Quiver Yangian from crystal melting
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Springer
Received: August 18, 2020 Accepted: September 24, 2020 Published: November 10, 2020
Wei Lia and Masahito Yamazakib a
Institute of Theoretical Physics, Chinese Academy of Sciences, 100190 Beijing, P.R. China b Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan
E-mail: [email protected], [email protected] Abstract: We find a new infinite class of infinite-dimensional algebras acting on BPS states for non-compact toric Calabi-Yau threefolds. In Type IIA superstring compactification on a toric Calabi-Yau threefold, the D-branes wrapping holomorphic cycles represent the BPS states, and the fixed points of the moduli spaces of BPS states are described by statistical configurations of crystal melting. Our algebras are “bootstrapped” from the molten crystal configurations, hence they act on the BPS states. We discuss the truncation of the algebra and its relation with D4-branes. We illustrate our results in many examples, with and without compact 4-cycles. Keywords: Conformal and W Symmetry, D-branes, Quantum Groups, Supersymmetric Gauge Theory ArXiv ePrint: 2003.08909
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)035
JHEP11(2020)035
Quiver Yangian from crystal melting
Contents 1 Introduction
1
2 Review: BPS crystal melting 2.1 Quiver diagram and superpotential 2.2 Crystal as a lift of periodic quiver 2.3 Crystal melting and molten crystal
3 3 5 7 9 9 11 12
4 BPS quiver Yangian for general quivers 4.1 Parameters 4.2 Generators and relations 4.2.1 Relations in terms of fields 4.2.2 Relations in terms of modes 4.3 Some properties of the algebra 4.3.1 Grading and filtration 4.3.2 Spectral shift 4.3.3 Gauge-symmetry shift 4.4 Serre relations
13 13 14 15 16 18 18 19 19 21
5 Bootstrapping affine Yangian of gl1 from plane partitions 5.1 Ansatz 5.2 Analysis 5.2.1 Vacuum −→ level-1 5.2.2 Level-1 −→ level-2 5.2.3 Level-2 −→ level-3 5.2.4 General levels 5.3 Serre relation and state counting 5.3.1 Vacuum 5.3.2 One box 5.3.3 Two boxes 5.3.4 Three boxes 5.3.5 Four boxes and beyond 5.4 Summary
22 22 24 24 25 26 30 31 32 32 33 33 33 34
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JHEP11(2020)035
3 Review: plane partition and affine Yangian of gl1 3.1 Affine Yangian of gl1 3.2 Plane partition 3.3 Action of affine Yangian of gl1 on plane partitions
35 35 38 39 39 39 41 42 43 47 47 47 48 53 55
7 Truncations of quiver Yangians and D4-branes 7.1 Truncations of quiver Yangians 7.2 Multiple truncations and rational algebras 7.3 Relation with D4-branes
56 56 57 59
8 Examples: Calabi-Yau threefolds without compact 4-cycles 63 8.1 Simplification when no compact 4-cycles are present 63 8.1.1 Mode expansion 63 8.1.2 Initial conditions 63 8.1.3 Central element of the algebra 64 8.1.4 Identification between universal central term ψ0 and vacuum charge C 65 8.2 Quiver Yangians for (C2 /Zn ) × C and affine Yangian of gln 66 3 8.2.1 C and affine Yangian of gl1 66 8.2.2 (C2 /Z2 ) × C and affine Yangian of gl2 69 2 8.2.3 (C /Zn ) × C and a
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