We compute the quantum cohomology of symplectic flag manifolds. Symplectic flag manifolds can be described by non-abelian GLSMs with superpotential. Although the ring relations cannot be directly read off from the equations of motion on the Coulomb b
ranch due to complication introduced by the non-abelian gauge symmetry, it can be shown that they can be extracted from the localization formula in a gauge-invariant form. Our result is general for all symplectic flag manifolds, which reduces to previously established results on symplectic Grassmannians and complete symplectic flag manifolds derived by other means. We also explain why a (0,2) deformation of the GLSM does not give rise to a deformation of the quantum cohomology.
Given a gauged linear sigma model (GLSM) $mathcal{T}_{X}$ realizing a projective variety $X$ in one of its phases, i.e. its quantum Kahler moduli has a maximally unipotent point, we propose an emph{extended} GLSM $mathcal{T}_{mathcal{X}}$ realizing t
he homological projective dual category $mathcal{C}$ to $D^{b}Coh(X)$ as the category of B-branes of the Higgs branch of one of its phases. In most of the cases, the models $mathcal{T}_{X}$ and $mathcal{T}_{mathcal{X}}$ are anomalous and the analysis of their Coulomb and mixed Coulomb-Higgs branches gives information on the semiorthogonal/Lefschetz decompositions of $mathcal{C}$ and $D^{b}Coh(X)$. We also study the models $mathcal{T}_{X_{L}}$ and $mathcal{T}_{mathcal{X}_{L}}$ that correspond to homological projective duality of linear sections $X_{L}$ of $X$. This explains why, in many cases, two phases of a GLSM are related by homological projective duality. We study mostly abelian examples: linear and Veronese embeddings of $mathbb{P}^{n}$ and Fano complete intersections in $mathbb{P}^{n}$. In such cases, we are able to reproduce known results as well as produce some new conjectures. In addition, we comment on the construction of the HPD to a nonabelian GLSM for the Plucker embedding of the Grassmannian $G(k,N)$.
We propose Picard-Fuchs equations for periods of nonabelian mirrors in this paper. The number of parameters in our Picard-Fuchs equations is the rank of the gauge group of the nonabelian GLSM, which is eventually reduced to the actual number of K{a}h
ler parameters. These Picard-Fuchs equations are concise and novel. We justify our proposal by reproducing existing mathematical results, namely Picard-Fuchs equations of Grassmannians and Calabi-Yau manifolds as complete intersections in Grassmannians. Furthermore, our approach can be applied to other nonabelian GLSMs, so we compute Picard-Fuchs equations of some other Fano-spaces, which were not calculated in the literature before. Finally, the cohomology-valued generating functions of mirrors can be read off from our Picard-Fuchs equations. Using these generating functions, we compute Gromov-Witten invariants of various Calabi-Yau manifolds, including complete intersection Calabi-Yau manifolds in Grassmannians and non-complete intersection Calabi-Yau examples such as Pfaffian Calabi-Yau threefold and Gulliksen-Neg{aa}rd Calabi-Yau threefold, and find agreement with existing results in the literature. The generating functions we propose for non-complete intersection Calabi-Yau manifolds are genuinely new.
By studying the infra-red fixed point of an $mathcal{N}=(0,2)$ Landau-Ginzburg model, we find an example of modular invariant partition function beyond the ADE classification. This stems from the fact that a part of the left-moving sector is a new co
nformal field theory which is a variant of the parafermion model.
A graded quiver with superpotential is a quiver whose arrows are assigned degrees $cin {0, 1, cdots, m}$, for some integer $m geq 0$, with relations generated by a superpotential of degree $m-1$. Ordinary quivers ($m=1)$ often describe the open strin
g sector of D-brane systems; in particular, they capture the physics of D3-branes at local Calabi-Yau (CY) 3-fold singularities in type IIB string theory, in the guise of 4d $mathcal{N}=1$ supersymmetric quiver gauge theories. It was pointed out recently that graded quivers with $m=2$ and $m=3$ similarly describe systems of D-branes at CY 4-fold and 5-fold singularities, as 2d $mathcal{N}=(0,2)$ and 0d $mathcal{N}=1$ gauge theories, respectively. In this work, we further explore the correspondence between $m$-graded quivers with superpotential, $Q_{(m)}$, and CY $(m+2)$-fold singularities, ${mathbf X}_{m+2}$. For any $m$, the open string sector of the topological B-model on ${mathbf X}_{m+2}$ can be described in terms of a graded quiver. We illustrate this correspondence explicitly with a few infinite families of toric singularities indexed by $m in mathbb{N}$, for which we derive toric graded quivers associated to the geometry, using several complementary perspectives. Many interesting aspects of supersymmetric quiver gauge theories can be formally extended to any $m$; for instance, for one family of singularities, dubbed $C(Y^{1,0}(mathbb{P}^m))$, that generalizes the conifold singularity to $m>1$, we point out the existence of a formal duality cascade for the corresponding graded quivers.
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-twi
sted 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.
We study 2d $mathcal{N}=(0,2)$ supersymmetric quiver gauge theories that describe the low-energy dynamics of D1-branes at Calabi-Yau fourfold (CY$_4$) singularities. On general grounds, the holomorphic sector of these theories---matter content and (c
lassical) superpotential interactions---should be fully captured by the topological $B$-model on the CY$_4$. By studying a number of examples, we confirm this expectation and flesh out the dictionary between B-brane category and supersymmetric quiver: the matter content of the supersymmetric quiver is encoded in morphisms between B-branes (that is, Ext groups of coherent sheaves), while the superpotential interactions are encoded in the $A_infty$ algebra satisfied by the morphisms. This provides us with a derivation of the supersymmetric quiver directly from the CY$_4$ geometry. We also suggest a relation between triality of $mathcal{N}=(0,2)$ gauge theories and certain mutations of exceptional collections of sheaves. 0d $mathcal{N}=1$ supersymmetric quivers, corresponding to D-instantons probing CY$_5$ singularities, can be discussed similarly.
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 par
t 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.
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. Q
uantum 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.
