This thesis is devoted to a thorough study of chiral rings in two-dimensional (0,2) theories. We first discuss properties of chiral operators in general two-dimensional (0,2) nonlinear sigma models, both in theories twistable to the A/2 or B/2 model, as well as in non-twistable theories. As a special case, we study the quantum sheaf cohomology of Grassmannians as a deformation of the usual quantum cohomology. The deformation corresponds to a (0,2) deformation of the nonabelian gauged linear sigma model whose geometric phase is associated with the Grassmannian. Combined with the classical result, the quantum ring structure is derived from the one-loop effective potential. Supersymmetric localization is also applicable in this case, which proves to be efficient in computing A/2 correlation functions. We then compute chiral operators in general (0,2) nonlinear sigma models, and apply them to the Gadde-Gukov-Putrov triality proposal, which says that certain triples of (0,2) GLSMs should RG flow to nontrivial IR fixed points. As another application, we extend previous works to construct (0,2) Toda-like mirrors to the sigma model engineering Grassmannians. / Ph. D. / This thesis studies a mathematical concept called the chiral ring, which emerges from string theory. String theory is a conjectured theory that potentially unifies the existing fundamental physical laws. It has connections with many branches of mathematics, especially geometry. Spacetime is ten-dimensional in string theory, of which four dimensions are visible, and the other six are hidden at ordinary energy levels. The chiral ring encodes many geometric properties of the hidden part of spacetime. These properties can in turn affect the visible universe even at low energies. Research on chiral rings has primarily focused on a special class of geometries which have large symmetries and so are easier to handle. In order to tackle more general scenarios, we analyze the chiral rings corresponding to theories with only half the symmetry and give several new results and applications.

A composition algebra is a non-zero algebra endowed with a strictly non-degenerate, multiplicative quadratic form. Finite-dimensional composition algebras exist only in dimension 1, 2, 4 and 8 and are in general not associative or unital. Over the real numbers, such algebras are division algebras if and only if they are absolute valued, i.e. equipped with a multiplicative norm. The problem of classifying all absolute valued algebras and, more generally, all composition algebras of finite dimension remains unsolved. In dimension eight, this is related to the triality phenomenon. We approach this problem using a categorical language and tools from representation theory and the theory of algebraic groups. We begin by considering the category of absolute valued algebras of dimension at most four. In Paper I we determine the morphisms of this category completely, and describe their irreducibility and behaviour under the actions of the automorphism groups of the algebras. We then consider the category of eight-dimensional absolute valued algebras, for which we provide a description in Paper II in terms of a group action involving triality. Then we establish general criteria for subcategories of group action groupoids to be full, and applying this to the present setting, we obtain hitherto unstudied subcategories determined by reflections. The reflection approach is further systematized in Paper III, where we obtain a coproduct decomposition of the category of finite-dimensional absolute valued algebras into blocks, for several of which the classification problem does not involve triality. We study these in detail, reducing the problem to that of certain group actions, which we express geometrically. In Paper IV, we use representation theory of Lie algebras to completely classify all finite-dimensional absolute valued algebras having a non-abelian derivation algebra. Introducing the notion of quasi-descriptions, we reduce the problem to the study of actions of rotation groups on products of spheres. We conclude by considering composition algebras over arbitrary fields of characteristic not two in Paper V. We establish an equivalence of categories between the category of eight-dimensional composition algebras with a given quadratic form and a groupoid arising from a group action on certain pairs of outer automorphisms of affine group schemes