Instanton R-matrix and W $$ \mathcal{W} $$ -symmetry
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Springer
Received: July 17, 2019 Accepted: November 11, 2019 Published: December 12, 2019
Tom´ aˇs Proch´ azka Arnold Sommerfeld Center for Theoretical Physics, Ludwig Maximilian University of Munich, Theresienstr. 37, M¨ unchen D-80333, Germany
E-mail: [email protected] Abstract: We study the relation between W1+∞ algebra and Arbesfeld-SchiffmannTsymbaliuk Yangian using the Maulik-Okounkov R-matrix. The central object linking these two pictures is the Miura transformation. Using the results of Nazarov and Sklyanin we find an explicit formula for the mixed R-matrix acting on two Fock spaces associated to two different asymptotic directions of the affine Yangian. Using the free field representation we propose an explicit identification of Arbesfeld-Schiffmann-Tsymbaliuk generators with the generators of Maulik-Okounkov Yangian. In the last part we use the Miura transformation to give a conformal field theoretic construction of conserved quantities and ladder operators in the quantum mechanical rational and trigonometric Calogero-Sutherland models on which a vector representation of the Yangian acts. Keywords: Conformal and W Symmetry, Integrable Field Theories, Conformal Field Theory, Higher Spin Symmetry ArXiv ePrint: 1903.10372
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP12(2019)099
JHEP12(2019)099
Instanton R-matrix and W-symmetry
Contents 1 Introduction
2
2 Miura transformation
4
Other triality frames
5
2.2
Conformal transformations
7
3 R-matrix
11
3.1
R-matrices of the mixed type
12
3.2
Mode expansions
13
3.3
Expansion of R at large spectral parameter
15
3.4
Matrix elements of R between simple states
18
3.5
Yang-Baxter equation
19
3.6
Yangian generators
20
4 Single boson representation
23
4.1
Nazarov-Sklyanin operators
26
4.2
Ladder operators
28
4.3
Nazarov-Sklyanin II
32
5 R-matrix from fermions
33
6 Arbesfeld-Schiffmann-Tsymbaliuk presentation
37
7 Calogero-Moser-Sutherland models
40
8 Outlook
45
A Higher order expressions for R-matrix
47
B Fock representation of W1+∞
47
C Higher relations of Yangian algebra
49
C.1 [E, F] relations
49
C.2 [E, E] relations
50
–1–
JHEP12(2019)099
2.1
1
Introduction
–2–
JHEP12(2019)099
W-algebras are remarkable algebraic structures introduced first by Zamolodchikov [1]. The algebra that he was studying was an extension of the Virasoro algebra underlying twodimensional conformal field theory by an additional generator of spin 3. Since then there have been many applications of these algebras in various areas of mathematical physics. Among the oldest are applications to integrable hierarchies of partial differential equations [2], matrix integrals [3], topological strings [4] or in the quantum Hall effect [5, 6]. More recently there are two directions of research where W-algebras play a prominent role. The first one is the AdS3 /CFT2 duality with higher spin symmetries [7–10]. The cosmological Einstein gravity in three dimensions can be formulated as a Chern-Simons
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