Simplifying Multiple Sums in Difference Fields

In this survey article we present difference field algorithms for symbolic summation. Special emphasize is put on new aspects in how the summation problems are rephrased in terms of difference fields, how the problems are solved there, and how the derived results in the given difference field can be reinterpreted as solutions of the input problem. The algorithms are illustrated with the Mathematica package \SigmaP\ by discovering and proving new harmonic number identities extending those from (Paule and Schneider, 2003). In addition, the newly developed package \texttt{EvaluateMultiSums} is introduced that combines the presented tools. In this way, large scale summation problems for the evaluation of Feynman diagrams in QCD (Quantum ChromoDynamics) can be solved completely automatically.Comment: Uses svmult.cls, to appear as contribution in the book "Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions" (www.Springer.com

Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories

The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these short-comings. Its so-called 'embarrassing parallelizability' allows us to solve many problems and extract physical information which elude the symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.Comment: 36 page

Trading Order for Degree in Creative Telescoping

We analyze the differential equations produced by the method of creative telescoping applied to a hyperexponential term in two variables. We show that equations of low order have high degree, and that higher order equations have lower degree. More precisely, we derive degree bounding formulas which allow to estimate the degree of the output equations from creative telescoping as a function of the order. As an application, we show how the knowledge of these formulas can be used to improve, at least in principle, the performance of creative telescoping implementations, and we deduce bounds on the asymptotic complexity of creative telescoping for hyperexponential terms

A Periodic Table of Effective Field Theories

We systematically explore the space of scalar effective field theories (EFTs) consistent with a Lorentz invariant and local S-matrix. To do so we define an EFT classification based on four parameters characterizing 1) the number of derivatives per interaction, 2) the soft properties of amplitudes, 3) the leading valency of the interactions, and 4) the spacetime dimension. Carving out the allowed space of EFTs, we prove that exceptional EFTs like the non-linear sigma model, Dirac-Born-Infeld theory, and the special Galileon lie precisely on the boundary of allowed theory space. Using on-shell momentum shifts and recursion relations, we prove that EFTs with arbitrarily soft behavior are forbidden and EFTs with leading valency much greater than the spacetime dimension cannot have enhanced soft behavior. We then enumerate all single scalar EFTs in d < 6 and verify that they correspond to known theories in the literature. Our results suggest that the exceptional theories are the natural EFT analogs of gauge theory and gravity because they are one-parameter theories whose interactions are strictly dictated by properties of the S-matrix

Geodesics on Calabi-Yau manifolds and winding states in nonlinear sigma models

We conjecture that a non-flat $D$-real-dimensional compact Calabi-Yau manifold, such as a quintic hypersurface with D=6, or a K3 manifold with D=4, has locally length minimizing closed geodesics, and that the number of these with length less than L grows asymptotically as L^{D}. We also outline the physical arguments behind this conjecture, which involve the claim that all states in a nonlinear sigma model can be identified as "momentum" and "winding" states in the large volume limit.Comment: minor corrections, 43 pages, to appear in frontiers in mathematical physics. Frontiers in Physics, Dec 16, 201

Chirality Change in String Theory

It is known that string theory compactifications leading to low energy effective theories with different chiral matter content ({\it e.g.} different numbers of standard model generations) are connected through phase transitions, described by non-trivial quantum fixed point theories. We point out that such compactifications are also connected on a purely classical level, through transitions that can be described using standard effective field theory. We illustrate this with examples, including some in which the transition proceeds entirely through supersymmetric configurations.Comment: 50 pages, 2 figure

Real Enumerative Questions in Complex and Tropical Geometry

The workshop Real Enumerative Questions in Complex and Tropical Geometry was devoted to a wide discussion and exchange of ideas between the best experts representing various points of view on the subject. Enumeration of real curves largely motivated the development of the tropical geometry and led to the discovery of new interesting geometric phenomena and deep links between this problematic and algebraic geometry, symplectic geometry, topology, and mathematical physics