A note on homotopy complex surfaces with negative tangent bundles

Abstract The superpotential in four-dimensional heterotic effective theories contains terms arising from holomorphic Chern-Simons invariants associated to the gauge and tangent bundles of the compactification geometry. These effects are crucial for a number of key features of the theory, including vacuum stability and moduli stabilization. Despite their importance, few tools exist in the literature to compute such effects in a given heterotic vacuum. In this work we present new techniques to explicitly determine holomorphic Chern-Simons invariants in heterotic string compactifications. The key technical ingredient in our computations are real bundle morphisms between the gauge and tangent bundles. We find that there are large classes of examples, beyond the standard embedding, where the Chern-Simons superpotential vanishes. We also provide explicit examples for non-flat bundles where it is non-vanishing and non-integer quantized, generalizing previous results for Wilson lines.

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Verlinde formulae on complex surfaces: K-theoretic invariants

Forum of Mathematics Sigma ◽

10.1017/fms.2020.50 ◽

2021 ◽

Vol 9 ◽

Author(s):

L. Göttsche ◽

M. Kool ◽

R. A. Williams

Keyword(s):

Moduli Space ◽

K3 Surfaces ◽

Complex Surfaces ◽

Type Formula ◽

Verlinde Formula ◽

Donaldson Invariants

Abstract We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between K-theoretic Donaldson invariants studied by Göttsche and Nakajima-Yoshioka and K-theoretic Vafa-Witten invariants introduced by Thomas and also studied by Göttsche and Kool. We verify our conjectures in many examples (for example, on K3 surfaces).

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Conversion Between Cubic Bezier Curves and Catmull–Rom Splines

SN Computer Science ◽

10.1007/s42979-021-00770-x ◽

2021 ◽

Vol 2 (5) ◽

Author(s):

Soroosh Tayebi Arasteh ◽

Adam Kalisz

Keyword(s):

Computer Graphics ◽

Geometric Design ◽

The Other ◽

Control Points ◽

Complex Surfaces ◽

Linear Transformations ◽

Computer Aided Geometric Design ◽

Cubic Curves ◽

Original Curve ◽

Computer Aided

AbstractSplines are one of the main methods of mathematically representing complicated shapes, which have become the primary technique in the fields of Computer Graphics (CG) and Computer-Aided Geometric Design (CAGD) for modeling complex surfaces. Among all, Bézier and Catmull–Rom splines are the most common in the sub-fields of engineering. In this paper, we focus on conversion between cubic Bézier and Catmull–Rom curve segments, rather than going through their properties. By deriving the conversion equations, we aim at converting the original set of the control points of either of the Catmull–Rom or Bézier cubic curves to a new set of control points, which corresponds to approximately the same shape as the original curve, when considered as the set of the control points of the other curve. Due to providing simple linear transformations of control points, the method is very simple, efficient, and easy to implement, which is further validated in this paper using some numerical and visual examples.

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