This is a list of some of the questions that I hope to incorporate into my future research program. The (0,2) world is full of many fascinating puzzles, as well as rich full-fledged research directions, and these are just a few of the ones I find particularly exciting, or where I think progress can be made.

- Rigid (2,2) superconformal field
theories

Exact CFT constructions, using for example the minimal models, can yield many (2,2) SCFTs in two dimensions that have no marginal deformations. However, if we insist on having U(1) charge integrality for the left and right R-currents, the situation is very different. In this case we get an extended (2,2) superconformal algebra, and the central charges must be multiples of 3. These are of course exactly the CFTs relevant for supersymmetric string compactifications, and in all known examples they possess marginal deformations.**Do there exist any rigid extended (2,2) SCFTs? If not, why not? Can constraints from crossing or modular invariance shed light on this?**

- Smooth abelian gauged linear sigma
models

A large class of (2,2) and (0,2) superconformal theories can be constructed as IR fixed point flows from abelian gauge theories. For instance, in the (2,2) setting one can reproduce the largest known list of Calabi-Yau sigma models, those with a geometry a hypersurface in a toric variety (there is also an extension for every complete intersection in a toric variety; it is not known whether that yields a finite list). In this case we know the criterion that a gauged linear sigma model must satisfy to have a phase corresponding to a smooth geometry: the criterion, due to Batyrev, states that the combinatorial data that specifies the representations of the gauge group and the superpotential is determined by a reflexive polytope. There is a (0,2) generalization of these theories that should yield an enormous class of (0,2) geometries that includes holomorphic vector bundles over Calabi-Yau manifolds. However, even specializing to that last class, we do not know of a combinatorial smoothness criterion, and that means we really have no idea of how large a class of theories can be obtained!**What are the combinatorial constraints on the data of a (0,2) gauged linear sigma model that guarantee a smooth geometry?**

- Classification of (0,2) Landau-Ginzburg
theories

A simpler but still very rich class of RG flows with non-trivial fixed points is obtained from Landau-Ginzburg theories. In the UV these are Lagrangian field theories with superpotential interactions. The latter are specified by a quasi-homogeneous ideal in some number of complex variables. In order for the theory to have a well-defined normalizable vacuum, this ideal must be zero-dimensional. In the case of (2,2) LG theories, there exists a beautiful constructive classification of all superpotentials at fixed IR central charge.*Can we obtain a similar classification of (0,2) LG theories? What is the structure of RG flows that can be obtained between these models by considering relevant deformations? How do resulting fixed points fit with those known from gauged lin**ear sigma models?*

- Toric Calabi-Yau (0,2) theories

Toric Calabi-Yau manifolds have been a great source of examples and results for topological string theory. A similar rich structure is to be found for (0,2) theories with such target spaces, but these theories have been little explored.**What are the framing conditions on bundle data in order to have a well-defined local (0,2) geometry? What is the moduli space of the resulting theories? Can we generalize (2,2) mirror symmetry results in this setting?**

- Admissible heterotic geometries

Already in the context of (2,2) theories and Calabi-Yau geometries, we know that string theory leads to a generalization of many geometric concepts. For instance, the walls of the Kaehler cone of the target-space geometry may be crossed without any singularity in the conformal field theory. (0,2) theories have a much richer structure of deformations, and we also expect heterotic stringy geometry to yield many surprises.**What kinds of degenerations of classical geometric data correspond to smooth conformal field theories? Can we describe the conformal field theory discriminant locus? How do we construct local models of corresponding singular geometries? Are there also useful local models for genuine conformal field theory singularities?**

- Supersymmetric regularization of
conformal perturbation theory (I am pursuing this in
collaboration with Ronen Plesser)

Conformal perturbation theory is an important conceptual tool in the study of quantum field theory. In two dimensions it is particularly powerful due to the strong constraints of (super)conformal symmetry. To turn CPT into a quantitative tool and to explore its conceptual consequences further, it is important to develop techniques to regularize the expansion while preserving supersymmetry. This is particularly challenging in the case of chiral theories such as typical (0,2) theories.**Can we develop appropriate supersymmetric regularization techniques by using the geometry of super-Riemann surfaces and insights from string theory?**

- M-theory/heterotic string duality
(I am thinking about this together with Ruben Minasian and
Sav Sethi)

M-theory is an appealing corner of the web of string dualities since its low energy limit has the elegant description of 11-dimensional supergravity with bosonic fields simply the metric and the 3-form gauge field: hence construction of backgrounds is quite geometric. However, the classes of singularities that are allowed in M-theory are far from understood. A particularly rich class of compactifications exploits the geometry of K3xK3 together with a choice of four-form G-flux. These are known, in a number of cases to correspond to various compactifications of heterotic string theory to 3 dimensions.**Can we describe the heterotic duals precisely and determine a map of moduli for theories with 8 and 4 supercharges? Are there non-geometric heterotic backgrounds in this class of theories? What can we learn about admissible singular geometries in M-theory by exploiting the duality?**

General constraints on (0,2) SCFTs

Although they preserve a very small amount of supersymmetry, we know many key properties of these theories. For instance, there is a good understanding, based on representation theory of the superconformal algebra and unitarity, of the structure of first order marginal deformations, as well as possible obstructions. However, much remains to be learned, and, in particular, we expect that applying further constraints from crossing and modular invariance, we might find new a priori constraints on (0,2) theories.**Is it possible to constrain the allowed Kac-Moody symmetries and their representations? Are there further restrictions on the moduli beyond the Kaehler structure?**

(0,2) RG flows and bundle stability

The one-loop conditions for Weyl invariance of a (0,2) non-linear sigma model are well-known, and they include the requirement that the holomorphic bundle be equipped with a Hermitian-Yang-Mills connection. Using the results of Donaldson-Uhlenbeck-Yau, this can be translated into a notion of stability of the bundle. However, it is a notoriously difficult problem to construct stable bundles on a given geometry. A related problem is that given a gauged linear sigma model for some (0,2) geometry there are no effective methods to check for bundle stability.**What is the end point of RG flow given an unstable bundle? How is this flow related to the heat flow used by Donaldson and its non-linear modification that appears in Uhlenbeck-Yau?**

- Theories with a higher-dimensional
origin

In recent years, a number of techniques have been used to obtain new (0,2) theories. These include dimensional reduction of four-dimensional N=1 gauge theories on magnetized backgrounds, the 6-dimensional (2,0) theory compactified on various four-manifolds, as well as theories constructed via the AdS3/CFT2 correspondence.**What classes of theories can be obtained by such higher-dimensional constructions? Are there overlaps between known classes of (0,2) models? Having made appropriate identification of the two-dimensional theory, can we use our knowledge of its dynamics to constrain the higher-dimensional avatars?**