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تسجيل مستخدم جديدسماعات كوانتوم النسبية
Relative quantum cohomology
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نحن نقوم بإنشاء نظام من PDE، ويسمى open WDVV، الذي يقيد الإضافية المحولة بالجراف والثابتة المرتبطة بها من المضلع المتساوي الطول والعرض من $L subset X$ مع سلسلة محددة. في نفس الوقت، نحن نحدد الجبر الكومولوجي الكمي لـ $X$ نسبيًا إلى $L$ ونثبت أنه متصل. نحن نحدد أيضًا الاتصال الكمي النسبي ونثبت أنه مسطح. يتم توليد صيغة التصطدم الجدارية التي تسمح بالتبادل بين عوامل الحد النقطي وبعض العوامل الداخلية في الثابتة المرتبطة بالجراف المفتوح. نتيجة أخرى هي قانون الإنقاص للثابتة المرتبطة بالجراف المفتوح للمضلعات التي ليس لها تأثير هومولوجي بأكثر من عامل حد نقطي. في هذه الحالة، يتم إظهار أن الثابتة المرتبطة بالجراف المفتوح مع عامل حد نقطي واحد يسترجع بعض الثابتة المغلقة. من open WDVV وصيغة التصطدم الجداري، يتم توليد سلسلة من العلاقات التكرارية التي تحدد كليًا الثابتة المرتبطة بالجراف المفتوح لـ $(X,L) = (mathbb{C}P^n, mathbb{R}P^n)$ مع $n$ فردي، التي تم تحديدها في العمل السابق للمؤلفين. لذلك، نحصل على صيغ كمية واضحة للثابتة المحددة باستخدام نظرية Fukaya-Oh-Ohta-Ono للسلاسل المحددة.
We establish a system of PDE, called open WDVV, that constrains the bulk-deformed superpotential and associated open Gromov-Witten invariants of a Lagrangian submanifold $L subset X$ with a bounding chain. Simultaneously, we define the quantum cohomology algebra of $X$ relative to $L$ and prove its associativity. We also define the relative quantum connection and prove it is flat. A wall-crossing formula is derived that allows the interchange of point-like boundary constraints and certain interior constraints in open Gromov-Witten invariants. Another result is a vanishing theorem for open Gromov-Witten invariants of homologically non-trivial Lagrangians with more than one point-like boundary constraint. In this case, the open Gromov-Witten invariants with one point-like boundary constraint are shown to recover certain closed invariants. From open WDVV and the wall-crossing formula, a system of recursive relations is derived that entirely determines the open Gromov-Witten invariants of $(X,L) = (mathbb{C}P^n, mathbb{R}P^n)$ with $n$ odd, defined in previous work of the authors. Thus, we obtain explicit formulas for enumerative invariants defined using the Fukaya-Oh-Ohta-Ono theory of bounding chains.
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