Related papers: Landau Singularities and Higher-Order Roots
We show that direct Feynman-parametric loop integration is possible for a large class of planar multi-loop integrals. Much of this follows from the existence of manifestly dual-conformal Feynman-parametric representations of planar loop…
The existence or not of Landau poles is one of the oldest open questions in non-asymptotic quantum field theories. We investigate the Landau pole issue in two condensed matter systems whose long-wavelength physics is described by…
Gurau argued in [arXiv:1006.0714] that the gluing spaces arising as Feynman diagrams of three-dimensional group field theory are not all pseudo-manifolds. I dispute this conclusion: albeit not properly triangulated, these spaces are genuine…
In this paper, we study the singularities of Feynman integrals using homological techniques. We analyse the Feynman integrals by compactifying the integration domain as well as the ambient space by embedding them in higher-dimensional…
Feynman integrals are very often computed from their differential equations. It is not uncommon that the $\varepsilon$-factorised differential equation contains only dlog-forms with algebraic arguments, where the algebraic part is given by…
We present a new link between the Invariant Theory of infinitesimal singular Riemannian foliations and Jordan algebras. This, together with an inhomogeneous version of Weyl's First Fundamental Theorems, provides a characterization of the…
In the computation of Feynman integrals which evaluate to multiple polylogarithms one encounters quite often square roots. To express the Feynman integral in terms of multiple polylogarithms, one seeks a transformation of variables, which…
We define the rigidity of a Feynman integral to be the smallest dimension over which it is non-polylogarithmic. We argue that massless Feynman integrals in four dimensions have a rigidity bounded by 2(L-1) at L loops, and we show that this…
Parametric representations of Feynman integrals have a key property: many, frequently all, of the Landau singularities appear as endpoint divergences. This leads to a geometric interpretation of the singularities as faces of Newton…
Consider a polynomial $f$ defined over a field $k$, the multiplicity is perhaps the most naive measurement of the singularities of $f$. This paper describes the first steps toward understanding a much more subtle measure of singularities…
We apply a recently suggested new strategy to solve differential equations for Feynman integrals. We develop this method further by analyzing asymptotic expansions of the integrals. We argue that this allows the systematic application of…
We consider F/M/Type IIA theory compactified to four, three, or two dimensions on a Calabi-Yau four-fold, and study the behavior near an isolated singularity in the presence of appropriate fluxes and branes. We analyze the vacuum and…
We describe a family of finite, four-dimensional, $L$-loop Feynman integrals that involve weight-$(L+1)$ hyperlogarithms integrated over $(L-1)$-dimensional elliptically fibered varieties we conjecture to be Calabi-Yau. At three loops, we…
Closed geodesics associated with indefinite binary quadratic forms, or equivalently with real quadratic irrationals, have long been studied as geometric $\mathrm{SL}_2(\mathbb{Z})$-invariants. Building on the Birman-Williams approach to…
Single scale Feynman integrals in quantum field theories obey difference or differential equations with respect to their discrete parameter $N$ or continuous parameter $x$. The analysis of these equations reveals to which order they…
We consider Lorentzian correlators of local operators. In perturbation theory, singularities occur when we can draw a position-space Landau diagram with null lines. In theories with gravity duals, we can also draw Landau diagrams in the…
We study the related questions: (i) when Feynman amplitudes in massless $\phi^4$ theory evaluate to multiple zeta values, and (ii) when their underlying motives are mixed Tate. More generally, by considering configurations of singular…
The integrand of any multi-loop integral is characterised after Feynman parametrisation by two polynomials. In this review we summarise the properties of these polynomials. Topics covered in this article include among others: Spanning trees…
We propose that Feynman integral reduction is controlled by solutions of the Landau equations. We study integral relations with prescribed propagator powers using syzygy methods and discuss how syzygies can be expressed as a sum over…
Let X be a normal variety such that $K_X$ is Q-Cartier, and let $f: X \rightarrow X$ be a finite surjective morphism of degree at least two. We establish a close relation between the irreducible components of the locus of singularities that…