ADE quiver theories and quantum integrable systems

Transcription

1 ADE quiver theories and quantum integrable systems Vasily Pestun, IAS Texas A&M, Mitchel Ins2tute, April 16, 2013

2 Non- perturba7ve QFT O(X) = DXe 1 g 2 S(X) O(X) S(x) Feynman diagrams Perturba7ve: x x 0 1 O = g k O k Non- perturba7ve: other cri7cal points contribute...e S(X 1 ) g 2 S(X) =S(X 0 )+ 1 2 S (X 0 )(X X 0 )

3 Why? Physics: Non- perturba7ve real QCD: L UV but in experiment we measure hadron states in IR N =2 SQFT QFT Dynamical supersymmetry breaking: might explain the hierarchy problem [WiQen] May be supersymmetry in UV? May be N =2 / N =1hybrid model (In a hidden sector and/or in UV) (Also mo7va7ons in string/brane construc7ons) [Antoniadis & others]

4 Mathematical physic, statistical mechanics and condensed matter: classical and quantum mechanical integrable systems (Toda and others) [ITEP] exactly solvable lauce models and spin chains (Yang- Baxter, Bethe ansatz) 2d conformal field theory [Nekrasov- Shatashvili] Z N =2 (S 4 ) = V 1...V n [V. P. ] [AGT] [Dorn OQo, Zamolodchikov- Zamolodchikov] [WiQen]

5 Mathematics: Quantum geometry of various moduli spaces: instantons, monopoles, Hitchin systems [A7yah, Hitchin, Donaldson, Nakajima & others] Geometric representa7on theory Mirror symmetry & top strings; GW, SW, GV, DT invariants Quantum groups [Faddeev, Drinfeld, Jimbo] Wall crossing, BPS states coun7ng [KS, GMN,CV] Cluster algebras, Y- systems [Zamolodchikov, Fomin- Zelevisnky]

6 N =2 {Q i α,q j β} =2P µ σ µ α β δij N =2 helicity Adj (G) 1 A N =2 helicity Rep(G) q 1/2 1/2 λ λ h h 0 0 Φ q - 1/2 N =1 N =1 N =1 N =1 1 2g 2 YM trf F trψ Dψ

7 The space of vacua Supersymmetric ground states, or vacua, are zero energy states u such that H u =0 In fact: zero energy state <=> supersymmetric state u H u = u Q Q u = Q u 2 [WiQen] N =2 theories have infinitely degenerate ground state! (without any obvious symmetry rela7ng different ground states, e.g. the spectrum of excita7ons is different) Define: M = u also known as the space of vacua [Seiberg- WiQen]

8 Classical space of vacua scalar poten7al: V =tr[φ, Φ ] 2 V =0 [Φ, Φ ]=0 Φ t G maximal commu7ng subalgebra of Lie(G) M = t G /W Weyl group

9 Quantum picture Classical space vacua M = t G /W Quantum space of vacua M Classical singulari7es are resolved (But new quantum singulari7es appear!)

10 Non- abelian UV gauge theory N =2 G N =2 Rep(G) rk(g) =r UV L UV = 1 2g 2 tr(f µν F µν + iψ Dψ +...) Abelian IR non- linear sigma model Maps(R 3,1, M) U(1) r metric on M ds 2 =Imτ ij da i dā j is special Kahler: special coordinates on M τ ij (a) = 2 F(a) a i a j prepoten2al IR [Seiberg- WiQen]

11 What is F(a)? Typically: F(a) = 1 2 τa2 + a 2 log a + classical one- loop k=0 F k (a)q k instanton (non- perturba7ve) instanton expansion parameter q =exp(2πiτ) =e 8π 2 g 2 +iθ complexified gauge coupling constant contribu7on of charge k instanton τ = 4πi g 2 F k (a) + θ 2π

12 Algebraic integrable system: hyperkahler fibra7on [Donagi- WiQen] fibers: complex tori (abelian variety) P dim C P =2r dim C ( ) =r a i = d 1 ω A i a D i = d 1 ω B i dim C M = r base M: quantum space of vacua a i are coordinates on M a D i are dual coordinates on M a D i = F a i

13 Seiberg- WiQen curve [Seiberg- WiQen] a and a D are periods of Seiberg- WiFen differen2al a i = A i λ SW a D i = B i λ SW quantum space of vacua M

14 Goal: Given the N =2 QCD (G,R) in UV, compute (sum up all instantons): The prepoten7al F(a) The SW curve and the differen7al λ SW The algebraic integrable system P M The following is based on a joint work with Nikita Nekrasov (2012) arxiv:

15 Quiver gauge theories We solve this class of N=2 theories: Gauge group is G = i I SU(N i ) MaQer in fundamental, an7- fundamental, bi- fundamental or adjoint representa7ons The theory is asympto7cally free or conformal in UV, i.e. the UV beta- func7on is nega7ve or zero ( top strings: [KMV] )

16 Quiver graph SU(5) N f 1 SU(9) N f 2 SU(6) SU(5) SU(3) N f =0 N f =0 N f 1 good UV theory: simple gauge group factors: nodes of the graph bifundamental maqer: connects two nodes β i 0 2N ( ) i N ( ) i + j:<ij>=0 N ( ) j

17 Chiral ring operators O k =trφ k correla7on func7on factorize O k O l = O k O l But, because of contact terms the naive algebraic rela7ons are corrected by instantons. For example, for G = SU(2) trφ 4 = 1 2 trφ2 2 + k c k q k

18 Density func7ons Density func7ons all powers trφ k ρ(x) ρ(x) trφ k = capture expecta7on values of x k ρ(x)dx support of density func7on Φ = diag(a 1,a 2,...) Density poten7als a 1 a 2 x y(x) =exp log(x x )ρ(x )dx

19 Density poten7als y(x) trφ k = 1 2πi x k d log y(x) density poten7als capture all chiral ring structure y(x; u) =x N exp k=1 x k k trφk u u M The Riemann surface of the analy2c poten2al y(x; u) is Seiberg- WiFen curve at a point u M of the quantum space of vacua

20 Density poten7als y(x) : The associated Riemann surface is the SW curve: G = SU(N) A- cycles x- plane density support intervals are the branch cuts for y(x) mirror x- plane: glued to x- plane at the cuts

21 What is the SW curve, exactly? What is generaliza7on for non- simple gauge groups such as in SU(Ni ) quiver theories?

22 Cross- cut equa7ons x- plane y + (x) y - (x) In quiver theory there is density ρ i (x) for each SU(N i ) factor trφ n y i (x) =exp log(x x )ρ i (x )dx i = x n ρ i (x)dx Let P i (x) encode the fund maqer and couplings q i : ( ) Ni P i (x) =q i (x m i,f ) f=1 The master equa2ons y + i (x)y i (x) =P i(x) y j (x + m ij ) j:<ij>=0

23 Cross- cut transforma7ons If quiver graph is acyclic, represent bi- fund masses as and redefine m ij = m i m j y i (x) y(x + m i ) The cross- cut equa7ons can be thought as cross- cut transforma7ons: x- plane y i (x) y 1 (x)p i i (x) j:<ji>=0 y j (x)

24 Cross- cut invariants The density poten7als y i (x) jump across the cuts. We shall construct cut- crossing invariants from y i (x) and then solve for y i (x) algebraically. For example, for simple gauge group G = SU(N) across the cut: y(x) P (x) y(x) Hence χ[y(x)] y(x)+ P (x) y(x) is cross cut invariant! Hence it is analy7cal func7on on en7re x- plane In addi7on, y(x) x N +... as x hence χ[y(x)] is degree N polynomial in x

25 SW curve for SU(N) with fundamental maqer Hence, we have shown where and u 0 = y(x)+ P (x) y(x) = T (x) P (x) =q N ( ) f=1 (x m f ) T (x) =u 0 x N + u 1 x N u N 1, N ( ) < 2N 1+q, N ( ) =2N Choice of vacua u M

26 Cross- cut invariants for quiver theories y i (x) y 1 (x)p i i (x) j:<ji>=0 y j (x) x- plane How we get invariants of the above transform for quivers? SU(5) SU(9) We shall use the deep fact: the N=2 quiver graphs fall in the famous ADE classifica7on N f 1 N f 2 SU(6) SU(5) N f 1 SU(3) N f =0 N f =0

27 Recall the assump7ons: ADE classifica7on Gauge group is G = i I SU(N i ) MaQer in fundamental, an7- fundamental, bi- fundamental or adjoint representa7ons β i 0 Theorem: The quiver graph is the ADE or affine ADE [KMV] Dynkin graph Adjacency matrix Cartan Proof: Non- posi7ve beta- func7on implies C ij N j 0 where C ij =2δ ij A ij then proceed as in Coxeter or Kac classifica7ons

28 Polyhedron Γ T a,b,c Affine Dynkin graph γ with labels i(a i ) Lie g Z r+1 (r=5) T r,1, A r (r=5) BD r 2 (r=7) T r 2,2, D r (r=7) 01 BT T 3,3,2 ( T 3,3,3 ) E 6 22 BO T 4,3,2 ( T 4,4,2 ) E 7 23 BI T 5,3,2 ( T 6,3,2 ) E 8 McKay s observation is that γ is affine ADE Dynkin diagram, with the trivial representation R 0 associated with the affine node 0. The finite Dynkin graph γ fin is always a tri-star graph T a,b,c with one trivalent vertex and three legs containing a, b, c vertices (where in the counting we included the center trivalent vertex). Hence the rank of the finite quiver γ fin is a + b + c 2. Infacta, b, c have simple interpretation:

29 Cross cut invariants for N = 2 quivers are ADE Weyl invairiants the transforma7on y i (x) y 1 (x)p i i (x) j:<ji>=0 y j (x) is equivalent to the i- th Weyl reflec7on on the ADE quiver group element: g(x) = i I y i (x) α i P i (x) Λ i here α i, Λ i : C T G are coroots (coweights) of the ADE quiver group A 1 example: y(x)/ P (x) 0 g A1 (x) = 0 P (x)/y(x)

30 SW curve for ADE quiver theories Take for as the cross- cut invariants the system of fundamental characters χ i = j P (Λ i,λ j ) tr (g(x)) = y j Λi i + P i y i <ji>=0 y j +... The SW curve is defined by the system of equa7ons {χ i [(y j ) j I ]=T i (x; u), i I} where χ i are ADE characters Main result [N.Nekrasov, V.P. 2012] and T i (x;u) are polynomials in x of degree N i, with coefficients u parametrizing moduli space M

31 Method To derive the cross- cut equa7ons on the density poten7als y i (x) we compute the par77on func7on of N = 2 theories in a twisted space- 7me R 4 1, 2 R 4 1, = 2 R 2 1 R then we take the limit 1, 2 0 [LMNS, Nekrasov, Nekrasov- Okounkov] R 4 1/r,1/r S4 r [V.P.]

32 The par77on func7on Z = q λ Z(λ) λ in {2d arrays of Young diagrams indexed by simple group factor and the color; the size of Young For example, for G = SU(3) x SU(4) a possible contribu7ng λ at instanton charge (10,11) is diagram is the instanton charge} λ =, We sum over all arrays of Young diagrams!

33 The dominant Young diagram profile In the limit 1, 2 0 the par77on func7on Z is dominated by the Young diagrams of the certain limi7ng profile

34 The densi7es ρ i (x) are second deriva7ves of the limi7ng profile For example, for G = SU(3) x SU(4) the array of cri7cal profiles determines densi7es schema7cally as x(, ) (ρ 1 (x), ρ 2 (x)) = D 2

35 Dominant profiles 1 = 2 =0 Seiberg- WiQen curve y(x) sa7sfies algebraic equa7on 1 =0, 2 =0 Seiberg- WiQen curve y(x) sa7sfies difference equa7on Nekrasov, V.P., Shatashvili ADE G quiver > Yangian(G quiver ) [Drinfeld, Jimbo] ADE characters > Yangian characters [Knight] SW curve > Baxter equa7on classical integrable system > quantum spin chain turning points sa7sfy Bethe ansatz equa7on

36 Quantum SW curve 1 =0, 2 =0 is a quan2za2on of classical algebraic integrable system associated to N=2 ADE quiver theory: ADE spin chain This relates to [AGT] correspondence cluster algebras, opers, W- algebras geometric Langlands Nekrasov, V.P., Shatashvili quan7za7on of moduli space of instantons, monopoles, Hitchin systems integrability for spectrum of anomalous dimensions in N=4 SYM many other exci7ng topics [Fomin- Zelevinsky, Frenkel- Reshe7khin]

37 Stochas7c par77ons The dominant profile problem can be visualized as as a limi7ng shape problem of a stochas7c process of falling snow in a corner : [Kerov- Vershik]

38 Thank you!