Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userQuantum sheaf cohomology on Grassmannians
118
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper we study the quantum sheaf cohomology of Grassmannians with deformations of the tangent bundle. Quantum sheaf cohomology is a (0,2) deformation of the ordinary quantum cohomology ring, realized as the OPE ring in A/2-twisted theories. Quantum sheaf cohomology has previously been computed for abelian gauged linear sigma models (GLSMs); here, we study (0,2) deformations of nonabelian GLSMs, for which previous methods have been intractable. Combined with the classical result, the quantum ring structure is derived from the one-loop effective potential. We also utilize recent advances in supersymmetric localization to compute A/2 correlation functions and check the general result in examples. In this paper we focus on physics derivations and examples; in a companion paper, we will provide a mathematically rigorous derivation of the classical sheaf cohomology ring.
rate research
Read More
Let the vector bundle $mathcal{E}$ be a deformation of the tangent bundle over the Grassmannian $G(k,n)$. We compute the ring structure of sheaf cohomology valued in exterior powers of $mathcal{E}$, also known as the polymology. This is the first part of a project studying the quantum sheaf cohomology of Grassmannians with deformations of the tangent bundle, a generalization of ordinary quantum cohomology rings of Grassmannians. A companion physics paper [arXiv:1512.08586] describes physical aspects of the theory, including a conjecture for the quantum sheaf cohomology ring, and numerous examples.
We study the quantum sheaf cohomology of flag manifolds with deformations of the tangent bundle and use the ring structure to derive how the deformation transforms under the biholomorphic duality of flag manifolds. Realized as the OPE ring of A/2-twisted two-dimensional theories with (0,2) supersymmetry, quantum sheaf cohomology generalizes the notion of quantum cohomology. Complete descriptions of quantum sheaf cohomology have been obtained for abelian gauged linear sigma models (GLSMs) and for nonabelian GLSMs describing Grassmannians. In this paper we continue to explore the quantum sheaf cohomology of nonabelian theories. We first propose a method to compute the generating relations for (0,2) GLSMs with (2,2) locus. We apply this method to derive the quantum sheaf cohomology of products of Grassmannians and flag manifolds. The dual deformation associated with the biholomorphic duality gives rise to an explicit IR duality of two A/2-twisted (0,2) gauge theories.
For a class of monadic deformations of the tangent bundles over nef-Fano smooth projective toric varieties, we study the correlators using quantum sheaf cohomology. We prove a summation formula for the correlators, confirming a conjecture by McOrist and Melnikov in physics literature. This generalizes the Szenes-Vergne proof of Toric Residue Mirror Conjecture for hypersurfaces.
In this paper we discuss physical derivations of the quantum K theory rings of symplectic Grassmannians. We compare to standard presentations in terms of Schubert cycles, but most of our work revolves around a proposed description in terms of two other bases, involving shifted Wilson lines and lambda_y classes, which are motivated by and amenable to physics, and which we also provide for ordinary Grassmannians.
Quantum Hall Effects (QHEs) on the complex Grassmann manifolds $mathbf{Gr}_2(mathbb{C}^N)$ are formulated. We set up the Landau problem in $mathbf{Gr}_2(mathbb{C}^N)$ and solve it using group theoretical techniques and provide the energy spectrum and the eigenstates in terms of the $SU(N)$ Wigner ${cal D}$-functions for charged particles on $mathbf{Gr}_2(mathbb{C}^N)$ under the influence of abelian and non-abelian background magnetic monopoles or a combination of these thereof. In particular, for the simplest case of $mathbf{Gr}_2(mathbb{C}^4)$ we explicitly write down the $U(1)$ background gauge field as well as the single and many-particle eigenstates by introducing the Pl{u}cker coordinates and show by calculating the two-point correlation function that the Lowest Landau Level (LLL) at filling factor $ u =1$ forms an incompressible fluid. Our results are in agreement with the previous results in the literature for QHE on ${mathbb C}P^N$ and generalize them to all $mathbf{Gr}_2(mathbb{C}^N)$ in a suitable manner. Finally, we heuristically identify a relation between the $U(1)$ Hall effect on $mathbf{Gr}_2(mathbb{C}^4)$ and the Hall effect on the odd sphere $S^5$, which is yet to be investigated in detail, by appealing to the already known analogous relations between the Hall effects on ${mathbb C}P^3$ and ${mathbb C}P^7$ and those on the spheres $S^4$ and $S^8$, respectively.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
