Phys. 2017, 132 (2017). https://doi.org/10.1007/JHEP03(2017)132
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2017)132