Women at the Intersection of Mathematics and High Energy Physics

6 Mar 2017, 08:00 → 10 Mar 2017, 19:00 Europe/Berlin

02.430 (Mainz Institute for Theoretical Physics, Johannes Gutenberg University)

In this meeting, we plan to bring together mathematical physicists who are specialized in mathematical aspects of quantum field theory, quantum gravity, and string theory. The workshop will focus on four intertwined pillars:

1. Locality and observables,
2. Symmetry and duality,
3. String compactifications,
4. Numbers and singularities.

Besides providing an opportunity to exchange knowledge at the frontier between theoretical high energy physics and mathematics, this Topical Workshop will bring together a still underrepresented minority of excellent women, with the aim to foster equal opportunities by networking and improving their visibility within the scientific community.

Executive Summary (PDF) Women2.pdf

Group Photo Group Photo

Participants (PDF) TN.pdf

Poster (high resolution)

Monday 6 March

1 Welcome

2 Physical predictions through string compactification: challenges and hopes

String theory remains to date our best framework to construct a quantum theory that unifies gravity and the other three fundamental forces of Nature. Although it has emerged as a fantastic arena in which mathematics and physics ideas constantly cross-fertilize, its credibility relies on the power of its testable predictions. In this talk, I will review progress made since the defining work of Candelas, Horowitz, Strominger and Witten in 1985 on Calabi-Yau string compactification, in the quest to predict the mass of quarks and leptons, which mathematically requires the explicit form of the Ricci flat metric on non-trivial Calabi-Yau manifolds.

Speaker: Anne Taormina (Durham)

11:00 Coffee Break

3 Moduli space of heterotic string compactifications

I review the geometry of heterotic string compactifications leading to supersymmetric gauge theories in 4 and 3 dimensions. The data of these compactifications are specified by a quadruple (Y, V, TY, H) where X is a 6 or 7 dimensional manifold with a G-sturcture (a certain SU(3) structure in 6 dimensions or an integrable G2 structure in 7), V is a vector bundle over X with a Yang Mills connection which satisfies instanton constraints, TY is the tangent bundle over Y with an instanton connection, and H is a three form on Y defined in terms of the B-field and the Chern-Simons forms for the connections on V and TY (the so called anomaly condition). We recast all the constraints on the geometry of these compactifications in terms of an extension bundle Q over Y which admits a differential which squares to zero. We show that the tangent space of the moduli space is then given in terms of the first cohomology group with values in Q. Time permitting, we discuss the fact that all our results can be reproduced from a superpotential. We find a Kahler metric on the moduli space which is a natural inner product on the moduli space, with a Kahler potential taking a remarkably simple form, and as in type II special geometry, it is quasi-topological.

Speaker: Xenia de la Ossa

13:00 Lunch Break

4 String compactifications on string-size tori from double field theory

Compactifications on manifolds whose size is of the order of the string length reveal purely stringy phenomena such as very light modes corresponding to strings winding around the space. As a consequence, the usual low-energy description of string compactifications given by field theory breaks down. In this talk I will review the basic features of compactifications on tori of string size, and we will see how one can get a low-energy effective action from the so-called double field theory.

Speaker: Mariana Grana

Slides Grana-6March2017.pdf

5 Generalised Geometry in String Theory

Speaker: Michela Petrini

Slides Petrini_6March2017.pdf

17:30 Reception

Tuesday 7 March

6 Calabi-Yaus and supersymmetric gauge theory

The mathematics of Calabi-Yau spaces plays a crucial role in the context of string compactifications. In this talk I discuss the gauged linear sigma model (GLSM), which is a supersymmetric gauge theory in two dimensions that encodes information about Calabi-Yaus and their moduli spaces. After a basic introduction to GLSMs, I will discuss new methods for computing quantum corrections in string compactifications using supersymmetric localization. I will focus in particular on the hemisphere partition function, which computes the quantum corrected central charge of D-branes on Calabi-Yaus.

Speaker: Johanna Knapp

notes Knapp_7March2017.pdf

11:00 Coffee Break

7 Singularities in F-theory compactifications, algebraic geometry and topology

I will discuss various aspects of the mathematics and the physics, especially of F-theory compactifications, on Calabi-Yau threefolds with singularities.

Speaker: Antonella Grassi

12:30 Lunch Break

8 Exact results in N=2 Super Yang-Mills Theories

We study the non-perturbative behaviour of superconformal gauge theories with rigid N=2 supersymmetry in four dimensions, in particular N=2* theories, and discuss the relation between their S-duality properties and the possibility of computing exact quantum observables. For these theories in fact, the prepotential function, that encodes the low-energy effective dynamics on the Coulomb branch of moduli space, and the chiral correlators obey a modular anomaly equation whose validity is related to S-duality. This fact allow one to write them in terms of (quasi)-modular forms, thus resumming all instanton contributions. The results can be checked against the microscopic multi-instanton calculus in the case of classical algebras, but are valid also for the exceptional algebras, where direct computations are not available. We also comment on the extension of these techniques to configuration of 4-dimensional N=2 gauge theories in presence of 2-dimensional defects.

Speaker: Marialuisa Frau

Slides Frau_7March2017.pdf

9 Moduli of heterotic G2 compactifications

I will discuss the moduli of heterotic compactifications on seven-dimensional manifolds with G2 structure with instanton bundles. I will also discuss how such compactifications can be used for model building in string theory.

Speaker: Magdalena Larfors

Slides Larfors_7March2017.pdf

Wednesday 8 March

10 Hopf algebra gauge theories on embedded graphs

We explain how the concept of a lattice gauge theory with values in a group can be generalised to a gauge theory with values in a Hopf algebra on a graph embedded into a surface. We give an axiomatic description of Hopf algebra gauge theories and show that they include the quantum algebra of observables obtained by the combinatorial quantisation of Chern-Simons theory as an example. We relate Hopf algebra gauge theories to lattice models from condensed matter physics. More specifically, we show that Kitaev's lattice model for a finite-dimensional semisimple Hopf algebra H is equivalent to a Hopf algebra gauge theory for its Drinfeld double D(H).

Speaker: Catherine Meusburger

10:30 Coffee Break

11 Locality of observables in quantum field theory

The aim of quantum field theory (QFT) is to unify quantum theory with the principles of relativity. The problem of a consistent mathematical description, beyond the level of formal perturbation theory, is still open to date, in particular for many physically interesting models describing interaction among relativistic particles. One mathematical framework for the description of QFT on Minkowski space is based on the Wightman axioms, which deal with the notion of quantum fields and with unbounded operators; another framework is the abstract Haag-Kastler setting, where algebras of bounded operators associated with space-time regions are the fundamental objects. This lecture gives an introduction to these two frameworks, and presents an example of such a construction in a class of interacting quantum field theories on 1+1-dimensional Minkowski space, the so called quantum integrable models.

Speaker: Daniela Cadamuro

notes Cadamuro-8March2017.pdf

12:30 Lunch Break

12 Quantum Einstein Equations of Loop Quantum Gravity

In this talk we give a brief review on the conceptual and mathematical framework underlying loop quantum gravity with a focus on its dynamics encoded in the so called quantum Einstein equations. We will discuss how the notion of Dirac observables in the context of general relativity can be used to derive a reduced phase space quantization for loop quantum gravity. Furthermore, we explain how the quantum Einstein equations can be formulated in this context and discuss some recent progress and results.

Speaker: Kristina Giesel

13 Worldsheet string theory in AdS/CFT: perturbation theory and beyond

String sigma-models relevant in the AdS/CFT correspondence are highly non-trivial two-dimensional field theories, for which predictions at finite coupling assume integrability and/or the correspondence itself. After having discussed general features of the perturbative approach, I will present progress on how to extract finite coupling information via the use of lattice field theory methods.

Speaker: Valentina Forini

Slides Forini_8March2017.pdf

Public Lecture: Wie Gravitationswellen unser Bild vom Universum erweitern

Speaker: Dr. Anda Degeratu

Convener: Anda Degeratu

slides Degeratu-Mainz.pdf

Thursday 9 March

14 Numbers and patterns in Feynman graphs

I will set the stage for and explain what periods of Feynman graphs are and why they should be interesting both to mathematicians and physicists. From there I will focus on some graph theoretic tools for understanding the resulting number and their symmetries.

Speaker: Karen Yeats

notes Yeats_9March2017.pdf

10:30 Coffee Break

15 Scattering amplitudes and their singularities

I will discuss the physical context in which amplitude calculations are required and describe some techniques in active development. I focus particularly on singularities and discontinuities as key tools for exploration and computation. I will describe their physical interpretations and ways to embed them in algebraic frameworks which can aid in constructing the full amplitudes.

Speaker: Ruth Britto

12:30 Lunch Break

16 From de Jonquières' counts to cohomological field theory

Enumerative geometry is an old subject with roots in the 19th century whose aim is to count the number of geometric objects of a certain type that satisfy given conditions. Advances in both mathematics, and unexpectedly, mathematical physics have led to the resolution of many of its conjectures and have highlighted new deep connections between mathematics and string theory. In this talk I will describe a classical enumerative problem, namely de Jonquières' count of certain prescribed hyperplane tangency conditions to a smooth curve embedded in projective space. I will then attempt to explain how this problem relates to certain ergodic dynamical systems and ultimately cohomological field theories.

Speaker: Mara Ungureanu

Slides Ungureanu_9March2017.pdf

17 An elliptic generalisation of polylogarithms for the sunrise and the kite integral

Feynman integrals are one of the most important tools of pertubation theory for high precision calculations in particle physics. Due to the presence of ultraviolet or infrared divergences these integrals may require regularisation where the dimensional regularisation is commonly used (the regularisation parameter ε denotes the deviation from the number of space-time dimensions). The result for a Feynman integral is then presented as a Laurent series in ε. An interesting question to ask is which kind of functions appears in the contributions of different ε-order. In the ε^0-term of one-loop integrals the logarithm and the dilogarithm occur. Many multi-loop integrals can be expressed in terms of generalisations of the logarithm and the dilogarithm, the so-called multiple polylogarithms. But there are some Feynman integrals which cannot be expressed within this class of functions of which the sunrise integral is the simplest one. In this talk, we show how the multiple polylogarithms can be generalised to express ε-terms of the Laurent expansion of the sunrise integral (for arbitrary masses) around two and four space-time dimensions. For the two-dimensional equal mass case we will also explain an algorithm to compute an arbitrary ε-order of the Laurent expansion. Recently, it has been shown that also the equal mass kite integral around four space-time dimensions with two massless and three massive propagators can be expressed in terms of these generalisations.

Speaker: Luise Adams

Friday 10 March

18 Motives from graphs

In this talk, I give a different graphical representatioin (unrelated to Feynman diagrams) for the numbers that arise as amplitude calculations in QFTs, i.e. mixed Tate motives. The graphs in this talk are chosen to better illuminate the symmetries underlying the mixed Tate motives themselves, with the eventual goal of understanding their structure.

Speaker: Susama Agarwala

11:00 Coffee Break

19 Continuous and Discrete Gauge Symmetries in F-Theory

We present recent developments in F-theory compactifications and focus on advances in constructions of globally consistent F-theory compactifications with Abelian and discrete gauge symmetries, emphasizing technical advances and insights into higher-rank gauge symmetries. We also present recent studies of the origin of Abelian and discrete symmetries in Heterotic/F-theory duality.

Speaker: Mirjam Cvetic

Slides Cvetic_10March2017.pdf