arXiv subjects

Siqi He

Publications and source records attributed to Siqi He.

34 records · Page 2Linked to original sources

arXiv2023

On the spectral variety for rank two Higgs bundles

In this article, we study the Hitchin morphism over a smooth projective variety $X$. The Hitchin morphism is a map from the moduli space of Higgs bundles to the Hitchin base, which in general not surjective when the dimension of X is greater than one. Chen-Ngô introduced the spectral base, which is a closed subvariety of the Hitchin base. They conjectured that the Hitchin morphism is surjective to the spectral base and also proved that the surjectivity is equivalent to the existence of finite Cohen-Macaulayfications of the spectral varieties. For rank two Higgs bundles over a projective manifold $X$, we explicitly construct a finite Cohen-Macaulayfication of the spectral variety as a double branched covering of $X$, thereby confirming Chen-Ngô's conjecture in this case. Moreover, using this Cohen-Macaulayfication, we can construct the Hitchin section for rank two Higgs bundles, which allows us to study the rigidity problem of the character variety and also to explore a generalization of the Milnor-Wood type inequality.

math.AG↗

arXiv2023

The Algebraic and Analytic Compactifications of the Hitchin Moduli Space

Following the work of Mazzeo-Swoboda-Weiss-Witt and Mochizuki, there is a map $\overlineΞ$ between the algebraic compactification of the Dolbeault moduli space of $\mathsf{SL}(2,\mathbb{C})$ Higgs bundles on a smooth projective curve coming from the $\mathbb{C}^\ast$ action, and the analytic compactification of Hitchin's moduli space of solutions to the $\mathsf{SU}(2)$ self-duality equations on a Riemann surface obtained by adding solutions to the decoupled equations, known as ``limiting configurations''. This map extends the classical Kobayashi-Hitchin correspondence. The main result of this paper is that $\overlineΞ$ fails to be continuous at the boundary over a certain subset of the discriminant locus of the Hitchin fibration. This suggests the possibility of a third, refined compactification which dominates both.

math.DG↗

arXiv2023

A large area, high counting rate micromegas-based neutron detector for BNCT

Beam monitoring and evaluation are very important to boron neutron capture therapy (BNCT), and a variety of detectors have been developed for these applications. However, most of the detectors used in BNCT only have a small detection area, leading to the inconvenience of the full-scale 2-D measurement of the beam. Based on micromegas technology, we designed a neutron detector with large detection area and high counting rate. This detector has a detection area of 288 mm multiples 288 mm and can measure thermal, epithermal, and fast neutrons with different detector settings. The BNCT experiments demonstrated that this detector has a very good 2-D imaging performance for the thermal, epithermal, fast neutron and gamma components, a highest counting rate of 94 kHz/channel, and a good linearity response to the beam power. Additionally, the flux fraction of each component can be calculated based on the measurement results. The Am-Be neutron source experiment indicates that this detector has a spatial resolution of approximately 1.4 mm, meeting the requirements of applications in BNCT. It is evident that this micromegas-based neutron detector with a large area and high counting rate capability has great development prospects in BNCT beam monitoring and evaluation applications.

physics.ins-det↗

arXiv2023

Using Fano factors to determine certain types of gene autoregulation

The expression of one gene might be regulated by its corresponding protein, which is called autoregulation. Although gene regulation is a central topic in biology, autoregulation is much less studied. In general, it is extremely difficult to determine the existence of autoregulation with direct biochemical approaches. Nevertheless, some papers have observed that certain types of autoregulations are linked to noise levels in gene expression. We generalize these results by two propositions on discrete-state continuous-time Markov chains. These two propositions form a simple but robust method to infer the existence of autoregulation in certain scenarios from gene expression data. This method only depends on the Fano factor, namely the ratio of variance and mean of the gene expression level. Compared to other methods for inferring autoregulation, our method only requires non-interventional one-time data, and does not need to estimate parameters. Besides, our method has few restrictions on the model. We apply this method to four groups of experimental data and find some genes that might have autoregulation. Some inferred autoregulations have been verified by experiments or other theoretical works.

q-bio.MN↗

arXiv2022

Inference on autoregulation in gene expression

Some genes can promote or repress their own expressions, which is called autoregulation. Although gene regulation is a central topic in biology, autoregulation is much less studied. In general, it is extremely difficult to determine the existence of autoregulation with direct biochemical approaches. Nevertheless, some papers have observed that certain types of autoregulations are linked to noise levels in gene expression. We generalize these results by two propositions on discrete-state continuous-time Markov chains. These two propositions form a simple but robust method to infer the existence of autoregulation from gene expression data. This method only needs to compare the mean and variance of the gene expression level. Compared to other methods for inferring autoregulation, our method only requires non-interventional one-time data, and does not need to estimate parameters. Besides, our method has few restrictions on the model. We apply this method to four groups of experimental data and find some genes that might have autoregulation. Some inferred autoregulations have been verified by experiments or other theoretical works.

q-bio.MN↗

arXiv2022

The branched deformations of the special Lagrangian submanifolds

The branched deformations of immersed compact special Lagrangian submanifolds are studied in this paper. If there exists a nondegenerate $\mathbb{Z}_2$ harmonic 1-form over a special Lagrangian submanifold $L$, we construct a family of immersed special Lagrangian submanifolds $\tilde{L}_t$, that are diffeomorphic to a branched covering of $L$ and $\tilde{L}_t$ convergence to $2L$ as current. This answers a question suggested by Donaldson. Combining with the work of Abouzaid and Imagi, we obtain constraints for the existence of nondegenerate $\mathbb{Z}_2$ harmonic 1-forms.

math.DG↗

arXiv2022

Existence of nondegenerate $\mathbb{Z}_2$ harmonic 1-forms via $\mathbb{Z}_3$ symmetry

Using $\mathbb{Z}_3$ symmetry, we present a topological condition for the existence of the $\mathbb{Z}_2$ harmonic 1-forms over Riemannian manifold. As a corollary, if $L$ is an oriented link on $S^3$ with determinant zero, then there exists a nondegenerate $\mathbb{Z}_2$ harmonic 1-form over the 3-cyclic branched covering of $L$. Furthermore, we found infinite number of rational homology 3-spheres that admit a nondegenerate $\mathbb{Z}_2$ harmonic 1-form.

math.DG↗

arXiv2021

Hecke modifications of Higgs bundles and the extended Bogomolny equation

We establish a Kobayashi-Hitchin correspondence between solutions of the extended Bogomolny equation with a Dirac type singularity and Hecke modifications of Higgs bundles. This correspondence was conjectured by Witten and plays an important role in the physical description of the the geometric Langlands program in terms of S-duality for N=4 super Yang-Mills theory in four dimensions.

math.DG↗

arXiv2020

On the Moduli Space of the Octonionic Nahm's Equations

In this paper, we study some basic properties of the octonionic Nahm's equations over $[0,1]$. We prove that the moduli space of the smooth solutions to the octonionic Nahm's equations over $[0,1]$ is a star-shaped smooth manifold with a complete metric. In addition, for any commuting triples of the cotangent bundle of a complex Lie group, we construct solutions to the octonionic Nahm's equations. Moreover, we introduce extra symmetry and study a decoupled version of the octonionic Nahm's equations over $[0,1]$. We prove a Kempf-Ness theorem for the meromorphic solutions to the decoupled octonionic Nahm's equations.

math.DG↗

arXiv2019

Opers and the twisted Bogomolny equations

In this paper, we study the dimensionally reduced twisted Kapustin-Witten equations on the product of a compact Riemann surface $Σ$ with $\mathbb{R}^+$. The main result is a Kobayashi-Hitchin type correspondence between the space of tilted Nahm pole solutions and the moduli space of Beilinson-Drinfeld opers. This corroborates a prediction of Gaiotto and Witten \cite[p.971]{gaiotto2012knot}.

math.DG↗

arXiv2019

Classification of Nahm pole solutions of the Kapustin-Witten equations on $S^1\times Σ\times \mathbb{R}^+$

In this note, we classify all solutions to the $\mathrm{SU(n)}$ Kapustin-Witten equations on $S^1\timesΣ\times \mathbb{R}^+$, where $Σ$ is a compact Riemann surface, with Nahm pole singularity at $S^1\timesΣ\times \{0\}$. We provide a similar classification of solutions with generalized Nahm pole singularities along a simple divisor (a "knot") in $S^1\timesΣ\times \{0\}$.

math.DG↗

arXiv2018

The Expansions of the Nahm Pole Solutions to the Kapustin-Witten Equations

For a 3-manifold $X$ and compact simple Lie group $G$, we study the expansions of polyhomogeneous Nahm pole solutions to the Kapustin-Witten equations over $X\times (0,+\infty)$. Let $y$ be the coordinate of $(0,+\infty)$, we prove that the sub-leading terms of a polyhomogeneous Nahm pole solution is smooth to the boundary when $y\to 0$ if and only if $X$ is an Einstein 3-manifold.

math.DG↗

arXiv2018

The extended Bogomolny equations with generalized Nahm pole boundary conditions, II

We develop a Kobayashi-Hitchin correspondence for the extended Bogomolny equations, i.e., the dimensionally reduced Kapustin-Witten equations, on the product of a compact Riemann surface $Σ$ with ${\mathbb R}^+_y$, with generalized Nahm pole boundary conditions at $y=0$. The correspondence is between solutions of these equations satisfying these singular boundary conditions and also limiting to flat connections as $y \to \infty$, and certain holomorphic data consisting of effective triplets $(\mathcal E, φ, L)$ where $(\mathcal E, φ)$ is a stable $\mathrm{SL}(n+1,\mathbb C)$ Higgs pair and $L \subset \mathcal E$ is a holomorphic line bundle. This corroborates a prediction of Gaiotto and Witten, and is an extension of our earlier paper \cite{HeMazzeo2017} which treats only the $\mathrm{SL}(2,\mathbb R)$ case.

math.DG↗

arXiv2017

The Extended Bogomolny Equations and Generalized Nahm Pole Boundary Condition

In this paper we develop a Kobayashi-Hitchin type correspondence between solutions of the extended Bogomolny equations on $Σ\times \RP$ with Nahm pole singularity at $Σ\times \{0\}$ and the Hitchin component of the stable $SL(2,\mathbb{R})$ Higgs bundle; this verifies a conjecture of Gaiotto and Witten. We also develop a partial Kobayashi-Hitchin correspondence for solutions with a knot singularity in this program, corresponding to the non-Hitchin components in the moduli space of stable $SL(2,\mathbb{R})$ Higgs bundles. We also prove existence and uniqueness of solutions with knot singularities on $\mathbb{C}\ti\RP$.

math.DG↗

arXiv2017

A Gluing Theorem for the Kapustin-Witten Equations with a Nahm Pole

In the present paper, we establish a gluing construction for the Nahm pole solutions to the Kapustin-Witten equations over manifolds with boundaries and cylindrical ends. Given two Nahm pole solutions with some convergence assumptions on the cylindrical ends, we prove that there exists an obstruction class for gluing the two solutions together along the cylindrical end. In addition, we establish a local Kuranishi model for this gluing picture. As an application, we show that over any compact four-manifold with $S^3$ or $T^3$ boundary, there exists a Nahm pole solution to the obstruction perturbed Kapustin-Witten equations. This is also the case for a four-manifold with hyperbolic boundary under some topological assumptions.

math.DG↗

arXiv2016

Rotationally Invariant Singular Solutions to the Kapustin-Witten Equations

In the present paper, we find a system of non-linear ODEs that gives rotationally invariant solutions to the Kapustin-Witten equations in 4-dimensional Euclidean space. We explicitly solve these ODEs in some special cases and find decaying rational solutions, which provide solutions to the Kapustin-Witten equations. The imaginary parts of the solutions are singular. By rescaling, we find some limit behavior for these singular solutions. In addition, for any integer $k$, we can construct a 5$|k|$ dimensional family of $C^1$ solutions to the Kapustin-Witten equations on Euclidean space, again with singular imaginary parts. Moreover, we find solutions to the Kapustin-Witten equations over $S^3\times (0,+\infty)$ with the Nahm pole boundary condition.

math.DG↗