An official website of the United States government
Abstract Let $$\textsf {X}$$ and $$\textsf {X}^{!}$$ be a pair of symplectic varieties dual with respect to 3D mirror symmetry. The $$K$$-theoretic limit of the elliptic duality interface is an equivariant $$K$$-theory class $$\mathfrak {m} \in K(\textsf {X}\times \textsf {X}^{!})$$. We show that this class provides correspondences $$ \begin{align*} & \Phi_{\mathfrak{m}}: K(\textsf{X}) \leftrightarrows K(\textsf{X}^{!}) \end{align*}$$mapping the $$K$$-theoretic stable envelopes to the $$K$$-theoretic stable envelopes. This construction allows us to relate various representation theoretic objects of $$K(\textsf {X})$$, such as action of quantum groups, quantum dynamical Weyl groups, $$R$$-matrices, etc., to those for $$K(\textsf {X}^{!})$$. In particular, we relate the wall $$R$$-matrices of $$\textsf {X}$$ to the $$R$$-matrices of the dual variety $$\textsf {X}^{!}$$. As an example, we apply our results to $$\textsf {X}=\textrm {Hilb}^{n}({{\mathbb {C}}}^2)$$—the Hilbert scheme of $$n$$ points in the complex plane. In this case, we arrive at the conjectures of Gorsky and Negut from [10].
more »
« less
Categorical Primitive Forms and Gromov–Witten Invariants of An Singularities
Abstract We introduce a categorical analogue of Saito’s notion of primitive forms. For the category $$\textsf{MF}(\frac{1}{n+1}x^{n+1})$$ of matrix factorizations of $$\frac{1}{n+1}x^{n+1}$$, we prove that there exists a unique, up to non-zero constant, categorical primitive form. The corresponding genus zero categorical Gromov–Witten invariants of $$\textsf{MF}(\frac{1}{n+1}x^{n+1})$$ are shown to match with the invariants defined through unfolding of singularities of $$\frac{1}{n+1}x^{n+1}$$.
more »
« less
- Award ID(s):
- 1811925
- PAR ID:
- 10128884
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract Suppose that $$\Sigma ^{n}\subset \mathbb{S}^{n+1}$$ is a closed embedded minimal hypersurface. We prove that the first non-zero eigenvalue $$\lambda _{1}$$ of the induced Laplace–Beltrami operator on $$\Sigma $$ satisfies $$\lambda _{1} \geq \frac{n}{2}+ a_{n}(\Lambda ^{6} + b_{n})^{-1}$$, where $$a_{n}$$ and $$b_{n}$$ are explicit dimensional constants and $$\Lambda $$ is an upper bound for the length of the second fundamental form of $$\Sigma $$. This provides the first explicitly computable improvement on Choi and Wang’s lower bound $$\lambda _{1} \geq \frac{n}{2}$$ without any further assumptions on $$\Sigma $$.more » « less
-
The third Painlevé equation in its generic form, often referred to as Painlevé-III($$D_6$$), is given by $$ \frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left(\frac{{\rm d}u}{{\rm d}x} \right)^2-\frac{1}{x} \frac{{\rm d}u}{{\rm d}x} + \frac{\alpha u^2 + \beta}{x}+4u^3-\frac{4}{u}, \qquad \alpha,\beta \in \mathbb C. $$ Starting from a generic initial solution $$u_0(x)$$ corresponding to parameters $$\alpha$$, $$\beta$$, denoted as the triple $$(u_0(x),\alpha,\beta)$$, we apply an explicit Bäcklund transformation to generate a family of solutions $$(u_n(x),\alpha + 4n,\beta + 4n)$$ indexed by $$n \in \mathbb N$$. We study the large $$n$$ behavior of the solutions $$(u_n(x), \alpha + 4n, \beta + 4n)$$ under the scaling $x = z/n$ in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann-Hilbert representation of the solution $$u_n(z/n)$$. Our main result is a proof that the limit of solutions $$u_n(z/n)$$ exists and is given by a solution of the degenerate Painlevé-III equation, known as Painlevé-III($$D_8$$), $$ \frac{{\rm d}^2U}{{\rm d}z^2} =\frac{1}{U}\left(\frac{{\rm d}U}{{\rm d}z}\right)^2-\frac{1}{z} \frac{{\rm d}U}{{\rm d}z} + \frac{4U^2 + 4}{z}.$$ A notable application of our result is to rational solutions of Painlevé-III($$D_6$$), which are constructed using the seed solution $(1,4m,-4m)$ where $$m \in \mathbb C \setminus \big(\mathbb Z + \frac{1}{2}\big)$$ and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at $z = 0$ when it is well defined, and by its monodromy data in the general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlevé-III, both $$D_6$$ and $$D_8$$ at $z = 0$. We also deduce the large $$n$$ behavior of the Umemura polynomials in a neighborhood of $z = 0$.more » « less
-
Subfactor standard invariants encode quantum symmetries. The small index subfactor classification program has been a rich source of interesting quantum symmetries. We give the complete classification of subfactor standard invariants to index , which includes , the first interesting composite index.more » « less
-
Abstract We present a new elementary algorithm that takes $$ \textrm{time} \ \ O_\epsilon \left( x^{\frac{3}{5}} (\log x)^{\frac{8}{5}+\epsilon } \right) \ \ \textrm{and} \ \textrm{space} \ \ O\left( x^{\frac{3}{10}} (\log x)^{\frac{13}{10}} \right) $$ time O ϵ x 3 5 ( log x ) 8 5 + ϵ and space O x 3 10 ( log x ) 13 10 (measured bitwise) for computing $$M(x) = \sum _{n \le x} \mu (n),$$ M ( x ) = ∑ n ≤ x μ ( n ) , where $$\mu (n)$$ μ ( n ) is the Möbius function. This is the first improvement in the exponent of x for an elementary algorithm since 1985. We also show that it is possible to reduce space consumption to $$O(x^{1/5} (\log x)^{5/3})$$ O ( x 1 / 5 ( log x ) 5 / 3 ) by the use of (Helfgott in: Math Comput 89:333–350, 2020), at the cost of letting time rise to the order of $$x^{3/5} (\log x)^2 \log \log x$$ x 3 / 5 ( log x ) 2 log log x .more » « less
