Related papers: Unique factorization in perturbative QFT
In this paper, we provide the notions of connection $1$-forms and curvature $2$-forms on graphs. We prove a Weitzenb\"ock formula for connection Laplacians in this setting. We also define a discrete Yang-Mills functional and study its…
In this paper general abelian gauge field theories interacting with matter fields are quantized on a closed and orientable Riemann surface $\Sigma$. The approach used is that of small perturbations around topologically nontrivial classical…
Although in general there is no meaningful concept of factorization in fields, that in free associative algebras (over a commutative field) can be extended to their respective free field (universal field of fractions) on the level of…
We show that every nontrivial finite or infinite connected directed graph with loops and at least one vertex without a loop is uniquely representable as a Cartesian or weak Cartesian product of prime graphs. For finite graphs the…
Feynman diagrams are the foremost tool in the perturbative study of quantum field theory. In gauge theories, the full potential of this tool is revealed when it is combined with the Slavanov-Taylor identities associated with the local gauge…
In this talk we discuss mathematical structures associated to Feynman graphs. Feynman graphs are the backbone of calculations in perturbative quantum field theory. The mathematical structures -- apart from being of interest in their own…
The well-known Loday-Quillen-Tsygan theorem calculates the Lie algebra homology of the infinite general linear Lie algebra $\mathfrak{gl}(A)$ over an unital associative algebra $A$. We generalize the Loday-Quillen-Tsygan theorem to an…
We relate graph complexes, Calabi-Yau $A_\infty$-categories and Kontsevich's cocycle construction. Our main result produces a commutative square of shifted Poisson algebras; one of its edges is the Loday-Quillen-Tsygan map, generalized to…
The Connes-Kreimer Hopf algebra of rooted trees, its dual, and the Foissy Hopf algebra of of planar rooted trees are related to each other and to the well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair of…
We argue that for a large class of N=1 supersymmetric gauge theories the effective superpotential as a function of the glueball chiral superfield is exactly given by a summation of planar diagrams of the same gauge theory. This perturbative…
A general diffeomorphism invariant SU(2) gauge theory is a gravity theory with two propagating polarizations of the graviton. We develop this description of gravity, in particular for future applications to the perturbative quantization.…
Pion scattering amplitudes were recently found to vanish on specific kinematic loci, and to factorise close to these loci into a product of two lower-point amplitudes of an extended theory. We propose a diagrammatic representation of pion…
We consider the structure of renormalizable quantum field theories from the viewpoint of their underlying Hopf algebra structure. We review how to use this Hopf algebra and the ensuing Hochschild cohomology to derive non-perturbative…
We prove an algebraic version of the Gauge-Invariant Uniqueness Theorem, a result which gives information about the injectivity of certain homomorphisms between ${\mathbb Z}$-graded algebras. As our main application of this theorem, we…
Application of intersection theory to construction of n-point finite-difference equations associated with classical integrable systems is discussed. As an example, we present a few exact discretizations of one-dimensional cubic and quintic…
Truncated Fourier, Gauss, Kummer and exponential sums can be used to factorize numbers: for a factor these sums equal unity in absolute value, whereas they nearly vanish for any other number. We show how this factorization algorithm can…
A covariant path-integral quantization is proposed for the non-Lagrangian gauge theory described by the Donaldson-Uhlenbeck-Yau equation. The corresponding partition function is shown to admit a nice path-integral representation in terms of…
We review work on `decomposition,' a property of two-dimensional theories with 1-form symmetries and, more generally, d-dimensional theories with (d-1)-form symmetries. Decomposition is the observation that such quantum field theories are…
We argue that one can factorize the difference equation of hypergeometric type on the nonuniform lattices in general case. It is shown that in the most cases of q-linear spectrum of the eigenvalues this directly leads to the dynamical…
In recent years enormous progress has been made in perturbative quantum field theory by applying methods of algebraic geometry to parametric Feynman integrals for scalar theories. The transition to gauge theories is complicated not only by…