scholarly journals MIRROR SYMMETRY AND STRING VACUA FROM A SPECIAL CLASS OF FANO VARIETIES

1996 ◽  
Vol 11 (17) ◽  
pp. 3049-3096 ◽  
Author(s):  
ROLF SCHIMMRIGK
Keyword(s):  
Chern Class ◽  
Mirror Symmetry ◽  
Central Charge ◽  
Mean Field ◽  
Fano Varieties ◽  
Complex Dimension ◽  
Crucial Information ◽  
First Chern Class ◽  
The One ◽  
Calabi Yau Manifolds

Because of the existence of rigid Calabi-Yau manifolds, mirror symmetry cannot be understood as an operation on the space of manifolds with vanishing first Chern class. In this article I continue to investigate a particular type of Kähler manifolds with positive first Chern class which generalize Calabi-Yau manifolds in a natural way and which provide a framework for mirrors of rigid string vacua. This class comprises Fano manifolds of a special type which encode crucial information about ground states of the superstring. It is shown in particular that the massless spectra of (2, 2)-supersymmetric vacua of central charge ĉ=D crit can be derived from special Fano varieties of complex dimension D crit +2(Q−1), Q>1, and that in certain circumstances it is even possible to embed Calabi-Yau manifolds into such higher dimensional spaces. The constructions described here lead to new insight into the relation between exactly solvable models and their mean field theories on the one hand and their corresponding Calabi-Yau manifolds on the other. Furthermore it is shown that Witten’s formulation of the Landau-Ginzburg/Calabi-Yau relation can be applied to the present framework as well.

scholarly journals The Kodaira Problem for Kähler Spaces with Vanishing First Chern Class

2021 ◽  
Vol 9 ◽  
Author(s):  
Patrick Graf ◽  
Martin Schwald
Keyword(s):  
Chern Class ◽  
Cohomology Group ◽  
Canonical Bundle ◽  
Deformation Space ◽  
Projective Varieties ◽  
Quotient Singularities ◽  
Finite Quotient ◽  
First Chern Class ◽  
Small Deformations

Abstract Let X be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that X admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if X has toroidal singularities, if X has finite quotient singularities and if the cohomology group ${\mathrm {H}^{2} \!\left ( X, {\mathscr {T}_{X}} \right )}$ vanishes.

scholarly journals Two-term expansion of the ground state one-body density matrix of a mean-field Bose gas

2021 ◽  
Vol 60 (3) ◽  
Author(s):  
Phan Thành Nam ◽  
Marcin Napiórkowski
Keyword(s):  
Density Matrix ◽  
Ground State ◽  
Mean Field ◽  
Interaction Strength ◽  
Bose Gas ◽  
Body Density ◽  
The Mean ◽  
The One ◽  
Mean Field Regime ◽  
Number Of Particles

AbstractWe consider the homogeneous Bose gas on a unit torus in the mean-field regime when the interaction strength is proportional to the inverse of the particle number. In the limit when the number of particles becomes large, we derive a two-term expansion of the one-body density matrix of the ground state. The proof is based on a cubic correction to Bogoliubov’s approximation of the ground state energy and the ground state.

scholarly journals Monodromy defects in free field theories

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Lorenzo Bianchi ◽  
Adam Chalabi ◽  
Vladimír Procházka ◽  
Brandon Robinson ◽  
Jacopo Sisti
Keyword(s):  
Entanglement Entropy ◽  
Free Field ◽  
Operator Product ◽  
Dirac Fermions ◽  
Maxwell Theory ◽  
Conformal Field ◽  
Defect Operator ◽  
Crucial Information ◽  
The One ◽  
Group Flow

Abstract We study co-dimension two monodromy defects in theories of conformally coupled scalars and free Dirac fermions in arbitrary d dimensions. We characterise this family of conformal defects by computing the one-point functions of the stress-tensor and conserved current for Abelian flavour symmetries as well as two-point functions of the displacement operator. In the case of d = 4, the normalisation of these correlation functions are related to defect Weyl anomaly coefficients, and thus provide crucial information about the defect conformal field theory. We provide explicit checks on the values of the defect central charges by calculating the universal part of the defect contribution to entanglement entropy, and further, we use our results to extract the universal part of the vacuum Rényi entropy. Moreover, we leverage the non-supersymmetric free field results to compute a novel defect Weyl anomaly coefficient in a d = 4 theory of free $$ \mathcal{N} $$ N = 2 hypermultiplets. Including singular modes in the defect operator product expansion of fundamental fields, we identify notable relevant deformations in the singular defect theories and show that they trigger a renormalisation group flow towards an IR fixed point with the most regular defect OPE. We also study Gukov-Witten defects in free d = 4 Maxwell theory and show that their central charges vanish.

scholarly journals A MEAN FIELD GAME ANALYSIS OF SIR DYNAMICS WITH VACCINATION

2020 ◽  
pp. 1-18
Author(s):  
Josu Doncel ◽  
Nicolas Gast ◽  
Bruno Gaujal
Keyword(s):  
Mean Field ◽  
Vaccination Strategy ◽  
Game Analysis ◽  
Total Cost ◽  
Mean Field Game ◽  
Vaccination Behavior ◽  
Switching Curve ◽  
The One ◽  
The Cost ◽  
Vaccination Period

We analyze a mean field game model of SIR dynamics (Susceptible, Infected, and Recovered) where players choose when to vaccinate. We show that this game admits a unique mean field equilibrium (MFE) that consists in vaccinating at a maximal rate until a given time and then not vaccinating. The vaccination strategy that minimizes the total cost has the same structure as the MFE. We prove that the vaccination period of the MFE is always smaller than the one minimizing the total cost. This implies that, to encourage optimal vaccination behavior, vaccination should always be subsidized. Finally, we provide numerical experiments to study the convergence of the equilibrium when the system is composed by a finite number of agents ( $N$ ) to the MFE. These experiments show that the convergence rate of the cost is $1/N$ and the convergence of the switching curve is monotone.

scholarly journals A systematic design approach for objectifying Building with Nature solutions

2021 ◽  
Vol 7 ◽  
pp. 29-50
Author(s):  
Mindert De Vries ◽  
Mark Van Koningsveld ◽  
Stefan Aarninkhof ◽  
Huib De Vriend
Keyword(s):  
Building Blocks ◽  
Hydraulic Engineering ◽  
Systematic Design ◽  
The Social ◽  
Building With Nature ◽  
Engineering Designs ◽  
Hydraulic Infrastructure ◽  
Crucial Information ◽  
Development Phases ◽  
The One

Hydraulic engineering infrastructure is supposed to keep functioning for many years and is likely to interfere with both the natural and the social environment at various scales. Due to its long life-cycle, hydraulic infrastructure is bound to face changing environmental conditions as well as changes in societal views on acceptable solutions. This implies that sustainability and adaptability are/should be important attributes of the design, the development and operation of hydraulic engineering infrastructure. Sustainability and adaptability are central to the Building with Nature (BwN) approach. Although nature-based design philosophies, such as BwN, have found broad support, a key issue that inhibits a wider mainstream implementation is the lack of a method to objectify BwN concepts. With objectifying, we mean turning the implicit into an explicit engineerable ‘object’, on the one hand, and specifying clear design ‘objectives’, on the other. This paper proposes the “Frame of Reference” approach as a method to systematically transform BwN concepts into functionally specified engineering designs. It aids the rationalisation of BwN concepts and facilitates the transfer of crucial information between project development phases, which benefits the uptake, acceptance and eventually the successful realisation of BwN solutions. It includes an iterative approach that is well suited for assessing status changes of naturally dynamic living building blocks of BwN solutions. The applicability of the approach is shown for a case that has been realised in the Netherlands. Although the example is Dutch, the method, as such, is generically applicable.

Connectivity, Dynamics, and Memory in Reservoir Computing with Binary and Analog Neurons

2010 ◽  
Vol 22 (5) ◽  
pp. 1272-1311 ◽  
Author(s):  
Lars Büsing ◽  
Benjamin Schrauwen ◽  
Robert Legenstein
Keyword(s):  
Analog Circuits ◽  
Memory Function ◽  
Mean Field ◽  
Network Connectivity ◽  
Network Models ◽  
Reservoir Computing ◽  
Computational Performance ◽  
Kernel Quality ◽  
The One ◽  
Long Timescales

Reservoir computing (RC) systems are powerful models for online computations on input sequences. They consist of a memoryless readout neuron that is trained on top of a randomly connected recurrent neural network. RC systems are commonly used in two flavors: with analog or binary (spiking) neurons in the recurrent circuits. Previous work indicated a fundamental difference in the behavior of these two implementations of the RC idea. The performance of an RC system built from binary neurons seems to depend strongly on the network connectivity structure. In networks of analog neurons, such clear dependency has not been observed. In this letter, we address this apparent dichotomy by investigating the influence of the network connectivity (parameterized by the neuron in-degree) on a family of network models that interpolates between analog and binary networks. Our analyses are based on a novel estimation of the Lyapunov exponent of the network dynamics with the help of branching process theory, rank measures that estimate the kernel quality and generalization capabilities of recurrent networks, and a novel mean field predictor for computational performance. These analyses reveal that the phase transition between ordered and chaotic network behavior of binary circuits qualitatively differs from the one in analog circuits, leading to differences in the integration of information over short and long timescales. This explains the decreased computational performance observed in binary circuits that are densely connected. The mean field predictor is also used to bound the memory function of recurrent circuits of binary neurons.

Properties of Relativistic Mean-Field Theory

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Mean Field Theory ◽  
Mean Field ◽  
Field Method ◽  
Basis Sets ◽  
Electron Interaction ◽  
Relativistic Mean Field ◽  
Dirac Equations ◽  
Electrons And Positrons ◽  
The One ◽  
Atomic Structure Calculations

We have previously seen how the Dirac equation for one particle requires some rather special consideration and interpretation in order to arrive at a form that is able to treat electrons and positrons on an equal footing. These problems persist also when we go to systems with more than one electron. One might think that the extension to several electrons should not introduce dramatic changes. After all, we noted that even the one-electron problem must be viewed as a many-electron (and -positron) system in order to arrive at a consistent description. The problem with introducing more electrons is that electron–electron interactions that were previously small—for the one-electron case typically arising from vacuum polarization and self-interaction—now occur to the same order as the kinetic energy and the interaction with the potential. So while a perturbative approach such as QED can use the solutions of the one-electron Dirac equations as a very good starting approximation to a more accurate description of the full system, the same would not work for a system with more electrons because it would mean neglecting interactions of the same magnitude as the zeroth-order energy. For applications to quantum chemistry, the treatment of the entire electron–electron interaction as a perturbation would be hopelessly impractical, as it is even in manyelectron relativistic atomic structure calculations. The technique for dealing with this problem is well known from nonrelativistic calculations on many-electron systems. One-particle basis sets are developed by considering the behavior of the single electron in the mean field of all the other electrons, and while this neglects a smaller part of the interaction energy, the electron correlation, it provides a suitable starting point for further variational or perturbational treatments to recover more of the electron–electron interaction. It is only natural to pursue the same approach for the relativistic case. Thus one may proceed to construct a mean-field method that can be used as a basis for the perturbation theory of QED.

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