動機化 DT/PT 對應

Abstract

本論文證明了在假設半穩定物件的模疊具有好模空間的情況下，卡拉比丘三維複流形上 Donaldson–Thomas／Pandharipande–Thomas（DT/PT）對應的動機版本。我們採用Toda所構造的 t-結構心臟與弱穩定條件，在此架構下定義動機 DT 與 PT 不變量。本文的主要結果為一組動機等式，揭示兩類不變量之間的關係。證明方法延續 Bridgeland 與 Toda 的穿牆架構，並依賴 卜辰璟近期證明的動機積分恆等式。本研究為數值 DT/PT 對應提供了動機上的提升。

This thesis proves a motivic version of the Donaldson–Thomas/Pandharipande–Thomas (DT/PT) correspondence on Calabi–Yau threefolds under the assumption on the existence of some good moduli spaces. Working with the heart constructed and a weak stability condition constructed by Toda, we define motivic DT and PT invariants via vanishing cycles. The main result establishes a motivic identity between these invariants, following Bridgeland and Toda's wall-crossing approach. The argument relies on the motivic integral identity recently proved by Bu. This provides a refinement of the numerical DT/PT correspondence and supports further development of categorified curve-counting theories.