arXiv subjects

Wenjun Niu

Publications and source records attributed to Wenjun Niu.

15 recordsLinked to original sources

arXiv2025

Double Yangian, Factorization, and qKZ-equation for Cotangent Lie Algebras

In this paper, we construct the dual $Y^*_\hbar(\mathfrak d)$ and double $DY_\hbar (\mathfrak d)$ of the Yangian $Y_\hbar (\mathfrak d)$ associated with a cotangent Lie algebra $\mathfrak d=T^*\mathfrak g$. We define a coherent factorization algebra version of the dual Yangian $Y_\hbar^*(\mathfrak d)^{\mathrm{co-op}}$ with opposite coproduct. Furthermore, we define a quantum vertex algebra structure on the quantum vacuum module $V_{\hbar,k}(\mathfrak d)$ of central extensions $\widehat{DY}_{\hbar,\ell} (\mathfrak d)$ of this double Yangian and show that its conformal blocks satisfy quantum KZ equations. We discuss examples of $\mathfrak d$ that arise from 3d $N=4$ gauge theories via the work of Costello-Gaiotto. These examples include Takiff Lie algebras $T^*\mathfrak g$, whose affine VOA is a large subalgebra of the chiral differential operator algebra of $G$, as well as the smallest type-A Lie superalgebra $\mathfrak{gl} (1|1)$.

math.QA↗

arXiv2025

Higgs Branches in the Omega-background via the Category of Line Operators

The vacuum manifold $\mathcal{M}$ of a topological twist of a 3d $\mathcal{N}=4$ gauge theory is a hyper-K\"ahler variety; deformations and quantizations of $\mathcal{M}$ can be constructed in the framework of 3 dimensional topological quantum field theories. In particular, based on physics arguments, turning on an omega-background results in the quantization of $\mathbb{C}[\mathcal{M}]$ as a Poisson algebra. In this paper, we implement this idea mathematically, in the context of the B-twist of 3d $\mathcal{N}=4$ gauge theories, namely for Higgs branches. Our strategy is to implement omega-background in the category of line operators via the choice of a ribbon twist, and obtain the quantum Hamiltonian reduction as derived endomorphisms in the equivariant category, the category where the ribbon twist acts trivially. We apply quadratic Koszul duality to perform this computation.

math-ph↗

arXiv2025

Line Operators in 3d Holomorphic QFT: Meromorphic Tensor Categories and dg-Shifted Yangians

We study line operators and their OPE's in perturbative 3d holomorphic-topological QFT's, including holomorphic-topological twists (quarter-BPS sectors) of 3d $N=2$ theories. In particular, we develop the representation theory of the category $C$ of perturbative line operators and its chiral tensor product, by generalizing techniques introduced by Costello and collaborators. We argue that lines are equivalent to modules for an $A_\infty$ algebra $A^!$ that's Koszul-dual to bulk local operators. We further establish a non-renormalization theorem for the OPE's of lines in a large class of theories (dubbed quasi-linear), allowing an exact resummation of quantum corrections. Based on physics arguments, we propose axioms for the full algebraic structure on $A^!$, calling it a "dg-shifted Yangian," which controls the OPE of lines. A key part of the structure is a Maurer-Cartan element $r(z)\in A^!\otimes A^!(\!(z^{-1})\!)$ that satisfies an $A_\infty$ generalization of the Yang-Baxter equation. As examples, we consider 3d $N=2$ gauge theories with arbitrary Chern-Simons levels, linear matter, and superpotential, and explicitly compute 1) perturbative bulk local operators (as $A_\infty$-chiral algebras); and 2) the Koszul-duals $A^!$ (proving they are dg-shifted Yangians).

hep-th↗

arXiv2025

D-convolution categories and Hopf algebras

For a smooth affine algebraic group $G$, one can attach various D-module categories to it that admit convolution monoidal structure. We consider the derived category of D-modules on $G$, the stack $G/G_{ad}$ and the category of Harish-Chandra bimodules. Combining the work of Beilinson-Drinfeld on D-modules and Hecke patterns with the recent work of the author with Dimofte and Py, we show that each of the above categories (more precisely the equivariant version) is monoidal equivalent to a localization of the DG category of modules of a graded Hopf algebra. As a consequence, we give an explicit braided monoidal structure to the derived category of D-modules on $G/G_{ad}$, which when restricted to the heart, recovers the braiding of Bezrukavnikov-Finkelberg-Ostrik.

math.RT↗

arXiv2025

1-shifted Lie bialgebras and their quantizations

In this paper, we define (cohomologically) 1-shifted Manin triples and 1-shifted Lie bialgebras, and study their properties. We derive many results that are parallel to those found in ordinary Lie bialgebras, including the double construction and the existence of a 1-shifted $r$-matrix satisfying the classical Yang-Baxter equation. Turning to quantization, we first construct a canonical quantization for each 1-shifted metric Lie algebra $\mathfrak{g}$, producing a deformation to the symmetric monoidal category of $\mathfrak{g}$ modules over a formal variable $\hbar$. This quantization is in terms of a curved differential graded algebra. Under a further technical assumption, we construct quantizations of transverse Lagrangian subalgebras of $\mathfrak{g}$, which is a pair of DG algebras connected by Koszul duality, and give rise to monoidal module categories of the quantized double. Finally, we apply this to Manin triples arising from Lie algebras of loop groups, and construct 1-shifted meromorphic $r$-matrices. The resulting quantizations are the cohomologically-shifted analogue of Yangians.

math.QA↗

arXiv2024

Quantum groupoids from moduli spaces of $G$-bundles

In a previous work, we have constructed the Yangian $Y_\hbar (\mathfrak{d})$ of the cotangent Lie algebra $\mathfrak{d}=T^*\mathfrak{g}$ for a simple Lie algebra $\mathfrak{g}$, from the geometry of the equivariant affine Grassmanian associated to $G$ with $\mathfrak{g}=\mathrm{Lie}(G)$. In this paper, we construct a quantum groupoid $\Upsilon_\hbar^\sigma (\mathfrak{d})$ associated to $\mathfrak{d}$ over a formal neighbourhood of the moduli space of $G$-bundles and show that it is a dynamical twist of $Y_\hbar(\mathfrak{d})$. Using this dynamical twist, we construct a dynamical quantum spectral $R$-matrix, which essentially controls the meromorphic braiding of $\Upsilon_\hbar^\sigma (\mathfrak{d})$. This construction is motivated by the Hecke action of the equivariant affine Grassmanian on the moduli space of $G$-bundles in the setting of coherent sheaves. Heuristically speaking, the quantum groupoid $\Upsilon_\hbar^\sigma (\mathfrak{d})$ controls this action at a formal neighbourhood of a regularly stable $G$-bundle. From the work of Costello-Witten-Yamazaki, it is expected that this Hecke action should give rise to a dynamical integrable system. Our result gives a mathematical confirmation of this and an explicit $R$-matrix underlying the integrability.

math.QA↗

arXiv2024

Tannakian QFT: from spark algebras to quantum groups

We propose a nonperturbative construction of Hopf algebras that represent categories of line operators in topological quantum field theory, in terms of semi-extended operators (spark algebras) on pairs of transverse topological boundary conditions. The construction is a direct implementation of Tannakian formalism in QFT. Focusing on d=3 dimensional theories, we find topological definitions of R-matrices, ribbon twists, and the Drinfeld double construction for generalized quantum groups. We illustrate our construction in finite-group gauge theory, and apply it to obtain new results for B-twisted 3d $\mathcal{N}=4$ gauge theories, a.k.a. equivariant Rozansky-Witten theory, or supergroup BF theory (including ordinary BF theory with compact gauge group). We reformulate our construction mathematically in terms of abelian and dg tensor categories, and discuss connections with Koszul duality.

hep-th↗

arXiv2024

Quantum Groups and Symplectic Reductions

Let $G$ be a reductive algebraic group with Lie algebra $\mathfrak{g}$ and $V$ a finite-dimensional representation of $G$. Costello-Gaiotto studied a graded Lie algebra $\mathfrak{d}_{\mathfrak{g}, V}$ and the associated affine Kac-Moody algebra. In this paper, we show that this Lie algebra can be made into a sheaf of Lie algebras over $T^*[V/G]=[\mu^{-1}(0)/G]$, where $\mu: T^*V\to \mathfrak{g}^*$ is the moment map. We identify this sheaf of Lie algebras with the tangent Lie algebra of the stack $T^*[V/G]$. Moreover, we show that there is an equivalence of braided tensor categories between the bounded derived category of graded modules of $\mathfrak{d}_{\mathfrak{g}, V}$ and graded perfect complexes of $[\mu^{-1}(0)/G]$.

math.RT↗

arXiv2024

Yangian for cotangent Lie algebras and spectral $R$-matrices

In this paper, we present a canonical quantization of Lie bialgebra structures on the formal power series $\mathfrak{d}[\![t]\!]$ with coefficients in the cotangent Lie algebra $\mathfrak{d} = T^*\mathfrak{g} = \mathfrak{g} \ltimes \mathfrak{g}^*$ to a simple complex Lie algebra $\mathfrak{g}$. We prove that these quantizations produce twists to the natural analog of the Yangian for $\mathfrak{d}$. Moreover, we construct spectral $R$-matrices for these twisted Yangians as compositions of twisting matrices. The motivation for the construction of these twisted Yangians over $\mathfrak{d}$ comes from certain 4d holomorphic-topological gauge theory. More precisely, we show that pertubative line operators in this theory can be realized as representations of these Yangians. Moreover, the comultiplications of these Yangians correspond to the monodial structure of the category of line operators.

math.QA↗

arXiv2024

Kazhdan-Lusztig Correspondence for Vertex Operator Superalgebras from Abelian Gauge Theories

We prove the Kazhdan-Lusztig correspondence for a class of vertex operator superalgebras which, via the work of Costello-Gaiotto, arise as boundary VOAs of topological B twist of 3d $\mathcal{N}=4$ abelian gauge theories. This means that we show equivalences of braided tensor categories of modules of certain affine vertex superalgebras and corresponding quantum supergroups. We build on the work of Creutzig-Lentner-Rupert to this large class of VOAs and extend it since in our case the categories don't have projective objectives and objects can have arbitrary Jordan H\"older length. Our correspondence significantly improves the understanding of the braided tensor category of line defects associated to this class of TQFT, by realizing line defects as modules of a Hopf algebra. In the process, we prove a case of the conjecture of Semikhatov-Tipunin, relating logarithmic CFTs to Nichols algebras of screening operators.

hep-th↗

arXiv2023

3d mirror symmetry of braided tensor categories

We study the braided tensor structure of line operators in the topological A and B twists of abelian 3d $\mathcal{N}=4$ gauge theories, as accessed via boundary vertex operator algebras (VOA's). We focus exclusively on abelian theories. We first find a non-perturbative completion of boundary VOA's in the B twist, which start out as certain affine Lie superalebras; and we construct free-field realizations of both A and B-twist VOA's, finding an interesting interplay with the symmetry fractionalization group of bulk theories. We use the free-field realizations to establish an isomorphism between A and B VOA's related by 3d mirror symmetry. Turning to line operators, we extend previous physical classifications of line operators to include new monodromy defects and bound states. We also outline a mechanism by which continuous global symmetries in a physical theory are promoted to higher symmetries in a topological twist -- in our case, these are infinite one-form symmetries, related to boundary spectral flow, which structure the categories of lines and control abelian gauging. Finally, we establish the existence of braided tensor structure on categories of line operators, viewed as non-semisimple categories of modules for boundary VOA's. In the A twist, we obtain the categories by extending modules of symplectic boson VOA's, corresponding to gauging free hypermultiplets; in the B twist, we instead extend Kazhdan-Lusztig categories for affine Lie superalgebras. We prove braided tensor equivalences among the categories of 3d-mirror theories. All results on VOA's and their module categories are mathematically rigorous; they rely strongly on recently developed techniques to access non-semisimple extensions.

hep-th↗

arXiv2023

Line Operators in $U(1|1)$ Chern-Simons Theory

We analyze the non-semisimple category of line operators in Chern-Simons gauge theories based off the Lie superalgebra $\mathfrak{gl}(1|1)$. Our proposal is that the category of line operators $\mathcal{C}$ can be identified with the derived category of modules for a boundary vertex operator algebra $\mathcal{V}$ realized as a certain infinite-order simple current extension of the affine current algebra $V(\mathfrak{gl}(1|1))$ by boundary monopole operators. By translating this simple current extension of $V(\mathfrak{gl}(1|1))$ to the unrolled, restricted quantum group $\overline{U}^E(\fgl(1|1))$, we show that our category of line operators admits a second description in terms of a quasi-quantum group $\mathcal{A}$ realized by uprolling. We also compare our results across an expected physical duality with the cyclic orbifold of a free, $B$-twisted hypermultiplet and find a slight discrepancy at the level of braiding and associator. We end with a detailed analysis of coupling to background flat $GL(1, \C)$ connections and the resulting category of non-genuine line operators.

hep-th↗

arXiv2022

3d Mirror Symmetry and the $βγ$ VOA

We study the simplest example of mirror symmetry for 3d $\mathcal N=4$ SUSY gauge theories: the A-twist of a free hypermultiplet and the B-twist of SQED. We particularly focus on the category of line operators in each theory. Using the work of Costello-Gaiotto, we define these categories as appropriate categories of modules for the boundary vertex operator algebras present in each theory. For the A-twist of a free hyper, this will be a certain category of modules for the $βγ$ VOA, properly containing the category previously studied by Allen-Wood. Applying the work of Creutzig-Kanade-McRae and Creutzig-McRae-Yang, we show that the category of line operators on the A side possesses the structure of a braided tensor category, extending the result of Allen-Wood. In addition, we prove that there is a braided tensor equivalence between the categories of line operators on the A side and B side, completing a non-trivial check of the 3d mirror symmetry conjecture. We derive explicit fusion rules as a consequence of this equivalence and obtain interesting relations with associated quantum group representations.

hep-th↗

arXiv2022

Local Operators of 4d $\mathcal{N}=2$ Gauge Theories From the Affine Grassmannian

We give a new, fully mathematical, construction of the space of local operators in the holomorphic-topological twist of 4d $\mathcal N=2$ gauge theories. It is based on computations of morphism spaces in the DG category of line operators, which (by work of Kapustin-Saulina and Cautis-Williams) may be represented as ind-coherent sheaves on the affine Grassmannian and, more generally, on the $\mathcal R_{G,N}$ spaces of Braverman-Finkelberg-Nakajima. We prove that characters of our morphisms spaces reproduce the Schur indices of 4d $\mathcal N=2$ theories, and that the spaces themselves agree with the 4d $\mathcal N=2$ vertex algebras of Beem-Lemos-Liendo-Peelaers-Rastelli-Van Rees, Oh-Yagi, Butson and Jeong. We also generalize our construction to local operators at junctions of Wilson-'t Hooft lines, and compare the Euler character of the morphism spaces to the Schur indices in the work of Cordova-Gaiotto-Shao.

math.AG↗

arXiv2018

On Behaviors of Maximal Dimension

In this paper, we investigate behaviors of Maximal Dimension, a group invariant involving certain configuration of maximal subgroups, which we denote by MaxDim. We prove that in some special cases, MaxDim(G\times H) = MaxDim(G) + MaxDim(H). We also prove a conjecture stated by Ellie Thieu which shows that groups with m < MaxDim can be constructed from groups with m < i.

math.GR↗