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The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over

Protection of gauge invariance in experimental realizations of lattice gauge theories based on energy-penalty schemes has recently stimulated impressive efforts both theoretically and in setups of quantum synthetic matter. A major challenge is the reliability of such schemes in non-abelian gauge theories where local conservation laws do not commute. Here, we show through exact diagonalization (ED) that non-abelian gauge invariance can be reliably controlled using gauge-protection terms that energetically stabilize the target gauge sector in Hilbert space, suppressing gauge violations due to unitary gauge-breaking errors. We present analytic arguments that predict a volume-independent protection strength V , which when sufficiently large leads to the emergence of an adjusted gauge theory with the same local gauge symmetry up to least a timescale ∝ V / V 0 3 . Thereafter, a renormalized gauge theory dominates up to a timescale ∝exp( V / V 0 )/ V 0 with V 0 a volume-independent energy factor, similar to the case of faulty abelian gauge theories. Moreover, we show for certain experimentally relevant errors that single-body protection terms robustly suppress gauge violations up to all accessible evolution times in ED, and demonstrate that the adjusted gauge theory emerges in this case as well. These single-body protection terms can be readily implemented with fewer engineering requirements than the ideal gauge theory itself in current ultracold-atom setups and noisy intermediate-scale quantum (NISQ) devices.

Non-Abelian gauge theories, such as quantum chromodynamics (QCD) or electroweak theory, are best studied with the aid of Green's functions that are gauge-invariant off-shell, but unlike for the photon in quantum electrodynamics, conventional graphical constructions fail. The Pinch Technique provides a systematic framework for constructing such Green's functions, and has many useful applications. Beginning with elementary one-loop examples, this book goes on to extend the method to all orders, showing that the Pinch Technique is equivalent to calculations in the background field Feynman gauge. The Pinch Technique Schwinger-Dyson equations are derived, and used to show how a dynamical gluon mass arises in QCD. Applications are given to the center vortex picture of confinement, the gauge-invariant treatment of resonant amplitudes, the definition of non-Abelian effective charges, high-temperature effects, and even supersymmetry. This book is ideal for elementary particle theorists and graduate students.

A bstract We use the 5-sphere partition functions of supersymmetric Yang-Mills theories to explore the (2, 0) superconformal theory on S 5 × S 1 . The 5d theories can be regarded as Scherk-Schwarz reductions of the 6d theory along the circle. In a special limit, the perturbative partition function takes the form of the Chern-Simons partition function on S 3 . With a simple non-perturbative completion, it becomes a 6d index which captures the degeneracy of a sector of BPS states as well as the index version of the vacuum Casimir energy. The Casimir energy exhibits the N 3 scaling at large N . The large N index for U( N ) gauge group also completely agrees with the supergravity index on AdS 7 × S 4 .

A bstract We extend recent work on the relation of 4 d and 3 d IR dualities of supersymmetric gauge theories with four supercharges to the case of orthogonal gauge groups. The distinction between different SO( N ) gauge theories in 4 d plays an important role in this relation. We show that the 4 d duality leads to a 3 d duality between an SO( N c ) gauge theory with N f flavors and an SO( N f − N c + 2) theory with N f flavors and extra singlets, and we derive its generalization in the presence of Chern-Simons terms. There are two different O ( N ) theories in 3 d , which we denote by O ( N ) ± , and we also show that the O ( N c ) − gauge theory is dual to a Spin( N f − N c + 2) theory, and derive from 4 d the known duality between O ( N c ) + and O ( N f − N c + 2) + . We verify the consistency of these 3 d dualities by various methods, including index computations.

The modern description of elementary particles, as formulated in the standard model of particle physics, is built on gauge theories 1 . Gauge theories implement fundamental laws of physics by local symmetry constraints. For example, in quantum electrodynamics Gauss’s law introduces an intrinsic local relation between charged matter and electromagnetic fields, which protects many salient physical properties, including massless photons and a long-ranged Coulomb law. Solving gauge theories using classical computers is an extremely arduous task 2 , which has stimulated an effort to simulate gauge-theory dynamics in microscopically engineered quantum devices 3 – 6 . Previous achievements implemented density-dependent Peierls phases without defining a local symmetry 7 , 8 , realized mappings onto effective models to integrate out either matter or electric fields 9 – 12 , or were limited to very small systems 13 – 16 . However, the essential gauge symmetry has not been observed experimentally. Here we report the quantum simulation of an extended U(1) lattice gauge theory, and experimentally quantify the gauge invariance in a many-body system comprising matter and gauge fields. These fields are realized in defect-free arrays of bosonic atoms in an optical superlattice of 71 sites. We demonstrate full tunability of the model parameters and benchmark the matter–gauge interactions by sweeping across a quantum phase transition. Using high-fidelity manipulation techniques, we measure the degree to which Gauss’s law is violated by extracting probabilities of locally gauge-invariant states from correlated atom occupations. Our work provides a way to explore gauge symmetry in the interplay of fundamental particles using controllable large-scale quantum simulators. Quantum simulation in a 71-site optical lattice certifies gauge invariance, showing how this essential property of lattice gauge theories can be maintained across a quantum phase transition.

A bstract We focus on the issue of proper definition of entanglement entropy in lattice gauge theories, and examine a naive definition where gauge invariant states are viewed as elements of an extended Hilbert space which contains gauge non-invariant states as well. Working in the extended Hilbert space, we can define entanglement entropy associated with an arbitrary subset of links, not only for abelian but also for non-abelian theories. We then derive the associated replica formula. We also discuss the issue of gauge invariance of the entanglement entropy. In the Z N gauge theories in arbitrary space dimensions, we show that all the standard properties of the entanglement entropy, e.g. the strong subadditivity, hold in our definition. We study the entanglement entropy for special states, including the topological states for the Z N gauge theories in arbitrary dimensions. We discuss relations of our definition to other proposals.

Soft machines are a promising design paradigm for human-centric devices 1 , 2 and systems required to interact gently with their environment 3 , 4 . To enable soft machines to respond intelligently to their surroundings, compliant sensory feedback mechanisms are needed. Specifically, soft alternatives to strain gauges—with high resolution at low strain (less than 5 per cent)—could unlock promising new capabilities in soft systems. However, currently available sensing mechanisms typically possess either high strain sensitivity or high mechanical resilience, but not both. The scarcity of resilient and compliant ultra-sensitive sensing mechanisms has confined their operation to laboratory settings, inhibiting their widespread deployment. Here we present a versatile and compliant transduction mechanism for high-sensitivity strain detection with high mechanical resilience, based on strain-mediated contact in anisotropically resistive structures (SCARS). The mechanism relies upon changes in Ohmic contact between stiff, micro-structured, anisotropically conductive meanders encapsulated by stretchable films. The mechanism achieves high sensitivity, with gauge factors greater than 85,000, while being adaptable for use with high-strength conductors, thus producing sensors resilient to adverse loading conditions. The sensing mechanism also exhibits high linearity, as well as insensitivity to bending and twisting deformations—features that are important for soft device applications. To demonstrate the potential impact of our technology, we construct a sensor-integrated, lightweight, textile-based arm sleeve that can recognize gestures without encumbering the hand. We demonstrate predictive tracking and classification of discrete gestures and continuous hand motions via detection of small muscle movements in the arm. The sleeve demonstration shows the potential of the SCARS technology for the development of unobtrusive, wearable biomechanical feedback systems and human–computer interfaces. Strain gauges with both high sensitivity and high mechanical resilience, based on strain-mediated contact in anisotropically resistive structures, are demonstrated within a sensor-integrated, textile-based sleeve that can recognize human hand motions via muscle deformations.

The accuracy of water level gauge data for high-head multi-stage ship locks is directly related to the accuracy and safety of the water conveyance control of the ship lock. The water level gauge of the ship lock is affected by the installation environment and other factors, and the water level data may deviate. During the operation of the ship lock, the water level gauge in the lock chamber will be calibrated regularly according to regulations. This article introduces the calculation and countermeasures for the calibration error of water level gauges in a high-head multi-stage ship lock, which can effectively control the errors caused by the calibration of water level gauges.

A bstract We study the low energy behaviour of supersymmetric gauge theories in 1 + 1 dimensions, with orthogonal and symplectic gauge groups and matters in the fundamental representation. We observe supersymmetry breaking in super-Yang-Mills theory and in theories with small numbers of flavors. For larger numbers of flavors, we discover duality between regular theories with different gauge groups and matter contents, where regularity refers to absence of quantum Coulomb branch. The result is applied to study families of superconformal field theories that can be used for superstring compactifications, with corners corresponding to three-dimensional Calabi-Yau manifolds. This work is motivated by recent development in mathematics concerning equivalences of derived categories.