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We give a description of the moment graph for Bott-Samelson varieties in arbitrary Lie type. We use this, along with curve neighborhoods and explicit moduli space computations, to compute a presentation for the small quantum cohomology ring of a particular Bott-Samelson variety in Type A. / Ph. D. / Since the early 1990’s, the study of quantum cohomology has been a fascinating, and fruitful field of research with connections to physics, representation theory, and combinatorics. The quantum cohomology of a space X encodes enumerative information about how many curves intersect certain subspaces of X; these counts are called Gromov-Witten invariants. For some spaces X, including the class of spaces we consider here, this count is only ”virtual” and negative Gromov-Witten invariants may arise. In this dissertation, we study the quantum cohomology of Bott-Samelson varieties. These spaces arise frequently in applications to representation theory and combinatorics, however their quantum cohomology was previously unexplored. The first of our three main theorems describes the moment graph for Bott-Samelson varieties. This is a description of what all the possible curves, stable under certain symmetries, exist in a Bott-Samelson variety. Our second main theorem is a technical result which enables us to compute some GromovWitten invariants directly. Finally, our third main theorem is a description of the quantum cohomology for a certain three-dimensional Bott-Samelson variety.

The odd symplectic Grassmannian IG := IG(k, 2n + 1) parametrizes k dimensional subspaces of C^2n+1 which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on IG with two orbits, and IG is itself a smooth Schubert variety in the submaximal isotropic Grassmannian IG(k, 2n + 2). We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of IG, i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case k = 2, and it gives an algorithm to calculate any quantum multiplication in the equivariant quantum cohomology ring. / Ph. D. / The thesis studies a problem in the general area of Combinatorial Algebraic Geometry. The goal of Algebraic Geometry is to study solutions to systems to polynomial equations. Such systems are ubiquitous in scientific research. We study a problem in enumerative geometry on a space called the odd symplectic Grassmannian. The problem seeks to find the number of curves which are incident to certain subspaces of the given Grassmannian. Due to subtle geometric considerations, the count is sometimes virtual, meaning that some curves need to be counted negatively. The rigorous context of such questions is that of Gromov-Witten theory, a subject with roots in physics. Our space affords a large number of symmetries, and the given counting problems translate into significant amount of combinatorial manipulations. The main result in the dissertation is a combinatorial algorithm to perform the virtual curve counting in the odd-symplectic Grassmannian.

This thesis studies N=(2,2) gauged linear sigma models (GLSMs) and three-dimensional N=2 Chern-Simons-matter theories and their mathematical applications. After a brief review of GLSMs, we systematically study nonabelian GLSMs for symplectic and orthogonal Grassmannians, following up a proposal in the math community. As consistency checks, we have compared global symmetries, Witten indices, and Calabi-Yau conditions to geometric expectations. We also compute their nonabelian mirrors following the recently developed nonabelian mirror symmetry. In addition, for symplectic Grassmannians, we use the effective twisted superpotential on the Coulomb branch of the GLSM to calculate the ordinary and equivariant quantum cohomology of the space, matching results in the math literature. Then we discuss 3d gauge theories with Chern-Simons terms. We propose a complementary method to derive the quantum K-theory relations of projective spaces and Grassmannians from the corresponding 3d gauge theory with a suitable choice of the Chern-Simons levels. In the derivation, we compare to standard presentations in terms of Schubert cycles, and also propose a new description in terms of shifted Wilson lines, which can be generalized to symplectic Grassmannians. Using this method, we are able to obtain quantum K-theory relations, which match known math results, as well as make predictions. / Doctor of Philosophy / In this thesis, we study two specific models of supersymmetric gauge theories, namely two-dimensional N=(2,2) gauged linear sigma models (GLSMs) and three-dimensional N=2 Chern-Simons-matter theories. These models have played an important role in quantum field theory and string theory for decades, and generated many fruitful results, improving our understanding of Nature by drawing on many branches in mathematics, such as complex differential geometry, intersection theory, quantum cohomology/quantum K-theory, enumerative geometry, and many others. Specifically, this thesis is devoted to studying their applications in quantum cohomology and quantum K-theory. In the first part of this thesis, we systematically study two-dimensional GLSMs for symplectic and orthogonal Grassmannians, generalizing the study for ordinary Grassmannians. By analyzing the Coulomb vacua structure of the GLSMs for symplectic Grassmannians, we are able to obtain the ordinary and equivariant quantum cohomology for these spaces. A similar methodology applies to 3d Chern-Simons-matter theories and quantum K-theory, for which we propose a new description in terms of shifted Wilson lines.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19259号 / 理博第4114号 / 新制||理||1592(附属図書館) / 32261 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 入谷 寛, 教授 加藤 毅, 教授 吉川 謙一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20885号 / 理博第4337号 / 新制||理||1623(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 望月 拓郎, 教授 中島 啓, 教授 小野 薫 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

Toric varieties are a class of geometric objects with a combinatorial structure encoded in polytopes. P1 x P2 is a well known variety and its polytope is the triangular prism. Studying the symmetries of the triangular prism and its truncations can lead to symmetries of the variety. Many of these symmetries permute the elements of the cohomology ring nontrivially and induce nontrivial relations. We discuss some toric symmetries of P1 x P2, and describe the geometry of the polytope of the corresponding blowups, and analyze the induced action on the cohomology ring. We exhaustively compute the toric symmetries of P1 x P2.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第25088号 / 理博第4995号 / 新制||理||1713(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 入谷 寛, 教授 塚本 真輝, 教授 吉川 謙一 / 学位規則第4条第1項該当 / Doctor of Agricultural Science / Kyoto University / DGAM

We use toric symmetry and blowups to study relationships in the Gromov-Witten theories of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$. These two spaces are birationally equivalent via the common blowup space, the permutohedral variety. We prove an equivalence of certain invariants on blowups at only points of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$ by showing that these invariants descend from the blowup. Further, the permutohedral variety has nontrivial automorphisms of its cohomology coming from toric symmetry. These symmetries can be forced to descend to the blowups at just points of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$. Enumerative consequences are discussed.

Les grassmanniennes symplectiques impaires sont une famille d'espaces quasi-homogènes très proches des grassmanniennes symplectiques de par leur construction et leurs propriétés. Dans ce travail, j'étudie leur cohomologie classique et quantique. Pour les grassmanniennes symplectiques impaires de droites, j'obtiens une règle de Pieri quantique ainsi qu'une présentation de l'anneau de cohomologie quantique. J'en déduis la semi-simplicité de cet anneau et je détermine une collection exceptionnelle complète pour la catégorie dérivée, ce qui me permet de vérifier pour cet exemple une conjecture de Dubrovin. Dans le cas général, je démontre un principe quantique-classique pour certains invariants de Gromov-Witten de degré un. Sous réserve de l'énumérativité des invariants de degré supérieur, je prouve que la règle de Pieri quantique est entièrement déterminée par le calcul des invariants de degré un. / Odd symplectic Grassmannians are a family of quasi-homogeneous spaces that are closely related to symplectic Grassmannians by their construction and properties. The goal of this work is to study their classical and quantum cohomology. For odd symplectic Grassmannians of lines, I obtain a quantum Pieri rule and a presentation of the quantum cohomology ring. I prove the semisimplicity of this ring and determine a full exceptional collection for the derived category, which enables me to check a conjecture of Dubrovin in this example. In the general case, I prove a quantum-to-classical principle for some degree one Gromov-Witten invariants. Assuming higher-dimensional Gromov-Witten invariants are enumerative, I conclude that the quantum Pieri rule is entirely determined by the knowledge of degree one invariants.