Vertex operator realization of some symmetric functions

Transcription

1 Vertex operator realization of some symmetric functions Jie Yang School of Mathematical Sciences, Capital Normal University USTC, November 14th, 2013

2 Motivation To study counting problem from many different view points From statistical physics From combinatorics From string and M-theory From gauge theory From conformal field theory From algebraic geometry From Lie algebra and representation theory 2/44

3 Outline 1 A brief review of some symmetric functions 2 Vertex operator realization of some symmetric functions 3 Applications Free fermion for topological string theory on C 3 Free fermion for resolved conifold Free fermion and 4d gauge theory The topological vertex B-model free fermion and topological vertex 4 Future work 3/44

4 Free fermion and 2d Yang-Mills theory arxiv: hep-th/ Douglas Quantum mechanics on a group manifold U(N) has a wave function which corresponds to N free fermions on a circle The wave function of an arbitrary excitation of those free fermions is capturized by a Weyl s character formula χ n ( z) = det 1 i,j N z n j i det 1 i,j N z j 1 n F i (1) There is a natural correspondence between this formula and Schur symmetric function 4/44

5 Bosonic excitation and 2D Young diagram 1 1 q = P(n)q n n n=1 n=1 n P(n) Young diagram states a a 2 1 0, a 2 0 5/44

6 Fermionic excitation and 2D Young diagram P(n) has a fermionic excitation description. a2 a1 λ λ = d(λ) i=1 ψ (ai )ψ (b i ) 0 a3 b2 b3 a i = λ i i + 1 2, b i = λ t i i b1 We have the inverse of the generating function in terms of free fermion r N+1/2 where the Virasoro generator is L 0 = (1 q r ) = Tr( ) F q L 0 r Z+1/2 r : ψ r ψ r : 6/44

7 Outline 1 A brief review of some symmetric functions 2 Vertex operator realization of some symmetric functions 3 Applications Free fermion for topological string theory on C 3 Free fermion for resolved conifold Free fermion and 4d gauge theory The topological vertex B-model free fermion and topological vertex 4 Future work 6/44

8 Schur functions Schur function is a generating series for semistandard Young tableaux , 1 2 2, 1 3 2, 1 2 3, 1 1 3, 1 3 3, s (2,1) (x 1,x 2,x 3 ) = x 2 1x 2 +x 1 x 2 2 +x 1 x 2 x 3 +x 1 x 2 x 3 +x 2 1x 3 +x 1 x 2 3 +x 2 2x 3 +x 2 x 2 3. Several definitions:, s λ = a λ+δ a δ, a α = detx α j i, a δ = i<j(x i x j ) (2) s λ = deth λi +j i = dete λ t i +j i (3) where h r and e r are complete and elementary symmetric functions. where (α i β i ) are the Frobenius symbols for λ. s λ = det(s (αi β j )) (4). 7/44

9 Skew Schur function Skew Schur function s λ/µ is a generating function for shape λ µ λ = µ= λ µ = s λ/µ = ν c λ µνs ν where c λ µν is the number of semistandard tableau of shape λ µ weight ν and of lattice word. s µ s ν = λ c λ µνs λ 8/44

10 Macdonald functions Orthogonal basis and kernel functions s λ (x) s λ (x) s µ (y) s µ t(y) P λ (x;q,t) Q µ (y;q,t) i,j i,j i,j 1 1 x i y j (1+x i y j ) n=0 A special case is called Jack function when t = q α. 1 tx i y j q n 1 x i y j q n 9/44

11 Outline 1 A brief review of some symmetric functions 2 Vertex operator realization of some symmetric functions 3 Applications Free fermion for topological string theory on C 3 Free fermion for resolved conifold Free fermion and 4d gauge theory The topological vertex B-model free fermion and topological vertex 4 Future work 9/44

12 Vertex operators Miwa, Jimbo, Date, Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras It began with the study of symmetries hidden in some integrable systems. Vertex operator was discovered from looking for symmetry from n-soliton solution to (n+1)-soliton solution of KdV hierarchy j=0 τ n+1 = e ǫx τ n where { { } X(k) = exp 2 k 2j+1 1 x 2j+1 }exp 2 (2j +1)k 2j+1 x 2j+1 j=0 10/44

13 Boson-fermion correspondence Bosonic field φ(z) = a 0 lnz +i n 0 a n n z n, [a n,a m ] = nδ n+m,0 Fermionic field ψ(z) = ψ r z r 1 2, ψ (z) = ψ sz s 1 2, {ψr,ψ s} = δ r+s,0 r Z+ 1 2 s Z+ 1 2 or where φ(z) =: ψψ : (z) Ψ(z) =: e φ(z) : B, Ψ (z) =: e φ(z) : B, { { Ψ(z) = exp z n t n }exp } z n1 n t n=1 n=1 n z a 0 11/44

14 The bosonic and fermionic Fock spaces correspond to each other via the Φ map and Φ : a n u = nt n u(t), n > 0 Φ : a n u = t n u(t) Φ(ψ(z) u ) = Ψ(z)Φ( u ), Φ(ψ (z) u ) = Ψ (z)φ( u ). Therefore Φ is a map from bosonic and fermionic states to polynomials. 12/44

15 Partition function in B-F correspondence Jacobi triple identity (1 q n ) n=1 j N+1/2(1+q j z)(1+q j z 1 ) = q l2 /2 z l l Z Charge 0 state partition function gives rise to Coeff z 0 (1+q j z)(1+q j z 1 ) = 1 1 q n n=1 j N+1/2 9/2 7/2 5/2 3/2 1/2 1/2 3/2 5/2 7/2 11/2 13/2 13/44

16 Vertex and symmetric functions Define an Hamiltonian H(t) = t n a n n=1 [a n,ψ r ] = ψ n+r, [a n,ψ s] = ψ n+s e H(t) ψ(z)e H(t) = e ξ(t,z) ψ(z) { } e ξ(t,z) = exp t n z n = 1+ f j (t)z i n=1 j=1 When t n = pn(x) n, p n (x) = i xn i is the power sum and f j = s (j) (x) = ρ j 1 z ρ χ (j) ρ p ρ (x) is the one-row Schur function. 14/44

17 State-symmetric function correspondence We can build up a map for states and symmetric functions where Φ : u = Φ( u ) Φ( u ) = z l ( l e H(t) u ) where l denotes the charge of u. ( l e H(t) u ) is a symmetric function in Schur basis when we let t n = p n(x) n. 15/44

18 Schur function From the Hamiltonian we can calculate { } i 0 exp xn i a n µ = s µ (x). n n=1 Therefore define the vertex operator as { } x n Γ ± (x) = exp n a ±n We can get a generating function of charge 0 Fock space F 0 with coefficient the corresponding Schur function (x i ) 0 = i=1γ s λ (x i ) λ λ n=1 16/44

19 Outline 1 A brief review of some symmetric functions 2 Vertex operator realization of some symmetric functions 3 Applications Free fermion for topological string theory on C 3 Free fermion for resolved conifold Free fermion and 4d gauge theory The topological vertex B-model free fermion and topological vertex 4 Future work 16/44

20 Outline 1 A brief review of some symmetric functions 2 Vertex operator realization of some symmetric functions 3 Applications Free fermion for topological string theory on C 3 Free fermion for resolved conifold Free fermion and 4d gauge theory The topological vertex B-model free fermion and topological vertex 4 Future work 16/44

21 Topological string on C 3 : the partition function BPS states partition function in C 3 is capturized by the statistical melted cubic crystal model. BPS states counting gives rise to the same partition function as the 3D Young diagram q π = 1+q +3q 2 + π 17/44

22 3D Young diagram Y 3 3D Young diagram Stability condition: the molecules must be melted from outside to inside. Weakly decreasing numbers fill 2D Young diagram in rows and columns MacMahon function is the generating function of Y 3 n=1 1 (1 q n ) = q π n π all Y 3 (5) 18/44

23 2D and 3D relation The diagonal slicing y x t = 0 1 Interlacing condition of the diagonal slices { λ(t) λ(t +1) t > 0 λ(t) λ(t 1) t < 0 where (6) λ µ λ 1 µ 1 λ 2 µ 2 (7) 19/44

24 Vertex operators Γ ± (x) = exp { n=1 } x n n a ±n (8) a ±n are the bosonic creation and annihilation operators. Γ can be treated as the creation operator while Γ + the annihilation of 2D Young diagrams, i.e. Γ + (1) µ = λ µ λ µ The have Weyl commutation relation λ, (9) Γ + (x)γ (y) = 1 1 xy Γ (y)γ + (x) (10) 20/44

25 Crystal melting partition function for C 3 The statistics of a melting cubic crystal is identified with the partition function of D0-D6 bound states of C 3 MacMahon partition function arxiv: hep-th/ Okounkov, Reshetikhin, Vafa n=1 1 (1 q n ) n = (11) q L 0 Γ + (1) q L 0 Γ + (1) q L 0 Γ }{{} (1)q L0 Γ (1)q L 0 }{{} where we perform commutation relation of vertex operators. λ( 2) λ( 1) λ(0) λ(1) λ(2) 21/44

26 Outline 1 A brief review of some symmetric functions 2 Vertex operator realization of some symmetric functions 3 Applications Free fermion for topological string theory on C 3 Free fermion for resolved conifold Free fermion and 4d gauge theory The topological vertex B-model free fermion and topological vertex 4 Future work 21/44

27 Vertex operators Toric diagram Γ ± Q Γ± { } x n Γ ± (x) = exp n α ±n n { } Γ ( 1) n 1 x n ±(x) = exp α ±n n Γ (1) µ = λ, λ t µ t Γ +(1) µ = λ, λ t µ t n 22/44

28 Refined wall crossing arxiv: collaborated with Haitao Liu Arrow diagrams for chamber n are [ ] Γ+ q i 1 1 ( Q) 1 2 [ ] Γ + q i 1 2 +n 1 q ( Q) 1 2 [ ] Γ q j+n 2 ( Q) 1 2 [ ] Γ q q j ( Q) 1 2 Figure : Arrow diagrams for chamber n of the conifold Stone diagrams with arrows are n=0 n=1 23/44

29 Follow the arrow we can get partition function [ ] [ ] Z crystal = 0 Γ + q i 1 1 ( Q) 1 2 Γ + q i 1 2 +n 1 q ( Q) 1 2 i=1 ] Γ [q 2 ( Q) 1 2 ] Γ [q2( Q) Γ + Γ + Γ [q2( Q) n 1 2 j=1 [ [ q n q 1 2 q n q 1 2 ] 2 ( Q) 1 2 ] 2 ( Q) 1 2 ] [ ] Γ + q q ( Q) 1 2 [ ] [ Γ q j+n 2 ( Q) 1 2 Γ q q j ( Q) 1 2 = M δ=1 (q 1,q 2 )M δ= 1 (q 1,q 2 ) i+j>n+1 i,j 1 (1 q i q j Q 1 ) (1 q i 1 2 i,j=1 ] 0 1 q j Q) 24/44

30 Generalized conifold Γ + Γ + Γ + Γ + Γ + Γ + We have some progress on constracting the free fermion formalism of crystal melting for NCDT chamber and DT chamber and computing the partition function of D0-D2-D6 bound states. 25/44

31 Outline 1 A brief review of some symmetric functions 2 Vertex operator realization of some symmetric functions 3 Applications Free fermion for topological string theory on C 3 Free fermion for resolved conifold Free fermion and 4d gauge theory The topological vertex B-model free fermion and topological vertex 4 Future work 25/44

32 Free fermion in Nekrasov s instanton counting arxiv: hep-th/ Nekrasov and Okounkov For the Nekrasov partition function in the special case ǫ 2 = ǫ 1 =, the instanton part of the partition function is Z inst = k Λ 2N k µ 2 k µ k = (l,i) (n,j) a l a n + (k l,i k n,j +j i) a l a n +j i (12) where l,n = 1,,N, and k correspond to N 2D Young diagrams. For N=1 a l = a n µ λ = i<j λ i λ j +j i j i = x λ 1 h(x) (13) 26/44

33 The operator formalizm of Nekrasov partition function { } { } 1 1 Z = p exp TrE + (z)j(z) exp TrH(z)J(z) Λ 2L 0 { exp 1 } TrE (z)j(z) p (14) where p is the charge p state, Λ is the energy scale, J(z) is the Û(1) 1 current, L 0 is the Virasoro generator, and the matrices of Lie algebra are E + (z) = ze N,1 + i=2 NE i 1,i H(z) = N ξ i E i,i i=1 E (z) = 1 z E1,N + i=2 NE i,i 1 27/44

34 N = 1 Nekrasov s formula for U(1) gauge theory, when ǫ 2 = ǫ 1 =, [ ] 2 Λ 2 λ 1. h(x) λ x λ If we make the identification Λ 2 Q and let q 1 in [ ] 2 Q λ q 2n(λ)+ λ 1, 1 q h(x) λ x λ they are the same. The latter one can be rewritten as Q λ s λ (q 1/2,q 3/2, )s λ (q 1/2,q 3/2, ). λ This is the partition function for topological string on C 3 /Z 2. More complicated examples come from glueing of the topological vertices. 28/44

35 For SU(N) group we need N-component fermion field where ξ r = 1 N+1 (r ). N 2 ψ (r) k = ψ N(k+ξr), ψ (s) l = ψ N(l ξ s) (15) {ψ (r) k,ψ(s) l } = δ r,s δ k+l,0 (16) The sets of charged Maya diagrams of µ and λ (r) satisfy {x i (µ)+n;i 1} = {N(x ir (λ (r) )+p r +ξ r );i 1} (17) arxiv: hep-th/ Maeda, Nakatsu, Takasaki and Tamakoshi It somehow corresponds to the N-core, N-quotient of a Young diagram λ in combinatorics. 29/44

36 Outline 1 A brief review of some symmetric functions 2 Vertex operator realization of some symmetric functions 3 Applications Free fermion for topological string theory on C 3 Free fermion for resolved conifold Free fermion and 4d gauge theory The topological vertex B-model free fermion and topological vertex 4 Future work 29/44

37 A-model Chern-Simons knot theory and topological vertex Witten s famous Jones polynomial paper has given the following formula for a Wilson loop so-called Hopf link vev. Z(S 3 ;L(R i ;R j,r k )) = i j k = S ijs ik S 0i S 3 In the paper he calculated the vev. for U(2) gauge group. Later a general polynomial HOMFLY was discovered for U(N) group and when N Schur polynomial comes into the story. 30/44

38 Zhou s identity Jian Zhou, A conjecture on Hodge integrals, math/ s ν (E µ (t)) = ( 1) ν q κν/2 ρ q ρ s µ/ρ(1,q, )s ν/ρ (1,q, ) s µ (1,q, ) where the formal series for the elementary polynomials is E µ (t) = (1+q µ j j t) j=1 31/44

39 Using Zhou s identity to relate Hopf link to topological vertex Aganagic, Klemm, Mariño, Vafa, The topological vertex i S 3 j k R i R j R k 32/44

40 C λµν ν µ λ 33/44

41 Iqbal-Kozçaz-Vafa refined topological vertex Z ν = m Z+ 1 2 Γ ǫ (x ǫ m ) = i,j= q i t j 1 1 q a(s)+1 t l(s) where ǫ = ± correspond to different colored dots and x m denote their coordinates and Γ are the same vertex operators. s ν 34/44

42 Refined topological vertex in another approach Heisenberg algebra over Q(q, t) [a m,a n ] = m 1 q m 1 t m δ m+n,0. (18) Besides Γ ± we also introduce the generalized vertex operators ( ) 1 1 t n G + (λ i,v) = exp n1 q na n(q 1 2 λ i t i 1 ) n v n, (19) n=1 and ( ) 1 1 t n G (λ i,v) = exp n1 q na n(q λ 1 i+ 2 t i ) n v n. (20) n=1 35/44

43 Vertex realization (collaborated with Liqiang Cai, Lifang Wang, Ke Wu) The first formula i,j=1 Γ 1 + (q 1 2 t k v 1 )Γ 1 + (q 3 2 t k v 1 )Γ 1 + (q 5 2 t k v 1 ) G+ 1 (λ 1,v)G+ 1 (λ 2,v)G+ 1 (λ 3,v) G+ 1 (λ k,v) G (λ 1,v)G (λ 2,v)G (λ 3,v) G (λ k,v) Γ 1 (q 1 2 t k v)γ 1 (q 3 2 t k v)γ 1 (q 5 2 t k v). After normal ordering we get a factor 1 1 q i t j 1 (s λ) 1 1 q a(s)+1 t l(s) (s λ) 1 1 q a(s) t l(s) 1 (21) Hope to relate to the generating function of k ϕk χ y (Hilb k [C 2 ]) χ y (Hilb k [C 2 ]) = (1 yt 1+a(s) q l(s) )(1 yt a(s) q 1 l(s) ) ν (1 t 1+a(s) q l(s) )(1 t a(s) q 1 l(s) ) s ν ν=k which corresponds to the partition function of the refined topological string over the following toric geometry. 36/ 44

44 For q = t we have the second formula Γ 1 + (q 3 2 +k v 1 )Γ 1 + (q 1 2 +k v 1 ) Γ 1 (q 1 2 k v)γ 1 (q 3 2 k v) Γ 1 (q λ k+ 1 2 k v)g (λ k,v)g 1 + (λ k,v) λ 1 1 λ 2 2 k k 1 k 2 k+1 k+2 λ k 2 k +6 = λ k 3 k +3 λ k 1 k +1 λ k k 1 λ k k 37/44

45 The third formula gives rise to the refined topological vertex m=1 1 (1 q m ) kγ 1 + (q 1 2 t k v 1 )Γ 1 + (q 3 2 t k v 1 )Γ 1 + (q 5 2 t k v 1 ) Γ 1 (q 1 2 t k v)γ 1 (q 3 2 t k v) G (λ k,v)g+ 1 (λ k,v)g (λ k 1,v)G+ 1 (λ k 1,v) G (λ 2,v)G+ 1 (λ 2,v)G (λ 1,v)G+ 1 (λ 1,v). (22) 38/44

46 Outline 1 A brief review of some symmetric functions 2 Vertex operator realization of some symmetric functions 3 Applications Free fermion for topological string theory on C 3 Free fermion for resolved conifold Free fermion and 4d gauge theory The topological vertex B-model free fermion and topological vertex 4 Future work 38/44

47 Ward identity In B-model D-branes are free fermion insertions on the Riemann surfaces. For C 3 there are 3 fermion insertions which give rise to 3 patches on a sphere. W 1+ is the symmetry group which preserves the symplectic form. A current of W algebra has Ward identity or total residue 0 on the Riemann surface W i C i = 0 (23) 39/44

48 Bogoliubov transformation u φ(u)e nu + v φ(v)e n[iπ v vφ(v)] + w φ(w)e n wφ(w) = 0 u v w (24) From a 0-vacuum to an -vacuum, there is a Bogoliubov transformation. Define fermion vacuum state vac, vac = m 0ψ m+1/2 ψ m+1/2 0 (25) A Bogoliubov transformation of the fermion vacuum state is { } V = 0 exp a mn ψ m+1/2 ψn+1/2 m,n 0 (26) Ward identity and the information of free fermion insertions determine a mn 40/44

49 Topological vertex in B-model When V is determined, the topological vertex for C 3 is then C λµν = V λ (1) µ (2) ν (3) (27) where λ,µ,ν are 2d Young diagrams. From the construction of V, in principle we can compute C λµν. 41/44

50 Outline 1 A brief review of some symmetric functions 2 Vertex operator realization of some symmetric functions 3 Applications Free fermion for topological string theory on C 3 Free fermion for resolved conifold Free fermion and 4d gauge theory The topological vertex B-model free fermion and topological vertex 4 Future work 41/44

51 Integrable systems Symmetric functions in integrable systems It was originated by Jimbo, Miwa, and collaborators that symmetric functions are related to KdV, KP hierarchy and they also provided vertex realization to study the symmetries in those systems. Later in some quantum integrable system symmetric functions are eigenfunctions of those Hamiltonian system, such as Calogero-Sutherland, some spin chain system. Symmetric functions and refinement The relation of Macdonald function and which kind of integrable system 42/44

52 Wall crossing Quantum dilogarithm n>0 Invariants and symmetries exp x n[n] = n=0 1 1 xq n+1/2 ûˆv = qˆvû E(û) = (1+q n 1/2 û) n=1 E(û)E(ˆv) = E(ˆv)E(q 1/2 ûˆv)e(û) How is Heisenberg algebra categorification going to enter this story? Namely, may the symmetry of the categorification inspire us on how we can find operators which satisfy Kontsevich-Soibelman motivic wall-crossing formula? 43/44

53 Thank you! 44/44