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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]

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 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]

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 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.

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.

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]

Recently, Bazhanov and Sergeev have described an Ising-type integrable model which can be identified as a sinh-Gordon-type model with an infinite number of states but with a real parameter q. This model is the subject of Sklyanin's Functional Bethe Ansatz. We develop in this paper the whole technique of the FBA which includes: (1) Construction of eigenstates of an off-diagonal element of a monodromy matrix. The most important ingredients of these eigenstates are the Clebsh-Gordan coefficients of the corresponding representation. (2) Separately, we discuss the Clebsh-Gordan coefficients, as well as the Wigner's 6j symbols, in details. The later are rather well known in the theory of 3 D indices. Thus, the Sklyanin basis of the quantum separation of variables is constructed. The matrix elements of an eigenstate of the auxiliary transfer matrix in this basis are products of functions satisfying the Baxter equation. Such functions are called usually the Q-operators. We investigate the Baxter equation and Q-operators from two points of view. (3) In the model considered the most convenient Bethe-type variables are the zeros of a Wronskian of two well defined particular solutions of the Baxter equation. This approach works perfectly in the thermodynamic limit. We calculate the distribution of these roots in the thermodynamic limit, and so we reproduce in this way the partition function of the model. (4) The real parameter q, which is the standard quantum group parameter, plays the role of the absolute temperature in the model considered. Expansion with respect to q (tropical expansion) gives an alternative way to establish the structure of the eigenstates. In this way we classify the elementary excitations over the ground state. [ABSTRACT FROM AUTHOR]

We demonstrate that quantum error correction is realized by the renormalization group in scalar field theories. We construct q -level states by using coherent states in the IR region. By acting on them the inverse of the unitary operator U that describes the renormalization group flow of the ground state, we encode them into states in the UV region. We find the situations in which the Knill–Laflamme condition is satisfied for operators that create coherent states. We verify this to the first order in the perturbation theory. This result suggests a general relationship between the renormalization group and quantum error correction and should give insights into understanding the role played by them in the gauge/gravity correspondence. [ABSTRACT FROM AUTHOR]

Previous research suggests that sharing emotionally intense experiences with others, for example by undergoing dysphoric collective rituals together, can lead to "identity fusion," a visceral feeling of oneness that predicts group cohesion and self-sacrifice for the group. In this pre-registered research, we provide the first quantitative investigation of identity fusion following participation in a national funeral, surveying 1632 members of the British public. As predicted, individuals reporting intense sadness during Queen Elizabeth II's funeral exhibited higher levels of identity fusion and pro-group commitment, as evidenced by generosity pledges to a British Monarchist charity. Also consistent with our hypotheses, feelings of unity in grief and emotional sharedness during the event mediated the relationship between sadness intensity and pro-group commitment. These findings shed light on importance of collective rituals in fostering group cohesion, cooperation, and the dynamics of shared emotional experiences within communities. [ABSTRACT FROM AUTHOR]

Among a plethora of mixtures, the methane (CH4) and hydrogen (H2) mixture has garnered considerable attention for multiple reasons, especially in the framework of energy production and industrial processes as well as ecological considerations. Despite the fact that the CH4/H2 mixture performs many critical tasks, the presence of other gases, such as carbon dioxide, sulfur compounds like H2S, and water vapor, leads to many undesirable consequences. Thus purification of this mixture from these gases assumes considerable relevance. In the current research, first-principle calculations in the frame of density functional theory are carried out to propose a new functional group for vertically aligned carbon nanotubes (VA-CNTs) interacting preferentially with polar molecules rather than CH4 and H2 in order to obtain a more efficient methane and hydrogen separations The binding energies associated with the interactions between several chemical groups and target gases were calculated first, and then a functional group formed by a modified ethylene glycol and acetyl amide was selected. This functional group was attached to the CNT edge with an appropriate diameter, and hence the binding energies with the target gases and steric hindrance were evaluated. The binding energy of the most polar molecule (H2O) was found to be more than six times higher than that of H2, indicating a significant enhancement of the nanotube tip's affinity toward polar gases. Thus, this functionalization is beneficial for enhancing the capability of highly packed functionalized VA-CNT membranes to purify CH4/H2 gas mixtures. [ABSTRACT FROM AUTHOR]

Quantum simulation has the potential to be an indispensable technique for the investigation of non-perturbative phenomena in strongly-interacting quantum field theories (QFTs). In the modern quantum era, with Noisy Intermediate Scale Quantum (NISQ) simulators widely available and larger-scale quantum machines on the horizon, it is natural to ask: what non-perturbative QFT problems can be solved with the existing quantum hardware? We show that existing noisy quantum machines can be used to analyze the energy spectrum of several strongly-interacting 1+1D QFTs, which exhibit non-perturbative effects like 'quark confinement' and 'false vacuum decay'. We perform quench experiments on IBM's quantum simulators to compute the energy spectrum of 1+1D quantum Ising model with a longitudinal field. Our results demonstrate that digital quantum simulation in the NISQ era has the potential to be a viable alternative to numerical techniques such as density matrix renormalization group or the truncated conformal space methods for analyzing QFTs. Recent work has shown that digital quantum simulations may be well suited for simulating non-perturbative quantum field theories. Here the authors use a superconducting quantum computer to obtain the energy spectrum of a strongly interacting quantum field theory mapped onto a quantum spin chain model. [ABSTRACT FROM AUTHOR]

Quantum groups occupy a significant position in both mathematics and physics, contributing to progress in these fields. It is interesting to obtain new quantum groups by the quantization of Lie bialgebras. In this paper, the quantization of the rank two Heisenberg–Virasoro algebra by Drinfel'd twists is presented, Lie bialgebra structures of which have been investigated by the authors recently. [ABSTRACT FROM AUTHOR]

This article examines quantum group symmetry using the Potts model. The transformation of the Potts model into a polynomial knot state on Kaufman square brackets is analyzed. It is shown how a dichromatic polynomial for a planar graph can be obtained using Temperley–Lieb operator algebra. The proposed work provides insight into the 7 4 knot-partition function of Takara Musubi using a strain factor that represents the particles in the lattice knots of the above-mentioned model. As far as theoretical physics is concerned, this statement provides a correct explanation of the connection between the Potts model and the similar square lattice of knot and link invariants. [ABSTRACT FROM AUTHOR]

We study the boundary integrability problem of the massless sector of AdS3 × S3 × T4 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 FROM AUTHOR]

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 FROM AUTHOR]

We study the boundary integrability problem of the massless sector of AdS3 × S3 × T4 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 FROM AUTHOR]

We consider quantum (skew) polynomial rings and observe that their graded automorphisms coincide with those of quantum exterior algebras. This allows us to define a quantum determinant that gives a homomorphism of groups acting on quantum polynomial rings. We use quantum subdeterminants to classify the resulting Drinfeld Hecke algebras for the symmetric group, other infinite families of Coxeter and complex reflection groups, and mystic reflection groups (which satisfy a version of the Shephard–Todd–Chevalley theorem). This direct combinatorial approach replaces the technology of Hochschild cohomology used by Naidu and Witherspoon over fields of characteristic zero and allows us to extend some of their results to fields of arbitrary characteristic and also locate new deformations of skew group algebras. [ABSTRACT FROM AUTHOR]

We study relative amenability and amenability of a right coideal ÑP ​ ⊆ℓ ∞ (G) of a discrete quantum group in terms of its group-like projection P. We establish a notion of a P-left invariant state and use it to characterize relative amenability. We also develop a notion of coamenability of a compact quasi-subgroup Nω ⊆L ∞(Ĝ) that generalizes coamenability of a quotient as defined by Kalantar, Kasprzak, Skalski, and Vergnioux (2022), where Ĝ is the compact dual of G. In particular, we establish that the coamenable compact quasi-subgroups of G are in one-to-one correspondence with the idempotent states on the reduced C* -algebra Cr​(Ĝ). We use this work to obtain results for the duality between relative amenability and amenability of coideals in ℓ∞ (Ĝ) and coamenability of their codual coideals in L∞(Ĝ), making progress towards a question of Kalantar et al. [ABSTRACT FROM AUTHOR]

Generalizing the quantum sine-Gordon and sausage models, we construct exact S-matrices for higher spin representations with quantum U q su 2 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 FROM AUTHOR]

We explore the difference Langlands correspondence using the four dimensional 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 , making it the wavefunction of a N-site gl 2 spin chain with bi-infinite spin modules. We construct the Q- and Q ~ -surface observables which are believed to be the Q-operators on the bi-infinite module over the Yangian Y gl 2 , and compute their eigenvalues, the Q-functions, as vevs of the surface observables. [ABSTRACT FROM AUTHOR]

Alternative mathematical explorations in quantum computing can be of great scientific interest, especially if they come with penetrating physical insights. In this paper, we present a critical revisitation of our application of geometric (Clifford) algebras (GAs) in quantum computing as originally presented in [C. Cafaro and S. Mancini, Adv. Appl. Clifford Algebras 21, 493 (2011)]. Our focus is on testing the usefulness of geometric algebras (GAs) techniques in two quantum computing applications. First, making use of the geometric algebra of a relativistic configuration space (namely multiparticle spacetime algebra or MSTA), we offer an explicit algebraic characterization of one- and two-qubit quantum states together with a MSTA description of one- and two-qubit quantum computational gates. In this first application, we devote special attention to the concept of entanglement, focusing on entangled quantum states and two-qubit entangling quantum gates. Second, exploiting the previously mentioned MSTA characterization together with the GA depiction of the Lie algebras SO 3 ; R and SU 2 ; C depending on the rotor group Spin + 3 , 0 formalism, we focus our attention to the concept of universality in quantum computing by reevaluating Boykin's proof on the identification of a suitable set of universal quantum gates. At the end of our mathematical exploration, we arrive at two main conclusions. Firstly, the MSTA perspective leads to a powerful conceptual unification between quantum states and quantum operators. More specifically, the complex qubit space and the complex space of unitary operators acting on them merge in a single multivectorial real space. Secondly, the GA viewpoint on rotations based on the rotor group Spin + 3 , 0 carries both conceptual and computational advantages compared to conventional vectorial and matricial methods. [ABSTRACT FROM AUTHOR]

We explored the impact of acupuncture (ACUP) in conjunction with a quantum lipid-lowering device (Quantum) on the blood lipids and gut microbiota in hyperlipidemic rats, focusing on the adenosine monophosphate- (AMP)-activated protein kinase (AMPK) signaling pathway. Fifty Sprague-Dawley rats were randomly allocated into five groups: Normal, Model, Acup + Quantum, Acup, and Quantum. Hyperlipidemic models were established in all groups except Normal. The Model group did not receive any intervention after modeling. The Acup + Quantum group received both treatments, the Acup group received only acupuncture, and the Quantum group received only the quantum lipid-lowering device. We used ELISA to measure serum lipid and liver enzyme levels, hematoxylin and eosin (HE) staining for liver pathology, Western blot for protein expression, and 16S rRNA sequencing to analyze intestinal microbiota diversity in rats. Elisa results showed that compared with the model group, Acup + Quantum group could reduce the levels of total cholesterol (TC), triglycerides (TG), low-density lipoprotein-cholesterol (LDL-C), aspartate transaminase (AST) and aspartate transaminase (ALT) in rats with hyperlipidemia (P<0.01), and increase the level of high-density lipoprotein-cholesterol (HDL-C) (P<0.01). HE staining results showed that compared with the model group, the hepatocytes of rats in the Acup + Quantum group looked round and full, the liver plates were arranged regularly and neatly, and there was no obvious abnormality in the liver sinusoids. Western blot results showed that compared with the model group, the Acup + Quantum group inhibited AMPK activation, increased P-AMPK/AMPK protein expression (P<0.05), and decreased phospho-acetyl-CoA carboxylases (P-ACC/ACC), Sterol regulatory element-binding transcription factor-1C (SREBP-1C), and FAS protein expression (P<0.05; P<0.01; P<0.01), which resulted in lipid-lowering effect. The results of intestinal flora showed that Acup + Quantum group improved the intestinal microbial microenvironment of hyperlipidemic rats by regulating the structure of intestinal microflora, increasing the abundance of Firmicutes flora, and decreasing the abundance of harmful bacteria, such as Bacteroidetes and Proteobacteria. Acupuncture combined with quantum lipid-lowering device can improve the blood lipid and liver function levels and regulate the intestinal microbial microenvironment of hyperlipidemic rats. This therapeutic outcome is likely achieved through the activation of the AMPK pathway and the beneficial modulation of the intestinal microbiota of rats. [ABSTRACT FROM AUTHOR]

In 1991, Gelfand and Retakh embodied the idea of a noncommutative Dieudonne determinant for a generating matrix of RTT algebra, namely, they found a representation of the quantum determinant of RTT algebra in the form of a product of principal quasi-determinants. In this note we construct an analogue of the above statement for the RE-algebra corresponding to the Drinfeld R-matrix for the order n=2,3. Namely, we have found a family of quasi-determinants that are principal with respect to the antidiagonal, commuting among themselves, whose product turns out to be the quantum determinant of this algebra. This family generalizes the construction of integrals of the full Toda system due to Deift et al. for the quantum case of RE-algebras. In our opinion, this result also clarifies the role of RE-algebras as a quantum homogeneous spaces and can be used to construct effective quantum field theories with a boundary.

Abstract We explore the quantum algebraic formalism of the gauge origami system in ℂ 4, where D2/D4/D6/D8-branes are present. We demonstrate that the contour integral formulas have free field interpretations, leading to the operator formalism of qq-characters associated with each D-brane. The qq-characters of D2 and D4-branes correspond to screening charges and generators of the affine quiver W-algebra, respectively. On the other hand, the qq-characters of D6 and D8-branes represent novel types of qq-characters, where monomial terms are characterized by plane partitions and solid partitions. The composition of these qq-characters yields the instanton partition functions of the gauge origami system, eventually establishing the BPS/CFT correspondence. Additionally, we demonstrate that the fusion of qq-characters of D-branes in lower dimensions results in higher-dimensional D-brane qq-characters. We also investigate quadratic relations among these qq-characters. Furthermore, we explore the relationship with the representations, q-characters, and the Bethe ansatz equations of the quantum toroidal gl 1 $$ {\mathfrak{gl}}_1 $$ . This connection provides insights into the Bethe/Gauge correspondence of the gauge origami system from both gauge-theoretic and quantum-algebraic perspectives. We finally present conjectures regarding generalizations to general toric Calabi-Yau four-folds. These generalizations imply the existence of an extensive class of qq-characters, which we call BPS qq-characters. These BPS qq-characters offer a new systematic approach to derive a broader range of BPS/CFT correspondence and Bethe/Gauge correspondence.

Abstract BPS states in supersymmetric theories can admit additional algebro-geometric structures in their spectra, described as quiver Yangian algebras. Equivariant fixed points on the quiver variety are interpreted as vectors populating a representation module, and matrix elements for the generators are then defined as Duistermaat-Heckman integrals in the vicinity of these points. The well-known wall-crossing phenomena are that the fixed point spectrum establishes a dependence on the stability (Fayet-Illiopolous) parameters ζ, jumping abruptly across the walls of marginal stability, which divide the ζ-space into a collection of stability chambers — “phases” of the theory. The standard construction of the quiver Yangian algebra relies heavily on the molten crystal model, valid in a sole cyclic chamber where all the ζ-parameters have the same sign. We propose to lift this restriction and investigate the effects of the wall-crossing phenomena on the quiver Yangian algebra and its representations — starting with the example of affine super-Yangian $${\text{Y}}\left({\widehat{\mathfrak{g}\mathfrak{l}}}_{1\left|1\right.}\right)$$ . In addition to the molten crystal construction more general atomic structures appear, in other non-cyclic phases (chambers of the ζ-space). We call them glasses and also divide in a few different classes. For some of the new phases we manage to associate an algebraic structure again as a representation of the same affine Yangian $${\text{Y}}\left({\widehat{\mathfrak{g}\mathfrak{l}}}_{1\left|1\right.}\right)$$ . This observation supports an earlier conjecture that the BPS algebraic structures can be considered as new wall-crossing invariants.

Abstract We consider matter correlators in the double-scaled SYK (DSSYK) model. It turns out that matter correlators have a simple expression in terms of the doubled Hilbert space $$\mathcal{H}\otimes \mathcal{H}$$ , where $$\mathcal{H}$$ is the Fock space of q-deformed oscillator (also known as the chord Hilbert space). In this formalism, we find that the operator which counts the intersection of chords should be conjugated by certain “entangler” and “disentangler”. We explicitly demonstrate this structure for the two- and four-point functions of matter operators in DSSYK.

Abstract Defining quantum information quantities directly in bulk quantum gravity is a difficult problem due to the fluctuations of spacetime. Some progress was made recently in [1], which provided a bulk interpretation of the Bekenstein Hawking formula for two sided BTZ black holes in terms of the entanglement entropy of gravitational edge modes. We generalize those results to give a bulk entanglement entropy interpretation of the quantum extremal surface formula in AdS3 gravity, as applied to a single interval in the boundary theory. Our computation further supports the proposal that AdS3 gravity can be viewed as a topological phase in which the bulk gravity edge modes are anyons that transform under the quantum group $${{\text{SL}}}_{q}^{+}\left(2,{\mathbb{R}}\right)$$ . These edge modes appear when we cut open the Euclidean path integral along bulk co-dimension 2 slices, and satisfies a shrinkable boundary condition which ensures that the Gibbons-Hawking calculation gives the correct state counting.

BPS states in supersymmetric theories can admit additional algebro-geometric structures in their spectra, described as quiver Yangian algebras. Equivariant fixed points on the quiver variety are interpreted as vectors populating a representation module, and matrix elements for the generators are then defined as Duistermaat-Heckman integrals in the vicinity of these points. The well-known wall-crossing phenomena are that the fixed point spectrum establishes a dependence on the stability (Fayet-Illiopolous) parameters ζ, jumping abruptly across the walls of marginal stability, which divide the ζ-space into a collection of stability chambers — “phases” of the theory. The standard construction of the quiver Yangian algebra relies heavily on the molten crystal model, valid in a sole cyclic chamber where all the ζ-parameters have the same sign. We propose to lift this restriction and investigate the effects of the wall-crossing phenomena on the quiver Yangian algebra and its representations — starting with the example of affine super-Yangian . In addition to the molten crystal construction more general atomic structures appear, in other non-cyclic phases (chambers of the ζ-space). We call them glasses and also divide in a few different classes. For some of the new phases we manage to associate an algebraic structure again as a representation of the same affine Yangian . This observation supports an earlier conjecture that the BPS algebraic structures can be considered as new wall-crossing invariants. [ABSTRACT FROM AUTHOR]

In the present article, we build the excitedcoherent states associated with deformed s u (1 , 1) algebra (DSUA), called photon-added deformed Perelomov coherent states (PA-DPCSs). The constructed coherent states are obtained by using an alterationof the Holstein–Primakoff realization (HPR) for DSUA. A general method to resolve of the problem of the unitary operator was developed for these kinds of quantum states. The Mandel parameter is considered to examine the statistical properties of PA-DPCSs. Furthermore, we offer a physical method to generate the PA-DPCSs in the framework of interaction among fields and atoms. Finally, we introduce the concept of entangled states for PA-DPCSs and examine the entanglement properties for entangled PA-DPCSs. [ABSTRACT FROM AUTHOR]

Gradient Flow Exact Renormalization Group (GF-ERG) is a framework to define the renormalization group flow of Wilsonian effective action utilizing coarse-graining along the diffusion equations. We apply it for Scalar Quantum Electrodynamics and derive flow equations for the Wilsonian effective action with the perturbative expansion in the gauge coupling. We focus on the quantum corrections to the correlation functions up to the second order of the gauge coupling and discuss the gauge invariance of the GF-ERG flow. We demonstrate that the anomalous dimension of the gauge field agrees with the standard perturbative computation and that the mass of the photon keeps vanishing in general spacetime dimensions. The latter is a noteworthy fact that contrasts with the conventional Exact Renormalization Group formalism in which an artificial photon mass proportional to a cutoff scale is induced. Our results imply that the GF-ERG can give a gauge-invariant renormalization group flow in a non-perturbative way. [ABSTRACT FROM AUTHOR]

We explore the quantum algebraic formalism of the gauge origami system in ℂ4, where D2/D4/D6/D8-branes are present. We demonstrate that the contour integral formulas have free field interpretations, leading to the operator formalism of qq-characters associated with each D-brane. The qq-characters of D2 and D4-branes correspond to screening charges and generators of the affine quiver W-algebra, respectively. On the other hand, the qq-characters of D6 and D8-branes represent novel types of qq-characters, where monomial terms are characterized by plane partitions and solid partitions. The composition of these qq-characters yields the instanton partition functions of the gauge origami system, eventually establishing the BPS/CFT correspondence. Additionally, we demonstrate that the fusion of qq-characters of D-branes in lower dimensions results in higher-dimensional D-brane qq-characters. We also investigate quadratic relations among these qq-characters. Furthermore, we explore the relationship with the representations, q-characters, and the Bethe ansatz equations of the quantum toroidal gl 1 . This connection provides insights into the Bethe/Gauge correspondence of the gauge origami system from both gauge-theoretic and quantum-algebraic perspectives. We finally present conjectures regarding generalizations to general toric Calabi-Yau four-folds. These generalizations imply the existence of an extensive class of qq-characters, which we call BPS qq-characters. These BPS qq-characters offer a new systematic approach to derive a broader range of BPS/CFT correspondence and Bethe/Gauge correspondence. [ABSTRACT FROM AUTHOR]

Quantum simulation qubit models of electronic Hamiltonians rely on specific transformations in order to take into account the fermionic permutation properties of electrons. These transformations (principally the Jordan–Wigner transformation (JWT) and the Bravyi–Kitaev transformation) correspond in a quantum circuit to the introduction of a supplementary circuit level. In order to include the fermionic properties in a more straightforward way in quantum computations, we propose to use methods issued from Geometric Algebra (GA), which, due to its commutation properties, are well adapted for fermionic systems. First, we apply the Witt basis method in GA to reformulate the JWT in this framework and use this formulation to express various quantum gates. We then rewrite the general one and two-electron Hamiltonian and use it for building a quantum simulation circuit for the Hydrogen molecule. Finally, the quantum Ising Hamiltonian, widely used in quantum simulation, is reformulated in this framework. [ABSTRACT FROM AUTHOR]

Gradient refractive index (GRIN) metamaterial lenses can achieve multi-band fusion infrared imaging and has the characteristics of integration, light weight, and achromaticity, compared with existing curved refractive lenses group. Constructing a three-dimensional (3D) GRIN lens for dual-band (3.0–5.0 and 7.5–9.2 µm) mid-infrared imaging is challenging due to the difficulty of fabricating the desired 3D GRIN materials with continuously changing linewidths. Here, we present a 3D self-focusing GRIN lens with a cylindrical symmetry configuration in the mid-infrared band. Such a 3D GRIN lens is designed with gradient woodpile polymer metamaterials based on effective medium theory and fabricated with high fidelity by grayscale two-photon lithography. Simulated and experimental results simultaneously exhibit a 3D GRIN lens possessing dual-band, achromatic, near-diffraction-limit focusing on the wavelengths of 4.5 and 7.5 µm. The protocol for developing the 3D GRIN lens with dual-band fusion imaging would prompt potential applications in integrated light-coupled devices and lightweight infrared imaging devices. [ABSTRACT FROM AUTHOR]

We consider matter correlators in the double-scaled SYK (DSSYK) model. It turns out that matter correlators have a simple expression in terms of the doubled Hilbert space , where is the Fock space of q-deformed oscillator (also known as the chord Hilbert space). In this formalism, we find that the operator which counts the intersection of chords should be conjugated by certain “entangler” and “disentangler”. We explicitly demonstrate this structure for the two- and four-point functions of matter operators in DSSYK. [ABSTRACT FROM AUTHOR]

The running of the top quark mass (mt) is probed at the next-to-next-to-leading order in quantum chromodynamics for the first time. The result is obtained by comparing calculations in the modified minimal subtraction () renormalisation scheme to the CMS result on differential measurement of the top quark-antiquark () production cross section at = 13 TeV. The scale dependence of mt is extracted as a function of the invariant mass of the system, up to an energy scale of about 0.5 TeV. The observed running is found to be in good agreement with the three-loop solution of the renormalisation group equations on quantum chromodynamics. [ABSTRACT FROM AUTHOR]

Defining quantum information quantities directly in bulk quantum gravity is a difficult problem due to the fluctuations of spacetime. Some progress was made recently in [1], which provided a bulk interpretation of the Bekenstein Hawking formula for two sided BTZ black holes in terms of the entanglement entropy of gravitational edge modes. We generalize those results to give a bulk entanglement entropy interpretation of the quantum extremal surface formula in AdS3 gravity, as applied to a single interval in the boundary theory. Our computation further supports the proposal that AdS3 gravity can be viewed as a topological phase in which the bulk gravity edge modes are anyons that transform under the quantum group . These edge modes appear when we cut open the Euclidean path integral along bulk co-dimension 2 slices, and satisfies a shrinkable boundary condition which ensures that the Gibbons-Hawking calculation gives the correct state counting. [ABSTRACT FROM AUTHOR]