本论文利用超对称场论半球配分函数 Z_B 研究了 N= (2, 2) 非交换规范线性西格玛模型（除特别指出，我们省略非交换并下称 GLSM）之 B 类超对称边界条件（下称 B-膜）在弦化凯勒模空间（Stringy Kahler moduli, 下称 SKM）中的输运问题。低能场论之 SKM 结构可由 GLSM 中的 Fayet-Illiopoulos 参数空间得到，其上奇点来自于对形变作用量的一圈量子修正。B-膜的膜中心荷可由 Z_B 计算，其是 SKM上的广义超几何函数。因此，SKM 中不同大尺度极限间的 B-膜输运可与 Z_B 的解析延拓相联系。运输的关键在于函数收敛性中称为窗口（window）的约束，其使得仅受分次约束定则（grade restriction rule）限制的 B-膜满足延拓条件。猜想每一个满足这些条件的 B-膜集合均构成凝聚层导出范畴的生成元。交换 GLSM 中的窗口是已知的，而非交换 GLSM 的则目前没有一般的结论。总而言之，我们根据其配分函数的行为分别讨论了无反常 GLSM（也称满足卡拉比-丘条件）和反常 GLSM中的窗口和分次约束问题。我们以一类 U(1) × U(k) 为规范群、称为 PAX 模型的无反常 GLSM 为例，其可实现作为行列式簇的 CY。当 k = 2，这类 CY 由 Gullinksen-Negard 首先发现，其镜对象未知。我们构建了其上 B-膜范畴生成元并精确计算 Z_B，给出了镜映射。我们据 Z_B 的性质推得其窗口是格拉斯曼簇导出范畴上的 Kapranov 生成元。其次，我们用窗口平移法计算了 SKM 中的单值化矩阵。这类矩阵由在 SKM 奇点上消失膜的球状旋转（spherical twist）给出，包含了 B-膜范畴对称性的信息。最后，这类单值化矩阵在此 GLSM 中由奇点消解的链群所支配，我们给出了这类链群并由此验证了单值化矩阵满足 SKM 基本群的结构。此结论推广了对一般消失膜的猜想。反常 GLSM 中的窗口分为大、小窗口两部分，后者中的 B-膜将构成希格斯分支上的导出范畴生成元。通过数值验证小窗口中的生成元在特定的积分路径下收敛且避开了 Z_B 积分的鞍点，我们研究并求解了各类 U(2)GLSM 中的小窗口，其给出了格拉斯曼簇导出范畴上的 Kuznetsov 子生成元。其次，我们利用这些小窗口研究了一类二参数反常 GLSM，其在大尺度极限下实现为法诺簇，但在其他相中为 CY 希格斯分支和库伦分支的混合相。利用小窗口及分次约束定则，我们研究了这些 CY 混合分支之间的 B-膜输运并推广了对反常 GLSM 中导出等价的研究。
This thesis studied the B-brane transport in stringy Kahler moduli (SKM) of N =(2, 2) non-abelian gauged linear sigma model (GLSM) using hemisphere partition function Z_B. Discriminant in the SKM in infrared theories is conducted by the 1-loop corrected Fayet-Illiopoulos-parameter term in GLSM. The central charge of B-brane is exactly computed by the hemisphere partition function Z_B, which is the generalized hypergeometric function on SKM. Thus, the transportation of B-branes between different large volume limit points is closely related to the analytic continuation of their Z_B. The key is the window from the convergent condition of Z_B, in where only those grade restricted B-branes can be transported by continuation. It is conjectured that each window contains an exceptional collection for the derived category of coherent sheaves. The window in abelian GLSM has been analyzed carefully, however, that in non-abelian GLSM is unsolved generally. Nevertheless, we discussed window and grade restrction rule in anomalous and non-anomalous GLSM using Z_B. We explored the PAX model in U(1) × U(k) GLSM which realizes determinantal CY. The Gullinksen-Negard CY in k = 2 has unknown mirror. We constructed the generating set for B-brane category and computed the Z_B that gives the mirror map. By the convergence of Z_B, the window is computed as Kapranov’s collection for the derived category of grassmannian varieites. Next, we computed the monodromy matrices in SKM using window shift proposal. This matrix as the auto-equivalence of derived category is given by the spherical twist of massless brane on discriminant, which depends on the link group of resolution of discriminant singularity. We tested this group and generalized the conjecture of other massless brane. Windows in anomalous GLSM are devided by big and small, the later one constitutes the derived category of Higgs branch. We numerically tested the small window in various U(2) GLSM to comfirm that generators in it are convergence and saddle-point free. They constitute the subset of Kuznetsov’s collection of grassmannian varieties. Next, a family of 2-parameter GLSM which realizes Fano varieties in large volum but CY mixed branch in other phases is studied using these small windows. By the grade restricion rule between CY mixed branches we studied B-brane transport and generalized the study of derived equivalence among anomalous GLSM.
