Vasily Pestun! Strings 2016 Beijing, August 4, 2016

Transcription

1 Moduli spaces, instantons, monopoles! and Quantum Algebras Vasily Pestun!! Strings 2016 Beijing, August 4,

2 4d gauge theories What can we do beyond perturbation theory?! Are there hidden algebraic structures?! What are the exactly computable quantities? 2

3 2d CFT integrable lattice models quantum integrable systems conserved quantities! holomorphic factorization! conformal blocks, OPE! Kac-Moody, Virasoro, W-algebra! quantum groups, R-matrix 3

4 What of these structures appear in gauge theories? For N = vacua sector = integrable system! holomorphic factorization! emergence of quantum algebras! CFT type (Virasoro, affine Kac-Moody)! Spin chain type (R-matrix, quantum group) 4

5 What of these structures appear in gauge theories? For N = vacua sector = integrable system! holomorphic factorization! emergence of quantum algebras! CFT type (Virasoro, affine Kac-Moody)! Spin chain type (R-matrix, quantum group) 5

6 vacua = integrable system N = base of algebraic! integrable system Seiberg-Witten 94! GKMMM 95! Donagi-Witten 95! Seiberg-Witten 96 6

7 vacua = integrable system M R S phase space (dim=2r) (HyperKahler) Lagrangian fibration M R U base (dim = r) (Special Kahler) 7

8 What we know? For N = vacua sector = integrable system! holomorphic factorization! emergence of quantum algebras! Kac-Moody/Virasoro/W-algebra type! Quantum groups (R-matrix, spin chains) 8

9 holomorphic factorization S = ( ) ( )! VP 07! Z(a) = partition function on R, Nekrasov 02, ( ) = 2d CFT conformal block Alday-Gaiotto-Tachikawa 08 9

10 holomorphic factorization 700 pages review book coming out to arxiv next week Vasily Pestun, Maxim Zabzine, Francesco Benini, Tudor Dimofte, Thomas T. Dumitrescu,! Kazuo Hosomichi, Seok Kim, Kimyeong Lee, Bruno Le Floch, Marcos Marino, Joseph A. Minahan, David R. Morrison, Sara Pasquetti, Jian Qiu, Leonardo Rastelli,! Shlomo S. Razamat, Silvu S. Pufu, Yuji Tachikawa, Brian Willett, Konstantin Zarembo! 10

11 , ( ) is conformal block of what algebra? Where does it come from? 11

12 emergence of quantum algebras If 4d theory of class Sg(C) then! Z(a) = conformal block of W(g) algebra! Alday-Gaiotto-Tachikawa 08 on 2d surface C W(sl2) = Vir Class Sg(C) is the 4d theory obtained by! reduction of 6d (0,2) type g on C M R = M Hitchin(, g) 12

13 emergence of quantum algebras M Hitchin (, g) W algebra, ( ) Alday-Gaiotto-Tachikawa 08! Nekrasov-Witten 12! Teschner 13! 13

14 emergence of quantum algebras M Hitchin (, g) W, ( ) algebra Alday-Gaiotto-Tachikawa 08! Nekrasov-Witten 12! Teschner 13! M What replaces W(g)-algebra for generic? 14

15 emergence of quantum algebras Proposition M, (M) M /, ( ) =C [ / ] Quantization (Planck constant) HyperKahler twistor space parameter Nekrasov-Witten, Bellinson-Drinfeld! Teschner, Kashaev! Kontsevich-Soibelman, Gaiotto-Moore-Neitzke! Fock-Goncharov,! Neitzke, Vafa, Cecotti-Neitzke-Vafa! Nekrasov-Rosly-Shatashvili, Gaiotto! Kontsevich, Gukov-Witten!! 15

16 Quantization on HK (X,ω) real symplectic manifold! Quantization has one parameter:!!! ˆ ˆ ˆ ˆ = {, }! +... Dirac! Fedosov! Kontsevich!! (, HK manifold!,, ) Quantization has two complex parameters:! 1)!! 2) (, CP twistor C ) holomorphic symplectic! Kontsevich! Cattaneo-Felder! Kapustin-Orlov! Gukov-Witten!! quantized by non-commutative open B-model 16

17 , (M) associative! algebra 1. Seiberg-Witten limit, (M) =C[U] functions on u-plane! Coulomb chiral ring in 4d! classical commuting Hamiltonians 2. Nekrasov-Shatashvili limit, (M)! Ring of quantum commuting Hamiltonians! The eigenvalues of elements in, (M) is spectrum! of quantum integrable system 17 classical! commutative! ring quantum! commutative! ring

18 , (M), Two large classes of 4d theories 1. class Sg 6d (0,2) g=ade reduced on C M Witten 96! Gaiotto 07! x x AGT 08! GMN Γ-quiver 6d quiver theories reduced on M 18 Douglas-Moore 96! Cherkis-Kapustin 00! Katz-Mayer-Vafa 97! Nekrasov-VP-(Shatashvili) 12, 13! Aganagic 15! Nekrasov 15, Kimura-VP 15! (2d flat)

19 class Sg theories 6d (0,2) g=ade reduced on C M x x M =, M flat (, ), (M) is quantum cluster algebra! Cheng s talk!, (M) (algebra of Verlinde loop operators) = C is W(g) algebra Fock-Goncharov! Teschner! Gaiotto-Moore-Neitzke! Alday-Gaiotto-Tachikawa 19 Drinfeld-Sokolov! Zamolodchikov! Feigin-Frenkel! (integrals of motion in Toda 2d field theory) Belinson-Drinfeld

20 Quiver theories 6d Γ-quiver gauge theories reduced on Γ= D4 GΓ=SO(8) M SU(n 3 ) SU(n 1 ) SU(n 2 ) SU(n 4 ) (group valued analogue of Hitchin system) [ ] = ( ) -polynomial =, = Nekrasov-VP 12!!!! Nekrasov-VP-Shatashvili 13!!!! Nekrasov 15, Kimura-VP 15! =, = =, = [ ( )]= tr ( ( )) character 20 [ ( )] = q-character = Wq-algebra (gγ) qq-character = Wqq-algebra (gγ) ( ) Frenkel-Reshetikhin 98 Frenkel-Reshetikhin 97

21 UV-observables Y i (x) =det(1 x 1 e i + 1 J J 2 ) Ei qq-characters are proper IR-observables! made from Yi(x) Nekrasov s talk 21

22 qq-characters 14,16! 15 U(n) Example A1 SU(2)-charge monopoles of charge n on CxS 1 good in UV Y i (x) =det(1 x 1 e i + 1 J J 2 ) Ei good in IR 1 = Y 1 (x)+y 1 1 (q 1 x) q = q 1 q 2 q i = e i 22

23 characters 12! 1. classical limit (ε1=ε2=0) 1[y] =tr y(x) 0 0 y(x) 1 = y(x)+ 1 y(x) q-character =T-polynomial is classical SW spectral curve y +1/y = T n (x) =x n + u 1 x n u n is spectral curve of SU(2) monopoles! of charge n on C x S 1 23

24 q-characters Nekrasov-VP-Shatashvili quantum integrable system (NS-limit ε1=h, ε2=0) Y (x) =Q(x)/Q(q 1 x) q-character =T-polynomial is quantum SW spectral curve Q(qx)+Q(q 1 x)=q(x)t (x) (aka TQ-Baxter or analytic Bethe anzats) T n (x) =x n + u 1 x n u n quantum commuting Hamiltonians 24

25 q-characters Nekrasov-VP-Shatashvili 13 for generic quiver we find traces of R-matrices! of rational, trigonometric and elliptic spin-chain! types for any ADE and their affinization (toroidal) Yang(g),U q (ĝ),e g,c (g) Gerasimov-Kharchev- Lebedev-Oblezin 04! The reason is that the phase space,! the monopole moduli space M on CxS 1! is a symplectic leaf of the Poisson-Lie loop group {g(x)} with classical r-matrix Poisson bracket! The quantization produces quantum R-matrix. x C 25 c.f. Costello s & Witten's talks

26 q-characters=classical W-algebra q-characters generate commutative Wq(g) algebra Nekrasov-VP-Shatashvili 13 Alternatively (after T-duality on the fibers of integrable system) this classical Poisson Wq(g) algebra can be obtained by group version of Drinfeld-Sokolov reduction on group-valued q- opers (q-connections)! {g(x)}/{g(x) s(x)g(x)s 1 (qx)} Steinberg 65! Semenov Tian Shansky 99 Sevostiyanov 00 C q S 1 q-twisted over 26

27 qq-characters=quantum W-algebra qq-characters generate associative Wqq(g) algebra: quantization of Wq(g) W qq (g) =C 1 [M G,n mono(c 2 S 1 ] {g(x)}/{g(x) s(x)g(x)s 1 (qx)} C q S 1 q-twisted over 27

28 Free field realization Kimura-VP 15! of quiver Wqq-algebra Consider quiver gauge theory! extended Nekrasov partition function Nekrasov- Marshakov! Z(t) = M exp i p=1 t i,p chy [p] i which depends on infinite set of parameters {t} = {t i,p } p =1, 2,...,1 i 0 (generating function of intersection of all Chern characters of Y-bundle over instanton moduli space) 28

29 Free field realisation of quiver Wqq-algebra Interpret Z{t} is a state in Fock space C[[t]] of Heisenberg algebra generated by {t i,p Kimura-VP 15:! c.f. Aganagic talk Zi = Y x2x Ŝ(x) 1i Z Ŝ(x) = d xs(x) q2 X ( S i,x =: s i, p x p + s i,0 x + s i,0 + s i,p x p) : p>0 p>0 29

30 S i,x =: ( s i, p p>0 p>0 ( ) = (1 q1)t p p>0 i,p, s i,0 = t i,0, s i,p = 1 p Here the screening current X s i, p x p + s i,0 x + s i,0 + p>0 is realised in terms of the free fields: s i,p x p) : )) 1 1 q p 2 cij is deformed Cartan matrix of quiver c ij =(δ ji e:j i µ 1 e )+(q 1 δ ji e:i j Elliptic version and non-simply laced version! 30 are defined similarly for generic quiver q 1 µ e ) c [p] ij i,p A1 : Kozcaz s talk

31 qq-characters = free field realisation of Wq1,q2(g) U(n) Example A1 T 1 (x) =Y 1 (x)+y1 1 (q 1 x) ( ) where i,p i,x =: ( i,p p>0 y i, p x p + y i,0 + p>0 y i,p x p) : y i, p =(1 q1)(1 p q2)( c p [ p] ) ij t j,p, y i,0 = c [0] ij t i,0 q 2 1 T 1 (x) Z = n y i,p = 1 p i,p 31

32 The construction of Wq1,q2(Γ) in Kimura-VP 15 is more general then Wq1,q2(g) in Frenkel-Reshetikhin 97 and applicable to any quiver, not necessarily of finite Dynkin type. It depends on parameters Hyperbolic example T 1,x = 1,x + q 1 1 : 2,µ 1 + q 2 1 [ x 2 2,µ 1 3 x x 1 2,µ 1 1,q 1 x : c = ( ) S(µ 1 µ 2 2 )S(µ 1 µ 1 3 ): 1,µ 1 1 µ 2q 1 1,µ 1 1 µ 3q 1 2,µ 1 x 2 2,µ 1 3 x 2,µ 1 1 q 1 x ( µ H 1 (, C ) S(w) = (1 q 1w)(1 q 2 w) (1 qw)(1 w) : +S(µ 2 µ 2 1 )S(µ 2 µ 1 3 ): 1,µ 1 2 µ 1q 1 1,µ 1 2 µ 3q 1 2,µ 1 x 1 2,µ 1 3 x 2,µ 1 2 q 1 x +S(µ 3 µ 2 1 )S(µ 3 µ 1 1 ): 1,µ 1 3 µ 1q 1 1,µ 1 3 µ 2q 1 2,µ 1 x 1 2,µ 1 2 x 2,µ 1 3 q 1 x + 32 : : ]

33 Zgauge=Ztop! W-algebra of (`refined topological strings) What are higher times {t}? Elliptic version and non-simply laced version are defined similarly for generic quiver A1 Kozcaz s talk 33

34 Generalization of Nakajima s results on equivariant K-theory of quiver variety and representation theory of Uq(Lg) to! affine Kac-Moody g with non-zero central charge Quantum q-geometric Langlands is transparent and completely geometric: exchange of ε1 vs ε2 Rich algebraic structures coming! from quantization of HyperKahler moduli spaces 34

35 Thanks to my collaborators!! N. Nekrasov! S. Shatashvili! T. Kimura!! 35

36 Thank you 36