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An Huang

Publications and source records attributed to An Huang.

At least 37 records · Page 2Linked to original sources
arXiv2020

Bounds on the Ricci curvature and solutions to the Einstein equations for weighted graphs

This is a preliminary study of the equation of motion of Euclidean classical gravity on a graph, based on the Lin-Lu-Yau Ricci curvature on graphs. We observe that the constant edge weights configuration gives the unique solution on an infinite tree w.r.t. the asymptotically constant boundary condition. We study the minimum and maximum of the action w.r.t. certain boundary conditions, on several types of graphs of interest. We also exhibit a new class of solutions to the equations of motion on the infinite regular tree.

math-ph↗
arXiv2020

From $p$-adic to Archimedean Physics: Renormalization Group Flow and Berkovich Spaces

We introduce the $p$-adic particle-in-a-box as a free particle with periodic boundary conditions in the $p$-adic spatial domain. We compute its energy spectrum, and show that the spectrum of the Archimedean particle-in-a-box can be recovered from the $p$-adic spectrum via an Euler product formula. This product formula arises from a flow equation in Berkovich space, which we interpret as a space of theories connected by a kind of renormalization group flow. We propose that Berkovich spaces can be used to relate $p$-adic and Archimedean quantities generally.

hep-th↗
arXiv2019

Existence of Solutions to Mean Field Equations on Graphs

In this paper, we prove two existence results of solutions to mean field equations $$Δu+e^u=ρδ_0$$ and $$Δu=λe^u(e^u-1)+4 π\sum_{j=1}^{M}{δ_{p_j}}$$ on an arbitrary connected finite graph, where $ρ>0$ and $λ>0$ are constants, $M$ is a positive integer, and $p_1,...,p_M$ are arbitrarily chosen vertices on the graph.

math.AP↗
arXiv2018

Differential zeros of period integrals and generalized hypergeometric functions

In this paper, we study the zero loci of local systems of the form $δΠ$, where $Π$ is the period sheaf of the universal family of CY hypersurfaces in a suitable ambient space $X$, and $δ$ is a given differential operator on the space of sections $V^\vee=Γ(X,K_X^{-1})$. Using earlier results of three of the authors and their collaborators, we give several different descriptions of the zero locus of $δΠ$. As applications, we prove that the locus is algebraic and in some cases, non-empty. We also give an explicit way to compute the polynomial defining equations of the locus in some cases. This description gives rise to a natural stratification to the zero locus.

math.AG↗
arXiv2018

Jacobian rings for homogenous vector bundles and applications

In this note, we examine the Jacobian ring description of the Hodge structure of zero loci of vector bundle sections on a class of ambient varieties. We consider a set of cohomological vanishing conditions that imply such a description, and we verify these conditions for some new cases. We also observe that the method can be directly extended to log homogeneous varieties. We apply the Jacobian ring to study the null varieties of period integrals and their derivatives, generalizing a result in [9] for projective spaces. As an additional application, we prove the Hodge conjecture for very generic hypersurfaces in certain generalized flag varieties.

math.AG↗
arXiv2018

Hasse-Witt matrices, unit roots and period integrals

Motivated by the work of Candelas, de la Ossa and Rodriguez-Villegas [6], we study the relations between Hasse-Witt matrices and period integrals of Calabi-Yau hypersurfaces in both toric varieties and partial flag varieties. We prove a conjecture by Vlasenko [23] on higher Hasse-Witt matrices for toric hypersurfaces following Katz's method of local expansion [14, 15]. The higher Hasse-Witt matrices also have close relation with period integrals. The proof gives a way to pass from Katz's congruence relations in terms of expansion coefficients [15] to Dwork's congruence relations [8] about periods.

math.AG↗
arXiv2018

Period integrals of local complete intersections and tautological systems

Tautological systems developed in [6],[7] are Picard-Fuchs type systems to study period integrals of complete intersections in Fano varieties. We generalize tautological systems to local complete intersections, which are zero loci of global sections of vector bundles over Fano varieties. In particular, we obtain similar criterion as [6, 7] about holonomicity and regularity of the system. We also prove solution rank formulas and geometric realizations of solutions following the work of hypersurfaces [5, 4].

math.AG↗
arXiv2017

Holonomic Systems for Period Mappings

Period mappings were introduced in the sixties [G] to study variation of complex structures of families of algebraic varieties. The theory of tautological systems was introduced recently [LSY,LY] to understand period integrals of algebraic manifolds. In this paper, we give an explicit construction of a tautological system for each component of a period mapping.

math.AG↗
arXiv2015

Chain Integral Solutions to Tautological Systems

We give a new geometrical interpretation of the local analytic solutions to a differential system, which we call a tautological system $τ$, arising from the universal family of Calabi-Yau hypersurfaces $Y_a$ in a $G$-variety $X$ of dimension $n$. First, we construct a natural topological correspondence between relative cycles in $H_n(X-Y_a,\cup D-Y_a)$ bounded by the union of $G$-invariant divisors $\cup D$ in $X$ to the solution sheaf of $τ$, in the form of chain integrals. Applying this to a toric variety with torus action, we show that in addition to the period integrals over cycles in $Y_a$, the new chain integrals generate the full solution sheaf of a GKZ system. This extends an earlier result for hypersurfaces in a projective homogeneous variety, whereby the chains are cycles. In light of this result, the mixed Hodge structure of the solution sheaf is now seen as the MHS of $H_n(X-Y_a,\cup D-Y_a)$. In addition, we generalize the result on chain integral solutions to the case of general type hypersurfaces. This chain integral correspondence can also be seen as the Riemann-Hilbert correspondence in one homological degree. Finally, we consider interesting cases in which the chain integral correspondence possibly fails to be bijective.

math.AG↗
arXiv2015

Graph invariants from ideas in physics and number theory

We study free scalar field theory on a graph, which gives rise to a modified version of discrete Green's function on a graph studied in \cite{CY}. We show that this gives rise to a graph invariant, which is closely related to the 2-dim Weisfeiler-Lehman algorithm for graph isomorphism testing. We complement this invariant by another type of graph invariants, coming from viewing graphs as quadratic forms over the integers. We explain that the combination of these two ideas give rise to an interesting approach to the graph isomorphism problem.

math.CO↗
arXiv2014

Period Integrals and the Riemann-Hilbert Correspondence

A tautological system, introduced in [16][17], arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold $X$, equipped with a suitable Lie group action. A geometric formula for the holonomic rank of such a system was conjectured in [4], and was verified for the case of projective homogeneous space under an assumption. In this paper, we prove this conjecture in full generality. By means of the Riemann-Hilbert correspondence and Fourier transforms, we also generalize the rank formula to an arbitrary projective manifold with a group action.

math.AG↗
arXiv2014

On cohomology theory of (di)graphs

To a digraph with a choice of certain integral basis, we construct a CW complex, whose integral singular cohomology is canonically isomorphic to the path cohomology of the digraph as introduced in \cite{GLMY}. The homotopy type of the CW complex turns out to be independent of the choice of basis. After a very brief discussion of functoriality, this construction immediately implies some of the expected but perhaps combinatorially subtle properties of the digraph cohomology and homotopy proved very recently \cite{GLMY2}. Furthermore, one gets a very simple expected formula for the cup product of forms on the digraph. On the other hand, we present an approach of using sheaf theory to reformulate (di)graph cohomologies. The investigation of the path cohomology from this framework, leads to a subtle version of Poincare lemma for digraphs, which follows from the construction of the CW complex.

math.CO↗
arXiv2013

On the Holonomic Rank Problem

A tautological system, introduced in \cite{LSY}\cite{LY}, arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold $X$, equipped with a suitable Lie group action. In this article, we introduce two formulas -- one purely algebraic, the other geometric -- to compute the rank of the solution sheaf of such a system for CY hypersurfaces in a generalized flag variety. The algebraic version gives the local solution space as a Lie algebra homology group, while the geometric one as the middle de Rham cohomology of the complement of a hyperplane section in $X$. We use both formulas to find certain degenerate points for which the rank of the solution sheaf becomes 1. These rank 1 points appear to be good candidates for the so-called large complex structure limits in mirror symmetry. The formulas are also used to prove a conjecture of Hosono, Lian and Yau on the completeness of the extended GKZ system when $X$ is $¶^n$.

math.AG↗
arXiv2010

A Note on Nahm's Conjecture in Rank 2 Case

The aim of this paper is to get a complete list of positive definite symmetric matrices with integer entries $\a&b\b&d\$ such that all complex solutions to the system of equations $1-x_1=x_1^ax_2^b\ 1-x_2=x_1^bx_2^d$ are real. This result is related to Nahm's conjecture in rank 2 case.

math-ph↗
arXiv2010

On Twisted Virasoro Operators and Number Theory

We explore some axioms of divergent series and their relations with conformal field theory. As a consequence we obtain another way of calculating $L(0,χ)$ and $L(-1,χ)$ for $χ$ being a Dirichlet character. We hope this discussion is also of interest to physicists doing renormalization theory for a reason indicated in the Introduction section. We introduce a twist of the oscillator representation of the Virasoro algebra by a group of Dirichlet characters and use this to give a 'physical interpretation' of why the values of certain divergent series should be given by special L values. Furthermore, we use this to show that some fractional powers which are crucial for some infinite products to have peculiar modular transformation properties are expressed explicitly by certain linear combinations of $L(-1, χ)$'s for appropriately chosen $χ$'s, and can be understood physically as a kind of 'vacuum Casimir engergy' in our settings. We also note a relation between field theory and our twisted operators. Lastly we give an attempt to reinterpret Tate's thesis by a sort of conformal field theory on a number field.

math-ph↗
arXiv2010

On S-duality and Gauss reciprocity law

We review the interpretation of Tate's thesis by a sort of conformal field theory on a number field in \cite{1}. Based on this and the existence of a hypothetical 3-dimensional gauge theory, we give a physical interpretation of the Gauss quadratic reciprocity law by a sort of S-duality.

math-ph↗