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We present a general and practical theoretical framework to investigate how energy is dissipated in open quantum system dynamics. This is performed by quantifying the contributions of individual bath components to the overall dissipation of the system. The framework is based on the Nakajima–Zwanzig projection operator technique, which allows us to express the rate of energy dissipation into a specific bath degree of freedom by using traces of operator products. The approach captures system-bath interactions to all orders, but is based on second-order perturbation theory on the off-diagonal subsystem's couplings and a Markovian description of the bath. The usefulness of our theory is demonstrated by applying it to various models of open quantum systems involving harmonic oscillators or spin baths and connecting the outcomes to existing results such as our previously reported formula derived for locally coupled harmonic baths [Kim and Franco, J. Chem. Phys. 154, 084109 (2021)]. We also prove that the dissipation calculated by our theory rigorously satisfies thermodynamic principles such as energy conservation and detailed balance. Overall, the strategy can be used to develop the theory and simulation of dissipation pathways to interpret and engineer the dynamics of open quantum systems. [ABSTRACT FROM AUTHOR]

With the development of quantum computers, traditional communication security based on classical computers can no longer guarantee unconditional security. At the same time, a series of encrypted communication solutions that rely on quantum mechanical properties have been spawned. This article focuses on the field of quantum signatures, based on the entangled Greenberger–Horne–Zeilinger state, combining group signatures (any member of the group can sign on behalf of the group) and blind signatures (the signer cannot know the accuracy of the signature content) proposed a new quantum group blind signature scheme that utilizes quantum key distribution to achieve unconditional security. By splitting the original message, the scheme strengthens the relevance of the signature participants and makes the signature to have stronger binding. Then, the security of the scheme against internal and external attacks is analyzed. In this scheme, a pleasing efficiency has been achieved, and the non-repudiation of signatures has been enhanced. At the same time, disputed signatures can be traced back to the signer. [ABSTRACT FROM AUTHOR]

An exciton–phonon (ex–ph) model based on our recently developed block interaction product basis framework is introduced to simulate the distal charge separation (CS) process in aggregated perylenediimide (PDI) trimer incorporating the quantum dynamic method, i.e., the time-dependent density matrix renormalization group. The electronic Hamiltonian in the ex–ph model is represented by nine constructed diabatic states, which include three local excited (LE) states and six charge transfer (CT) states from both the neighboring and distal chromophores. These diabatic states are automatically generated from the direct products of the leading localized neutral or ionic states of each chromophore's reduced density matrix, which are obtained from ab initio quantum chemical calculation of the subsystem consisting of the targeted chromophore and its nearest neighbors, thus considering the interaction of the adjacent environment. In order to quantum-dynamically simulate the distal CS process with massive coupled vibrational modes in molecular aggregates, we used our recently proposed hierarchical mapping approach to renormalize these modes and truncate those vibrational modes that are not effectively coupled with electronic states accordingly. The simulation result demonstrates that the formation of the distal CS process undergoes an intermediate state of adjacent CT, i.e., starts from the LE states, passes through an adjacent CT state to generate the intermediates (∼ 200 fs), and then formalizes the targeted distal CS via further charge transference (∼ 1 ps). This finding agrees well with the results observed in the experiment, indicating that our scheme is capable of quantitatively investigating the CS process in a realistic aggregated PDI trimer and can also be potentially applied to exploring CS and other photoinduced processes in larger systems. [ABSTRACT FROM AUTHOR]

In the past decade, the quantum chemical version of the density matrix renormalization group method has established itself as the method of choice for strongly correlated molecular systems. However, despite its favorable scaling, in practice, it is not suitable for computations of dynamic correlation. Several approaches to include that in post-DMRG methods exist; in our group, we focused on the tailored coupled cluster (TCC) approach. This method works well in many situations; however, in exactly degenerate cases (with two or more determinants of equal weight), it exhibits a bias toward the reference determinant representing the Fermi vacuum. Although sometimes it is possible to use a compensation scheme to avoid this bias for energy differences, it is certainly a drawback. In order to overcome this bias of the TCC method, we have developed a Hilbert-space multireference version of tailored CC, which can treat several determinants on an equal footing. We have implemented and compared the performance of three Hilbert-space multireference coupled cluster (MRCC) variants—the state universal one and the Brillouin–Wigner and Mukherjee's state specific ones. We have assessed these approaches on the cyclobutadiene and tetramethyleneethane molecules, which are both diradicals with exactly degenerate determinants at a certain geometry. We have also investigated the sensitivity of the results on the orbital rotation of the highest occupied and lowest unoccupied molecular orbital (HOMO–LUMO) pair, as it is well known that Hilbert-space MRCC methods are not invariant to such transformations. [ABSTRACT FROM AUTHOR]

Using light as an external stimulus to control (bio)chemical processes offers many distinct advantages. Most importantly, it allows for spatiotemporal control simply through operating the light source. Photocleavable protecting groups (PPGs) are a cornerstone class of compounds that are used to achieve photocontrol over (bio)chemical processes. PPGs are able to release a payload of interest upon light irradiation. The successful application of PPGs hinges on their efficiency of payload release, captured in the uncaging Quantum Yield (QY). Heterolytic PPGs efficiently release low pKa payloads, but their efficiency drops significantly for payloads with higher pKa values, such as alcohols. For this reason, alcohols are usually attached to PPGs via a carbonate linker. The self‐immolative nature of the carbonate linker results in concurrent release of CO2 with the alcohol payload upon irradiation. We introduce herein novel PPGs containing sulfites as self‐immolative linkers for photocaged alcohol payloads, for which we discovered that the release of the alcohol proceeds with higher uncaging QY than an identical payload released from a carbonate‐linked PPG. Furthermore, we demonstrate that uncaging of the sulfite‐linked PPGs results in the release of SO2 and show that the sulfite linker improves water solubility as compared to the carbonate‐based systems. [ABSTRACT FROM AUTHOR]

For a finite dimensional representation V of a finite reflection group W , we consider the rational Cherednik algebra H t , c (V , W) associated with (V , W) at the parameters t ≠ 0 and c. The Dunkl total angular momentum algebra O t , c (V , W) arises as the centraliser algebra of the Lie superalgebra osp (1 | 2) containing a Dunkl deformation of the Dirac operator, inside the tensor product of H t , c (V , W) and the Clifford algebra generated by V. We show that when dim ⁡ V ≥ 3 and for every value of the parameter c , the centre of O t , c (V , W) is isomorphic to a univariate polynomial ring. Notably, the generator of the centre changes depending on whether or not (− 1) V is an element of the group W. Using this description of the centre, and using the projection of the pseudo scalar from the Clifford algebra into O t , c (V , W) , we establish results analogous to "Vogan's conjecture" for a family of operators depending on suitable elements of the double cover W ˜. [ABSTRACT FROM AUTHOR]

We show, by introducing purely auxiliary gluinos and scalars, that the quantum path integral for a class of 3D interacting non-supersymmetric gauge theories localises. The theories in this class all admit a ‘Manin gauge theory’ formulation, that we introduce; it is obtained by enhancing the gauge algebra of the theory to a Dirac structure inside a Manin pair. This machinery allows us to do localisation computations for every theory in this class at once, including for 3D Yang-Mills theory, and for its Third Way deformation; the latter calculation casts the Third Way path integral into an almost 1-loop exact form. [ABSTRACT FROM AUTHOR]

We prove the existence of a universal braided compact quantum group acting on a graph C ∗ -algebra in the category of - C ∗ -algebras with a twisted monoidal structure, in the spirit of the seminal work of Wang. To achieve this, we construct a braided analogue of the free unitary quantum group and study its bosonization. As a concrete example, we compute this universal braided compact quantum group for the Cuntz algebra. [ABSTRACT FROM AUTHOR]

Ever since new tetrads in four-dimensional Lorentzian curved spacetimes were introduced, many outstanding properties and results have been proven regarding Riemannian geometry, gauge theory or group theory. These tetrads might carry spacetime-kinematic information about particle multiplets. Among other properties, these new tetrads, locally and covariantly diagonalize stress–energy tensors. In the case where particles have an electromagnetic field, the local planes of gauge symmetry served as a tool in order to propose a new gravitational-kinematic interpretation of particle multiplets. The core reason stems from the mathematical fact that all local gauge symmetries associated to the Standard Model have been proven to be isomorphic to local groups of tetrad transformations in four-dimensional Lorentzian spacetimes. Therefore, the different stress–energy tensors have been proven to determine the local gauge geometry as a natural part of Riemannian geometry. The stress–energy tensors determine locally the orthogonal planes such that the rotation on either of them of the local tetrad vectors that span these planes is isomorphic to local tetrad gauge transformations. All these properties put together allow for a reinterpretation presented previously of particle states as particle gravitational-kinematic-spacetime tetrad states in the asymptotically flat limit. We proceed in this paper to study the case where there are T , U and V isospin subalgebras or submultiplets within the context of this new Riemannian interpretation of gauge symmetries. [ABSTRACT FROM AUTHOR]

In [21] , we give a Ringel-Hall algebra approach to the canonical bases in the symmetric affine cases. In this paper, we extend the results to general symmetrizable affine cases by using Ringel-Hall algebras of representations of a valued quiver. We obtain a bar-invariant basis B ′ = { C (c , t λ) | (c , t λ) ∈ G a } in the generic composition algebra C ⁎ and prove that B ′ = B ′ ⊔ (− B ′) coincides with Lusztig's signed canonical basis B. Moreover, in type B ˜ n , C ˜ n , B ′ is the canonical basis B. [ABSTRACT FROM AUTHOR]

The practical implementation of aqueous zinc‐ion batteries (AZIBs) has encountered obstacles stemming from the limited reversibility of the zinc anode, primarily due to dendrite proliferation and water‐induced reactions occurring. In this investigation, a novel bifunctional interphase is proposed by integrating nitrogen and oxygen group graphene quantum dot (N‐O‐GQD) additives into the electrolyte. Experimental results and theoretical calculations demonstrate that the amphipathic N‐O‐GQD additive enhances the stability of the electrode by forming a protection layer on Zn surface. The zincophilic and hydrophobic nitrogen function groups stick to the surface of Zn electrodes to form a hydrophobic layer that shields water molecules from the electrode and promotes the uniform deposition of Zn. The hydrophilic hydroxyl function groups are exposed to the electrolyte to improve the compatibility at the electrode/electrolyte interface. As a result, the amphipathic N‐O‐GQD additive enables a robust cycling performance at high depth of discharge (DOD). Significantly, cells incorporating N‐O‐GQDs demonstrate a remarkable Coulombic efficiency of 99.7% over 900 cycles and sustain dendrite‐free cycling for 564 h (DOD = 51%). Particularly noteworthy is the performance of the modified Zn||ZnVO full cell with robust cycling behavior, enduring 4 000 cycles at 10 A g−1. [ABSTRACT FROM AUTHOR]

Wilson–Fisher expansion near upper critical dimension has proven to be an invaluable conceptual and computational tool in our understanding of the universal critical behavior in the ϕ4 field theories that describe low-energy physics of the canonical models, such as Ising, XY, and Heisenberg. Here, I review its application to a class of the Gross–Neveu–Yukawa (GNY) field theories, which emerge as possible universal description of a number of quantum phase transitions in electronic two-dimensional systems such as graphene and d-wave superconductors. GNY field theories may be viewed as minimal modifications of the ϕ4 field theories in which the order parameter is coupled to relativistic Dirac fermions through Yukawa term and which still exhibit critical fixed points in the suitably formulated Wilson–Fisher ϵ-expansion. I discuss the unified GNY field theory for a set of different symmetry-breaking patterns, with focus on the semimetal-Néel-ordered-Mott insulator quantum phase transition in the half-filled Hubbard model on the honeycomb lattice, for which a comparison between the state-of-the-art ϵ-expansion, quantum Monte Carlo, large N, and functional renormalization-group calculations can be made. [ABSTRACT FROM AUTHOR]

Since the discovery of quantum mechanics, the fact that the wavefunction is defined on the 3 n -dimensional configuration space rather than on the 3-dimensional space has seemed uncanny to many, including Schrödinger, Lorentz, and Einstein. Even today, this continues to be seen as a significant issue in the foundations of quantum mechanics. In this article, it will be shown that the wavefunction is, in fact, a genuine object on space. While this may seem surprising, the wavefunction does not possess qualitatively new features that were not previously encountered in objects known from Euclidean geometry and classical physics. The methodology used involves finding equivalent reinterpretations of the wavefunction exclusively in terms of objects from the geometry of space. The result is that we will find the wavefunction to be equivalent to geometric objects on space in the same way as was always the case in geometry and physics. This will be demonstrated to hold true from the perspective of Euclidean geometry, but also within Felix Klein's Erlangen Program, which naturally fits into the classification of quantum particles by the representations of spacetime isometries, as realized by Wigner and Bargmann, adding another layer of confirmation. These results lead to clarifications in the debates about the ontology of the wavefunction. From an empirical perspective, we already take for granted that all quantum experiments take place in space. I suggest that the reason why this works is that they can be interpreted naturally and consistently with the results presented here, showing that the wavefunction is an object on space. [ABSTRACT FROM AUTHOR]

We apply Takesaki’s and Connes’s ideas on structure analysis for type III factors to the study of links (a short term of Markov kernels) appearing in asymptotic representation theory. [ABSTRACT FROM AUTHOR]

We introduce a natural generalization of the notion of strongly approximately transitive (SAT) states for actions of locally compact quantum groups. In the case of discrete quantum groups of Kac type, we show that the existence of unique stationary SAT states entails rigidity results concerning injective extensions of quantum group von Neumann algebras. [ABSTRACT FROM AUTHOR]

Under a suitable hypothesis, we construct a full set of pairwise orthogonal maximal vectors in V ⊗ n , where V = V (1) is the simple module of highest weight 1 for the quantized enveloping algebra U (sl 2). We give a number of applications, one of which is an orthogonal basis of the simple modules for the Temperley–Lieb algebra TL n. We relate this new orthogonal basis to the standard cellular basis. [ABSTRACT FROM AUTHOR]

We show that there exists a unique crystal base of a parabolic Verma module over a quantum orthosymplectic superalgebra, which is induced from a q -analogue of a polynomial representation of a general linear Lie superalgebra. [ABSTRACT FROM AUTHOR]

In 1998, Georgia Benkart and Tom Roby introduced the down-up algebra A. The algebra A is associative, noncommutative, and infinite-dimensional. It is defined by two generators A , B and two relations called the down-up relations. In the present paper, we introduce the Z 3 -symmetric down-up algebra A. We define A by generators and relations. There are three generators A , B , C and any two of these satisfy the down-up relations. We describe how A is related to some familiar algebras in the literature, such as the Weyl algebra, the Lie algebras sl 2 and sl 3 , the sl 3 loop algebra, the Kac-Moody Lie algebra A 2 (1) , the q -Weyl algebra, the quantized enveloping algebra U q (sl 2) , and the quantized enveloping algebra U q (A 2 (1)). We give some open problems and conjectures. [ABSTRACT FROM AUTHOR]

We construct a family of cogroupoids associated to preregular forms and recover the Morita-Takeuchi equivalence for Artin-Schelter regular algebras of dimension two, observed by Raedschelders and Van den Bergh. Moreover, we study the 2-cocycle twists of pivotal analogues of these cogroupoids, by developing a categorical description of preregularity in any tensor category that has a pivotal structure. [ABSTRACT FROM AUTHOR]

Motivated by the fact that the Hopf-cyclic (co)homologies of function algebras over Lie groups and universal enveloping algebras over Lie algebras capture the Lie group and Lie algebra (co)homologies, we hereby upgrade the classical van Est isomorphism to ones between the Hopf-cyclic (co)homologies of quantized algebras of functions and quantized universal enveloping algebras, both in h-adic and q-deformation frameworks. [ABSTRACT FROM AUTHOR]

The Yang–Baxter–like matrix equation plays an important role in quantum group theory, knot theory and braid groups, which has received great attention from physicists and mathematicians. A difficult and important open problem is to find all the solutions of the Yang–Baxter–like matrix equation. As a matter of fact, it is difficult to find all of the solutions even when the coefficient matrix is a $ 3\times 3 $ 3 × 3 matrix. To the best of our knowledge, when the coefficient matrix is a $ 3\times 3 $ 3 × 3 diagonalizable complex matrix with three distinct nonzero eigenvalues, finding all the solutions of the Yang–Baxter–like matrix equation is still an open problem. In order to fill-in this gap, we first present a sufficient and necessary condition for the commuting solutions of the Yang–Baxter–like matrix equation, and give all the commuting solutions of the matrix equation. Second, with the help of a simplified matrix equation, we derive all the non–commuting solutions of the Yang–Baxter–like matrix equation by discussing whether the off–diagonal elements of the solutions are zero or not. [ABSTRACT FROM AUTHOR]

We establish a Schur-Weyl duality between a shifted quantum affine algebra and an Ariki-Koike algebra. Then, we realize a cyclotomic q -Schur algebra in the context of the Schur-Weyl duality. [ABSTRACT FROM AUTHOR]

In this expository paper, we present a brief introduction to the geometrical modeling of some quantum computing problems. After a brief introduction to establish the terminology, we focus on quantum information geometry and Z X -calculus, establishing a connection between quantum computing questions and quantum groups, i.e. Hopf algebras. [ABSTRACT FROM AUTHOR]

Quantum computing is an exciting field with high disruptive potential, but very difficult to access. For this reason, many approaches to teaching quantum computing are being developed worldwide. This always raises questions about the didactic concept, the content actually taught, and how to measure the success of the teaching concept. In 2022 and 2023, the authors taught a total of nine two-week MOOCs (massive open online courses) with different possible learning paths on the Hasso Plattner Institute's OpenHPI platform. The purpose of the platform is to make computer science education available to everyone free of charge. The nine quantum courses form a self-contained curriculum. A total of more than 17,000 course attendances have been taken by about 7400 natural persons, and the number is still rising. This paper presents the course concept and evaluates the anonymized data on the background of the participants, their behaviour in the courses, and their learning success. This paper is the first to analyze such a large dataset of MOOC-based quantum computing education. The summarized results are a heterogeneous personal background of the participants biased towards IT professionals, a majority following the didactic recommendations, and a high success rate, which is strongly correlatated with following the didactic recommendations. The amount of data from such a large group of quantum computing learners provides many avenues for further research in the field of quantum computing education. The analyses show that the MOOCs are a low-threshold concept for getting into quantum computing. It was very well received by the participants. The concept can serve as an entry point and guide for the design of quantum computing courses. [ABSTRACT FROM AUTHOR]

We develop a theory of stated SL(n$n$)‐skein modules, Sn(M,N)$\mathcal {S}_n(M,\mathcal {N})$, of 3‐manifolds M$M$ marked with intervals N$\mathcal {N}$ in their boundaries. These skein modules, generalizing stated SL(2)‐modules of the first author, stated SL(3)‐modules of Higgins', and SU(n)‐skein modules of the second author, consist of linear combinations of framed, oriented graphs, called n$n$‐webs, with ends in N$\mathcal {N}$, considered up to skein relations of the Uq(sln)$U_q(sl_n)$‐Reshetikhin–Turaev functor on tangles, involving coupons representing the anti‐symmetrizer and its dual. We prove the Splitting Theorem asserting that cutting of a marked 3‐manifold M$M$ along a disk resulting in a 3‐manifold M′$M^{\prime }$ yields a homomorphism Sn(M)→Sn(M′)$\mathcal {S}_n(M)\rightarrow \mathcal {S}_n(M^{\prime })$ for all n$n$. That result allows to analyze the skein modules of 3‐manifolds through the skein modules of their pieces. The theory of stated skein modules is particularly rich for thickened surfaces M=Σ×(−1,1)$M=\Sigma \times (-1,1)$, in whose case, Sn(M)$\mathcal {S}_n(M)$ is an algebra, denoted by Sn(Σ)$\mathcal {S}_n(\Sigma)$. One of the main results of this paper asserts that the skein algebra of the ideal bigon is isomorphic with Oq(SL(n))$\mathcal {O}_q(SL(n))$ and it provides simple geometric interpretations of the product, coproduct, counit, the antipode, and the cobraided structure on Oq(SL(n))$\mathcal {O}_q(SL(n))$. (In particular, the coproduct is given by a splitting homomorphism.) We show that for surfaces with boundary Σ$\Sigma$ every splitting homomorphism is injective and that Sn(Σ)$\mathcal {S}_n(\Sigma)$ is a free module with a basis induced from the Kashiwara–Lusztig canonical bases. Additionally, we show that a splitting of a thickened bigon near a marking defines a right Oq(SL(n))$\mathcal {O}_q(SL(n))$‐comodule structure on Sn(M)$\mathcal {S}_n(M)$, or dually, a left Uq(sln)$U_q(sl_n)$‐module structure. Furthermore, we show that the skein algebra of surfaces Σ1,Σ2$\Sigma _1, \Sigma _2$ glued along two sides of a triangle is isomorphic with the braided tensor product Sn(Σ1)⊗̲Sn(Σ2)$\mathcal {S}_n(\Sigma _1){\underline{\otimes }}\mathcal {S}_n(\Sigma _2)$ of Majid. These results allow for geometric interpretation of further concepts in the theory of quantum groups, for example, of the braided products and of Majid's transmutation operation. Building upon the above results, we prove that the factorization homology with coefficients in the category of representations of Uq(sln)$U_q(sl_n)$ is equivalent to the category of left modules over Sn(Σ)$\mathcal {S}_n(\Sigma)$ for surfaces Σ$\Sigma$ with ∂Σ=S1$\partial \Sigma =S^1$. We also establish isomorphisms of our skein algebras with the quantum moduli spaces of Alekseev–Schomerus and with the internal algebras of the skein categories for these surfaces and g=sl(n)$\mathfrak {g}=sl(n)$. [ABSTRACT FROM AUTHOR]

Let g be a simple complex Lie algebra of a classical type and U q g the corresponding Drinfeld–Jimbo quantum group at q not a root of unity. With every point t of the fixed maximal torus T of an algebraic group G with Lie algebra g we associate an additive category O q t of U q g -modules that is stable under tensor product with finite-dimensional quasiclassical U q g -modules. We prove that O q t is essentially semi-simple and use it to explicitly quantize equivariant vector bundles on the conjugacy class of t. [ABSTRACT FROM AUTHOR]

We introduce a notion of representation for a class of generalised quivers known as Coxeter quivers. These representations are built using fusion categories associated to U q (s l 2) at roots of unity and we show that many of the classical results on representations of quivers can be generalised to this setting. Namely, we prove a generalised Gabriel's theorem for Coxeter quivers that encompasses all Coxeter–Dynkin diagrams—including the non-crystallographic types H and I. Moreover, a similar relation between reflection functors and Coxeter theory is used to show that the indecomposable representations correspond bijectively to the (extended) positive roots of Coxeter root systems over fusion rings. [ABSTRACT FROM AUTHOR]

For an essentially small hereditary abelian category A , we associate an algebra H Δ (A) , called the Δ-Hall algebra of A. The basis of H Δ (A) is the isomorphism classes of objects in A , and the Δ-Hall numbers calculate certain three-cycles of exact sequences in A. We show that the Δ-Hall algebra H Δ (A) is isomorphic to the 1-periodic derived Hall algebra associated to A. By taking suitable extension and twisting, we can obtain the ı Hall algebra and the semi-derived Hall algebra related to A respectively. When applied to the nilpotent representation category A = re p nil (k Q) for an arbitrary quiver Q without loops, the (resp. extended) Δ-Hall algebra provides a realization of the (resp. universal) ı quantum group associated to Q. [ABSTRACT FROM AUTHOR]

We define quantum automorphisms and isomorphisms of Hadamard matrices. We show that every Hadamard matrix of size N\ge 4 has quantum symmetries and that all Hadamard matrices of a fixed size are mutually quantum isomorphic. These results pass also to the corresponding Hadamard graphs. We also define quantum Hadamard matrices acting on quantum spaces and bring an example thereof over matrix algebras. [ABSTRACT FROM AUTHOR]

We use the crystallised C ∗ -algebra C (S U q (2)) at q = 0 to obtain a unitary that gives an approximate equivalence involving the GNS representation on the L 2 space of the Haar state of the quantum SU(2) group and the direct integral of all the infinite dimensional irreducible representations of the C ∗ -algebra C (S U q (2)) for nonzero values of the parameter q. This approximate equivalence gives a KK class via the Cuntz picture in terms of quasihomomorphisms as well as a Fredholm representation of the dual quantum group S U q (2) ^ with coefficients in a C ∗ -algebra in the sense of Mishchenko. [ABSTRACT FROM AUTHOR]

Abstract We show, by introducing purely auxiliary gluinos and scalars, that the quantum path integral for a class of 3D interacting non-supersymmetric gauge theories localises. The theories in this class all admit a ‘Manin gauge theory’ formulation, that we introduce; it is obtained by enhancing the gauge algebra of the theory to a Dirac structure inside a Manin pair. This machinery allows us to do localisation computations for every theory in this class at once, including for 3D Yang-Mills theory, and for its Third Way deformation; the latter calculation casts the Third Way path integral into an almost 1-loop exact form.

Abstract AGT conjecture reveals a connection between 4D N $$ \mathcal{N} $$ = 2 gauge theory and 2D conformal field theory. Though some special instances have been proven, others remain elusive and the attempts on its full proof never stop. When the Ω background parameters satisfy −ϵ 1/ϵ 2 ≡ β = 1, the story can be simplified a bit. A proof of the correspondence in the case of A 1 gauge group was given in 2010 by Mironov et al., while the A n extension is verified by Matsuo and Zhang in 2011, with an assumption on the Selberg integral of n + 1 Schur polynomials. Then in 2020, Albion et al. obtained the rigorous result of this formula. In this paper, we show that the conjecture on the Selberg integral of Schur polynomials is formally equivalent to their result, after applying a more complicated complex contour, thus leading to the proof of the A n case at β = 1. To perform a double check, we also directly start from this formula, and manage to show the identification between the two sides of AGT correspondence.

Abstract We show that quantum circuit complexity for the unitary time evolution operator of any time-independent Hamiltonian is bounded by linear growth at early times, independent of any choices of the fundamental gates or cost metric. Deviations from linear early-time growth arise from the commutation algebra of the gates and are manifestly negative for any circuit, decreasing the linear growth rate and leading to a bound on the growth rate of complexity of a circuit at early times. We illustrate this general result by applying it to qubit and harmonic oscillator systems, including the coupled and anharmonic oscillator. By discretizing free and interacting scalar field theories on a lattice, we are also able to extract the early-time behavior and dependence on the lattice spacing of complexity of these field theories in the continuum limit, demonstrating how this approach applies to systems that have been previously difficult to study using existing techniques for quantum circuit complexity.

Abstract In this paper, we give a proof of 5D A n AGT conjecture at β = 1, where the gauge theory side is one dimension higher than the original 4D case, and corresponds to the q-deformation of the 2D conformal field theory side. We define a q-deformed A n Selberg integral, which generalizes the A n Selberg integral and the q-deformed A 1 Selberg integral in the literature. A q-deformed A n Selberg average formula with n + 1 Schur polynomials is proposed and proved to complete the proof.

Abstract Cluster states are crucial resources for measurement-based quantum computation (MBQC). It exhibits symmetry-protected topological (SPT) order, thus also playing a crucial role in studying topological phases. We present the construction of cluster states based on Hopf algebras. By generalizing the finite group valued qudit to a Hopf algebra valued qudit and introducing the generalized Pauli-X operator based on the regular action of the Hopf algebra, as well as the generalized Pauli-Z operator based on the irreducible representation action on the Hopf algebra, we develop a comprehensive theory of Hopf qudits. We demonstrate that non-invertible symmetry naturally emerges for Hopf qudits. Subsequently, for a bipartite graph termed the cluster graph, we assign the identity state and trivial representation state to even and odd vertices, respectively. Introducing the edge entangler as controlled regular action, we provide a general construction of Hopf cluster states. To ensure the commutativity of the edge entangler, we propose a method to construct a cluster lattice for any triangulable manifold. We use the 1d cluster state as an example to illustrate our construction. As this serves as a promising candidate for SPT phases, we construct the gapped Hamiltonian for this scenario and provide a detailed discussion of its non-invertible symmetries. We demonstrate that the 1d cluster state model is equivalent to the quasi-1d Hopf quantum double model with one rough boundary and one smooth boundary. We also discuss the generalization of the Hopf cluster state model to the Hopf ladder model through symmetry topological field theory. Furthermore, we introduce the Hopf tensor network representation of Hopf cluster states by integrating the tensor representation of structure constants with the string diagrams of the Hopf algebra, which can be used to solve the Hopf cluster state model.

Abstract We extend our consideration of commutative subalgebras (rays) in different representations of the W 1+∞ algebra to the elliptic Hall algebra (or, equivalently, to the Ding-Iohara-Miki (DIM) algebra U q , t gl ̂ ̂ 1 $$ {U}_{q,t}\left({\hat{\hat{\mathfrak{gl}}}}_1\right) $$ ). Its advantage is that it possesses the Miki automorphism, which makes all commutative rays equivalent. Integrable systems associated with these rays become finite-difference and, apart from the trigonometric Ruijsenaars system not too much familiar. We concentrate on the simplest many-body and Fock representations, and derive explicit formulas for all generators of the elliptic Hall algebra e n,m . In the one-body representation, they differ just by normalization from z n q m D ̂ $$ {z}^n{q}^{m\hat{D}} $$ of the W 1+∞ Lie algebra, and, in the N -body case, they are non-trivially generalized to monomials of the Cherednik operators with action restricted to symmetric polynomials. In the Fock representation, the resulting operators are expressed through auxiliary polynomials of n variables, which define weights in the residues formulas. We also discuss q, t-deformation of matrix models associated with constructed commutative subalgebras.

Arbitrary connections on a generic Hopf algebra H are studied and shown to extend to connections on tensor fields. On this ground a general definition of metric compatible connection is proposed. This leads to a sufficient criterion for the existence and uniqueness of the Levi-Civita connection, that of invertibility of an H -valued matrix. Provided invertibility for one metric, existence and uniqueness of the Levi-Civita connection for all metrics conformal to the initial one is proven. This class consists of metrics which are neither central (bimodule maps) nor equivariant, in general. For central and bicoinvariant metrics the invertibility condition is further simplified to a metric independent one. Examples include metrics on S L q (2). [ABSTRACT FROM AUTHOR]

In a recent article Projective geometries, Q-polynomial structures, and quantum groups Terwilliger (arXiv:2407.14964) defined a certain weighted adjacency matrix, depending on a free (positive real) parameter, associated with the projective geometry, and showed (among many other results) that it is diagonalizable, with the eigenvalues and their multiplicities explicitly written down, and that it satisfies the Q -polynomial property (with respect to the zero subspace). In this note we • Write down an explicit eigenbasis for this matrix. • Evaluate the adjacency matrix-eigenvector products, yielding a new proof for the eigenvalues and their multiplicities. • Evaluate the dual adjacency matrix-eigenvector products and directly show that the action of the dual adjacency matrix on the eigenspaces of the adjacency matrix is block-tridiagonal, yielding a new proof of the Q -polynomial property. [ABSTRACT FROM AUTHOR]

We consider the representation theory of the Ariki-Koike algebra, a q -deformation of the group algebra of the complex reflection group C r ≀ S n. We define the addition of a runner full of beads for the abacus display of a multipartition and investigate some combinatorial properties of this operation. We focus our attention on the v -decomposition numbers, i.e. the polynomials arising from the Fock space representation of the quantum group U v ( sl ˆ e). Using Fayers' LLT-type algorithm for Ariki-Koike algebras, we relate v -decomposition numbers for different values of e for the class of e -multiregular multipartitions, by adding a full runner of beads to each component of the abacus displays for the labelling multipartitions. [ABSTRACT FROM AUTHOR]

We define the notion of quantized function algebras at q = 0 or crystallization of the q deformations of the type A n compact Lie groups at the C ∗ -algebra level. The C ∗ -algebra A n (0) is defined as a universal C ∗ -algebra given by a finite set of generators and relations. We obtain these relations by looking at the irreducible representations of the quantized function algebras for q > 0 and taking limit as q → 0 + after rescaling the generating elements appropriately. We then prove that in the n = 2 case the irreducible representations A 2 (0) are precisely the q → 0 + limits of the irreducible representations of the C ∗ -algebras A 2 (q) . [ABSTRACT FROM AUTHOR]

Abstract In this note, we aim to review algorithms for constructing crystal representations of quiver Yangians in detail. Quiver Yangians are believed to describe an action of the BPS algebra on BPS states in systems of D-branes wrapping toric Calabi-Yau three-folds. Crystal modules of these algebras originate from molten crystal models for Donaldson-Thomas invariants of respective three-folds. Despite the fact that this subject was originally at the crossroads of algebraic geometry with effective supersymmetric field theories, equivariant toric action simplifies applied calculations drastically. So the sole pre-requisite for this algorithm’s implementation is linear algebra. It can be easily taught to a machine with the help of any symbolic calculation system. Moreover, these algorithms may be generalized to toroidal and elliptic algebras and exploited in various numerical experiments with those algebras. We illustrate the work of the algorithms in applications to simple cases of Y sl 2 $$ \textrm{Y}\left({\mathfrak{sl}}_2\right) $$ , Y gl ̂ 1 $$ \textrm{Y}\left({\hat{\mathfrak{gl}}}_1\right) $$ and Y gl ̂ 1 1 $$ \textrm{Y}\left({\hat{\mathfrak{gl}}}_{\left.1\right|1}\right) $$ .

Abstract We construct the elliptic Miura transformation and use it to obtain the expression of the currents of elliptic corner VOA. We subsequently prove a novel combinatorial formula that is essential for deriving the quadratic relations of the currents. In addition, we give a conjecture that relates the correlation function of the currents of elliptic corner VOA to a certain family of partially symmetric polynomials. The elliptic Macdonald polynomials, constructed recently by Awata-Kanno- Mironov-Morozov-Zenkevich, and Fukuda-Ohkubo-Shiraishi, can be obtained as a particular case of this family.

Abstract We study the boundary integrability problem of the massless sector of AdS 3 × S 3 × T 4 string theory. Exploiting the difference-form of the massless scattering theory, we find a very simple and exhaustive list of reflection matrices for all the possible boundary coideal subalgebras — singlet and vector representations, right and left boundary — and check basic properties of our solutions, primarily the boundary Yang-Baxter equation, for all possible combinations of scattering particles.

Abstract We investigate the multifusion generalization of string-net ground states and lattice Hamiltonians, delving into their associated weak Hopf symmetries. For the multifusion string-net, the gauge symmetry manifests as a general weak Hopf algebra, leading to a reducible vacuum string label; the charge symmetry, serving as a quantum double of gauge symmetry, constitutes a connected weak Hopf algebra. This implies that the associated topological phase retains its characterization by a unitary modular tensor category (UMTC). The bulk charge symmetry can also be captured by a weak Hopf tube algebra. We offer an explicit construction of the weak Hopf tube algebra structure and thoroughly discuss its properties. The gapped boundary and domain wall models are extensively discussed, with these 1d phases characterized by unitary multifusion categories (UMFCs). We delve into the gauge and charge symmetries of these 1d phases, as well as the construction of the boundary and domain wall tube algebras. Additionally, we illustrate that the domain wall tube algebra can be regarded as a cross product of two boundary tube algebras. As an application of our model, we elucidate how to interpret the defective string-net as a restricted multifusion string-net.

Abstract Generalizing the quantum sine-Gordon and sausage models, we construct exact S-matrices for higher spin representations with quantum U q su 2 $$ {\mathcal{U}}_{\textrm{q}}\left({\mathfrak{su}}_2\right) $$ symmetry, which satisfy unitarity, crossing-symmetry and the Yang-Baxter equations with minimality assumption, i.e. without any unnecessary CDD factor. The deformation parameter q is related to a coupling constant. Based on these S-matrices, we derive the thermodynamic Bethe ansatz equations for q a root of unity in terms of a universal kernel where the nodes are connected by graphs of non-Dynkin type. We solve these equations numerically to find out Hagedorn-like singularity in the free energies at some critical scales and find a universality in the critical exponents, all near 0.5 for different values of the spin and the coupling constant.

Abstract We explore the difference Langlands correspondence using the four dimensional N $$ \mathcal{N} $$ = 2 super-QCD. Surface defects and surface observables play the crucial role. As an application, we give the first construction of the full set of quantum integrals, i.e. commuting differential operators, such that the partition function of the so-called regular monodromy surface defect is their joint eigenvectors in an evaluation module over the Yangian Y gl 2 $$ \left(\mathfrak{gl}(2)\right) $$ , making it the wavefunction of a N-site gl 2 $$ \mathfrak{gl}(2) $$ spin chain with bi-infinite spin modules. We construct the Q- and Q ~ $$ \overset{\sim }{\textbf{Q}} $$ -surface observables which are believed to be the Q-operators on the bi-infinite module over the Yangian Y gl 2 $$ \left(\mathfrak{gl}(2)\right) $$ , and compute their eigenvalues, the Q-functions, as vevs of the surface observables.

This paper is a continuation of Ref. 7. In this paper, we give an explicit construction of a non-Bichon type co-action on a multigraph that is, it preserves quantum symmetry of (V,E) in our sense but not always in Bichon’s sense.2 This construction itself is motivated from automorphisms of quantum graphs. [ABSTRACT FROM AUTHOR]

We construct the elliptic Miura transformation and use it to obtain the expression of the currents of elliptic corner VOA. We subsequently prove a novel combinatorial formula that is essential for deriving the quadratic relations of the currents. In addition, we give a conjecture that relates the correlation function of the currents of elliptic corner VOA to a certain family of partially symmetric polynomials. The elliptic Macdonald polynomials, constructed recently by Awata-Kanno- Mironov-Morozov-Zenkevich, and Fukuda-Ohkubo-Shiraishi, can be obtained as a particular case of this family. [ABSTRACT FROM AUTHOR]

This paper reviews Holevo's contributions to quantum information theory during the 20 century. At that time, he mainly studied three topics, classical-quantum channel coding, quantum estimation with Craméro–Rao approach and quantum estimation with the group covariant approach. This paper addresses these three topics. [ABSTRACT FROM AUTHOR]

We introduce the notion of an action of a discrete or compact quantum group on an operator system, and study equivariant operator system injectivity. We then prove a duality result that relates equivariant injectivity with dual injectivity of associated crossed products. As an application, we give a description of the equivariant injective envelope of the reduced crossed product built from an action of a discrete quantum group on an operator system. [ABSTRACT FROM AUTHOR]