Small Quantum Cohomology as Quantum Integrable System

Transcription

1 Small Quantum Cohomology as Quantum Integrable System Christian Korff Reader and University Research Fellow of the Royal Society School of Mathematics & Statistics, University of Glasgow ck/ Integrability in Topological Field Theory, 6 2 April 22

2 Outline Outline Reminder: (small) quantum cohomology Quantum Kostka numbers and toric Schur functions Vicious and osculating walkers on the cylinder Free fermion description

3 Related literature Related literature ( ck/publications.html) quantum cohomology/wznw fusion rings as integrable models: C. K. and C. Stroppel, Adv Math 225 (2) 2-68; arxiv: (type A: g = û(n) k and ŝu(n) k ) C. K. A combinatorial derivation of the Racah-Speiser algorithm for Gromov-Witten invariants; arxiv: C. K. J Phys A 3 (2) 32; arxiv:6.7 C. K. QC via vicious and osculating walkers; arxiv:2.9 Bethe vectors as idempotents: discrete quantum NLS model C. K. RIMS Kokyuroku Bessatsu B28 (2) pp. 2-53; arxiv:6.532 (Section 5, Cor 5.3 and Section 7) C. K. Cylindric Macdonald functions and a deformation of the Verlinde algebra. preprint; arxiv:.6356 (Section 7, Prop 7.5)

4 Introduction The small quantum cohomology ring Gr n,n+k Grassmannian of n-planes in C n+k small quantum cohomology ring [Siebert-Tian 997] qh (Gr n,n+k ) = Z[q][e,...,e n ]/ h k+,...,h n+k,h n+k +q( ) n where h r = det(e i+j ) i,j r and a vector space basis is given by {s λ := det(e λ i i+j) i,j k } with λ {partitions with Young diagram in n k box } Fusion ring of û(n) k Wess-Zumino-Novikov-Witten model [Gepner, Intriligator, Vafa, Witten] and [Agnihotri]: F Z n,k = qh (Gr n,n+k )/ q F n,k := F Z n,k Z C is called Verlinde algebra.

5 Introduction Reminder Schubert varieties and rational maps Given a flag F F 2 F n+k = C n+k the Schubert variety Ω λ (F) is defined as Ω λ (F) = {V Gr n,n+k dim(v F k+i λi ) i, i =,...n}. Definition of 3-point Gromov-Witten invariants C ν,d λ,µ = # of rational f : P Gr n,n of degree d which meet Ω λ (F), Ω µ (F ), Ω ν (F ) for general flags F,F,F modulo automorphisms in P. If there is an number of such maps, set C ν,d λ,µ =. Poincaré duality: ν = (k ν n,...,k ν ) Schubert class: [Ω λ ] s λ

6 Introduction Quantum Kostka numbers and Gromov-Witten invariants Quantum Kostka numbers and Gromov-Witten invariants [Bertram, Ciocan-Fontanine, Fulton]: s µ s λ s λr = s µ s ( λ ) s ( λr) = d,ν (n,k) d,ν (n,k) q d s ν K ν/d/µ,λ q d s ν K ν /d/µ,λ Quantum Giambelli formula [Bertram]: s λ = det(s λi i+j) s µ s λ = d,ν (n,k) 3-point, genus Gromov-Witten invariants q d C ν,d λµ s ν, d = λ + µ ν n+k

7 Introduction Quantum Kostka numbers and Gromov-Witten invariants Proposition (intersection pairing) F n,k is a commutative Frobenius algebra with η(s λ,s µ ) = δ λ µ. A Frobenius algebra A is a finite-dimn l, unital, assoc algebra with non-degenerate bilinear form η(a b,c) = η(a,b c), a,b,c A. Id: A A : k A η: A A k m: A A A Δ: A A A Topological quantum field theories [Witten, Segal, Atiyah] Commutative Frobenius algebras are categorically equivalent to 2D topological quantum field theories. [Dijkgraaf]

8 Introduction Frobenius structures and toric Schur functions Computation of the coproduct Frobenius isomorphism Φ : s λ η(s λ, ) F n,k Φ F n,k F n,k F n,k Φ Φ m F n,k F n,k Proposition (generalised skew Schur function) s ν = d,µ s ν/d/µ s µ, s ν/d/µ := λ C ν,d λµ s λ.

9 Introduction Frobenius structures and toric Schur functions Toric Schur function [Postnikov]: s ν/d/µ (x,...,x n ) = λ = λ C ν,d λµ s λ(x,...,x n ) K ν/d/µ,λ m λ (x,...,x n ), -many variables: cylindric Schur functions [Gessel-Krattenthaler] [McNamara] [Lapointe-Morse] [Lam] Fusion ring as quantum integrable model (Korff-Stroppel 2) Identify the toric Schur functions as partition functions and the fusion ring as the quantum integrals of motion. Other example: XXX Bethe algebra [Varchenko et al]

10 Vicious and osculating walkers -words and partitions Reminder: correspondence between -words and partitions n = k = w = w V= w' = Rot(w) = The following bijections induce symmetries of GW invariants: w w = w N...w 2 w w w = ( w N ) ( w 2 )( w ) w Rot(w) := w 2 w 3...w N w Position of -letters: l i (λ) = λ n+ i +i

11 Vicious and osculating walkers Non-intersecting lattice paths Vicious and osculating walkers on the cylinder Statistical models on n N and k N square lattice with periodic boundary conditions in the horizontal direction (N = n+k). 2 3 N 2 3 N l (μ) l 2(μ) l n(μ) l (μ) l 2(μ) l n(μ) 2 2 n k l (ν) l 2(ν) l n(ν) vicious walkers l (ν) l 2(ν) l n(ν) osculating walkers

12 Vicious and osculating walkers Non-intersecting lattice paths Allowed vertex configurations and their weights x i indeterminate assigned to the i th lattice row. weight: x i x i time (percolation c.f. [Brak][Fisher][Forrester][Guttmann et al][wu])

13 Vicious and osculating walkers transfer matrices Row partition functions Fix start/end positions via -words w(µ),w(ν) of length N. Definition (transfer matrices) Weighted sums over row configurations: E(x i ) ν,µ := H(x i ) ν,µ := # of outer edges # of horizontal edges q 2 x i osc row config # of outer edges # of horizontal edges q 2 x i vicious row config Proposition (integrability commuting transfer matrices) E(x)E(y) = E(y)E(x), H(x)H(y) = H(y)H(x), E(x)H(y) = H(y)E(x)

14 Vicious and osculating walkers Weighted path counting Theorem (Generating function for Gromov-Witten invariants) The partition functions have the following expansions, (H(x n ) H(x 2 ) H(x )) ν,µ = d q d s ν/d/µ (x,...,x n ) (E(x k ) E(x 2 ) E(x )) ν,µ = d q d s ν /d/µ (x,...,x k ) Let h(s),c(s) be hook length and content of s λ. Corollary (Sum rule for Gromov-Witten invariants) Set x i = q = for all i n. Then H n ν,µ = d,λ C ν,d λµ s λ n+c(s) h(s) = E n ν,µ

15 Vicious and osculating walkers Proof Cylindric loops λ[r] := (...,λ n +r +k r n,λ +r r+ 2 λ[],...,λ n +r r+n k 3 λ[2] 5,λ +r k,...) r+n+...

16 Vicious and osculating walkers Proof Cylindric skew tableaux λ/d/µ := { i,j Z Z/(n, k)z λ[d] i j > µ[] i } n k 3 2 λ[2] 5... μ[] 3 2 2

17 Vicious and osculating walkers Proof Proposition Vicious/osculating paths are in bijection with cylindric tableaux. 2 3 N n Level-rank duality: τ H = E τ with τ : λ λ K ν/d/µ,λ = # of cylindric tableaux of weight λ [BCF][Postnikov]

18 Vicious and osculating walkers Quantum integrals of motion: XX spin-chain Quantum integrals of motion Define matrices S (a b) via the expansion H(x)E(y) = +(x +y) x a y b S (a b) a,b Definition (Fusion matrices) Let λ = (α,...,α r β,...,β r ) with λ,λ < N. S λ := det(s (αi β j )) i,j r Proposition (Functional relation) H(x)E( x) = +( ) n qx n+k

19 Vicious and osculating walkers The algebraic Bethe ansatz Theorem (Korff-Stroppel 2) There exists an orthogonal basis {e λ } λ (n,k) such that the matrices H,E and the S λ s are diagonal. 2 mapping the e λ s onto the idempotents of F n,k yields an algebra isomorphism, in particular S λ S µ = q d C ν,k λµ S ν. d,ν (n,d) XX-Heisenberg spin chain The transfer matrices H, E commute with the Hamiltonian of the so-called quantum XX-Heisenberg spin chain.

20 Free fermion description Clifford algebra Fermion creation and annihilation Fix N = n+k and consider the vector space (Fock space) F = N F n,k, F n,k = CW n,k, n= where F,N = C{ } = C and w = is the vacuum. Let n i (w) = w + +w i be the number of -letters in [,i]. For i N define the (linear) maps ψ i,ψ i : F n,k F n±,k, { ψ ( ) ni (w) w, w i = and w j = w j +δ i,j i(w) :=, w i = { ( ) ni (w) w, w i = and w j = w j δ i,j ψ i (w) :=, w i =.

21 Free fermion description Clifford algebra Example Take n = k = and µ = (,3,3,). The boundary ribbon (shaded boxes) starts in the (3 n) = diagonal. Below the diagram the respective -words w(µ) and w(ψ 3 µ) are displayed.

22 Free fermion description Clifford algebra Proposition (Clifford algebra) The maps ψ i,ψ i : F n F n,k± yield an irred rep of the Clifford algebra, i.e. one has the relations (i,j =,...,N) ψ i ψ j +ψ j ψ i = ψ iψ j +ψ jψ i =, ψ i ψ j +ψ jψ i = δ ij. Introducing w,w = i δ w i,w i any w,w F. one has ψ i w,w = w,ψ i w for Bijections: partitions -words words in the Clifford algebra λ w(λ) ψ l (λ) ψ l n(λ) The Clifford algebra is the fundamental object in the description of quantum cohomology.

23 Free fermion description Return of the vicious walkers Connection with vicious walkers Expansion of the transfer matrix H(x) : F n,k F n,k, H(x) = N x r H r, H r = r= α =r ψ α N N uα N N uα (( )r qψ ) α N where u i = ψ i+ ψ i shifts one particle from site i to site i +. Proposition (commutation relation with fusion matrices) S λ (q)ψ i = ψ is λ ( q)+ l(λ) ψ i+r r= λ/µ=(r) S µ ( q) where ψ j+n = ( )n+ qψ j and n = particle number operator.

24 Free fermion description Fermion creation of quantum cohomology rings Fermion creation of quantum cohomology rings Corollary (Korff-Stroppel 2) The last commutation relation implies the product formula λ ψ i (µ) = S λ(q)ψ i (µ) = λ r=λ/ν=(r) where denotes the product with q replaced by q. Inductive algorithm ψ i+r (ν µ) One can successively generate the entire ring hierarchy {qh (Gr n,n+k )} n+k n= starting from n =.

25 Free fermion description Fermion creation of quantum cohomology rings Example Consider the ring qh (Gr 2,5 ). Via ψ i : qh (Gr,5 ) qh (Gr 2,5 ) one can compute the product in qh (Gr 2,5 ) through the product in qh (Gr,5 ): = ψ ( 2+2 = = qψ 2+2 ( ψ 2 ) ( ) +ψ 2+3 ( qψ 2+3 ) ( ) = q +q. )

26 Free fermion description Fermion creation of quantum cohomology rings Fermionic product formula Let λ,µ (n,k). Then w(λ) w(µ) := T ψ l (µ)+t n ψ l 2 (µ)+t n ψ l 3 (µ)+t n 2 ψ l (µ)+t n 3, where l i (µ) positions of -letters in w(µ) T = (semistandard) tableau of shape λ t i = number of entries i n in T ψ i = ψ i for i =,...,N and ψ i+n := ( )n+ qψ i, ψ i+n := ( ) n q ψ i, n := N ψ iψ i. i=

27 Free fermion description Fermion creation of quantum cohomology rings Example Set N = 7, n = N k = and λ = (2,2,,), µ = (3,3,2,). Step. Positions of -letters: l(µ) = (l,...,l ) = (2,,6,7). Step 2. Write down all tableaux of shape λ such that l = (l +t n,...,l n +t ) modn with l i l j for i j. 2, 3 (3,5,7,9) 2 3, (3,5,7,9) 2 2, (3,,8,9) 3, (,5,6,9) 3 2, 3 (3,6,7,8) 2 3 3, (3,6,7,8) Step 3. For each l i > N make the replacement ψ l , (3,6,8,7) 3 2, (,5,7,8) ψ l ( )n+ qψ n l ψ l i N ψ l n 2 3, (,5,7,8) Step. Let l be the reduced positions in [,N]. Choose permutation π S n s.t. l < < l n and multiply with ( )l(π) (,5,8,7).

28 Free fermion description Fermion creation of quantum cohomology rings The three tableaux 3 2, (,5,7,8) 2 3, (,5,7,8) (,5,8,7) yield the same -word w =, λ(w) = (3,2,2,) but with changing sign, ψ l +2 ψ l 2 +ψ l 3 + ψ l + = ψ l +2 ψ l 2 +ψ l 3 + ψ l + = We obtain the product expansion ψ l +2 ψ l 2 + ψ l 3 +2 ψ l = q ψ ψ ψ 5 ψ 7. = q +2q +q +q +q 2.

29 Free fermion description Fermion creation of quantum cohomology rings Corollary (Quantum Racah-Speiser Algorithm, C.K. 29) Let λ,µ,ν P n,k. Given a permutation π S n set α i (π) = (l i (ν) l π(i) (µ))modn d(π) = #{i l i (ν) l π(i) (µ) < }. Then one has the following identity for Gromov-Witten invariants, C ν,d λµ = ( ) l(π)+(n )d K λ,α(π), π S n,d(π)=d where K λµ are the Kostka numbers. Setting q = the formula specializes to the known Racah-Speiser algorithm for Littlewood-Richardson coefficients.

30 Free fermion description The End Thank you for your attention!

31 Kirillov-Reshetikhin crystals Basic notions of crystal theory Example KR crystal: g = ŝl N, B r = { -words with r -letters} e i w = f i w = { (w,...,w i =,w i =,...,w N ), w i =, w i =, else { (w,...,w i =,w i+ =,...,w N ), w i+ =, w i =, else

32 Kirillov-Reshetikhin crystals Basic notions of crystal theory Example (cont d) Affine nil Temperley-Lieb algebra f 5 f f i 2 = f i f i+ f i = f i+ f i f i+ =

33 Kirillov-Reshetikhin crystals Basic notions of crystal theory Tensor products of crystals B B is the set B B together with the maps, { e i (b b ei (b) b ) =, ε i (b) > ϕ i (b ) b e i (b ), else { f i (b b fi (b) b ) =, ε i (b) ϕ i (b ) b f i (b ), else where one sets b = and b =

34 Kirillov-Reshetikhin crystals Basic notions of crystal theory Things can get more complicated... Set N = 5 and consider the KR crystal B = B 2 B B : KR crystals are perfect crystals: they and their tensor products are always connected.

35 Kirillov-Reshetikhin crystals Basic notions of crystal theory The combinatorial R-matrix Theorem (Kashiwara et al) There exists a unique graph isomorphism R r,s : B r B s B s B r which preserves the crystal structure. Example (c.f. Nakayashiki-Yamada) Let N = 6 and r = 3,s = 2. Then we find R 3, = 2 6 3

36 Kirillov-Reshetikhin crystals Lattice paths as crystal vertices Lattice paths as crystal vertices T ν,µ (λ ) cylindric tableaux of shape ν /d/µ and weight λ. Define ι : T ν,µ (λ ) B λ B λ 2 B λ k as follows: 2 3 N k λ = 5,5,3,2

37 Kirillov-Reshetikhin crystals Main theorem For simplicity assume d > d min := max l { l i= (w i(ν) w i (µ))}. Theorem (Osculating walkers as crystal vertices) Let b B λ := B λ B λ k. The following are equivalent. (i) b ι(t ν,µ (λ )) (ii) ϕ(b) = i I ( w i+(µ))ω i, ε(b) = i I ( w i+(ν))ω i. (iii) R λ (b µ Rot b) = b b ν, where Rot is the ŝl N Dynkin diagram automorphism. Here R λ := R n,λ r R n,λ 2 R n,λ is the unique crystal graph isomorphism R λ : B n B λ B λ B n.

38 Kirillov-Reshetikhin crystals Proof: (i) (ii) b b 2... b r b r+ i i+ b b b 2 b r i... i+ i i+ b b b 2... b r i i+ b b 2... b r b r+

39 Kirillov-Reshetikhin crystals Proof: (ii) (iii) i i ε(b) 2 i n b φ(b) j j 2 j n w (ν) = i + r w (μ) = j + r Claim It follows from (ii) that ε(b µ Rot b) = ε(b b ν ) and ϕ(b µ Rot b) = ϕ(b b ν ). Claim + uniqueness of combinatorial R (iii)

40 Kirillov-Reshetikhin crystals Proof: (iii) (i) Osculating walks via the combinatorial R-matrix b μ - - Rot (b) Rot (b) R b ν (A) 2 N b b μ (B) b ν

41 Kirillov-Reshetikhin crystals Littlewood-Richardson coefficients The case of minimal degree d = d min := max l { l i= (w i(ν) w i (µ))}. Proposition (Fulton-Woodward, Yong, Postnikov) There exists a N such that K ν/d/µ,λ = K Rot a (ν)/rot a (µ),λ and C ν,d λ,µ = CRota (ν), λrot a (µ) Robinson-Schensted-Knuth correspondence: B λ = B(α) SST(α,λ ), α λ where B(α) is the irred sl N -crystal of highest weight α.

42 Kirillov-Reshetikhin crystals Littlewood-Richardson coefficients Previous example with N = 5 and B 2,, = B 2 B B :

43 Kirillov-Reshetikhin crystals Vicious walks and the combinatorial R-matrix Vicious walks τ : B n B N n, b λ b λ swaps zero and one-letters in -word b and then reverses its order. Define R r,s := ( τ)r N r,s (τ ) and R λ := R n,λ r R n,λ. Corollary Let b B λ. The following statements are equivalent. (i) b ι(t ν,µ (λ)) (ii) ϕ(b) = i I w i+(µ )ω i, ε(b) = i I w i+(ν )ω i (iii) R λ (b µ Rot b) = b b ν

44 Kirillov-Reshetikhin crystals Vicious walks and the combinatorial R-matrix Constructing vicious from osculating walks (A) (B) b μ b μ' Rot - (b) Rot - V (b ) b ν b ν'

45 Kirillov-Reshetikhin crystals Symmetries of quantum Kostka numbers Corollary We have the following symmetries of quantum Kostka numbers, K ν/d/µ,λ = K ν/d/µ,si λ = K Rot(ν)/d R /Rot(µ),λ = K µ /d/ν,λ, where s i λ = (...,λ i+,λ i,...) and d R = d +w (µ) w (ν). Proof Yang-Baxter equation: R 23 R 2 R 23 = R 2 R 23 R 2 Rotation (Dynkin diagram automorphism) Rot(b b 2 ) := Rot(b ) Rot(b 2 ), Rot R = R Rot. reversing -words (Lusztig involution) : B r B s B s B r with b b 2 b 2 b, f i = e N i, e i = f N i, R r,s = R s,r.

46 Conclusions Conclusions Additional results Eigenvectors of the transfer matrices = idempotents. Yang-Baxter algebras, affine nil Temperley-Lieb algebra and Schur polynomials. Algorithms to generate vicious/osculating walks. The ŝl N-Verlinde algebra and cylindric Macdonald functions. Outlook Combinatorial definition of GW invariants and positivity. Other Lie algebras. Quantum Horn conjecture. Categorification of integrable systems.