GLSMs for exotic Grassmannians
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Springer
Received: August 12, 2020 Accepted: September 25, 2020 Published: October 30, 2020
Wei Gu,a Eric Sharpeb and Hao Zoub a
Center for Mathematical Sciences, Harvard University, Cambridge, MA 02138, U.S.A. b Department of Physics, Virginia Tech, 850 West Campus Dr., Blacksburg, VA 24061, U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: In this paper we explore nonabelian gauged linear sigma models (GLSMs) for symplectic and orthogonal Grassmannians and flag manifolds, checking e.g. global symmetries, Witten indices, and Calabi-Yau conditions, following up a proposal in the math community. For symplectic Grassmannians, we check that Coulomb branch vacua of the GLSM are consistent with ordinary and equivariant quantum cohomology of the space. Keywords: Field Theories in Lower Dimensions, Sigma Models, Supersymmetric Gauge Theory ArXiv ePrint: 2008.02281
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)200
JHEP10(2020)200
GLSMs for exotic Grassmannians
Contents 1 Introduction
1 3 3 6 6 10 12 13 15 16 17
3 Orthogonal Grassmannians OG(k, n) 3.1 Background and GLSM realization 3.2 Mixed Higgs-Coulomb phases at r 0 3.3 Calabi-Yau condition 3.4 Orthogonal flag manifolds 3.5 Mirrors of orthogonal Grassmannians
18 18 21 24 25 25
4 Conclusions
28
A Symmetric polynomials
28
B Equivariant quantum cohomology B.1 Projective spaces B.2 Grassmannians
30 31 31
C Tensor product representation
34
D Equivariant quantum cohomology for SG(n, 2n)
35
E Simple examples of mixed Higgs-Coulomb branches
37
F Dualities and examples
39
References
39
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2 Symplectic Grassmannians SG(k, 2n) 2.1 Background and GLSM realization 2.2 Quantum cohomology of Lagrangian Grassmannians 2.2.1 Ordinary quantum cohomology 2.2.2 Equivariant quantum cohomology 2.3 Quantum cohomology for general symplectic Grassmannians 2.4 Witten indices 2.5 Calabi-Yau condition 2.6 Symplectic flag manifolds 2.7 Mirrors of symplectic Grassmannians
1
Introduction
• An : these are the Grassmannians G(k, n + 1) = SL(n + 1)/P , which have global symmetry U (n + 1) SU (n + 1) = P SU (n + 1) = . (1.1) U (1) Zn+1 • Bn : these are the orthogonal Grassmannians OG(n, 2n + 1) = SO(2n + 1, C)/P . • Cn : these are the symplectic and Lagrangian Grassmannians SG(k, 2n), LG(n, 2n) = Sp(2n, C)/P . • Dn : these are the orthogonal Grassmannians OG(n, 2n) = SO(2n, C)/P . (The various Grassmannians above are sometimes referred to as type A, B, C, D Grassmannians respectively, in reference to their symmetries.) In each of these cases, the global 1
In most of this paper, for simplicity we focus on Grassmannians, but analogues for flag manifolds do exist, and we discuss corresponding GLSMs later in this paper.
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JHEP10(2020)200
Gauged linear sigma models (GLSMs) [1] have proven to be extraordinary physical tools to examine a wide range of questions in string theory and string compactifications, ranging from global properties of moduli spaces of SCFTs for Calabi-Ya
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