Study spin chains and sigma models on flag manifolds, calculating spectra and geodesics.
problem Understanding the spectrum and geodesics of sigma models on flag manifolds.
method Connecting SU(n) spin chains to sigma models and calculating spectra and geodesics.
result Calculated the spectrum of the Laplace-Beltrami operator and geodesics for CP1 and F3. Generative neural samplers estimate quantum spin system properties.
problem Estimating observables for quantum spin systems.
method Autoregressive models using Suzuki-Trotter transformation.
result Results for energy, specific heat, and susceptibility are in good agreement with Monte Carlo methods.
We show that the Hilbert space formed from a block spin renormalization construction of a cyclic quantum spin chain (based on the Temperley-Lieb algebra) does not support a chiral conformal field theory whose Hamiltonian generates translation on the circle as a continuous limit of the rotations on the lattice.
Review of sigma models on flag manifolds, linking to spin chains and integrable theories.
problem Understanding phase transitions and anomalies in spin chains and sigma models.
method Analyzing topological angles, discrete 't Hooft anomalies, and integrable models.
result Gapless phases in certain spin chains can be explained by discrete anomalies in continuum theories.
This work connects point particles to spin chains using geometric methods.
problem Understanding dynamics of free point particles on Riemannian manifolds.
method Kirillov orbit method, geometric quantization, Lagrangian submanifolds.
result Establishes a spectral equivalence between Laplace-Beltrami operator and a spin Hamiltonian.
New quantum integrals discovered for a spin chain model.
problem Exploring quantum integrals for a spin chain model.
method Using surface defects and observables in 4D N=2 super-QCD. result First construction of quantum integrals and their joint eigenvectors.
Quantum systems on coadjoint orbits yield spectra matching Dolbeault and de Rham indices.
problem Finding exact spectra of generalized Laplace operators on coadjoint orbits.
method Truncation of 1D sigma models and nonlinear chiral multiplets.
result Exact spectra of spin chain Hamiltonians match Dolbeault and de Rham indices of target spaces.
Motion of curves and surfaces in R3 lead to nonlinear evolution equations which are often integrable. They are also intimately connected to the dynamics of spin chains in the continuum limit and integrable soliton systems through geometric and gauge symmetric connections/equivalence. Here we point out the fact that…
Solves geodesics and Laplace-Beltrami spectrum on flag manifolds.
problem Geodesics and Laplace-Beltrami spectrum on flag manifolds.
method Invariant metrics and finite-dimensional approximations.
result Explicit solutions for geodesics and spectrum.
TensorNetwork is an open source library for implementing tensor network algorithms in TensorFlow. We describe a tree tensor network (TTN) algorithm for approximating the ground state of either a periodic quantum spin chain (1D) or a lattice model on a thin torus (2D), and implement the algorithm using TensorNetwork. We…
Study isotropic embeddings of Lie group orbits and their application to magnetic geodesic flows.
problem Embedding coadjoint orbits and their equivalence to magnetic geodesic flows.
method Review and initiate study of isotropic and Lagrangian embeddings for SO and Sp cases, then apply to magnetic geodesic flows. result Equivalence between magnetic geodesic flows and certain spin chains.
We introduce a refinement of the Ozsvath-Szabo complex associated to a balanced sutured manifold (X,τ) by Juhasz. An algebra Aτ is associated to the boundary of a sutured manifold and a filtration of its generators by H2(X,∂X;Z) is defined. For a fixed Spin^c structure s over the manifold X′, which…
We show that the smooth geometry of a hyperbolic 3-manifold emerges from a classical spin system defined on a 2d discrete lattice, and moreover show that the process of this "dimensional oxidation" is equivalent with the dimensional reduction of a supersymmetric gauge theory from 4d to 3d. More concretely, we propose a…
Let L be a Lagrangian submanifold in a symplectic vector space which is closed, oriented and spin. Using virtual fundamental chains of moduli spaces of nonconstant pseudo-holomorphic disks with boundaries on L, one can define a Maurer-Cartan element of a Lie bracket operation in string topology (the loop bracket) d…
The Jones-Wenzl projectors play a central role in quantum topology, underlying the construction of SU(2) topological quantum field theories and quantum spin networks. We construct chain complexes whose graded Euler characteristic is the "classical" projector in the Temperley-Lieb algebra. We show that they are homotopy…
A spin model is used for simulations of financial markets. To determine return volatility in the spin financial market we use the GARCH model often used for volatility estimation in empirical finance. We apply the Bayesian inference performed by the Markov Chain Monte Carlo method to the parameter estimation of the GAR…
Study of surface defects in gauge theories leads to duality and separation of variables.
problem Understanding surface observables and their transitions in gauge theories.
method Utilized Fourier transformations and spectral problems to derive dualities and separation of variables.
result Exact duality between spectral problems of spin chains and Gaudin models.
Link Floer homology is an invariant for links which has recently been described entirely in a combinatorial way. Originally constructed with mod 2 coefficients, it was generalized to integer coefficients thanks to a sign refinement. In this paper, thanks to the spin extension of the permutation group we give an alterna…
Study non-negative curvature Markov chains, proving entropy contraction.
problem Prove entropy contraction for Markov chains with non-negative curvature.
method Prove 1-step contraction in Wasserstein distance implies 1-step contraction in relative entropy.
result Prove MLSI with constant equal to minimal rate increment for mean-field zero-range process.
The formation of price in a financial market is modelled as a chain of Ising spin with three fundamental figures of trading. We investigate the time behaviour of the model, and we compare the results with the real EURO/USD change rate. By using the test of local Poisson hypothesis, we show that this minimal model leads…
A theory explaining how deep learning works is yet to be developed. Previous work suggests that deep learning performs a coarse graining, similar in spirit to the renormalization group (RG). This idea has been explored in the setting of a local (nearest neighbor interactions) Ising spin lattice. We extend the discussio…
The periodic Floer homology of a surface symplectomorphism, defined by the first author and M. Thaddeus, is the homology of a chain complex which is generated by certain unions of periodic orbits, and whose differential counts certain embedded pseudoholomorphic curves in R cross the mapping torus. It is conjectured to …
Filiz et al. (2008) proposed a model for the pattern of defaults seen among a group of firms at the end of a given time period. The ingredients in the model are a graph, where the vertices correspond to the firms and the edges describe the network of interdependencies between the firms, a parameter for each vertex that…
Paper approximates fractional harmonic maps with numerical methods.
problem Approximating fractional harmonic maps with constraints and nonlocality.
method Weak compactness results and numerical methods for various PDEs.
result Convergence of numerical approximations for fractional harmonic maps.
New connection between dynamics and Heegaard Floer homology.
problem Understanding pseudo-Anosov flows and their dynamics.
method Using Heegaard Floer homology and veering branched surfaces, the paper constructs a chain complex to categorify the zeta function of a pseudo-Anosov flow.
result Generators of the chain complex correspond to closed multi-orbits of the flow, and their homology classes have dynamical significance.
Study of autocorrelation times in neural MCMC simulations for the 2D Ising model.
problem Estimating autocorrelation times in Neural Markov Chain Monte Carlo simulations.
method Analytical and empirical methods to estimate autocorrelation times, proposing new loss functions and training schemes.
result Proposed new loss functions and training schemes that improve autocorrelation times in neural MCMC simulations.
Study Floer theory of hyperbolic three-manifolds using Dirac spectral flow.
problem Computing Floer theory of hyperbolic three-manifolds with non-trivial homology.
method Combining geometric data with Fourier analytic tools and odd Selberg trace formulas.
result First computations of monopole Floer chain complexes for hyperbolic three-manifolds.
Solves sigma model on U(3)/U(1)^3, describing geodesics and spectrum.
problem Classical and quantum problems for 1D sigma model with specific target space.
method Mapping to Gaudin model, solving polynomial equations.
result Explicit description of geodesics and spectrum found.
Unified approach learns Ising models from various dynamics and data types.
problem Efficiently learning Ising model parameters from data under diverse conditions.
method Simple logistic regression approach, generalizing existing algorithms.
result Logistic regression succeeds in multiple new settings where assumptions are violated.
The compact exceptional Lie groups F4, E6, E7 and E8 have spinor groups as a subgroup as follows: E8 \supset Ss(16) \supset Spin(15) \supset Spin(14) \supset Spin(13), E7 \supset Spin(12) \supset Spin(11), E6 \supset Spin(10), F4 \supset Spin(9) \supset Spin(8) \supset Spin(7) \supset \cdot \cdot \cdot \supset Spin(1) …
New method makes machine learning approximations unbiased and efficient.
problem Efficient sampling of complex probability distributions.
method Uses autoregressive neural networks with cluster updates and physical symmetries.
result Shows unbiased and low-variance approximations for phase transitions.
The paper explores spin^h structures and their obstructions.
problem Understanding which manifolds are spin^h.
method Analyzing the fifth Stiefel-Whitney class and constructing generalised spin structures.
result Every compact orientable manifold of dimension 7 or lower is spin^h, and there are higher-dimensional manifolds that are not.
Study mSpin(7)-dDT connections on manifolds with mSpin(7)-structures.
problem Understanding moduli spaces of mSpin(7)-dDT connections. method Introduced and studied mSpin(7)-dDT connections using fully nonlinear PDEs. result Moduli space MmSpin(7)′ has finite expected dimension and smoothness under certain conditions. New Spin(7)-instantons constructed on Joyce's manifold.
problem Constructing Spin(7)-instantons on Joyce's compact manifold. method Gluing non-flat connections on local model spaces to a flat connection on the Spin(7)-orbifold. result More than 20,000 new four-parameter families of Spin(7)-instantons. This is an introduction to the construction of higher-dimensional knots by spinning methods. Simple spinning of classical knots was introduced by E. Artin in 1926, and several generalizations have followed. These include twist spinning, superspinning or p-spinning, frame spinning, roll spinning, and deform spinning. We…
New mSpinh-manifolds studied for their properties.
problem Understanding new manifold classifications.
method Exploring mSpinh-manifolds as a new category. result Highlights of mSpinh-manifolds discussed. Study on harmonic flow of Spin(7)-structures in 8D manifolds.
problem Comparing isometric Spin(7)-structures.
method Developed a notion of harmonicity for Spin(7)-structures and studied the harmonic flow.
result Analytical properties of the harmonic flow of Spin(7)-structures presented.
Study of higher spin Killing spinors on 3D manifolds, proving rigidity and providing explicit expressions.
problem Understanding higher spin Killing spinors on 3D manifolds.
method Definition and detailed study of higher spin Killing spinors in arbitrary dimension, focusing on 3D manifolds. Rigidity result and explicit expressions for 3-sphere and 3-hyperbolic space.
result Proved a rigidity result for 3D manifolds admitting higher spin Killing spinors and provided explicit expressions for these spinors.
The paper characterizes new invariant spinr spinors on projective spaces.
problem Characterizing new invariant spinr spinors on projective spaces. method Adapting spin representation via exterior forms to the generalised spinr context. result Complete description of the space of invariant spinr spinors for CPn, HPn, and OP2. Spin(7) geometry linked to multisymplectic geometry.
problem Understanding Spin(7) structures through multisymplectic geometry.
method Utilized Spin(7) identities to prove non-degeneracy of Cayley four-form in multisymplectic context.
result Spin(7) geometry is a special case of multisymplectic geometry.
We define spin frames, with the aim of extending spin structures from the category of (pseudo-)Riemannian manifolds to the category of spin manifolds with a fixed signature on them, though with no selected metric structure. Because of this softer requirements, transformations allowed by spin frames are more general tha…
We explore differential and algebraic operations on the exterior product of spinor representations and their twists that give rise to cohomology, the spin cohomology. A linear differential operator d is introduced which is associated to a connection ∇ and a parallel spinor ζ, ∇ζ=0, and the algebraic o…
Proves almost flat spin^c manifolds bound compact manifolds.
problem Proving almost flat spin^c manifolds bound compact manifolds.
method Long-standing conjecture of Farrell--Zdravkovska and S. T. Yau settled.
result Every almost flat spin^c manifold bounds a compact orientable manifold.
Rigidity of elliptic genera proven for non-spin manifolds with S1-action.
problem Rigidity of elliptic genera for non-spin manifolds with S1-action. method Analysis of universal covering spin condition and π2(M) for rigidity. result Rigidity of elliptic genera is proven for spin universal coverings but not for non-spin universal coverings.
First non-trivial examples of deformed Spin(7)-instantons constructed.
problem Constructing deformed Spin(7)-instantons and connections.
method Constructing on cotangent bundles of CP2 and cones over 3-Sasakian 7-manifolds. result First non-trivial examples of deformed Spin(7)-instantons.
This paper uses Heegaard Floer theory to study pseudo-Anosov flows and their periodic points.
problem Understanding the differential of Heegaard Floer chain complexes associated with pseudo-Anosov flows.
method Introduces a refined grading to analyze the homology of subcomplexes representing irreducible multi-orbits.
result The homology of these subcomplexes is 1-dimensional, providing insights into periodic points of pseudo-Anosov flows.
Study shows certain spin manifolds can't meet DEC condition.
problem Non-existence of spin fill-ins meeting DEC condition.
method Analyzes spin Riemannian manifolds and generalized mean curvature functions.
result Closed spin manifolds cannot satisfy DEC if curvature is large.
Spin TFTs created by gauging line defects in 3D.
problem Creating spin TFTs from oriented TFTs with framed line defects.
method Constructing a spin TFT from an oriented TFT with framed line defects and a commutative Frobenius algebra.
result Spin TFTs extend earlier classifications and reproduce abelian spin Chern-Simons theories.