Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRelative quantum cohomology
سماعات كوانتوم النسبية
540
0
0.0
(
0
)
Ask ChatGPT about the research
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.
rate research
Read More
We present some computations of relative symplectic cohomology, with the help of an index bounded contact form. For a Liouville domain with an index bounded boundary, we construct a spectral sequence which starts from its classical symplectic cohomology and converges to its relative symplectic cohomology inside a Calabi-Yau manifold.
We show that the exterior derivative operator on a symplectic manifold has a natural decomposition into two linear differential operators, analogous to the Dolbeault operators in complex geometry. These operators map primitive forms into primitive forms and therefore lead directly to the construction of primitive cohomologies on symplectic manifolds. Using these operators, we introduce new primitive cohomologies that are analogous to the Dolbeault cohomology in the complex theory. Interestingly, the finiteness of these primitive cohomologies follows directly from an elliptic complex. We calculate the known primitive cohomologies on a nilmanifold and show that their dimensions can vary depending on the class of the symplectic form.
We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered cohomologies give a two-sided resolution of Lefschetz maps, and thereby, they are directly related to the kernels and cokernels of the Lefschetz maps. We also introduce a novel, non-associative product operation on differential forms for symplectic manifolds. This product generates an A-infinity algebra structure on forms that underlies the filtered cohomologies and gives them a ring structure. As an application, we demonstrate how the ring structure of the filtered cohomologies can distinguish different symplectic four-manifolds in the context of a circle times a fibered three-manifold.
We construct a global geometric model for complex analytic equivariant elliptic cohomology for all compact Lie groups. Cocycles are specified by functions on the space of fields of the two-dimensional sigma model with background gauge fields and $mathcal{N} = (0, 1)$ supersymmetry. We also consider a theory of free fermions valued in a representation whose partition function is a section of a determinant line bundle. We identify this section with a cocycle representative of the (twisted) equivariant elliptic Euler class of the representation. Finally, we show that the moduli stack of $U(1)$-gauge fields carries a multiplication compatible with the complex analytic group structure on the universal (dual) elliptic curve, with the Euler class providing a choice of coordinate. This provides a physical manifestation of the elliptic group law central to the homotopy-theoretic construction of elliptic cohomology.
We compute the degree of the generalized Plucker embedding $kappa$ of a Quot scheme $X$ over $PP^1$. The space $X$ can also be considered as a compactification of the space of algebraic maps of a fixed degree from $PP^1$ to the Grassmanian $rm{Grass}(m,n)$. Then the degree of the embedded variety $kappa (X)$ can be interpreted as an intersection product of pullbacks of cohomology classes from $rm{Grass}(m,n)$ through the map $psi$ that evaluates a map from $PP^1$ at a point $xin PP^1$. We show that our formula for the degree verifies the formula for these intersection products predicted by physicists through Quantum cohomology~cite{va92}~cite{in91}~cite{wi94}. We arrive at the degree by proving a version of the classical Pieris formula on the variety $X$, using a cell decomposition of a space that lies in between $X$ and $kappa (X)$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
