Principal Investigator: Andrew Neitzke
This research is based on current developments at the interface between physics and geometry. In recent work with Davide Gaiotto and Greg Moore, the principal investigator has used techniques of supersymmetric gauge theory to attack the problem of counting stable geometric objects in Calabi-Yau threefolds. As an application they showed that the desired counts, known as "generalized Donaldson-Thomas invariants," are actually imprinted into hyperkahler metrics on certain auxiliary moduli spaces. This connection illuminates previously mysterious aspects of the counting problem; in particular it gives a new geometric understanding of the "wall-crossing formula" which governs how these invariants jump, i.e. how the relevant geometric objects can split and join. At the same time, it gives a totally new way of looking at the hyperkahler metrics in question. The proposed research builds on this recent work in various directions. Much of the program is part of a continuing collaboration between the PI, Davide Gaiotto and Greg Moore. First, they will apply their new construction to get more explicit information than was previously available about complete hyperkahler metrics, with the ultimate goal being a new description of the Ricci-flat metric on a K3 surface. Second, they will explore extensions of their construction to encompass metrics on moduli spaces of Higgs bundles associated to groups other than SU(2). Third, they will use the gauge theory perspective to study wall-crossing properties of conjectural new invariants which extend Donaldson-Thomas. In collaboration with Sergio Cecotti and Cumrun Vafa, the PI will also look for new restrictions on the Donaldson-Thomas invariants coming from their gauge-theoretic interpretation.
The crowning achievement of fundamental physics over the last century was the development of "quantum field theory", the toolkit which physicists use to describe the behavior of subatomic particles. Many of the methods of quantum field theory look radically different from the usual methods of mathematicians. Nevertheless it has been gradually appreciated that many of these ideas do have applications to problems of "pure" mathematics: for example, questions about geometry can sometimes be rephrased as questions about subatomic physics! In particular, recently it was discovered (by the PI together with collaborators Greg Moore and Davide Gaiotto) that by studying the behavior of certain four-dimensional and three-dimensional quantum systems at very low energies, one can get detailed information about the geometry of certain spaces ("hyperkahler spaces") which have been intensely studied by mathematicians in recent years. This appears to be the beginning of a much richer story: by using deeper properties of the quantum systems, the PI aims to get deeper information about the corresponding geometry. There are numerous applications to related areas of mathematics, including the "geometric Langlands program" which aims to create a bridge between geometry and number theory.
The work funded by this grant has significantly advanced the state of the art in our understanding of supersymmetric quantum field theory (a branch of high energy physics, the part of physics that deals with subatomic particles) and related geometric problems. In joint work with collaborators Davide Gaiotto and Greg Moore, the PI has developed new methods for answering questions such as: "If we know the fundamental laws of physics, how do we compute the number of stable particles and their masses and charges?" They discovered that in many cases this question can be answered by reducing it to a different-sounding, purely mathematical question, involving counting certain networks of paths on a 2-dimensional surface (like a sphere or a donut). This question is similar to ones which have been studied by mathematicians for years, but extends them -- in the past, mathematicians focused primarily on networks consisting just of a single path, but now it turns out that this was just the beginning of a much richer story. In turn, the problem of counting these networks can be solved using new techniques in algebraic geometry, introduced a few years ago by the mathematicians Maxim Kontsevich and Yan Soibelman. Other results funded by this grant include work on a new way of solving the famous "Einstein field equation," which describes the curvature of space-time in Einstein's theory of general relativity. This new approach had been introduced earlier by Gaiotto, Moore and the PI. This part of the work was conducted mainly by graduate student Cesar Garza, whose training and research was supported in part by funds from the grant. Garza has made progress on a particularly thorny bit of the problem, concerning the question of singularities -- places where the space-time becomes infinitely spiky or bent instead of smooth. Garza has shown in many examples that, if we construct a solution of the Einstein equation using Gaiotto, Moore and the PI's new method, then singularities in fact do not occur: the space-time is actually smooth. This is a very important point since for many applications (both in physics and mathematics) singularities cause serious complications.
