Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Higher Spin and Yangian Wei Li Institute of Theoretical Physics, Chinese Academy of Sciences Sanya, 2019/01/07 Wei Li Higher Spin and Yangian 1
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Stringy symmetry Reference 1. Higher Spins and Yangian Symmetries JHEP 1704 , 152 (2017), [arXiv:1702.05100] with Matthias Gaberdiel, Rajesh Gopakumar, and Cheng Peng 2. Twisted sectors from plane partitions JHEP 1609 , 138 (2016), [arXiv:1606.07070] with Shouvik Datta, Matthias Gaberdiel, and Cheng Peng 3. The supersymmetric affine yangian JHEP 1805 , 200 (2018), [arXiv:1711.07449] with Matthias Gaberdiel, Cheng Peng, and Hong Zhang 4. Twin plane partitions and N = 2 affine yangian JHEP 1811 , 192 (2018), [arXiv:1807.11304] with Matthias Gaberdiel and Cheng Peng Wei Li Higher Spin and Yangian 2
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Stringy symmetry There is a large hidden symmetry in string theory Symmetry ? Stringy Wei Li Higher Spin and Yangian 3
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Stringy symmetry Different manifestation of stringy symmetry Integrable structure Stringy Symmetry Higher spin symmetry Wei Li Higher Spin and Yangian 4
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Stringy symmetry Different manifestation of stringy symmetry Integrable structure ? Stringy Symmetry Higher spin symmetry Wei Li Higher Spin and Yangian 5
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Stringy symmetry Different manifestation of stringy symmetry Integrable structure ? ? Stringy Symmetry Higher spin symmetry Wei Li Higher Spin and Yangian 6
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Stringy symmetry Today Integrable structure ? Higher spin symmetry Wei Li Higher Spin and Yangian 7
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Stringy symmetry A concrete relation between HS and integrability Affine Yangian of gl(1) “Isomorphic” W symmetry Wei Li Higher Spin and Yangian 8
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Stringy symmetry Application: plane partition as representations of W ∞ Affine Yangian of gl(1) “Isomorphic” Representation W symmetry Plane partitions Representation Wei Li Higher Spin and Yangian 9
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Gluing Two questions 1. Supersymmetrize △ ? 2. △ as lego pieces for new VOA/affine Yangian? A surprising (partial) answer Glue two △ to get N = 2 version of △ Wei Li Higher Spin and Yangian 10
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Gluing N = 2 version? ? Representation ? N=2 W symmetry Representation Wei Li Higher Spin and Yangian 11
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Gluing New Yangian algebra from W algebra N=2 Affine Yangian of gl(1) Define Representation Twin N=2 W symmetry plane partitions Rrepresentation Wei Li Higher Spin and Yangian 12
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Corner chiral algebra Finite truncation of affine Yangian of gl 1 Fukuda Matsuo Nakamura Zhu ’15 Prochazka ’15 ◮ gives chiral algebra of Y-junction Gaiotto Rapcak ’17 ◮ Gluing of these finite truncations should give chiral algebra of Y-junction webs Rapcak Prochazka’17 Wei Li Higher Spin and Yangian 13
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Corner chiral algebra 5-brane junction with D3 brane interfaces Gaiotto Rapcak ’17 x 4 , x 5 , x 6 NS5 x 3 x 2 N D3 R 3 C × × x 0 , x 1 x 7 , x 8 , x 9 L D3 M D3 D5 (1,1) picture: Gaiotto Rapcak ’17 conjecture: VOA on the 2D junction of 4D QFT Wei Li Higher Spin and Yangian 14
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Corner chiral algebra Representation Plane partition VOA W/AffineYangian Vertex Web QFT from IIB 5 brane junction (p,q) web Geometry C3 Toric CY3 Topological Topological Topological String String Vertex Wei Li Higher Spin and Yangian 15
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Corner chiral algebra Representation Plane partition VOA W/AffineYangian Vertex Web QFT from IIB 5 brane junction (p,q) web Geometry C3 Toric CY3 Topological Topological Topological String String Vertex Wei Li Higher Spin and Yangian 16
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Corner chiral algebra Representation Plane partition VOA W/AffineYangian Vertex Web QFT from IIB 5 brane junction (p,q) web Geometry C3 Toric CY3 Topological Topological Topological String String Vertex Wei Li Higher Spin and Yangian 17
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Corner chiral algebra Network of Representation Plane partition plane partitions VOA VOA Web W/AffineYangian Vertex Web QFT from IIB 5 brane junction (p,q) web Geometry C3 Toric CY3 Topological Topological Topological String String Vertex Wei Li Higher Spin and Yangian 18
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Corner chiral algebra Outline Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary Wei Li Higher Spin and Yangian 19
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary W—Affine Yangian Relation between W algebra and affine Yangian Affine Yangian of gl(1) “Isomorphic” W symmetry Wei Li Higher Spin and Yangian 20
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary W—Affine Yangian Modes of W 1+ ∞ � W ( s ) W ( s ) ( z ) = n s = 1 , 2 , 3 , . . . z n + s n ∈ Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . spin-5 . . . X − 4 X − 3 X − 2 X − 1 X 0 X 1 X 2 X 3 X 4 . . . spin-4 . . . U − 4 U − 3 U − 2 U − 1 U 0 U 1 U 2 U 3 U 4 . . . spin-3 . . . W − 4 W − 3 W − 2 W − 1 W 0 W 1 W 2 W 3 W 4 . . . spin-2 . . . L − 4 L − 3 L − 2 L − 1 L 0 L 1 L 2 L 3 L 4 . . . spin-1 . . . J − 4 J − 3 J − 2 J − 1 J 0 J 1 J 2 J 3 J 4 . . . For λ = 0 and λ = 1 , prove isomorphism using free field realization. (fermion) (boson) For generic λ , check isomorphism up to spin-4 Enough to find the map between Yangian parameter ( h 1 , h 2 , h 3 ) and W 1+ ∞ parameter ( c, λ ) Wei Li Higher Spin and Yangian 21
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary W—Affine Yangian Regrouping the modes � W ( s ) W ( s ) ( z ) = n s = 1 , 2 , 3 , . . . z n + s n ∈ Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . spin-5 . . . X − 3 X − 2 X − 1 ∼ e 4 X 0 ∼ ψ 5 X 1 ∼ f 4 X 2 X 3 X 4 spin-4 . . . U − 3 U − 2 U − 1 ∼ e 3 U 0 ∼ ψ 4 U 1 ∼ f 3 U 2 U 3 U 4 spin-3 W − 1 ∼ e 2 W 0 ∼ ψ 3 W 1 ∼ f 2 . . . W − 3 W − 2 W 2 W 3 W 4 spin-2 L − 1 ∼ e 1 L 0 ∼ ψ 2 L 1 ∼ f 1 . . . L − 3 L − 2 L 2 L 3 L 4 spin-1 J − 1 ∼ e 0 J 0 ∼ ψ 1 J 1 ∼ f 0 . . . J − 3 J − 2 J 2 J 3 J 4 affine Yangian generators ∞ ∞ ∞ � � � e j ψ j f j e ( z ) = ψ ( z ) = 1 + σ 3 f ( z ) = z j +1 z j +1 z j +1 j =0 j =0 j =0 For λ = 0 and λ = 1 , prove isomorphism using free field realization. (fermion) (boson) Wei Li Higher Spin and Yangian 22
Introduction W—Affine Yangian—Plane Partition Gluing and N = 2 affine Yangian Summary W—Affine Yangian Affine Yangian of gl 1 Def: Associative algebra with generators e j , f j and ψ j , j = 0 , 1 , . . . ◮ Generators ∞ ∞ ∞ � � � ψ j e j f j ψ ( z ) = 1 + ( h 1 h 2 h 3 ) e ( z ) = f ( z ) = z j +1 z j +1 z j +1 j =0 j =0 j =0 ◮ Parameters ( h 1 , h 2 , h 3 ) with h 1 + h 2 + h 3 = 0 ◮ One S 3 invariant function ϕ ( z ) = ( z + h 1 )( z + h 2 )( z + h 3 ) ( z − h 1 )( z − h 2 )( z − h 3 ) ◮ Defining relations 1 ψ ( z ) − ψ ( w ) [ e ( z ) , f ( w )] = − z − w h 1 h 2 h 3 ψ ( z ) e ( w ) ∼ ϕ ( z − w ) e ( w ) ψ ( z ) ψ ( z ) f ( w ) ∼ ϕ ( w − z ) f ( w ) ψ ( z ) e ( z ) e ( w ) ∼ ϕ ( z − w ) e ( w ) e ( z ) f ( z ) f ( w ) ∼ ϕ ( w − z ) f ( w ) f ( z ) ϕ − 1 ϕ 3 (∆) 3 (∆) ϕ − 1 ϕ 3 (∆) 3 (∆) e ψ f Wei Li Higher Spin and Yangian 23
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