Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - HIGGSBNDL (Higgs bundles: Supersymmetric Gauge Theories and Geometry)

Teaser

String theory is first and foremost a theory that combines quantum mechanics with the theory of gravity into one unified theoretical framework. However its relevance surpasses far beyond the literal interpretation as a unified theory describing the standard model of particle...

Summary

String theory is first and foremost a theory that combines quantum mechanics with the theory of gravity into one unified theoretical framework. However its relevance surpasses far beyond the literal interpretation as a unified theory describing the standard model of particle physics or the cosmology underlying our universe, and reaches deep into our understanding of the fundamentals of quantum field theories, as well as mathematics. A remarkable connection that string theory induces is that of quantum field theories and geometric structures.
There are various points of contact how this connection can arise, but all of them in one way or another follow from studying a type of Hitchin system, whose solutions are so-called Higgs bundles. This connection is central to the research undertaken by the team of this ERC Consolidator grant.

One fundamental question for every quantum field theory is whether it has a consistent extension beyond its original realm of definition, such as an embedding into a consistent theory of quantum, such as string theory. One could argue that only theories that have an embedding into a theory of quantum gravity should be considered consistent theories.
In fact this approach furnishes a pathway to classify methodically consistent quantum field theories, e.g. conformally (i.e. scale-) invariant theories, using string theoretic methods.
The relation to geometric structures arises by the classic paradigm that string theories or extended objects within them, so-called branes, can be dimensionally reduced on compact spaces to lower-dimensional theories, whose properties crucially depend on the geometric data of the compatification space. In particular, the more specialized the geometry the more rigid the structure of the quantum field theory. Or in reverse, the more unconstrained the quantum field theory, the more complicated (or unconstrained) the geometries. The latter is in particular the case when the theories have minimal supersymmetry -- a symmetry, between bosonic and fermionic states of the theory. The more supersymmetries a theory has, the more constrained it is. One of the challenges in modern applications of string theory is thus to construct and classify quantum theories (usually with conformal invariance) with the minimal amount of supersymmetry -- usually denoted by N=1 supersymmetry.

The ERC project started off with two main objectives: understanding the geometry-quantum field theory relation for four dimensional N=1 supersymmetric theories, as well as theories in two and three dimensions with small amount or minimal supersymmetry.
As most string theoretic constructions require some amount of supersymmetry, this class of theories is the least constrained one that still allow some control when studying in a string theoretic context.
String theory or M-theory offers a multitude of pathways to study such theories. The first step is to find a embedding, or a realization, of theories with minimal supersymmetry in a string theoretic setting. This generically maps the problem of characterizing or even classifying such theories to one in geometry. Thus one of the key interdisciplinary points of contact of this ERC project has been that with experts on geometry, where there has been an intense exchange of ideas and research progress.

String theory usually does not only provide one description, but a multitude of different approaches to study one given quantum field theory.
One of the most powerful dualities is the holographic duality for conformally invariant quantum field theories: the so-called AdS/CFT correspondence gives an alternative description of a conformal field theory in a regime that is usually not accessible with standard quantum field theory methods, in terms of a theory of gravity or strings, in a particular spacetime geometry, which has an anti-de Sitter factor (which can be visualized in terms of a hyperboloid). The conformal field theory is thought to be realized at the

Work performed

The geometric setup that was studied in the first part of the ERC Consolidator grant project is the compactification of M-theory (a non-perturbative extension of string theory to eleven dimensions) on a seven-dimensional manifold, resulting in a four dimensional theory with N=1 supersymmetry. To retain supersymmetry in four dimensions, the seven-manifolds have to have a special property, namely, parallel transport on the tangent spaces to the manifolds cannot be a generic seven-dimensional rotation, but has to lie in a subgroup of these, the group G2. Manifolds of this type are called G2 holonomy manifolds.
Such geometries are known to be notoriously difficult to understand, and they provide a challenge to geometers.
In particular compact manifolds with G2 holonomy have until recently been very scarce. In the last few years there has been some mathematical progress in the construction of large classes of G2 holonomy manifolds, so-called twisted connected sums (TCS). From a 4d N=1 quantum field theory point of view, these geometries have one key short-coming: M-theory on TCS manifolds lacks one key property, namely chiral matter -- an essentially ingredient in both phenomenological applications but also conceptually a drastic shortcoming.
The goal of the first part of the ERC project was to pinpoint how this shortcoming could be remedied -- albeit not in a full compact G2 manifold setting, but in a subsector of the theory, which determines the quantum field theory part of the compactification. The key tool in this is the approach using Higgs bundles, which provide a precise correspondence between quantum field theory data and local geometric properties of the G2 manifold.
In a series of papers, the PI Schafer-Nameki together with other team members Braun, Cizel, Hubner and other collaborators, developed the foundations for the understanding of M-theory on TCS G2-manifolds, and most importantly, provided a Higgs bundle description of these geometries. In particular this allowed the team to pinpoint how the Higgs bundles for TCS manifolds have to be modified in order to include chiral matter in four dimensions. This exciting result has mathematical and physical implications which are the subject of ongoing research.

The second pillar of the ERC proposal is the study of quantum field theories from compactifications of branes, more precisely six dimensional M5-branes -- these are fundamental membranes in eleven-dimensional M-theory. Compactifying these a on d-dimensional manifold Md, with d<6, results (under certain constraints) in a supersymmetric quantum field theory in 6-d dimensions, whose properties are interlinked with those of the original M5-brane theory as well as the geometry and topology of Md. The PI and collaborators studied compactifications on four-manifolds, as well as three-manifolds to two and three dimensions, respectively. The two dimensional theories resulting from this analysis have the key property of being conformally invariant, and one of the exciting new outcomes of the research is that such theories have a description as quantum field theories, whose coupling constant has a spacetime dependence. Conformally invariant theories furthermore often have a strong coupling description in terms of a holographic dual, i.e. a description as a gravitational theory in anti-de Sitter (AdS) spacetime. Schafer-Nameki and collaborators proposed a new arena for holography or the AdS/CFT correspondence, for such theories with spatially dependent coupling. This was a surprising and very unexpected direction that the research led the team into and goes beyond the initially proposed research agenda.

In the idea set of compactification of M5-branes, the PI and two students, Eckhard and Wong, developed a new class of theories, which arise from M5-branes on three-dimensional manifolds M3, resulting in three dimensional N=1 supersymmetric theories. These are particularly interesting theories, as minimal supersymmetry in

Final results

The research of the ERC-project team has led to progress on three fronts:

the team, in particular the PI Schafer-Nameki and postdoc Braun, studied the physics of M-theory on TCS G2-manifolds. This work pushed the mathematical results into the direction of geometries that have some of the essential physical properties when utilized to compactify M-theory to four dimensions. This work culminated in a general description of the gauge theory sector of M-theory on G2-holonomy manifolds in terms of Higgs bundles. The central result of this work is a Higgs bundle description of the TCS geometries, including a precise description of the types of deformations that are necessary to include some of the so-far missing, but essential features (such as chiral matter) in the four dimensional physics. A challenging goal is to next develop the mathematics to incorporate such deformations in the actual compact G2 geometries.

With regards to the M5-brane compactifications, the strong expectation -- with many results already being completed -- is much more refined understanding of the correspondence for 3d N=1 theories, in particular sharpening the computational aspects as well as conceptual underpinning of this new construction of 3d N=1 theories and the relation to the geometries (again these are closely related to G2-holonomy spaces).
This work furthermore, in an unexpected turn of events, extended the realm of the holographic AdS/CFT correspondence to theories with spacetime dependent coupling, and added holography as a third pillar for the project.

One big open question that in particular the new postdocs in the team together with Schafer-Nameki are planning to focus their attention to is developing a classification of supersymmetric conformal theories using G2-holonomy manifolds as we well as Calabi-Yau manifolds (another class of reduced holonomy manifolds), in four and five dimensions, respectively. A concrete classification program in five dimensions has been already initiated with some of the team members and will appear shortly. This is a concerted effort drawing on the geometric (mostly algebraic geometry in this case) as well as field-theoretic expertise of the team members. A challenge thus far remains a classification of four-dimensional minimally supersymmetric conformal field theories. This can either be achieved using M-theory on G2-manifolds, which have scale invariance implemented, or in terms of what is called F-theory. Much exciting progress along this direction is expected in the next 30 months of the project.

Website & more info

More info: https://people.maths.ox.ac.uk/schafernamek/.