Dialogues Between Physics and Mathematics
C. N. Yang at 100
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Article
Open AccessMachine Learning Clifford Invariants of ADE Coxeter Elements
There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest i...
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Article
Open AccessMahler Measure for a Quiver Symphony
Adopting the Mahler measure from number theory, we introduce it to toric quiver gauge theories, and study some of its salient features and physical implications. We propose that the Mahler measure is a univers...
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Article
Open AccessCrystal melting, BPS quivers and plethystics
We study the refined and unrefined crystal/BPS partition functions of D6-D2-D0 brane bound states for all toric Calabi-Yau threefolds without compact 4-cycles and some non-toric examples. They can be written a...
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Article
Open AccessIntegrality, duality and finiteness in combinatoric topological strings
A remarkable result at the intersection of number theory and group theory states that the order of a finite group G (denoted |G|) is divisible by the dimension dR of any irreducible complex representation of G. W...
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Chapter and Conference Paper
From the String Landscape to the Mathematical Landscape: A Machine-Learning Outlook
We review the recent programme of using machine-learning to explore the landscape of mathematical problems. With this paradigm as a model for human intuition—complementary to and in contrast with the more form...
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Book
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Article
Open AccessDessins d’enfants, Seiberg-Witten curves and conformal blocks
We show how to map Grothendieck’s dessins d’enfants to algebraic curves as Seiberg-Witten curves, then use the mirror map and the AGT map to obtain the corresponding 4d
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Article
Open AccessChiral rings, Futaki invariants, plethystics, and Gröbner bases
We study chiral rings of 4d N \( \mathcal{N} \) = 1 supersymmetric gauge th...
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Chapter
Postscriptum
Thus concludes our excursion into the terra sancta of Calabi–Yau manifolds. From the compact landscape of CICYs and KS hypersurfaces to the non-compact vista of quiver representations and Sasaki–Einstein cones, f...
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Chapter
The Compact Landscape
The landscape of compact Calabi–Yau manifolds, from the torus as an elliptic curve to the quintic manifold. Emphasis is on how software such as Macaulay2 computes desired topological quantities. Links to vast ...
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Chapter
Machine-Learning the Landscape
This chapter is an elementary introduction of machine-learning to the mathematics and theoretical physics student. The emphasis is on using the data introduced in Chaps. 2
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Chapter
Prologus Terræ Sanctæ
An invitation to the world of complex geometry and Calabi–Yau manifolds, extended to the student of physics and to the student of mathematics. It assumes no prior knowledge of string theory or of algebraic geo...
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Chapter
The Non-Compact Landscape
The landscape of affine Calabi-Yau varieties, as an initiation to quiver representations. Emphasis is on how to use Macaulay2 and SageMath to compute quantities such as moduli spaces and Hilbert series.
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Chapter
Direct Integration for Multi-Leg Amplitudes: Tips, Tricks, and When They Fail
Direct hyperlogarithmic integration offers a strong alternative to differential equation methods for Feynman integration, particularly for multi-particle diagrams. We review a variety of results by the authors...
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Article
Open AccessQuiver gauge theories: beyond reflexivity
Reflexive polygons have been extensively studied in a variety of contexts in mathematics and physics. We generalize this programme by looking at the 45 different lattice polygons with two interior points up to...
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Chapter and Conference Paper
Rendering Non-Euclidean Space in Real-Time Using Spherical and Hyperbolic Trigonometry
We introduce a method of calculating and renderin...
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Article
Open AccessCalabi–Yau Volumes and Reflexive Polytopes
We study various geometrical quantities for Calabi–Yau varieties realized as cones over Gorenstein Fano varieties, obtained as toric varieties from reflexive polytopes in various dimensions. Focus is made on r...
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Article
Open AccessPatterns in Calabi–Yau Distributions
We explore the distribution of topological numbers in Calabi–Yau manifolds, using the Kreuzer–Skarke dataset of hypersurfaces in toric varieties as a testing ground. While the Hodge numbers are well-known to e...
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Article
Open AccessTesting R-parity with geometry
We present a complete classification of the vacuum geometries of all renormalizable superpotentials built from the fields of the electroweak sector of the MSSM. In addition to the Severi and affine Calabi-Yau ...
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Article
Open AccessAn algebraic approach to the scattering equations
We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scatterin...
