Charge Operator Research Articles

Impact of myristic acid on novel designed manifold assimilated with domestic solar water heater: An experimental approach

In this study, myristic acid as heat storage material was assimilated with a novel designed manifold of evacuated tube solar collector (EVTSC). The foremost advantage of this design is that the integration of myristic acid in the manifold does not affect the vacuum tube’s strength and does not occupy extra roof space. Also, the advantageous thermal and chemical properties of myristic acid improve the performance of the system. The proposed system was tested for two different charging operations of storage material i.e. half-day charging operation and whole-day charging operation for water flow of 0.066, 0.133, and 0.20 LPM (liter per minute). These charging operations were selected to optimize the performance of the proposed system for daytime and nighttime hot water demand. The heat transfer analysis of the proposed system during both charging operations is also determined and compared. In addition to this, the machine learning approach is used to predict the hot water temperature. Concerning experimental results, the daily efficiency during HD-CHO is almost 62.42 %, 69.17 %, and 79.12 % respectively for the water flow of 0.066, 0.133, and 0.20 LPM. Whereas, during WD-CHO, daily efficiency was approximately 71.49 %, 80.42 %, and 85.21 % respectively for corresponding flow rates. Furthermore, the daily efficiency of the proposed system during HD-CHO and WD-CHO is found to be 42.85 % and 61 % higher than conventional systems. The higher and lower flow rates during both charging operations are recommended in terms of high daily efficiency and high hot water temperature respectively. Hence, the proposed system is designed to improve thermal performance, fulfill the hot water demand during late evening time, and decrease the overheating concern as compared to the conventional one.

The unitary Fermi gas at large charge and large N

We study the unitary Fermi gas in a harmonic trapping potential starting from a microscopic theory in the limit of large charge and large number of fermion flavors N. In this regime, we present an algorithmic procedure for extracting data from perturbation theory, order-by-order, without the need for other assumptions. We perform a gradient expansion in the interior of the particle cloud, sufficiently far from the cloud edge where the particle density drops rapidly to zero. In this latter region we present the first microscopic computation characterizing the contribution of the edge terms. The microscopic theory reproduces the predictions of the superfluid eft, including the action, the form of the gap equation, and the energy of the system in a harmonic trap (which maps, via the non-relativistic state-operator correspondence, to the scaling dimension of the lowest operator of charge Q). We additionally give the Wilsonian coefficients at leading order in N up to nnlo in the large-charge expansion.

Scaling to practical pouch cell supercapacitor: Electrodes by electrophoretic deposition

The scale-up of supercapacitors by electrophoretic deposition (EPD) from coin cell to pouch cell with commercially relevant mass loadings and thicknesses is reported. The use of EPD in electrode fabrication mainly reduces the interfacial resistance and increases the mechanical flexibility of the electrodes. The cycling performance or conversion efficiency can also be improved due to the highly porous EPD coatings. An exemplary investigation of activated carbon (AC) electrodes with an electrolyte comprising of tetraethylammonium tetrafluoroborate in acetonitrile is carried out. According to the general literature, EPD of AC on metal substrates has not performed well for supercapacitor electrodes unless they were thinner and with lower mass loadings than commercial requirements. As a consequence, and to redress this research gap, all the electrodes prepared in this work demonstrate high mass loadings (8 mg cm−2) and practical layer thicknesses (125 µm) and contain polyvinylidene fluoride binders with electrically conductive carbon black particles. Research investigations include: (a) impact of EPD of AC onto small (10 cm2) and large areas (50 cm2) of aluminum foil current collectors, (b) scaling-up of coin to pouch cells, and (c) the preparation of electrode coatings on both sides of the current collector for the first time using EPD for pouch cell investigations. Our research learning shows the evidence of practical cell performance, including current loading (40 A g−1), tens of thousands of successive charge and discharge operation (150,000 cycles), power (30 kW kg−1) and energy densities (10 W h kg−1), capacitance (154 F g−1), capacitance retention (80%) and coulombic efficiency (relatively close to 100%). Based upon the success of the pouch cells investigated in this work, further research studies on the use of EPD for preparing energy storage electrodes for commercial cylindrical types of supercapacitors is envisaged.

Numerical tests of the large charge expansion

We perform Monte-Carlo measurements of two and three-point functions of charged operators in the critical O(2) model in 3 dimensions. Our results are compatible with the predictions of the large charge superfluid effective field theory. To obtain reliable measurements for large values of the charge, we improved the Worm algorithm and devised a measurement scheme which mitigates the uncertainties due to lattice and finite size effects.

Analytic Gradients for the Electrostatic Embedding QM/MM Model in Periodic Boundary Conditions Using Particle-Mesh Ewald Sums and Electrostatic Potential Fitted Charge Operators.

Long-range electrostatic effects are fundamental for describing chemical reactivity in the condensed phase. Here, we present the methodology of an efficient quantum mechanical/molecular mechanical (QM/MM) model in periodic boundary conditions (PBC) compatible with QM/MM boundaries at chemical bonds. The method combines electrostatic potential fitted charge operators and electrostatic potentials derived from the smooth particle-mesh Ewald (PME) sum approach. The total energy and its analytic first derivatives with respect to QM, MM, and lattice vectors allow QM/MM molecular dynamics (MD) in the most common thermodynamic ensembles. We demonstrate the robustness of the method by performing a QM/MM MD equilibration of methanol in water. We simulate the cis/trans isomerization free-energy profiles in water of proline amino acid and a proline-containing oligopeptide, showing a correct description of the reaction barrier. Our PBC-compatible QM/MM model can efficiently be used to study the chemical reactivity in the condensed phase and enzymatic catalysis.

Fractional-Order Equivalent-Circuit Model Identification of Commercial Lithium-Ion Batteries

The precise identification of electrical model parameters of Li-Ion batteries is essential for efficient usage and better prediction of the battery performance. In this work, the model identification performance of two metaheuristic optimization algorithms is compared. The algorithms in comparison are the Marine Predator Algorithm (MPA) and the Partial Reinforcement Optimizer (PRO) to find the optimal model parameter values. Three fractional-order (FO) electrical equivalent circuit models (ECMs) of Li-Ion batteries with different levels of complexity are used to fit the electrochemical impedance spectroscopy (EIS) data operating under different states of charge (SoC) and different operating temperatures. It is found that there is a tradeoff between ECM complexity, identification accuracy, and precision.

An FPGA-based balancing of capacitor voltage for multi-level inverter

ABSTRACT This paper introduces a novel cascaded multilevel inverter circuit configuration aimed at generating a five-level load voltage using a single DC source and a capacitor. In this inverter configuration, an additional switch is added to the charging circuit which helps to charge the capacitor from the DC source. Additionally, an inductor is incorporated to mitigate the substantial charging current during capacitor charging operation. A level-shifted pulse width modulation (LS-PWM) technique is implemented to generate gate pulses for both inverter and charging switches. The capacitor gets charged when an extra switch is enabled and discharged when it is connected to a load. Analysis of charging and discharging of capacitor is described thoroughly in this paper. Each operational mode of the inverter is thoroughly analysed. Furthermore, the proposed five-level inverter setup is designed, and its control logic is implemented using FPGA platform. Subsequently, simulation and experimental investigations are conducted under varied loads and different modulation indices to evaluate the efficacy of the proposed inverter configuration and the respective results are presented.

Pose estimation in robotic electric vehicle plug-in charging tasks using auto-annotation and deep learning-based keypoint detector

The rapid growth of electric vehicles and advancements in self-driving technologies necessitate the development of specialized infrastructure to support autonomy. Accurate pose estimation of electric vehicles sockets is crucial for efficient and reliable plug-in charging operations. This process is inherently complex due to several factors, including the textureless black color of the socket, environment-dependent lighting conditions, and the presence of small geometrical features. To address these challenges, we propose a comprehensive method that combines Deep Neural Network-based pose estimation and auto-annotation methodology. Auto-annotation facilitates the generation of diverse training data, enhancing the accuracy, robustness, and generalization capabilities of the deep learning model. The advantages of the proposed method were evaluated through a comprehensive series of experiments conducted in both simulation and controlled laboratory settings. In our evaluation, we implemented a robotic charging system consisting of a manipulator with a charging plug and a hand-eye monocular camera and conducted plug-in testing in three scenarios: (A) uniform lighting conditions, (B) dark, and (C) highly uneven illumination of the socket surface. The experimental results show that our method can achieve precise and reliable pose estimation with mean absolute errors less than 0.73 mm and 0.82 deg, and an average Insertion Success Rate of over 97.5%.

Electrochemical-Thermal-Mechanical Coupling Analysis of Lithium-Ion Batteries under Fast Charging

Thermal runaway and volume expansion caused by temperature rise under fast charging of lithium-ion batteries (LIBs) are the major reasons of battery explosion and cycle performance degradation. In this work, considering the radiation heat transfer on the battery surface, an electrochemical-thermal-mechanical coupling model of cylindrical LIBs under fast charging (state of charge (SOC) ≤80%) is developed in order to investigate the distributions of the temperature and stress. Then, choosing 18650 LIBs as the object, the charge efficiency, temperature, and stress distributions are compared between fast charging and galvanostatic operation under SOC≤80%, respectively. Finally, the influences of the thickness, particle radius, the maximum lithium-ion concentration of the cathode, and initial electrolyte concentration on the temperature, radial stress, and hoop stress of LIB during charge are explored, respectively. Numerical results show that fast charging improves 20.8% of the charge efficiency. Increasing the cathodic thickness and decreasing the cathodic maximum lithium-ion concentration or initial electrolyte concentration can reduce the temperature of LIB during the charge. The results of this work will provide some reference value for the design of LIBs under fast charging.

Enhancing Electric Vehicle Charger Performance with Synchronous Boost and Model Predictive Control for Vehicle-to-Grid Integration

This paper investigates optimizing the power exchange between electric vehicles (EVs) and the grid, with a specific focus on the DC-DC converters utilized in vehicle-to-grid (V2G) systems. It specifically explores using model predictive control (MPC) in synchronous boost converters to enhance efficiency and performance. Through experiments and simulations, this paper shows that replacing diodes with SIC MOSFETs in boost converters significantly improves efficiency, particularly in synchronous mode, by minimizing the deadtime of SIC MOSFETs during switching. Additionally, this study evaluates MPC’s effectiveness in controlling boost converters, highlighting its advantages over traditional control methods. Real-world validations further validate the robustness and applicability of MPC in V2G systems. This study utilizes TMS320F28379D, one of Texas Instruments’ leading digital signal processors, enabling the implementation of MPC with a high PWM frequency of up to 200 MHz. This processor features dual 32-bit CPUs and a 16-bit ADC, allowing for high-resolution readings from sensors. Leveraging digital signal processing technologies and advanced electronic circuits, this study advances the development of high-performance boost converters, achieving power outputs of up to 48 watts and output voltages of 24 volts. Electronic circuits (PCB boards) have been devised, implemented, and evaluated to showcase their significance in advancing efficient V2G integration.

More on symmetry resolved operator entanglement

The ‘operator entanglement’ of a quantum operator O is a useful indicator of its complexity, and, in one-dimension, of its approximability by matrix product operators. Here we focus on spin chains with a global U(1) conservation law, and on operators O with a well-defined U(1) charge, for which it is possible to resolve the operator entanglement of O according to the U(1) symmetry. We employ the notion of symmetry resolved operator entanglement (SROE) introduced in Rath et al (2023 PRX Quantum 4 010318) and extend the results of the latter paper in several directions. Using a combination of conformal field theory and of exact analytical and numerical calculations in critical free fermionic chains, we study the SROE of the thermal density matrix ρβ=e−βH and of charged local operators evolving in Heisenberg picture O=eitHOe−itH . Our main results are: i) the SROE of ρβ obeys the operator area law; ii) for free fermions, local operators in Heisenberg picture can have a SROE that grows logarithmically in time or saturates to a constant value; iii) there is equipartition of the entanglement among all the charge sectors except for a pair of fermionic creation and annihilation operators.

On the symmetries of elementary fermions

This paper deals with the symmetries of elementary fermions and their derivation from fundamental physical principles such as the Lorentz invariance and from the spinor-helicity formalism employed to three-vectors. The generators of the Lorentz group are discussed, and the physics of the associated chiral spins is used to establish the chiral symmetry SU(2). The helicity of spin is defined which, when being applied to the hadronic isospin, yields the symmetry group SU(4). The Kronecker product of these two basic symmetries defines a unified symmetry SU(2)⊗SU(4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$SU(2) \\otimes SU(4)$$\\end{document} of the basic light fermions, namely the chiral doublets of the single lepton and triple quarks of the first family. Breaking this symmetry, according to mechanisms that are used in the electroweak interactions of the Standard Model (SM) of quantum field theory, yields quantum electrodynamics (QED) with symmetry group U(1), weak interactions with symmetry group SU(2), and what we like to name quantum hadrodynamics (QHD, akin to quantum chromodynamics QCD of the SM) with symmetry group SU(3). The gauge bosons associated with QED and QHD remain massless, but the weak bosons and the V bosons, related to the transformation of the quarks into a lepton and vice versa, become massive by the Higgs mechanism. Their masses are defined by the Higgs vacuum and the two coupling constants involved in the unified model. The V-boson mass is predicted to be 35.4 GeV. Furthermore, a possible explanation of hadron confinement is given in terms of the two hadronic charge operators. Moreover, the concept of hypercharge as used in the SM is not needed. Various versions of the extended Dirac equation that include the above symmetry groups are derived.

RFID Based EV Charging Station (SOLAR)

As the number of Electric Vehicles (EVs) proliferates, the significance of charging infrastructure escalates, necessitating solutions that balance the demands of the local distribution grid and EV users. This paper proposes the integration of an RFID system for user identification and charging authorization within a smart charging infrastructure, facilitating charge monitoring and control. The RFID technology offers a cost-effective means to identify and authorize vehicles for charging, ensuring efficient charging operations while adhering to grid constraints and fulfilling the requirements of EV drivers. Charging protocols are voltage-based, and the system enables remote monitoring of charging levels via an Internet of Things (IoT) enabled server, providing accessibility from any location at any time.

Newton vs Coulomb in AdS/CFT and the weak gravity conjecture

We study (near-)circular orbits of charged particles in the background of charged black holes in asymptotically anti–de Sitter (AdS) spacetimes of arbitrary dimensionality. We calculate the energy and angular momentum of such particles in a large-radius limit. This allows us to compute the anomalous dimension of the dual charged double-twist operators in a large-spin expansion, making a prediction for the bootstrap analysis at large charge and spin. We relate our result to the weak gravity conjecture (WGC) for AdS black holes of all sizes. We also discuss the relation of WGC with the existence of the innermost stable circular orbit (ISCO) in any dimension. Published by the American Physical Society 2024

Ligand-channel-enabled ultrafast Li-ion conduction.

Li-ion batteries (LIBs) for electric vehicles and aviation demand high energy density, fast charging and awide operating temperature range, which are virtually impossible because they require electrolytes to simultaneously have high ionic conductivity, low solvation energyand low melting point and form ananion-derived inorganic interphase1-5. Here we report guidelines for designing such electrolytes by using small-sized solvents with low solvation energy. The tiny solvent in the secondary solvation sheath pulls out the Li+ in the primary solvation sheath to form a fast ion-conduction ligand channel to enhance Li+ transport, while the small-sized solvent with low solvation energy also allows the anion to enter the first Li+ solvation shell to form aninorganic-rich interphase. The electrolyte-design concept is demonstrated by using fluoroacetonitrile (FAN) solvent. The electrolyte of 1.3 M lithium bis(fluorosulfonyl)imide (LiFSI) in FAN exhibits ultrahigh ionic conductivity of 40.3 mS cm-1 at 25 °C and 11.9 mS cm-1 even at -70 °C, thus enabling 4.5-V graphite||LiNi0.8Mn0.1Co0.1O2 pouch cells (1.2 Ah, 2.85 mAh cm-2) to achieve high reversibility (0.62 Ah) when the cells are charged and discharged even at -65 °C. The electrolyte with small-sized solvents enables LIBs to simultaneously achieve high energy density, fast charging and awide operating temperature range, which is unattainable for the current electrolyte design but is highly desired for extreme LIBs. This mechanism is generalizable and can be expanded to other metal-ion battery electrolytes.

Long-term thermal performance investigation of phase change plates (PCPs) employing tunnel lining ground heat exchangers for cool storage

Combining tunnel lining ground heat exchangers (GHEs) and phase change plates (PCPs) to extract and store the geothermal energy is a novel and environment-friendly energy utilization method, and then PCPs can be used for space cooling. However, existing investigations have not focused on the long-term thermal performances of PCPs employing tunnel lining GHEs for cool storage. In the present work, a coupled finite element model of tunnel lining GHEs and PCPs was built to study long-term operation performances of PCPs during the cold energy charging operation period. Numerical modeling and programming (i.e., co-simulation) were employed to realize the continuous cold energy charging operation mode of PCPs employing tunnel lining GHEs for cool storage. The results indicate that the solidification efficiency (i.e., cold energy charging efficiency) of PCPs and phase interface evolution within the PCPs under the long-term cold energy charging mode are affected by the number of times for cool storage (N) and thermal conductivity of the surrounding rock (ksr) compared with the short-term mode. The drop rate of the liquid fraction and solidification efficiency of PCPs decrease with an increase in the number of times for cool storage. At N = 1, there is no obvious difference for the solidification efficiency of PCPs with increasing ksr. The solidification efficiency of PCPs under N = 10 improves by approximately 35.3% as ksr rises by 3 W/(m K). Additionally, the solidification time of PCPs increases linearly with N under ksr = 1.22 W/(m K). However, the solidification time is related parabolically to N under ksr = 2.22, 3.22, and 4.22 W/(m K). Finally, the long-term thermal response of the surrounding rock is analyzed, which also verifies the heat transfer characteristics of PCPs under the long-term cold energy charging mode.

An urban charging load forecasting model based on trip chain model for private passenger electric vehicles: A case study in Beijing

The rapid adoption of electric vehicles (EVs) has led to dramatic increase in charging demands that poses great challenges for efficient charging infrastructure rollout and operation. It is crucial to accurately assess charging demand in urban areas to optimize the siting and sizing of charging infrastructure. This paper proposes a novel urban charging load forecasting model for private passenger EVs based on massive operating data of EVs in Beijing. First, the characteristics of travel patterns for private passenger EVs, urban road network, functional area distribution and charging infrastructure distribution within the entire Beijing area are identified. Then a charging load forecasting model that can simultaneously simulate trip chains for EVs is constructed by considering the occupancy states of public charging piles and the interactions among different EVs. Finally, the effectiveness of the proposed charging load forecasting model is verified based on comprehensive test data. Our findings imply that the number of EVs at recharge and the charging power can be reliably predicted with the accuracy of over 84.73 % and 81.92 %, respectively. It provides the foundation for optimal charging infrastructure planning and charging scheduling.

Smart electric vehicle charging for reducing photovoltaic energy curtailment

The increased diffusion of Photovoltaic (PV) generation could be limited by the grid capacity to host the power input from PV systems, especially in mid-day hours on sunny days. This paper proposes a framework of analysis to determine the mitigation of the PV energy curtailment in the grid, based on a novel smart charging strategy for electric vehicles (EVs) that operate in the grid-to-vehicle mode. This framework adopts a combination of methods and techniques, with a clustering procedure to group the EV charging stations with some PV systems based on the electrical distance, a novel smart charging strategy for the EVs based on the definition of fuzzy weights, and the exploitation of the blockchain technology with a Proof-of-Authority consensus protocol to reinforce network security and achieve decentralization. The blockchain provides full traceability of EV charging operations and grid status monitoring during time. The proposed framework contains novel indicators of avoided PV power reduction and energy curtailment. To account for the EV-related uncertainty, these indicators are determined from a Monte Carlo-based assessment of the three-phase power flow results. The results are presented on a three-phase unbalanced 123-node system with given locations of PV systems and EV charging stations.

Robust Charging Network Planning for Metropolitan Taxi Fleets

We study the robust charging station location problem for a large-scale commercial taxi fleet. Vehicles within the fleet coordinate on charging operations but not on customer acquisition. We decide on a set of charging stations to open to ensure operational feasibility. To make this decision, we propose a novel solution method situated between the location routing problems with intraroute facilities and flow refueling location problems. Additionally, we introduce a problem variant that makes a station sizing decision. Using our exact approach, charging stations for a robust operation of citywide taxi fleets can be planned. We develop a deterministic core problem employing a cutting plane method for the strategic problem and a branch-and-price decomposition for the operational problem. We embed this problem into a robust solution framework based on adversarial sampling, which allows for planner-selectable risk tolerance. We solve instances derived from real-world data of the metropolitan area of Munich containing 1,000 vehicles and 60 potential charging station locations. Our investigation of the sensitivity of technological developments shows that increasing battery capacities shows a more favorable impact on vehicle feasibility of up to 10 percentage points compared with increasing charging speeds. Allowing for depot charging dominates both of these options. Finally, we show that allowing just 1% of operational infeasibility risk lowers infrastructure costs by 20%. Funding: This work was partially funded by the Deutsche Forschungsgemeinschaft [Project 277991500]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0207 .

Large charge ’t Hooft limit of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super-Yang-Mills

The planar integrability of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super-Yang-Mills (SYM) is the cornerstone for numerous exact observables. We show that the large charge sector of the SU(2) N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM provides another interesting solvable corner which exhibits striking similarities despite being far from the planar limit. We study non-BPS operators obtained by small deformations of half-BPS operators with R-charge J in the limit J → ∞ with λJ≡gYM2J/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\lambda}_J\\equiv {g}_{\ extrm{YM}}^2J/2 $$\\end{document} fixed. The dynamics in this large charge ’t Hooft limit is constrained by a centrally-extended psu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{psu} $$\\end{document}(2|2)2 symmetry that played a crucial role for the planar integrability. To the leading order in 1/J, the spectrum is fully fixed by this symmetry, manifesting the magnon dispersion relation familiar from the planar limit, while it is constrained up to a few constants at the next order. We also determine the structure constant of two large charge operators and the Konishi operator, revealing a rich structure interpolating between the perturbative series at weak coupling and the worldline instantons at strong coupling. In addition we compute heavy-heavy-light-light (HHLL) four-point functions of half-BPS operators in terms of resummed conformal integrals and recast them into an integral form reminiscent of the hexagon formalism in the planar limit. For general SU(N) gauge groups, we study integrated HHLL correlators by supersymmetric localization and identify a dual matrix model of size J/2 that reproduces our large charge result at N = 2. Finally we discuss a relation to the physics on the Coulomb branch and explain how the dilaton Ward identity emerges from a limit of the conformal block expansion. We comment on generalizations including the large spin ’t Hooft limit, the combined large N-large J limits, and applications to general N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 superconformal field theories.