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標題: | 箭袋理論的穩定性條件與過牆現象 Stability Conditions of Quiver Algebras and Wall-Crossing Phenomena |
作者: | Chien-Hsun Wang 王建勛 |
指導教授: | 賀培銘 |
關鍵字: | 箭袋,穩定性條件,導出範疇,卡拉比-丘,過牆現象, quiver,stability condition,derived category,Calabi-Yau,wall-crossing, |
出版年 : | 2018 |
學位: | 博士 |
摘要: | 在這篇論文,我們研究箭袋理論的穩定性條件以及卡拉比-丘流行的量子不變量的過牆現象。在第一部分,我們研究卡拉比-丘範疇的仿射型 A 的箭袋理論的穩定性條件。三角範疇的穩定性條件是由 Bridgeland 所提出,它是由中心荷與一組相位固定的子範疇購成,並且滿足一系列的公設。穩定性條件所形成的空間是複流形。我們考慮在球面上的亞純平方微分形式,它在原點和無窮遠點各別有冪次(N-2)p+2 和(N-2)(n-p)+2 的極點,並且在球面上有 n 個冪次 N-2 的相異零點。將兩個極點取實可定向拉開,得到在內邊與外邊有(N-2)p 和(N-2)(n-p)個標點的環形面。平方微分形式在環形面上連結標點軌跡給出環形面的多邊形化。我們可由此多邊形化構造出相對應的箭袋理論。
當改變此亞純平方微分形式,以至於穿越模空間的牆時,相對應的環形面多邊形化將突變成新的多邊形化。Torkildsen 證明了環形面的多邊形突變類與對應的箭袋理論的突變類是雙射的。我們可得出箭袋理論的導出範疇的格羅滕迪克群同構於平方微分形式的週期寫像。依據 Gadbled-Thiel-Wagner 的工作,我們可以證明球面捻函子形成的自守等價子群同構於仿射型 A 的編結群。根據此結論,我們證明了仿射型 A 箭袋理論的穩定性條件空間商掉此編結群,做為複跡形,同構於此亞純平方微分形式的模空間。在第二部分,我們計算了局部曲線卡拉比-丘流行的高秩 DT 不變量。此量子不變量對應於 D6-D2-D0 與 D4-D2-D0 膜的 BPS 態數目。我們推導了高秩 D6 與 D4的精細過牆公式並且確認了在非精細極限下,與已知的公式符合。透過此精細過牆公式,我們計算了(-1,-1)和(-2,0)局部有理曲線的精細不變量,同時也討論了 D4 膜組態的模性質。 In this thesis, we study stability conditions of quiver algebras and wall-crossing formula of certain Calabi-Yau manifolds. In part 1, we study the space of stability conditions on a Calabi-Yau-N category associated to an affine type A-quiver. Bridgeland introduced the notion of stability conditions on a triangulated category which consists of a group homomorphism from Grothendieck group to complex number and a family of full additive subcategories of semistable objects of phase satisfying certain axioms. We consider a meromorphic quadratic differential on the sphere which has a pole of order (N-2)p+2 at 0 and pole of order (N-2)(n-p)+2 at infinity and n zeroes of order N-2. Denote by Quad(N,n) the moduli space of the quadratic differential. Taking the real oriented blowup at poles gives us an annulus with p marked points on the interior boundary and n-p marked points on the external boundary. Trajectories called N-2 diagonals of the quadratic differential connecting marked points on annulus give rise to an N-angulation. We associate the coloured quiver Q of affine A type to the N-angulation. As varying the meromorphic differential, an N-angulation would be mutated to a new N-angulation when crossing some walls in Quad(N,n). Torkildsen shows that the mutation class of N-angulation of the annulus with marked points quotient by some relations is bijective to the mutation class of the associated coloured quiver of affine type A.We can obtain a bijection between N-2-diagonals and simple objects of a heart. This implies that there is an isomorphism between the Grothendieck group of the derived category of affine type A quiver algebras and periods of the differential. Following the method by Khovanov-Seidel-Thomas’ work and Gadbled-Thiel-Wagner, I showed that the categorical action of affine type A braid group on the derived category is faithful. By applying the framework of Bridgeland and Smith, I prove that there is a holomorphic isomorphism of complex manifolds between the space of stability conditions and Quad(N,n).In part 2, we present some computations of higher rank refined Donaldson-Thomas invariants on local curve geometries, corresponding to local D6-D2-D0 or D4-D2-D0 configurations. A refined wall-crossing formula for invariants with higher D6 or D4 ranks is derived and verified to agree with the existing formulas under the unrefined limit. Using the formula, refined invariants on the (-1,-1) and (-2,0) local rational curve with higher D6 or D4 ranks are computed. Some modularity properties of the higher D4 configuration are also discussed. |
URI: | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/69589 |
DOI: | 10.6342/NTU201801061 |
全文授權: | 有償授權 |
顯示於系所單位: | 物理學系 |
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