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Shing-Tung Yau

Publications and source records attributed to Shing-Tung Yau.

At least 127 records · Page 7Linked to original sources

Strictification and gluing of Lagrangian distributions on derived schemes with shifted symplectic forms

A strictification result is proved for isotropic distributions on derived schemes equipped with negatively shifted homotopically closed $2$-forms. It is shown that any derived scheme over $\mathbb{C}$ equipped with a $-2$-shifted symplectic structure, and having a Hausdorff space of classical points, admits a globally defined Lagrangian distribution as a dg $\mathbb{C}^{\infty}$-manifold.

math.AG↗

Global shifted potentials for moduli stacks of sheaves on Calabi-Yau four-folds

It is shown that there are globally defined Lagrangian distributions on the stable loci of derived Quot-stacks of coherent sheaves on Calabi--Yau four-folds. Dividing by these distributions produces perfectly obstructed smooth stacks with globally defined $-1$-shifted potentials, whose derived critical loci give back the stable loci of smooth stacks of sheaves in global Darboux form.

math.AG↗

On Upper Bounds in Dimension Gaps of CFT's

We consider CFT's arising from branes probing singularities of internal manifolds. We focus on holographic models with internal space including arbtirary Sasaki-Einstein manifolds coming from CY as well as arbitrary sphere quotients. In all these cases we show that there is a universal upper bound (depending only on the spacetime dimension) for the conformal dimension of the first non-trivial spin 2 operator in the dual CFT and a minimal diameter (in AdS units) for the internal space of the holographic dual and conjecture it holds for all CFT's.

hep-th↗

Special Lagrangian cycles and Calabi-Yau transitions

We construct special Lagrangian 3-spheres in non-Kähler compact threefolds equipped with the Fu-Li-Yau geometry. These non-Kähler geometries emerge from topological transitions of compact Calabi-Yau threefolds. From this point of view, a conifold transition exchanges holomorphic 2-cycles for special Lagrangian 3-cycles.

math.DG↗

Discrete Morse Theory on Digraphs

In this paper, we give a necessary and sufficient condition that discrete Morse functions on a digraph can be extended to be Morse functions on its transitive closure, from this we can extend the Morse theory to digraphs by using quasi-isomorphism between path complex and discrete Morse complex, we also prove a general sufficient condition for digraphs that the Morse functions satisfying this necessary and sufficient condition.

math.CO↗

PhyloTransformer: A Discriminative Model for Mutation Prediction Based on a Multi-head Self-attention Mechanism

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has caused an ongoing pandemic infecting 219 million people as of 10/19/21, with a 3.6% mortality rate. Natural selection can generate favorable mutations with improved fitness advantages; however, the identified coronaviruses may be the tip of the iceberg, and potentially more fatal variants of concern (VOCs) may emerge over time. Understanding the patterns of emerging VOCs and forecasting mutations that may lead to gain of function or immune escape is urgently required. Here we developed PhyloTransformer, a Transformer-based discriminative model that engages a multi-head self-attention mechanism to model genetic mutations that may lead to viral reproductive advantage. In order to identify complex dependencies between the elements of each input sequence, PhyloTransformer utilizes advanced modeling techniques, including a novel Fast Attention Via positive Orthogonal Random features approach (FAVOR+) from Performer, and the Masked Language Model (MLM) from Bidirectional Encoder Representations from Transformers (BERT). PhyloTransformer was trained with 1,765,297 genetic sequences retrieved from the Global Initiative for Sharing All Influenza Data (GISAID) database. Firstly, we compared the prediction accuracy of novel mutations and novel combinations using extensive baseline models; we found that PhyloTransformer outperformed every baseline method with statistical significance. Secondly, we examined predictions of mutations in each nucleotide of the receptor binding motif (RBM), and we found our predictions were precise and accurate. Thirdly, we predicted modifications of N-glycosylation sites to identify mutations associated with altered glycosylation that may be favored during viral evolution. We anticipate that PhyloTransformer may guide proactive vaccine design for effective targeting of future SARS-CoV-2 variants.

q-bio.QM↗

Green's Functions for Vladimirov Derivatives and Tate's Thesis

Given a number field $K$ with a Hecke character $χ$, for each place $ν$ we study the free scalar field theory whose kinetic term is given by the regularized Vladimirov derivative associated to the local component of $χ$. These theories appear in the study of $p$-adic string theory and $p$-adic AdS/CFT correspondence. We prove a formula for the regularized Vladimirov derivative in terms of the Fourier conjugate of the local component of $χ$. We find that the Green's function is given by the local functional equation for Zeta integrals. Furthermore, considering all places $ν$, the field theory two-point functions corresponding to the Green's functions satisfy an adelic product formula, which is equivalent to the global functional equation for Zeta integrals. In particular, this points out a role of Tate's thesis in adelic physics.

hep-th↗

A localized spacetime Penrose inequality and horizon detection with quasi-local mass

For an admissible class of smooth compact initial data sets with boundary, we prove a comparison theorem between the Wang/Liu-Yau quasi-local mass of the boundary and the Hawking mass of strictly minimizing hulls in the Jang graphs of the domain. Using this, we prove a quasi-local Penrose inequality that involves these quasi-local masses of the boundary and the area of an outermost marginally outer trapped surface (MOTS) in the domain or the area of minimizing minimal surface within the Jang graphs. Moreover, we obtain sufficient conditions for the (non)existence of a MOTS within a domain, in the spirit of the folklore hoop conjecture.

math.DG↗

Semiorthogonal decomposition of $\mathrm{D}^b(\mathrm{Bun}_2^L)$

We study the derived category of coherent sheaves on various versions of moduli space of vector bundles on curves by the Borel-Weil-Bott theory for loop groups and $Θ$-stratification, and construct a semiorthogonal decomposition with blocks given by symmetric powers of the curve.

math.AG↗

Soft noncommutative schemes via toric geometry and morphisms from an Azumaya scheme with a fundamental module thereto -- (Dynamical, complex algebraic) D-branes on a soft noncommutative space

A class of noncommutative spaces, named `soft noncommutative schemes via toric geometry', are constructed and the mathematical model for (dynamical/nonsolitonic, complex algebraic) D-branes on such a noncommutative space, following arXiv:0709.1515 [math.AG] (D(1)), is given. Any algebraic Calabi-Yau space that arises from a complete intersection in a smooth toric variety can embed as a commutative closed subscheme of some soft noncommutative scheme. Along the study, the notion of `soft noncommutative toric schemes' associated to a (simplicial, maximal cone of index $1$) fan, `invertible sheaves' on such a noncommutative space, and `twisted sections' of an invertible sheaf are developed and Azumaya schemes with a fundamental module as the world-volumes of D-branes are reviewed. Two guiding questions, Question 3.12 (soft noncommutative Calabi-Yau spaces and their mirror) and Question 4.2.14 (generalized matrix models), are presented.

math.AG↗

BMS charges without supertranslation ambiguity

The asymptotic symmetry of an isolated gravitating system, or the Bondi-Metzner-Sachs (BMS) group, contains an infinite-dimensional subgroup of supertranslations. Despite decades of study, the difficulties with the "supertranslation ambiguity" persisted in making sense of fundamental notions such as the angular momentum carried away by gravitational radiation. The issues of angular momentum and center of mass were resolved by the authors recently. In this paper, we address the issues for conserved charges with respect to both the classical BMS algebra and the extended BMS algebra. In particular, supertranslation ambiguity of the classical charge for the BMS algebra, as well as the extended BMS algebra, is completely identified. We then propose a new invariant charge by adding correction terms to the classical charge. With the presence of these correction terms, the new invariant charge is then shown to be free from any supertranslation ambiguity. Finally, we prove that both the classical and invariant charges for the extended BMS algebra are invariant under the boost transformations.

gr-qc↗

Super quantum cohomology I: Super stable maps of genus zero with Neveu-Schwarz punctures

In this article we define stable supercurves and super stable maps of genus zero via labeled trees. We prove that the moduli space of stable supercurves and super stable maps of fixed tree type are quotient superorbifolds. To this end, we prove a slice theorem for the action of super Lie groups on Riemannian supermanifolds and discuss superorbifolds. Furthermore, we propose a Gromov topology on super stable maps such that the restriction to fixed tree type yields the quotient topology from the superorbifolds and the reduction is compact. This would, possibly, lead to the notions of super Gromov-Witten invariants and small super quantum cohomology to be discussed in sequels.

math.DG↗

Ricci-flat graphs with maximum degree at most 4

A graph is called Ricci-flat if its Ricci curvatures vanish on all edges, here the definition of Ricci curvature on graphs was given by Lin-Lu-Yau. The authors in arXiv:1301.0102 and arXiv:1802.02982 obtained a complete characterization for all Ricci-flat graphs with girth at least five. In this paper, we completely determined all Ricci-flat graphs with maximum degree at most 4.

math.DG↗

$\ell$-adic Tautological Systems

Tautological systems was introduced in Lian-Yau as the system of differential equations satisfied by period integrals of hyperplane sections of some complex projective homogenous varieties. We introduce the $\ell$-adic tautological systems for the case where the ground field is of characteristic $p$.

math.AG↗

General relativity from $p$-adic strings

For an arbitrary prime number $p$, we propose an action for bosonic $p$-adic strings in curved target spacetime, and show that the vacuum Einstein equations of the target are a consequence of worldsheet scaling symmetry of the quantum $p$-adic strings, similar to the ordinary bosonic strings case. It turns out that certain $p$-adic automorphic forms are the plane wave modes of the bosonic fields on $p$-adic strings, and that the regularized normalization of these modes on the $p$-adic worldsheet presents peculiar features which reduce part of the computations to familiar setups in quantum field theory, while also exhibiting some new features that make loop diagrams much simpler. Assuming a certain product relation, we also observe that the adelic spectrum of the bosonic string corresponds to the nontrivial zeros of the Riemann Zeta function.

hep-th↗

Non-Holomorphic Cycles and Non-BPS Black Branes

We study extremal non-BPS black holes and strings arising in M-theory compactifications on Calabi-Yau threefolds, obtained by wrapping M2 branes on non-holomorphic 2-cycles and M5 branes on non-holomorphic 4-cycles. Using the attractor mechanism we compute the black hole mass and black string tension, leading to a conjectural formula for the asymptotic volumes of connected, locally volume-minimizing representatives of non-holomorphic, even-dimensional homology classes in the threefold, without knowledge of an explicit metric. In the case of divisors we find examples where the volume of the representative corresponding to the black string is less than the volume of the minimal piecewise-holomorphic representative, predicting recombination for those homology classes and leading to stable, non-BPS strings. We also compute the central charges of non-BPS strings in F-theory via a near-horizon $AdS_3$ limit in 6d which, upon compactification on a circle, account for the asymptotic entropy of extremal non-supersymmetric 5d black holes (i.e., the asymptotic count of non-holomorphic minimal 2-cycles).

hep-th↗

Stability of the tangent bundle through conifold transitions

Let $X$ be a compact, Kähler, Calabi-Yau threefold and suppose $X\mapsto \underline{X}\leadsto X_t$ , for $t\in Δ$, is a conifold transition obtained by contracting finitely many disjoint $(-1,-1)$ curves in $X$ and then smoothing the resulting ordinary double point singularities. We show that, for $|t|\ll 1$ sufficiently small, the tangent bundle $T^{1,0}X_{t}$ admits a Hermitian-Yang-Mills metric $H_t$ with respect to the conformally balanced metrics constructed by Fu-Li-Yau. Furthermore, we describe the behavior of $H_t$ near the vanishing cycles of $X_t$ as $t\rightarrow 0$.

math.DG↗

The Positive Mass Theorem with Arbitrary Ends

We prove a Riemannian positive mass theorem for manifolds with a single asymptotically flat end, but otherwise arbitrary other ends, which can be incomplete and contain negative scalar curvature. The incompleteness and negativity is compensated for by large positive scalar curvature on an annulus, in a quantitative fashion. In the complete noncompact case with nonnegative scalar curvature, we have no extra assumption and hence prove a long-standing conjecture of Schoen and Yau.

math.DG↗