Related papers: On Laporta's 4-loop sunrise formulae

This mostly expository paper centers on recently proved conjectures in two areas: A) A conjecture of A. Oppenheim on the values of real indefinite quadratic forms at integral points. B) Conjectures of Dani, Raghunathan, and Margulis on…

We study $\lambda$-hypersurfaces that are critical points of a Gaussian weighted area functional $\int_{\Sigma} e^{-\frac{|x|^2}{4}}dA$ for compact variations that preserve weighted volume. First, we prove various gap and rigidity theorems…

In this paper, we use Sakai's geometric framework to explore the profound interconnection between recurrence coefficients of the semiclassical Laguerre weight $w(x)=x^{\lambda}\mathrm{e}^{-x^2+sx}$, $x\in\mathbb{R}^+$, $\lambda>-1$,…

In this paper, we derive an interior Schauder estimate for the divergence form elliptic equation \begin{equation*} D_i(a(x)D_iu)=D_if_i \end{equation*} in $\mathbb{R}^2$, where $a(x)$ and $f_i(x)$ are piecewise H\"older continuous in a…

Let $l\geq 6$ be any integer, where $l\equiv 2$ mod $4$. Suppose that $\mu(\tau)d\tau$ is a measure with bounded variation and is supported on a compact subset of the complex plane, where…

We use the conformal invariance and the holographic correspondence to fully specify the dependence of entanglement entropy on the extrinsic geometry of the 2d surface $\Sigma$ that separates two subsystems of quantum strongly coupled…

Let $\Lambda = \mathrm{SL}_2(\Bbb Z)$ be the modular group and let $c_n(\Lambda)$ be the number of congruence subgroups of $\Lambda$ of index at most $n$. We prove that $\lim\limits_{n\to \infty} \frac{\log c_n(\Lambda)}{(\log n)^2/\log\log…

Comparison of three different regularization methods of calculating the one-loop effective Heisenberg-Euler Lagrangian of quantum electro-dynamics (QED) is employed to derive some interesting integrals involving the asymptotic expansion of…

In our companion papers, we managed to derive a connection formulation of Lorentzian General Relativity in D+1 dimensions with compact gauge group SO(D+1) such that the connection is Poisson commuting, which implies that Loop Quantum…

A non-perturbative and background-independent quantum formulation of quadratic gravity is provided. A canonical quantization procedure introduced in previous works, named after Dirac and Pauli, is here applied to quadratic gravity to…

We clarify the relationship between different multiple polylogarithms in weight~4 by writing suitable linear combinations of a given type of iterated integral I_{n_1,...,n_d}(z_1,...,z_d), in depth d>1 and weight \sum_i n_i=4 in terms of…

In the path integral formulation of the reduced phase space Loop Quantum Gravity (LQG), we propose a new approach to allow the spatial cubic lattice (graph) to change dynamically in the physical time evolution. The equations of motion of…

We introduce a new class of symmetric space orbital integrals important for applications in certain relative trace formulas appearing in the theory of automorphic representations. We verify a fundamental lemma for $U_2\times…

The spinor-helicity formalism has proven to be very efficient in the calculation of scattering amplitudes in quantum field theory, while the loop tree duality (LTD) representation of multi-loop integrals exhibits appealing and interesting…

Let $N$ be a closed spin manifold with positive scalar curvature and $D_N$ the Dirac operator on $N$. Let $M_1$ and $M_2$ be two Galois covers of $N$ such that $M_2$ is a quotient of $M_1$. Then the quotient map from $M_1$ to $M_2$…

We construct several towers of scalar quantum field theories with an $O(N)$ symmetry which have higher derivative kinetic terms. The Lagrangians in each tower are connected by lying in the same universality class at the $d$-dimensional…

We develop an action formulation of stochastic dynamics in the Hilbert space. By generalizing the Wiener process into 1+3-dimensional spacetime, we define a Lorentz-invariant random field. By coupling the random to quantum fields, we obtain…

We employ new calculational technique and present complete list of classical $r$-matrices for $D=4$ complex homogeneous orthogonal Lie algebra $\mathfrak{o}(4;\mathbb{C})$, the rotational symmetry of four-dimensional complex space-time.…

We give an expression for number of points for the family of Dwork K3 surfaces $$X_{\lambda}^4: \hspace{.1in} x_1^4+x_2^4+x_3^4+x_4^4=4\lambda x_1x_2x_3x_4$$ over finite fields of order $q\equiv 1\pmod 4$ in terms of Greene's finite field…

The system of 4 differential equations in the external invariant satisfied by the 4 master integrals of the general massive 2-loop sunrise self-mass diagram is solved by the Runge-Kutta method in the complex plane. The method, whose…