scholarly journals A uniqueness theorem for Frobenius manifolds and Gromov–Witten theory for orbifold projective lines

2015 ◽  
Vol 2015 (702) ◽  
Author(s):  
Yoshihisa Ishibashi ◽  
Yuuki Shiraishi ◽  
Atsushi Takahashi
Keyword(s):  
Uniqueness Theorem ◽  
Initial Conditions ◽  
Projective Line ◽  
Frobenius Manifolds ◽  
Witten Theory ◽  
Projective Lines

AbstractWe prove that the Frobenius structure constructed from the Gromov–Witten theory for an orbifold projective line with at most three orbifold points is uniquely determined by the Witten–Dijkgraaf–Verlinde–Verlinde equations with certain natural initial conditions.

scholarly journals Gromov–Witten theory and cycle-valued modular forms

2018 ◽  
Vol 2018 (735) ◽  
pp. 287-315 ◽  
Author(s):  
Todor Milanov ◽  
Yongbin Ruan ◽  
Yefeng Shen
Keyword(s):  
Field Theory ◽  
Modular Forms ◽  
Generating Functions ◽  
Mirror Symmetry ◽  
Complex Structure ◽  
Field Theories ◽  
Witten Theory ◽  
Cohomological Field Theories ◽  
Cohomological Field Theory ◽  
Projective Lines

AbstractIn this paper, we review Teleman’s work on lifting Givental’s quantization of{\mathcal{L}^{(2)}_{+}{\rm GL}(H)}action for semisimple formal Gromov–Witten potential into cohomological field theory level. We apply this to obtain a global cohomological field theory for simple elliptic singularities. The extension of those cohomological field theories over large complex structure limit are mirror to cohomological field theories from elliptic orbifold projective lines of weight(3,3,3),(2,4,4),(2,3,6). Via mirror symmetry, we prove generating functions of Gromov–Witten cycles for those orbifolds are cycle-valued (quasi)-modular forms.

scholarly journals Weighted Projective Lines and Rational Surface Singularities

2020 ◽  
Vol Volume 3 ◽  
Author(s):  
Osamu Iyama ◽  
Michael Wemyss
Keyword(s):  
Rational Surface ◽  
Projective Line ◽  
Degree Zero ◽  
Dual Graphs ◽  
Canonical Algebra ◽  
Tilting Bundle ◽  
Surface Singularities ◽  
Necessary And Sufficient ◽  
Projective Lines ◽  
Rational Surface Singularities

In this paper we study rational surface singularities R with star shaped dual graphs, and under very mild assumptions on the self-intersection numbers we give an explicit description of all their special Cohen-Macaulay modules. We do this by realising R as a certain Z-graded Veronese subring S^x of the homogeneous coordinate ring S of the Geigle-Lenzing weighted projective line X, and we realise the special CM modules as explicitly described summands of the canonical tilting bundle on X. We then give a second proof that these are special CM modules by comparing qgr S^x and coh X, and we also give a necessary and sufficient combinatorial criterion for these to be equivalent categories. In turn, we show that qgr S^x is equivalent to qgr of the reconstruction algebra, and that the degree zero piece of the reconstruction algebra coincides with Ringel's canonical algebra. This implies that the reconstruction algebra contains the canonical algebra, and furthermore its qgr category is derived equivalent to the canonical algebra, thus linking the reconstruction algebra of rational surface singularities to the canonical algebra of representation theory. Comment: Final version

THE TENSOR PRODUCT IN THE THEORY OF FROBENNIUS MANIFOLDS

1999 ◽  
Vol 10 (02) ◽  
pp. 159-206 ◽  
Author(s):  
RALPH M. KAUFMANN
Keyword(s):  
Tensor Product ◽  
Initial Conditions ◽  
Frobenius Manifold ◽  
Frobenius Manifolds

We introduce the operation of forming the tensor product in the theory of analytic Frobenius manifolds. Building on the results for formal Frobenius manifolds which we extend to the additional structures of Euler fields and flat identities, we prove that the tensor product of pointed germs of Frobenius manifolds exists. Furthermore, we define the notion of a tensor product diagram of Frobenius manifolds with factorizable flat identity and prove the existence of such a diagram and hence a tensor product Frobenius manifold. These diagrams and manifolds are unique up to equivalence. Finally, we derive the special initial conditions for a tensor product of semi-simple Frobenius manifolds in terms of the special initial conditions of the factors.

scholarly journals Applications of Mutations in the Derived Categories of Weighted Projective Lines to Lie and Quantum Algebras

2018 ◽  
Vol 2020 (19) ◽  
pp. 5814-5871
Author(s):  
Bangming Deng ◽  
Shiquan Ruan ◽  
Jie Xiao
Keyword(s):  
Projective Line ◽  
Derived Categories ◽  
Line Bundles ◽  
Hall Algebra ◽  
Moody Algebra ◽  
Weighted Projective Line ◽  
Coherent Sheaves ◽  
Enveloping Algebra ◽  
The Right ◽  
Projective Lines

Abstract Let $\textrm{coh}\ \mathbb{X}$ be the category of coherent sheaves over a weighted projective line $\mathbb{X}$ and let $D^b(\textrm{coh}\ \mathbb{X})$ be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in $D^b(\textrm{coh}\ \mathbb{X})$ attached to certain line bundles. As applications, we first show that these mutation functors give rise to simple reflections for the Weyl group of the star-shaped quiver ${Q}$ associated with $\mathbb{X}$. By further dealing with the Ringel–Hall algebra of $\mathbb{X}$, we show that these functors provide a realization for Tits’ automorphisms of the Kac–Moody algebra ${\mathfrak g}_{Q}$ associated with ${Q}$, as well as for Lusztig’s symmetries of the quantum enveloping algebra of ${\mathfrak g}_{Q}$.

scholarly journals On the transformation group of the second Painlevé equation

2000 ◽  
Vol 157 ◽  
pp. 15-46 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Weyl Group ◽  
Transformation Group ◽  
Blow Up ◽  
Initial Conditions ◽  
Painlevé Equation ◽  
Affine Weyl Group ◽  
Painleve Equation ◽  
Second Painlevé Equation ◽  
Projective Lines ◽  
Projective Surfaces

We show that for the second Painlevé equation y″ = 2y3 + ty + α, the Bäcklund transformation group G, which is isomorphic to the extended affine Weyl group of type Â1, operates regularly on the natural projectification χ(c)/ℂ(c, t) of the space of initial conditions, where c = α - 1/2. χ(c)/ℂ(c, t) has a natural model χ[c]/ℂ(t)[c]. The group G does not operate, however, regularly on χ[c]/ℂ(t)[c]. To have a family of projective surfaces over ℂ(t)[c] on which G operates regularly, we have to blow up the model χ[c] along the projective lines corresponding to the Riccati type solutions.

scholarly journals An existence and uniqueness theorem for the dynamics of flexural shells

2019 ◽  
Vol 25 (2) ◽  
pp. 317-336 ◽  
Author(s):  
Paolo Piersanti
Keyword(s):  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
A Priori ◽  
Elastic Shell ◽  
Time Dependent ◽  
Existence And Uniqueness Theorem ◽  
Dimensional Model ◽  
Unique Strong Solution ◽  
Membrane Shell

In this paper, we define, a priori, a natural two-dimensional model for a time-dependent flexural shell. As expected, this model takes the form of a set of hyperbolic variational equations posed over the space of admissible linearized inextensional displacements, and a set of initial conditions. Using a classical argument, we prove that the model under consideration admits a unique strong solution. However, the latter strategy makes use of function spaces, which are not amenable for numerically approximating the solution. We thus provide an alternate formulation of the studied problem using a suitable penalty scheme, which is more suitable in the context of numerical approximations. For the sake of completeness, in the final part of the paper, we also provide an existence and uniqueness theorem for the case where the linearly elastic shell under consideration is an elliptic membrane shell.

scholarly journals Knots in the Skyrme–Faddeev model

2007 ◽  
Vol 463 (2087) ◽  
pp. 3001-3020 ◽  
Author(s):  
Paul Sutcliffe
Keyword(s):  
Global Minimum ◽  
Initial Conditions ◽  
Dimensional Space ◽  
Minimum Energy ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Projective Line ◽  
Rational Maps ◽  
Torus Knots ◽  
Faddeev Model

The Skyrme–Faddeev model is a modified sigma model in three-dimensional space, which has string-like topological solitons classified by the integer-valued Hopf charge. Numerical simulations are performed to compute soliton solutions for Hopf charges up to 16, with initial conditions provided by families of rational maps from the three-sphere into the complex projective line. A large number of new solutions are presented, including a variety of torus knots for a range of Hopf charges. Often these knots are only local energy minima, with the global minimum being a linked solution, but for some values of the Hopf charge they are good candidates for the global minimum energy solution. The computed energies are in agreement with Ward's conjectured energy bound.

scholarly journals Quantum spectral curve for the Gromov–Witten theory of the complex projective line

2017 ◽  
Vol 2017 (726) ◽  
Author(s):  
Petr Dunin-Barkowski ◽  
Motohico Mulase ◽  
Paul Norbury ◽  
Alexander Popolitov ◽  
Sergey Shadrin
Keyword(s):  
Spectral Curve ◽  
Projective Line ◽  
Witten Theory ◽  
Complex Projective Line ◽  
Complex Projective

AbstractWe construct the quantum curve for the Gromov–Witten theory of the complex projective line.

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