New Publications by Valentino Tosatti

Title: Ricci-flat Metrics and Dynamics on K3 Surfaces

Abstract: We give an overview of some recent interactions between the geometry of K3 surfaces and their Ricci-flat Kähler metrics and the dynamical study of K3 automorphisms with positive entropy.

arXiv: 2003.06976

Title: Collapsing Calabi-Yau Manifolds

Dedicated to Professor Shing-Tung Yau on the occasion of his 70th birthday

Abstract: We survey some recent developments on the problem of understanding degenerations of Calabi-Yau manifolds equipped with their Ricci-flat K¨ahler metrics, with an emphasis on the case when the metrics are volume collapsing.

arXiv: 2003.00673

New and Published Work by Juven Wang

New preprint:

Title: Non-Liquid Cellular States

Abstract: The existence of quantum non-liquid states and fracton orders, both gapped and gapless states, challenges our understanding of phases of entangled matter. We generalize Wen’s cellular topological states to liquid or non-liquid cellular states. We propose a mechanism to construct more general non-abelian states by gluing gauge-symmetry-breaking vs gauge-symmetry-extension interfaces as extended defects in a cellular network, including defects of higher-symmetries. Our approach also includes the anyonic particle/string condensation and composite string (p-string)/membrane condensations. This also shows gluing the familiar extended topological quantum field theory or conformal field theory data via topology, geometry, and renormalization consistency criteria (via certain modified group cohomology or cobordism theory data) in a tensor network can still guide us to analyze the non-liquid states. (Part of the abelian construction can be understood from the K-matrix Chern-Simons theory approach and coupled-layer-by-junction constructions.) This may also lead us toward a unifying framework for quantum systems of both higher-symmetries and sub-system/sub-dimensional symmetries.

Continue reading New and Published Work by Juven Wang

Two New Publications by Aghil Alaee

Title: A localized spacetime Penrose inequality and horizon detection with quasi-local mass
Aghil Alaee, Martin Lesourd, Shing-Tung Yau

Abstract: Our setting is a simply connected bounded domain with a smooth connected boundary, which arises as an initial data set for the general relativistic constraint equations satisfying the dominant energy condition. Assuming the domain to be admissible in a certain precise sense, we prove a localized spacetime Penrose inequality for the Liu-Yau and Wang-Yau quasi-local masses and the area of an outermost marginally outer trapped surface (MOTS). On the basis of this inequality, we obtain sufficient conditions for the existence and non-existence of a MOTS (along with outer trapped surfaces) in the domain, and for the existence of a minimal surface in its Jang graph, expressed in terms of various quasi-local mass quantities and the boundary geometry of the domain.


Title: Stability of a quasi-local positive mass theorem for graphical hypersurfaces of Euclidean space
Aghil Alaee, Armando J. Cabrera Pacheco, Stephen McCormick
Abstract: We present a quasi-local version of the stability of the positive mass theorem. We work with the Brown–York quasi-local mass as it possesses positivity and rigidity properties, and therefore the stability of this rigidity statement can be studied. Specifically, we ask if the Brown–York mass of the boundary of some compact manifold is close to zero, must the manifold be close to a Euclidean domain in some sense?
Here we consider a class of compact n-manifolds with boundary that can be realized as graphs in ℝn+1, and establish the following. If the Brown–York mass of the boundary of such a compact manifold is small, then the manifold is close to a Euclidean hyperplane with respect to the Federer–Fleming flat distance.


New Publications by Juven Wang

Title: Higher Anomalies, Higher Symmetries, and Cobordisms III: QCD Matter Phases Anew

Zheyan Wan, Juven Wang

Abstract: We explore QCD4 quark matter, the μ-T (chemical potential-temperature) phase diagram, possible ‘t Hooft anomalies, and topological terms, via non-perturbative tools of cobordism theory and higher anomaly matching. We focus on quarks in 3-color and 3-flavor on bi-fundamentals of SU(3), then analyze the continuous and discrete global symmetries and pay careful attention to finite group sectors. We input constraints from T=CP or CT time-reversal symmetries, implementing QCD on unorientable spacetimes and distinct topology. Examined phases include the high T QGP (quark-gluon plasma/liquid), the low T ChSB (chiral symmetry breaking), 2SC (2-color superconductivity) and CFL (3-color-flavor locking superconductivity) at high density. We introduce a possibly useful but only approximate higher anomaly, involving discrete 0-form axial and 1-form mixed chiral-flavor-locked center symmetries, matched by the above four QCD phases. We also enlist as much as possible, but without identifying all of, ‘t Hooft anomalies and topological terms relevant to Symmetry Protected/Enriched Topological states (SPTs/SETs) of gauged SU(2) or SU(3) QCDd-like matter theories in general in any spacetime dimensions d=2,3,4,5 via cobordism.



Title: Higher Anomalies, Higher Symmetries, and Cobordisms II: Applications to Quantum Gauge Theories

 Zheyan Wan, Juven Wang, Yunqin Zheng

Abstract: We discuss the topological terms, the global symmetries and their ‘t Hooft anomalies of pure gauge theories in various dimensions, with dynamical gauge group G, the Lorentz symmetry group GLorentz, and the internal global symmetry Ge,[1]×Gm,[d−3] which consists of 1-form electric center symmetry Ge,[1] and (d−3) form magnetic symmetry Gm,[d−3]. The topological terms are determined by the cobordism invariants (Ωd)G′ where G′ is the group extension of GLorentz by G, which also characterize the invertible TQFTs or SPTs with global symmetry G′. The ‘t Hooft anomalies are determined by the cobordism invariants (Ωd+1)G′′ where G′′ is the symmetry extension of GLorentz by the higher form symmetry Ge,[1]×Gm,[d−3]. Different symmetry extensions correspond to different fractionalizations of GLorentz quantum numbers on the symmetry defects of Ge,[1]×Gm,[d−3]. We compute the cobordism groups/invariants described above for G= U(1), SU(2) and SO(3) in d≤5, thus systematically classifies all the topological terms and the ‘t Hooft anomalies of d dimensional quantum gauge theories with the above gauge groups.


Title: Non-Abelian Gauged Fractonic Matter Field Theory: New Sigma Models, Superfluids and Vortices

Juven Wang, Shing-Tung Yau

Abstract: By gauging a higher-moment polynomial global symmetry and a discrete charge conjugation (i.e., particle-hole) symmetry (mutually non-commutative) coupled to matter fields, we derive a new class of higher-rank tensor non-abelian gauge field theory with dynamically gauged matter fields: Non-abelian gauged matters interact with a hybrid class of higher-rank (symmetric or generic non-symmetric) tensor gauge theory and anti-symmetric tensor topological field theory, generalizing [arXiv:1909.13879, 1911.01804]’s theory. We also apply a quantum phase transition similar to that between insulator v.s. superfluid/superconductivity (U(1) symmetry disordered phase described by a topological gauge theory or a disordered Sigma model v.s. U(1) global/gauge symmetry-breaking ordered phase described by a Sigma model with a U(1) target space underlying Goldstone modes): We can regard our tensor gauge theories as disordered phases, and we transient to their new ordered phases by deriving new Sigma models in continuum field theories. While one low energy theory is captured by degrees of freedom of rotor or scalar modes, another side of low energy theory has vortices and superfluids – we explore non-abelian vortices (two types of vortices mutually interacting non-commutatively) beyond an ordinary group structure and their Cauchy-Riemann relation.


Title: Higher-Rank Non-Abelian Tensor Field Theory: Higher-Moment or Subdimensional Polynomial Global Symmetry, Algebraic Variety, Noether’s Theorem, and Gauge

Juven Wang, Kai Xu, Shing-Tung Yau

Abstract: With a view toward a theory of fracton and embeddon in condensed matter, we introduce a higher-moment polynomial degree-(m-1) global symmetry, acting on complex scalar/vector/tensor fields. We relate this higher-moment global symmetry of n-dimensional space, to a lower degree (either ordinary or higher-moment, e.g., degree-(m-1-ℓ)) subdimensional or subsystem global symmetry on layers of (n−ℓ)-submanifolds. These submanifolds are algebraic affine varieties (i.e., solutions of polynomials). The structure of layers of submanifolds as subvarieties can be studied via mathematical tools of embedding, foliation and algebraic geometry. We also generalize Noether’s theorem for this higher-moment polynomial global symmetry. We can promote the higher-moment global symmetry to a local symmetry, and derive a new family of higher-rank-m symmetric tensor gauge theory by gauging. By further gauging a discrete charge conjugation symmetry, we derive a new more general class of non-abelian rank-m tensor gauge field theory: a hybrid class of (symmetric or non-symmetric) higher-rank-m tensor gauge theory and anti-symmetric tensor topological field theory, generalizing [arXiv:1909.13879]’s theory interplaying between gapless and gapped sectors.




New Publication by Xinqi Gong

Image of the first pageTitle: Attention mechanism enhanced LSTM with residual architecture and its application for protein-protein interaction residue pairs prediction

Jiale Liu & Xinqi Gong

Recurrent neural network(RNN) is a good way to process sequential data, but the capability of RNN to compute long sequence data is inefficient. As a variant of RNN, long short term memory(LSTM) solved the problem in some extent. Here we improved LSTM for big data application in protein-protein interaction interface residue pairs prediction based on the following two reasons. On the one hand, there are some deficiencies in LSTM, such as shallow layers, gradient explosion or vanishing, etc. With a dramatic data increasing, the imbalance between algorithm innovation and big data processing has been more serious and urgent. On the other hand, protein-protein interaction interface residue pairs prediction is an important problem in biology, but the low prediction accuracy compels us to propose new computational methods.

In order to surmount aforementioned problems of LSTM, we adopt the residual architecture and add attention mechanism to LSTM. In detail, we redefine the block, and add a connection from front to back in every two layers and attention mechanism to strengthen the capability of mining information. Then we use it to predict protein-protein interaction interface residue pairs, and acquire a quite good accuracy over 72%. What’s more, we compare our method with random experiments, PPiPP, standard LSTM, and some other machine learning methods. Our method shows better performance than the methods mentioned above.

We present an attention mechanism enhanced LSTM with residual architecture, and make deeper network without gradient vanishing or explosion to a certain extent. Then we apply it to a significant problem– protein-protein interaction interface residue pairs prediction and obtain a better accuracy than other methods. Our method provides a new approach for protein-protein interaction computation, which will be helpful for related biomedical researches.

PDF of the paper

New Publication by Spiro Karigiannis

Title: Introduction to G2 geometry

 Spiro Karigiannis

Abstract: These notes give an informal and leisurely introduction to G2 geometry for beginners. A special emphasis is placed on understanding the special linear algebraic structure in 7 dimensions that is the pointwise model for G2 geometry, using the octonions. The basics of G2-structures are introduced, from a Riemannian geometric point of view, including a discussion of the torsion and its relation to curvature for a general G2-structure, as well as the connection to Riemannian holonomy. The history and properties of torsion-free G2 manifolds are considered, and we stress the similarities and differences with Ka ̈hler and Calabi-Yau manifolds. The notes end with a brief survey of three important theorems about compact torsion-free G2 manifolds.


New Publication by Enno Keßler, Artan Sheshmani and Shing-Tung Yau

Title: Super $J$-holomorphic Curves: Construction of the Moduli Space
Enno Keßler, Artan Sheshmani and Shing-Tung Yau

Abstract: Let $M$ be a super Riemann surface with holomorphic distribution $\mathcal{D}$ and $N$ a symplectic manifold with compatible almost complex structure $J$. We call a map $\Phi\colon M\to N$ a super $J$-holomorphic curve if its differential maps the almost complex structure on $\mathcal{D}$ to $J$. Such a super $J$-holomorphic curve is a critical point for the superconformal action and satisfies a super differential equation of first order. Using component fields of this super differential equation and a transversality argument we construct the moduli space of super $J$-holomorphic curves as a smooth subsupermanifold of the space of maps $M\to N$.

Arxiv: 1911.05607

Recent Publications by Juven Wang

Phys. Rev. D 100, 085012 – Published 21 October 2019

Title: Quantum 4d Yang-Mills theory and time-reversal symmetric 5d higher-gauge topological field theory

Zheyan Wan,  Juven Wang , and Yunqin Zheng

Abstract: We explore various 4d Yang-Mills gauge theories (YM) living as boundary conditions of 5d gapped short-/long-range entangled (SRE/LRE) topological states. Specifically, we explore 4d time-reversal symmetric pure YM of an SU(2) gauge group with a second-Chern-class topological term at θ=π [SU(2)θ=π YM], by turning on the background fields for both the time reversal (i.e., on unorientable manifolds) and the one-form center global symmetry. We find four siblings of SU(2)θ=π YM with distinct couplings to background fields, labeled by (K1,K2)K1=0, 1 specifies Kramers singlet/doublet Wilson line and new mixed higher ’t Hooft anomalies; K2=0, 1 specifies the boson/fermionic Wilson line and a new Wess-Zumino-Witten–like counterterm. Higher anomalies indicate that to realize all higher n-global symmetries locally on n simplices, the 4d theory becomes a boundary of a 5d higher-symmetry-protected topological state (SPTs, as an invertible topological quantum field theory (iTQFT) or a cobordism invariant in math, or as a 5d higher-symmetric interacting topological superconductor in condensed matter)… Continue reading Recent Publications by Juven Wang

Geometry and analysis of the Yang-Mills-Higgs-Dirac model

New Publication from Enno Keßler et al.:

Title: Geometry and analysis of the Yang-Mills-Higgs-Dirac model
Authors: Jürgen Jost, Enno Keßler, Ruijun Wu, Miaomiao Zhu

Abstract: The harmonic sections of the Kaluza-Klein model can be seen as a variant of harmonic maps with additional gauge symmetry. Geometrically, they are realized as sections of a fiber bundle associated to a principal bundle with a connection. In this paper, we investigate geometric and analytic aspects of a model that combines the Kaluza-Klein model with the Yang-Mills action and a Dirac action for twisted spinors. In dimension two we show that weak solutions of the Euler Lagrange system are smooth. For a sequence of approximate solutions on surfaces with uniformly bounded energies we obtain compactness modulo bubbles, namely, energy identities and the no-neck property hold.


More publications by Juven Wang in Mathematical Physics and Physical Review D

Journal of Mathematical Physics60, 052301 (2019);

Title: A new SU(2) anomaly

Abstract: A familiar anomaly affects SU(2) gauge theory in four dimensions: a theory with an odd number of fermion multiplets in the spin 1/2 representation of the gauge group, and more generally in representations of spin 2r + 1/2, is inconsistent. We describe here a more subtle anomaly that can affect SU(2) gauge theory in four dimensions under the condition that fermions transform with half-integer spin under SU(2) and bosons transform with integer spin. Such a theory, formulated in a way that requires no choice of spin structure, and with an odd number of fermion multiplets in representations of spin 4r + 3/2, is inconsistent. The theory is consistent if one picks a spin or spinc structure. Under Higgsing to U(1), the new SU(2) anomaly reduces to a known anomaly of “all-fermion electrodynamics.” Like that theory, an SU(2) theory with an odd number of fermion multiplets in representations of spin 4r + 3/2 can provide a boundary state for a five-dimensional gapped theory whose partition function on a closed five-manifold Y is (1)Yw2w3(−1)∫Yw2w3. All statements have analogs with SU(2) replaced by Sp(2N). There is also an analog in five dimensions.

Continue reading More publications by Juven Wang in Mathematical Physics and Physical Review D