Related papers: Copositive geometry of Feynman integrals

The moduli space of Frobenius manifolds carries a natural involutive symmetry, and a distinguished class - so-called modular Frobenius manifolds - lie at the fixed points of this symmetry. In this paper a classification of semi-simple…

We study the sunset Feynman integral defined as the scalar two-point self-energy at two-loop order in a two dimensional space-time. We firstly compute the Feynman integral, for arbitrary internal masses, in terms of the regulator of a class…

A geometrical approach to the calculation of N-point Feynman diagrams is reviewed. It is shown that the geometrical splitting yields useful connections between Feynman integrals with different momenta and masses. It is demonstrated how…

An alternative, geometrical proof of a known theorem concerning the decomposition of positive maps of the matrix algebra $M_{2}(\mathbb{C})$ has been presented. The premise of the proof is the identification of positive maps with operators…

In a previous paper [M.~Hanada, H.~Kawai and Y.~Kimura, Prog. Theor. Phys. 114 (2005), 1295] it is shown that a covariant derivative on any n-dimensional Riemannian manifold can be expressed in terms of a set of n matrices, and a new…

This talk reviews recent developments in the field of analytical Feynman integral calculations. The central theme is the geometry associated to a given Feynman integral. In the simplest case this is a complex curve of genus zero (aka the…

We review quantization of gauge fields using algebraic properties of 3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian gauge theory quantized in a covariant gauge from scalar integrands for connected 3-regular…

Given a closed, convex cone $K\subseteq \mathbb{R}^n$, a multivariate polynomial $f\in\mathbb{C}[\mathbf{z}]$ is called $K$-stable if the imaginary parts of its roots are not contained in the relative interior of $K$. If $K$ is the…

We introduce the notion of one-sided mapping cones of positive linear maps between matrix algebras. These are convex cones of maps that are invariant under compositions by completely positive maps from either the left or right side. The…

We consider the following question: How much of the combinatorial structure determining properties of $\overline{\mathcal{M}_{0, n}}$ is ``intrinsic'' and how much new information do we obtain from using properties specific to this space?…

In this paper we introduce a notion of Feynman geometry on which quantum field theories could be properly defined. A strong Feynman geometry is a geometry when the vector space of $A_\infty$ structures is finite dimensional. A weak Feynman…

This expository text is about using toric geometry and mirror symmetry for evaluating Feynman integrals. We show that the maximal cut of a Feynman integral is a GKZ hypergeometric series. We explain how this allows to determine the minimal…

We elucidate the vector space (twisted relative cohomology) that is Poincar\'e dual to the vector space of Feynman integrals (twisted cohomology) in general spacetime dimension. The pairing between these spaces - an algebraic invariant…

We introduce a type of graph integrals which are holomorphic analogs of configuration space integrals. We prove their (ultraviolet) finiteness by considering a compactification of the moduli space of graphs with metrics, and study their…

This is an introduction to some of the analytic (or integrable systems) aspects of quantum cohomology which have attracted much attention during the last few years. The small quantum cohomology algebra, regarded as an example of a Frobenius…

Many combinatorial properties of a point set in the plane are determined by the set of possible partitions of the point set by a line. Their essential combinatorial properties are well captured by the axioms of oriented matroids. In fact,…

In [1], Connes presented axioms governing noncommutative geometry. He went on to claim that when specialised to the commutative case, these axioms recover spin or spin^c geometry depending on whether the geometry is ''real'' or not. We…

The interaction between combinatorics and algebraic and differential geometry is very strong. While researching a problem of Hessian topology, we came across a series of identities of binomial coefficients, which are useful for proving a…

In the Symmetries of Feynman Integrals (SFI) approach, a diagram's parameter space is foliated by orbits of a Lie group associated with the diagram. SFI is related to the important methods of Integrations By Parts and of Differential…

In this paper we study the affine geometric structure of the graph of a polynomial $f \in \mathbb{R} [x,y]$. We provide certain criteria to determine when the parabolic curve is compact and when the unbounded component of its complement is…