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A<sc>bstract</sc> We construct Schubert line defects in the 3d$$ \mathcal{N}=2 $$ $N=2$ supersymmetric gauged linear sigma model (GLSM) with target space a partial flag manifoldX= Fl(k;n), generalizing our construction for complete flag manifolds given in a companion paper (part I) [1]. In the context of the 3d GLSM/quantum K-theory correspondence, the Schubert line defects are constructed as 1d$$ \mathcal{N}=2 $$ $N=2$ supersymmetric gauge theories coupled to the 3d field theory, and they flow to objects supported on Schubert varietiesX_w⊆Xin the quantum K-theory. The flavored Witten index of the 1d defect is expected to compute the Chern character of [𝒪_w] — more precisely, it gives us a polynomial representative of the Schubert class in the quantum K-theory ring. We give strong evidence for this claim by showing in examples that the Witten indices of Schubert defects indeed reproduce a recently-defined set of polynomials that represent the Schubert classes in the Whitney presentation, which we call the parabolic Whitney polynomials. Moreover, upon using the quantum ring relations, we can convert these polynomials into seemingly new polynomials in the Toda presentation, which we call the parabolic quantum Grothendieck polynomials. These new polynomials specialize to known polynomials in various limits, including to the quantum Grothendieck polynomials in the case of the complete flag. In the 2d limit, our construction also realizes the Schubert classes [X_w] in the quantum cohomology ring of the partial flag manifold, and the parabolic quantum Grothendieck polynomials then reduce to previously known parabolic quantum Schubert polynomials. 

A<sc>bstract</sc> We construct new half-BPS line defects in 3d$$ \mathcal{N}=2 $$ $N=2$ supersymmetric quiver gauge theories whose Higgs branches are complete flag manifoldsX= Fl(n). Upon circle compactification, the bulk theory flows to a non-linear sigma model (NLSM) with target spaceXand the line defects flow to objects supported on Schubert varietiesX_w⊆X. TheseSchubert line defectsform an important basis of the quantum K-theory ofX. They are realized as$$ \mathcal{N}=2 $$ $N=2$ supersymmetric quantum mechanics (SQM) quivers coupled to the 3d gauge theory. We show that the insertion of the Schubert line defect restricts the target space of the 3d gauged linear sigma model (GLSM) to the Schubert varietyX_w, with the 1d degrees of freedom physically realizing a Bott-Samelson resolution ofX_w. Moreover, we verify in examples that the 1d flavored Witten index of the quiver SQM reproduces the (equivariant) Chern character of the structure sheaf$$ {\mathcal{O}}_{X_w} $$ $O_{X_w}$ as a (double) quantum Grothendieck polynomial, generalizing previous results forXa Grassmannian manifold. Our construction thus provides a more direct realization of the 3d GLSM/quantum K-theory correspondence for complete flag manifolds. Finally, in the small-circle limit, we obtain a 0d-2d coupled system that realizes the Schubert classes [X_w] in the quantum cohomology ring ofX. 

A<sc>bstract</sc> In this article, we work out some variations on the discussion of the C-field flux densities in the Sati-Schreiber program. We start by explaining the need for global completion of the field content: the fluxes, the gauge potentials and the gauge transformations on the worldvolume of a single M5 brane in eleven-dimensional supergravity, and how this is encoded by the choice of flux quantization law. Assuming Hypothesis H that the 4-flux in M-Theory is flux quantized in a non-abelian cohomology theory called 4-cohomotopy, and the three-flux on the M5 brane worldvolume in (twisted) 3-cohomotopy, we generalize some previous calculations known in the literature to include twisting by background gravity and placing M5 branes on orbifolds. We show that the null concordances of cohomotopically charged fluxes give rise to the traditional gauge potentials and the null concordances of concordances give rise to the corresponding gauge transformations via surjections, in the cases of tangentially twisted cohomotopy, twistorial cohomotopy and equivariant twistorial cohomotopy. We construct the surjections explicitly for these cases and check the consistency relations with the corresponding Bianchi identities. Thus, we show how the traditional formulas for the local gauge potentials on M5 brane worldvolume on curved spacetimes and orbifolds are indeed reproduced by the homotopy theory and, as such, become amenable to global completion in cohomotopical charge quantization. 

A<sc>bstract</sc> In this paper, we propose a construction of GLSM defects corresponding to Schubert cycles in Lagrangian Grassmannians, following recent work of Closset-Khlaif on Schubert cycles in ordinary Grassmannians. In the case of Lagrangian Grassmannians, there are superpotential terms in both the bulk GLSM as well as on the defect itself, enforcing isotropy constraints. We check our construction by comparing the locus on which the GLSM defect is supported to mathematical descriptions, checking dimensions, and perhaps most importantly, comparing defect indices to known and expected polynomial invariants of the Schubert cycles in quantum cohomology and quantum K theory. 

A<sc>bstract</sc> In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions, extending our previous work. Specifically, in this work we discuss more general gauged noninvertible symmetries in which the noninvertible symmetry is not multiplicity free, and discuss the case of Rep(A₄) in detail. We realize Rep(A₄) gaugings for thec= 1 CFT at the exceptional point in the moduli space and find new self-duality under gauging a certain non-group algebra object, leading to a larger noninvertible symmetry Rep(SL(2, ℤ₃)). We also discuss more general examples of decomposition in two-dimensional gauge theories with trivially-acting gauged noninvertible symmetries. 

Abstract In this paper we discuss dilaton shifts (Euler counterterms) arising in decomposition of two-dimensional quantum field theories with higher-form symmetries. Relative shifts between universes are fixed by locality and take a universal form, reflecting underlying (noninvertible, quantum) symmetries. The first part of this paper constructs a general formula for such dilaton shifts, and discusses related computations. In the second part of this paper, we comment on the relation between decomposition and ensembles. 

A<sc>bstract</sc> In this paper, we test and extend a proposal of Gu, Pei, and Zhang for an application of decomposition to three-dimensional theories with one-form symmetries and to quantum K theory. The theories themselves do not decompose, but, OPEs of parallel one-dimensional objects (such as Wilson lines) and dimensional reductions to two dimensions do decompose, sometimes in two independent ways. We apply this to extend conjectures for quantum K theory rings of gerbes (realized by three-dimensional gauge theories with one-form symmetries) via both orbifold partition functions and gauged linear sigma models. 

A<sc>bstract</sc> In this paper, we discuss how gauging one-form symmetries in Chern-Simons theories is implemented in an A-twisted topological open string theory. For example, the contribution from a fixed H/Z bundle on a three-manifold M, arising in a BZ gauging of H Chern-Simons, for Z a finite subgroup of the center of H, is described by an open string worldsheet theory whose bulk is a sigma model with target a Z-gerbe (a bundle of one-form symmetries) over T^∗M, of characteristic class determined by the H/Z bundle. We give a worldsheet picture of the decomposition of one-form-symmetry-gauged Chern-Simons in three dimensions, and we describe how a target-space constraint on bundles arising in the gauged Chern-Simons theory has a natural worldsheet realization. Our proposal provides examples of the expected correspondence between worldsheet global higher-form symmetries, and target-space gauged higher-form symmetries. 

A<sc>bstract</sc> In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions. We specialize to certain gaugeable cases, specifically, fusion categories of the form$$ \textrm{Rep}\left(\mathcal{H}\right) $$ $\mathrm{Rep}\left(H\right)$ for$$ \mathcal{H} $$ $H$ a suitable Hopf algebra (which includes the special case Rep(G) forGa finite group). We also specialize to the case that the fusion category is multiplicity-free. We discuss how to construct a modular-invariant partition function from a choice of Frobenius algebra structure on$$ {\mathcal{H}}^{\ast } $$ $H^{\ast }$ . We discuss how ordinaryGorbifolds for finite groupsGare a special case of the construction, corresponding to the fusion category Vec(G) = Rep(ℂ[G]^*). For the cases Rep(S₃), Rep(D₄), and Rep(Q₈), we construct the crossing kernels for general intertwiner maps. We explicitly compute partition functions in the examples of Rep(S₃), Rep(D₄), Rep(Q₈), and$$ \textrm{Rep}\left({\mathcal{H}}_8\right) $$ $\mathrm{Rep}\left(H_8\right)$ , and discuss applications inc= 1 CFTs. We also discuss decomposition in the special case that the entire noninvertible symmetry group acts trivially. 

Given two otherwise decoupled $D$ -dimensional CFTs which possess a common (finite) symmetry subcategory, one can consider entangled boundary states of their ( $D+1$ )-dimensional symmetry topological field theories (SymTFTs). This roughly corresponds to performing a gauging of the tensor product of two CFTs, and we call this phenomena “SymTFT entanglement” (or “S entanglement” for short). In the case when these CFTs have semiclassical holographic duals, the S entanglement relates the bulk gauge charges between two otherwise disconnected AdS spacetimes as we highlight in several top-down examples. We show that taking partial traces of such S-entangled states leads to a streamlined approach to preparing ensemble-averaged CFTs in string theory. This ensemble averaging coincides with that generated by $\alpha$ states in the baby universe Hilbert space, and we propose a symmetry-enriched generalization of this Hilbert space via generalized global symmetries. We quantify how this symmetry-governed averaging violates holographic factorization and leads to the emergence of bulk global symmetries. We also consider the eternal (two-sided) AdS black hole geometries, where our SymTFT entanglement considerations imply that there exist refinements of the usual theromofield double state preparation of the system. We show that one may prepare the system in such a way that the total CFT data does not factorize into left and right copies. As anticipated by Marolf and Wall [], we highlight that such considerations are necessary to define the gauge charges of eternal black holes and, in certain cases, can imply that there exist extended bulk objects stretching across the wormhole which cannot be expressed in terms of a product of left and right CFT operators.