3D Mirror Research Articles

Higgs branch of heterotic little string theories: Hasse diagrams and generalized symmetries

We study the Higgs branches of the 6D (1, 0) little string theories that live on the world volume of NS5-branes probing an ADE-singularity in the heterotic E8×E8 and Spin(32)/Z2 string theories. On the E8×E8 side, such little string theories (LSTs) are obtained via fusion of orbi-instanton SCFTs. For the C2/ZK orbifolds, we determine a magnetic quiver for the Higgs branch from the alternative type IIA brane system engineering the LST; we show that the magnetic quiver obtained in this way is the same as the Coulomb gauging of the 3D mirrors associated with the orbi-instanton building blocks. Using quiver subtraction, we determine the Hasse diagram of Higgs branch renormalization group flows between the LSTs, and we analyze how the structure constants of the generalized global symmetries vary along the edges of the Hasse diagram. From the Hasse diagram of the Higgs branch, we are immediately able to identify LSTs with the same T-duality-invariant properties, and thus propose candidate T-dual pairs. We perform a similar analysis of the Higgs branch Hasse diagram and putative T-dual families for particular E6-orbifold LSTs by taking advantage of a duality between a rank zero orbi-instanton theory and a rank one conformal matter theory. Published by the American Physical Society 2024

Discrete global symmetries: gauging and twisted compactification

Discrete global symmetries of 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFTs are studied via two operations: gauging and twisted compactification. We consider gauging of discrete symmetries in several well-known 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFTs, including SU(n) SQCD with 2n flavors, theories of class S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{S} $$\\end{document} of type A2n−1, and Argyres-Douglas theories of type (AN, AN), as well as propose new 4d SCFTs as a result. The wreathing technique, which involves gauging a subgroup of the automorphism group of the quiver diagram of the corresponding 3d mirror theory, is exploited. This allows us to understand several properties of discretely gauged theories, including moduli spaces and how discrete gauging affects the mixed ’t Hooft anomaly between the 1-form symmetry and the 0-form flavor symmetry. Many examples are viewed through the lens of the Argyres-Seiberg duality and its generalization. We also examine discrete gauging of SU(2) SQCD with 4 flavors by various ℤ2 and ℤ2 × ℤ2 subgroups of the permutation group S4 using the superconformal index. Regarding compactification, we propose a magnetic quiver for 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SU(n) SQCD with 2n flavors compactified on a circle with a ℤ2 twist. The twisted compactification by non-invertible symmetries of the 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM theory with gauge group SU(N) is revisited. The non-invertible symmetry naturally gives rise to a ℤk action on the scalar fields parametrizing the moduli space. Upon examining the ℤk invariant chiral ring of the Higgs branch, we find that, in addition to the largest branch of the moduli space that is expected to be captured by the ABJ(M) theory, there exist in general nilpotent operators that lead to a branch of the moduli space which is a radical ideal.

Exploring Seiberg-like N-alities with eight supercharges

We show that a large subclass of 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 quiver gauge theories consisting of unitary and special unitary gauge nodes with only fundamental/bifundamental matter have multiple Seiberg-like IR duals. A generic quiver T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{T} $$\\end{document} in this subclass has a non-zero number of balanced special unitary gauge nodes and it is a good theory in the Gaiotto-Witten sense. We refer to this phenomenon as IR N-ality and the set of mutually IR dual theories as the N-al set associated with the quiver T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{T} $$\\end{document}. Starting from T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{T} $$\\end{document}, we construct a sequence of dualities by step-wise implementing a set of quiver mutations which act locally on the gauge nodes. The associated N-al theories can then be read off from this duality sequence. The quiver T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{T} $$\\end{document} generically has an emergent Coulomb branch global symmetry in the IR, such that the rank of the IR symmetry is always greater than the rank visible in the UV. We show that there exists at least one theory in the N-al set for which the rank of the IR symmetry precisely matches the rank of the UV symmetry. In certain special cases that we discuss in this work, the correct emergent symmetry algebra itself may be read off from this preferred theory (or theories) in addition to the correct rank. Finally, we give a recipe for constructing the 3d mirror associated with a given N-al set and show how the emergent Coulomb branch symmetry of T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{T} $$\\end{document} is realized as a UV-manifest Higgs branch symmetry of the 3d mirror. This paper is the second in a series of four papers on 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 Seiberg-like dualities, preceded by the work [1].

Generalized symmetries and anomalies of 3d $\mathcal{N}=4$ SCFTs

We study generalized global symmetries and their ’t Hooft anomalies in 3d \mathcal{N}=4𝒩=4 superconformal field theories (SCFTs). Following some general considerations, we focus on good quiver gauge theories, comprised of balanced unitary nodes and unbalanced unitary and special unitary nodes. While the global form of the Higgs branch symmetry group may be determined from the UV Lagrangian, the global form of Coulomb branch symmetry groups and associated mixed ’t Hooft anomalies are more subtle due to potential symmetry enhancement in the IR. We describe how Coulomb branch symmetry groups and their mixed ’t Hooft anomalies can be deduced from the UV Lagrangian by studying center charges of various types of monopole operators, providing a concrete and unambiguous way to implement ’t Hooft anomaly matching. The final expression for the symmetry group and ’t Hooft anomalies has a concise form that can be easily read off from the quiver data, specifically from the positions of the unbalanced and flavor nodes with respect to the positions of the balanced nodes. We provide consistency checks by applying our method to compute symmetry groups of 3d \mathcal{N}=4𝒩=4 theories corresponding to magnetic quivers of 4d Class S theories and 5d SCFTs. We are able to match these results against the flavor symmetry groups of the 4d and 5d theories computed using independent methods. Another strong consistency check is provided by comparing symmetry groups and anomalies of two theories related by 3d mirror symmetry.

Characterizing chiroptical properties of 2D/3D structures based on an improved coupled dipole theory.

To reveal the difference/connection between two-dimensional and three- dimensional (2D and 3D) chiroptical properties and their relation with 2D/3D symmetry/breaking, we develop an improved coupled dipole theory (ICDT) based on a model system of nanorod (NR) dimer. Our analytical ICDT can overcome the shortcoming of the traditional coupled dipole theory and points out the important role of scattering circular dichroism (SCD) in characterizing 2D chirality. The ICDT, supported by finite-difference time-domain (FDTD) simulation, reveals the physical origin of 2D chiroptical response: the interaction induced asymmetric effective polarizability for two identical NRs in a symmetry broken configuration. By tuning the NR's position/inter-particle distance, we find an optimal structure of maximum SCD due to the competition between geometric symmetry breaking and interaction. In addition, the interplay between 2D in-plane mirror symmetry breaking and three-dimensional (3D) mirror symmetry breaking leads to a symmetry broken system with zero SCD. The relation between chirality and reciprocity has also been addressed.

Exploring Seiberg-like dualities with eight supercharges

We propose a family of IR dualities for 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 U(N) SQCD with Nf fundamental flavors and P Abelian hypermultiplets i.e. P hypermultiplets in the determinant representation of the gauge group. These theories are good in the Gaiotto-Witten sense if the number of fundamental flavors obeys the constraint Nf≥ 2N − 1 with generic P ≥ 1, and in contrast to the standard U(N) SQCD, they do not admit an ugly regime. The IR dualities in question arise in the window Nf = 2N + 1, 2N, 2N − 1, with P = 1 in the first case and generic P ≥ 1 for the others. The dualities involving Nf = 2N ± 1 are characterized by an IR enhancement of the Coulomb branch global symmetry on one side of the duality, such that the rank of the emergent global symmetry group is greater than the rank of the UV global symmetry. The dual description makes the rank of this emergent global symmetry manifest in the UV. In addition, one can read off the emergent global symmetry itself from the dual quiver. We show that these dualities are related by certain field theory operations and assemble themselves into a duality web. Finally, we show that the U(N) SQCDs with Nf≥ 2N − 1 and P Abelian hypers have Lagrangian 3d mirrors, and this allows one to explicitly write down the 3d mirror associated with a given IR dual pair. This paper is the first in a series of four papers on 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 Seiberg-like dualities.

3D Mirror Symmetry for Instanton Moduli Spaces

We prove that the Hilbert scheme of k points on {mathbb {C}}^2 (hbox {Hilb}^k[{mathbb {C}}^2]) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the {mathbb {C}}^times _hbar -action. First, we find a two-parameter family X_{k,l} of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of hbox {Hilb}^k[{mathbb {C}}^2] is obtained via direct limit llongrightarrow infty and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted hbar -opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-N sheaves on {mathbb {P}}^2 with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.

Diffuse midline glioma invasion and metastasis rely on cell-autonomous signaling.

Diffuse midline gliomas (DMG) are pediatric tumors with negligible 2-year survival after diagnosis characterized by their ability to infiltrate the central nervous system. In the hope of controlling the local growth and slowing the disease, all patients receive radiotherapy. However, distant progression occurs frequently in DMG patients. Current clues as to what causes tumor infiltration circle mainly around the tumor microenvironment, but there are currently no known determinants to predict the degree of invasiveness. In this study, we use patient-derived glioma stem cells (GSCs) to create patient-specific 3D avatars to model interindividual invasion and elucidate the cellular supporting mechanisms. We show that GSC models in 3D mirror the invasive behavior of the parental tumors, thus proving the ability of DMG to infiltrate as an autonomous characteristic of tumor cells. Furthermore, we distinguished 2 modes of migration, mesenchymal and ameboid-like, and associated the ameboid-like modality with GSCs derived from the most invasive tumors. Using transcriptomics of both organoids and primary tumors, we further characterized the invasive ameboid-like tumors as oligodendrocyte progenitor-like, with highly contractile cytoskeleton and reduced adhesion ability driven by crucial over-expression of bone morphogenetic pathway 7 (BMP7). Finally, we deciphered MEK, ERK, and Rho/ROCK kinases activated downstream of the BMP7 stimulation as actionable targets controlling tumor cell motility. Our findings identify 2 new therapeutic avenues. First, patient-derived GSCs represent a predictive tool for patient stratification in order to adapt irradiation strategies. Second, autocrine and short-range BMP7-related signaling becomes a druggable target to prevent DMG spread and metastasis.

A toolkit for ortho-symplectic dualities

We propose new confining dualities in 3dN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 gauge theories with orthogonal gauge groups, with and without monopole superpotentials. Deriving some of those dualities requires a sequence of gauging and Higgsing for a ℤ2 symmetry. This prevents the gauge theory from developing a smooth quantum moduli space and affects the global structure of the gauge group, muting it from SO to O+.The confining dualities provide tools to deconfine fields transforming in the symmetric rank-2 representation of classical gauge groups. As an application, we propose and derive S-confining dualities for SO(N) (Sp(N)) gauge theories with an adjoint and 1 (2) fundamentals, respectively. From these S-confining dualities, we readily obtain various duality appetizers and the 3d mirror of A2N Argyres-Douglas theories.

3d $\mathcal{N}=4$ mirror symmetry with 1-form symmetry

The study of 3d mirror symmetry has greatly enhanced our understanding of various aspects of 3d \mathcal{N}=4𝒩=4 theories. In this paper, starting with known mirror pairs of 3d \mathcal{N}=4𝒩=4 quiver gauge theories and gauging discrete subgroups of the flavour or topological symmetry, we construct new mirror pairs with non-trivial 1-form symmetry. By providing explicit quiver descriptions of these theories, we thoroughly specify their symmetries (0-form, 1-form, and 2-group) and the mirror maps between them.

A tale of 2-groups: Dp(USp(2N)) theories

A 1-form symmetry and a 0-form symmetry may combine to form an extension known as the 2-group symmetry. We find the presence of the latter in a class of Argyres-Douglas theories, called Dp(USp(2N)), which can be realized by ℤ2-twisted compactification of the 6d mathcal{N} = (2, 0) of the D-type on a sphere with an irregular twisted puncture and a regular twisted full puncture. We propose the 3d mirror theories of general Dp(USp(2N)) theories that serve as an important tool to study their flavor symmetry and Higgs branch. Yet another important result is presented: we elucidate a technique, dubbed “bootstrap”, which generates an infinite family of {D}_p^b(G) theories, where for a given arbitrary group G and a parameter b, each theory in the same family has the same number of mass parameters, same number of marginal deformations, same 1-form symmetry, and same 2-group structure. This technique is utilized to establish the presence or absence of the 2-group symmetries in several classes of {D}_p^b(G) theories. In this regard, we find that the Dp(USp(2N)) theories constitute a special class of Argyres-Douglas theories that have a 2-group symmetry.

Three-Dimensional Mirror-Assisted and Concave Pyramid-Shaped Solar-Thermal Steam Generator for Highly Efficient and Stable Water Evaporation and Brine Desalination.

Although significant advances have been achieved in developing solar-driven water evaporators for seawater desalination, there is still room for simultaneously enhancing water evaporation efficiency, salt resistance, and utilization of solar energy. Herein, for the first time, we demonstrate a highly efficient three-dimensional (3D) mirror-assisted and concave pyramid-shaped solar-thermal water evaporation system for high-yield and long-term desalination of seawater and brine water, which consists of a 3D concave pyramid-shaped solar-thermal architecture on the basis of polypyrrole-coated nonwoven fabrics (PCNFs), a 3D mirror array, a self-floating polystyrene foam layer, and a tail-like PCNF for upward transport of water. The 3D concave pyramid-shaped solar-thermal architecture enables multiple solar light reflections to absorb more solar energy, while the 3D mirror-assisted solar light enhancement design can activate the solar-thermal energy conversion of the back side of the concave pyramid-shaped PCNF architecture to improve the solar-thermal energy conversion efficiency. Crucially, selective accumulation of the precipitated salts on the back side of the concave pyramid-shaped architecture is realized, ensuring a favorable salt-resistant feature. The 3D mirror-assisted and concave pyramid-shaped solar-driven water evaporation system achieves a record high water evaporation rate of 4.75 kg m-2 h-1 under 1-sun irradiation only and exhibits long-term desalination stability even when evaporating high-salinity brine waters, demonstrating its great applicability and reliability for high-yield solar-driven desalination of seawater and high-salinity brine water.

FI-flows of 3d N=4 theories

We study the 3d mathcal{N} = 4 RG-flows triggered by Fayet-Iliopoulos deformations in unitary quiver theories. These deformations can be implemented by a new quiver algorithm which contains at its heart a problem at the intersection of linear algebra and graph theory. When interpreted as magnetic quivers for SQFTs in various dimensions, our results provide a systematic way to explore RG-flows triggered by mass deformations and generalizations thereof. This is illustrated by case studies of SQCD theories and low rank 4d mathcal{N} = 2 SCFTs. A delightful by-product of our work is the discovery of an interesting new 3d mirror pair.

Higgs branches of Argyres-Douglas theories as quiver varieties

We present a general prescription for constructing 3d mathcal{N} = 4 Lagrangians for the IR SCFTs that arise from the circle reduction of a large class of Argyres-Douglas theories. The resultant Lagrangian gives a realization of the Higgs branch of the 4d SCFT as a quiver variety, up to a set of decoupled interacting SCFTs with empty Higgs branches. As representative examples, we focus on the families (Ap−N−1, AN−1) and Dp(SU(N)). The Lagrangian in question is generically a non-ADE-type quiver gauge theory involving only unitary gauge nodes with fundamental and bifundamental hypermultiplets, as well as hypermultiplets which are only charged under the U(1) subgroups of certain gauge nodes. Our starting point is the Lagrangian 3d mirror of the circle-reduced Argyres-Douglas theory, which can be read off from the class mathcal{S} construction. Using the toolkit of the S-type operations, developed in [1], we show that the mirror of the 3d mirror for any Argyres-Douglas theory in the aforementioned families is guaranteed to be a Lagrangian theory of the above type, up to some decoupled free sectors. We comment on the extension of this procedure to other families of Argyres-Douglas theories. In addition, for the case of Dp(SU(N)) theories, we compare these 3d Lagrangians to the ones found in [2] and propose that the two are related by an IR duality. We check the proposed IR duality at the level of the three-sphere partition function for specific examples. In contrast to the 3d Lagrangians in [2], which are linear chains involving unitary-special unitary nodes, we observe that the Coulomb branch global symmetries are manifest in the 3d Lagrangians that we find.

Line defects in three dimensional mirror symmetry beyond ADE quivers

Understanding the map of line defects in a Quantum Field Theory under a given duality is generically a difficult problem. This paper is the second in a series which aims to address this question in the context of 3d mathcal{N} = 4 mirror symmetry. A general prescription for constructing vortex defects and their mirror maps in quiver gauge theories beyond the A-type was presented by the author in an earlier paper [1], where specific examples involving D-type and affine D-type quivers were discussed. In this paper, we apply the aforementioned prescription to construct a family of vortex defects as coupled 3d-1d systems in quiver gauge theories beyond the ADE-type, and study their mirror maps. Specifically, we focus on a class of quiver gauge theories involving unitary gauge nodes with edge multiplicity greater than 1, i.e. two gauge nodes in these theories may be connected by multiple bifundamental hypermultiplets. Quiver gauge theories of this type arise as 3d mirrors of certain Argyres-Douglas theories compactified on a circle. Some of these quiver gauge theories are known to have a pair of 3d mirrors, which are themselves related by an IR duality, discussed recently in [2]. For a concrete example where a pair of 3d mirrors do exist, we study how the vortex defects constructed using our prescription map to Wilson defects in each mirror theory.

3D mirror symmetry and the βγ VOA

We study the simplest example of mirror symmetry for 3d [Formula: see text] 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 [Formula: see text] 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 nontrivial 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.

Higgs branch, Coulomb branch, and Hall-Littlewood index

The Higgs branch of 4D $\mathcal{N}=2$ superconformal field theories can be analyzed via the Hilbert series of the Higgs branch or, in special cases, by computing the Hall-Littlewood index. For any class $\mathcal{S}$ theory corresponding to a genus-zero Riemann surface, they are conjectured to be identical. We present several families of counterexamples. We find that for any class $\mathcal{S}$ theory with four or more ${\mathbb{Z}}_{2}$-twisted punctures, they do not match. We construct 3D mirrors for such theories and analyze their Coulomb branch Hilbert series to compute the Higgs branch Hilbert series of the 4D theory. We further construct $a=c$ theories in class $\mathcal{S}$ using the twisted punctures, and these theories, which includes the ${\stackrel{^}{D}}_{4}(SU(2n+1))$ theories, have Hall-Littlewood index different from the Hilbert series of the Higgs branch. We conjecture that this is the case for all $a=c$ theories with nonempty Higgs branch, including $\mathcal{N}\ensuremath{\ge}3$ superconformal field theories.

Quantum K-theory of toric varieties, level structures, and 3d mirror symmetry

We introduce a new version of 3d mirror symmetry for the toric quotient stack [Cn/K] where K=(C⁎)k and the torus action is determined by the given charge matrix. It is inspired by a 3d N=2 abelian mirror symmetry construction in physics. Given some toric data, we introduce the K-theoretic I-function with the effective level structure for the associated quotient stack. When a particular stability condition is chosen, it restricts to the I-function for the particular toric variety. The mirror of a GIT quotient stack is defined by the Gale dual of the original toric data. We then prove the mirror conjecture that the I-functions of a mirror pair coincide, under the mirror map, which switches Kähler and equivariant parameters, and changes q to q−1.

3d mirror symmetry of the cotangent bundle of the full flag variety

Aganagic and Okounkov proved that the elliptic stable envelope provides the pole cancellation matrix for the enumerative invariants of quiver varieties known as vertex functions. This transforms a basis of a system of q-difference equations holomorphic in variables $${\varvec{z}}$$ with poles in variables $${\varvec{a}}$$ to a basis of solutions holomorphic in $${\varvec{a}}$$ with poles in $${\varvec{z}}$$ . The resulting functions are expected to be the vertex functions of the 3d mirror dual variety. In this paper, we prove that the functions obtained by applying the elliptic stable envelope to the vertex functions of the cotangent bundle of the full flag variety are precisely the vertex functions for the same variety under an exchange of the parameters $$\text{\AA} \leftrightarrow {\varvec{z}}$$ . As a corollary of this, we deduce the expected 3d mirror relationship for the elliptic stable envelope.

3D mirror printing combined with minimally invasive percutaneous plate osteosynthesis in the treatment of adult median clavicle fracture

To retrospectively analyze clinical effect of 3D mirror printing combining with minimally invasive percutaneous plate osteosynthesis(MIPPO) technology in the treatment of adult clavicle fracture. The clinical data of 32 adult patients with collarbone AllmanⅠdisplacement of fracture who had undergone the combination of 3D mirror printing with MIPPO technology treatment were collected from October 2019 to October 2020. There were 19 males and 13 females, aged from 19 to 55 years old with an average of(34.12±16.42) years old. According to Allman classification, there were 14 cases of AllmanⅠb and 18 cases of AllmanⅠc. Fracture occurred on the left in 17 cases and on the right in 15 cases. Postoperative complications, operation time, incision length, intraoperative blood loss, the number of fluoroscopies, fracture healing time of all patients were retrospectively analyzed, and the curative effect was evaluated by Constant-Murley shoulder joint score before operation, 1 and 6 months after operation. All the 32 patients were followed up for 6 to 8 months, with an average of (6.67±2.13) months. All the patients did not have postoperative complications such as incision infection and skin numbness in the operating area. The operation time was (35.37±4.28) min. The incision length was (3.78±0.45) cm. Intraoperative blood loss was(37.23±4.76) ml. The number of fluoroscopies was(3.12±0.47) times. Fracture healing time was (8.18±2.58) weeks. Constant-Murley shoulder joint score of the affected side at 1 and 6 months after operation was 81.08±3.92 and 98.03±1.05 respectively, which had a significant difference compared with 54.62±5.25 before operation(P<0.05). According to Constant-Murley shoulder function score at 6 months after operation, 30 cases were excellent and 2 cases were good. 3D mirror printing can effectively assist MIPPO technique in the treatment of adult clavicle AllmanⅠdisplaced fracture. It not only has the advantages of less trauma and fewer complications of the MIPPO technique, but also can further shorten the operation time and improve the operation maneuverability. It is a better method in the treatment of adult clavicle AllmanⅠdisplaced fracture.