Catching integrability with fishnets and instantons Gregory - - PowerPoint PPT Presentation

• p. 1/26

Catching integrability with fishnets and instantons

Gregory Korchemsky

IPhT, Saclay Part 1: In collaboration with Nikolay Gromov, Vladimir Kazakov, Stefano Negro, Grigory Sizov arXiv:1706.04167 Part 2: In collaboration with Fernando Alday arXiv:1605.06346, 1609.08164, 1610.01425, 1704.00448

IGST 17, July 20, 2017

• p. 2/26

Integrability and fishnet graphs

✔ Fishnet graphs are four-dimensional scalar conformally invariant Feynman diagrams ✔ Define completely integrable lattice model

[Zamolodchikov’80]

✔ Appear everywhere in planar N = 4 SYM: ✗ Scattering amplitudes ✗ Correlation functions ✔ What is the relation between integrability of planar N = 4 SYM and fishnet Feynman diagrams? ✔ Simplified model: strongly twisted N = 4 SYM at weak coupling

[Joao Caetano, Dima Chicherin talks]

✗ A non-unitary ‘chiral’ (almost) CFT dominated by fishnet graphs ✗ Integrable in planar limit, related to conformal SU(2, 2) spin chain

This talk: use integrability to compute exactly the scaling dimensions of BMN operators

• p. 3/26

Strongly twisted N = 4 SYM

L = − 1 4F 2

µν − 1

2 Dµφ†

i Dµφi + i ¯

ψADψA + Lint

[Leigh,Strassler][Frolov]

Lint = g2 1 4 {φ†

i , φi}{φ† j, φj} − e−iǫijkγk φ† i φ† jφiφj

− e− i

2 γ−

j

¯ ψjφj ¯ ψ4 + e

i 2 γ−

j

¯ ψ4φj ¯ ψj + iǫijk e

i 2 ǫjkmγ+

m ¯

ψkφi ¯ ψj + c.c.

• Twist parameters γ±

1 = −(γ3 ± γ2)/2, γ± 2 = −(γ3 ± γ2)/2,

Is expected to be integrable in the planar limit Double scaling limit: strong twist + weak coupling

[Gurdogan,Kazakov]

g2 → 0 , γ1,2 = fixed , γ3 → −i ∞ , ξ2 = g2 e−iγ3 = fixed Gauge field, fermions and one scalar decouple L = − 1 2∂µφ†

1∂µφ1 − 1

2 ∂µφ†

2∂µφ2 + ξ2φ† 1φ† 2φ1φ2

Supersymmetry and R−symmetry is broken PSU(2, 2|4) → SU(2, 2) × U(1) × U(1)

• p. 4/26

Bi-scalar chiral CFT

L = Nc 1 2 ∂µφ†

1∂µφ1 + 1

2∂µφ†

2∂µφ2 + ξ2φ† 1φ† 2φ1φ2

• Non-unitary theory, chiral vertex

Feynman rules: φ1 φ2 φ†

1

φ†

2

No mass or vertex renormalization in the planar limit Is not complete at quantum level

[Fokken,Sieg,Wilhelm’14]

✔ Quantum corrections induce new interaction vertices given by double trace operators ✔ ξ2 does not run in the planar limit, but new couplings do run – conformal anomaly! ✔ Irrelevant in the planar limit for operator of length J ≥ 3

• p. 5/26

BMN vacuum

Operator with SU(2, 2) × U(1)2 charges (∆, 0, 0|J, 0) OJ(x) = tr[φJ

1 ]

Protected in undeformed N = 4 SYM but receive quantum corrections in bi-scalar chiral CFT OJ(x) ¯ OJ(0) ∼ 1 (x2)∆J (ξ2) , ∆J(ξ2) = J + γJ(ξ2) Globe-like Feynman diagrams OJ(x) ¯ OJ(0) ∼

x

γJ ∼ Anomalous dimension comes from integration in the vicinity of poles (wheel-like graphs) γJ = γ(1)

J ξ2J + γ(2) J ξ4J + . . .

Expansion runs in powers of ξ2J

• p. 6/26

Anomalous dimensions from wheel graphs

γJ = γ(1)

J ξ2J + γ(2) J ξ4J + . . .

Anomalous dimension at M wrappings γ(M)

J

= wheel graph with M frames γ(1)

J

= = −2 2J − 2 J − 1

• ζ2J−3 ,

γ(2)

J

=

[Broadhurst’85]

Double wrapping:

[Ahn,Bajnok,Bombardelli,Nepomechie’13]

γ(2)

J=3 =

• 189ζ7 − 144ζ32

[Panzer’15]

γ(2)

J=4 = 4

• 309ζ11 + 16ζ3,8 + 20ζ5,6 − 4ζ6,5 + 40ζ8,3 − 8ζ3,3,5 + 40(ζ3,5,3 + ζ5,3,3) − 200 ζ2

5

• γ(2)

J

= Sum of multiple zeta values ζi1,i2,...

[Gurdogan,Kazakov’15]

This talk: use integrability to find γJ(ξ) for J = 3 and arbitrary ξ

• p. 7/26

Wheel graphs and Heisenberg SU(2, 2) spin chain

Wheel graphs are generated by integral operator

[Gurdogan,Kazakov’15]

HJ(x, y) =

J

• i=1

1 (xi − yi)2(yi − yi+1)2 =

x1 x2 xJ y1 y2 yJ

TJ,M (x, y) = HJ ◦ HJ ◦ · · · ◦ HJ

• M

= Correlation function OJ(x) ¯ OJ(0) ∼

x

=

• M≥0

ξ2JM 0|HM

J |x = 0|

1 1 − ξ2JHJ |x

• p. 8/26

Wheel graphs and Heisenberg SU(2, 2) spin chain II

Fundamental transfer matrix for noncompact SU(2, 2) spin chain TJ(u) = tr0

• R01(u)R02(u) . . . R0J(u)
• R12(u)−integral operator, solution to Yang-Baxter equation for the principal series of the 4D

conformal group

[Derkachov,GK,Manashov’01], [Chicherin,Derkachov,Isaev’13]

Ru(x1, x2|y1, y2) = c(u) (x2

12)−u−1[(x1 − y2)2(x2 − y1)2]u+2(y2 12)−u+1 = x1 x1 x2 y1 y2

Diagrammatic representation of the transfer matrix TJ(u) =

x1 x1 x1 x1 x2 x2 xJ xJ y1 y1 y2 y2 yJ yJ yJ ... ... u=−1

− − − − → Fishnet generating operator is the special case of the transfer matrix HJ ∼ TJ(−1)

• p. 9/26

Baxter equation for length J = 3

✔ Baxter equation for the SU(2, 2) spin chain

A (u + i) q (u + 2i)−B

• u + i

2

• q (u + i)+C (u) q (u)−B
• u − i

2

• q (u − i)+A (u − i) q (u − 2i) = 0

A(u), B(u), C(u) transfer matrices, sum over local integrals of motion

✔ For the BMN operators of length J = 3, it factorizes into the product of 2nd order Baxter eqs

(∆ − 1)(∆ − 3) 4u2 − m u3 − 2

• q(u) + q(u + i) + q(u − i) = 0

∆−scaling dimension, m−integral of motion

✔ The quantization conditions for ∆ and m follow from the double scaling limit of Quantum

Spectral Curve for twisted N = 4 SYM

[Gromov,Kazakov,Leurent,Volin’15]

m2 = −ξ6 , q4(0, m)q2(0, −m) + q2(0, m)q4(0, −m) = 0 in terms of ‘pure’ solutions q2(u, m) and q4(u, m) with large u asymptotics q2(u, m) ∼ u∆/2−1/2 , q4(u, m) ∼ u−∆/2+3/2 , u → ∞

• p. 10/26

Numerical solution for J = 3

The scaling dimension ∆3 of the BMN operator O3 = tr(φ3

1) for arbitrary coupling ξ3

0.1 0.2 0.3 0.4 0.5 0.5 1.0 1.5 2.0 2.5 3.0

Re[

(

^3)]

0.1 0.2 0.3 0.4 0.5

• 1.5
• 1.0
• 0.5

0.5 1.0 1.5

Im[

(

^3)]

The scaling dimension becomes imaginary at ξ3

⋆ ≈ 0.2

∆3(ξ) = 2 + c

• ξ3

⋆ − ξ3

Weak coupling expansion has finite convergency radius Nonunitary CFT = ⇒ complex scaling dimensions, no unitary bounds on ∆’s At ξ = ξ⋆ the operator O∆ collides with its shadow O4−∆

• p. 11/26

Weak and strong coupling expansion at J = 3

✔ At weak coupling ξ < 1 the Baxter equation can be solved perturbatively in m2 = −ξ6

∆3 − 3 = −12ζ3ξ6 + ξ12 189ζ7 − 144ζ32 + ξ18 − 1944ζ8,2,1 − 3024ζ33 − 3024ζ5ζ32 + 6804ζ7ζ3 + 198π8ζ3 175 + 612π6ζ5 35 + 270π4ζ7 + 5994π2ζ9 − 925911ζ11 8

• + O(ξ24)

where ζi1,...,ik =

n1>···>nk>0 1/(ni1 1 . . . nik k ) are multiple zeta functions

✔ At strong coupling ξ ≫ 1 the Baxter equation can be solved semiclassically (see below)

∆3 − 2 = i 2 √ 2 ξ3/2

• 1 +

1 (16ξ3) + 55 2 1 (16ξ3)2 + 2537 2 1 (16ξ3)3 + + 830731 8 1 (16ξ3)4 + 98920663 8 1 (16ξ3)5 + 31690179795 16 1 (16ξ3)6 + O(ξ−21)

• .

Expansion coefficients grow factorially, asymptotic Borel nonsummable series Receives nonperturbative corrections ∆3 ∼ e−cξ3 (similar to the cusp in N = 4 SYM)

✔ Excellent agreement with numerical results ✔ Generalization to tr(φJ

1 ) with J > 3 (in progress)

• p. 12/26

Excited states with the U(1) charge J = 3

Operators with the same SU(2, 2) × U(1)2 charge (∆, 0, 0|J = 3, 0) but higher length On1,n2 = tr

• φ3

1 (φ1φ† 1)n1(φ2φ† 2)n2

+ permutations + derivatives ‘Length’ of the operator ∆(0) = 3 + 2n1 + 2n2 Protected operators On1,0 = tr

• φ3

1(φ1φ† 1)n1

Operators of length 5 O1 = tr(φ†

2φ3 1φ2) ,

O2 = tr(φ†

2φ2 1φ2φ1) ,

O3 = tr(φ†

2φ1φ2φ2 1) ,

O4 = tr(φ†

2φ2φ3 1)

Mixing to leading order φ1φ2 → φ2φ1 , φ†

2φ1 → φ1φ† 2 ,

φ†

2φ1φ2 → φ1 → φ2φ1φ† 2

φ1 φ1 φ1 φ2 φ†

2

O3 → O4 O3 → O4 O3 → O2

• p. 13/26

Logarithmic multiplet

Mixinig matrix µ d

dµ Oi(x) = VijOj(x) for operators of length 5

V =      4ξ2 O(ξ4) O(ξ6) 4ξ2 O(ξ4) −ξ4 4ξ2      = U−1      1 −2iξ3 2iξ3      U Nonunitary CFT = ⇒ Nonhermitian mixing matrix Lower block describes two conformal operators ∆± = 5 ± 2iξ3 Upper block is Jordan cell of rank r = 2 Logarithmic multiplet ˜ Oi(x) ˜ Oj(0) = 1 (x2)5

• 1

1 ln(x2µ2)

• The same pattern takes place for operators of higher length

Chiral bi-scalar CFT = Logarithmic CFT4

[Caetano’16]

The Baxter equation + QSC allows us to predict the scaling dimensions for arbitrary ξ2

• p. 14/26

Spectrum of states with the U(1) charge J = 3

Numerical solution to the quantization condition for operators of length ∆(0) = 3, 5, 7, 9, . . . Two different families of states: ✔ At J = 3, 7, 11 the dimensions ∆J(ξ) are real at weak coupling ∆7,A = 7 − ξ6 2 − 17ξ12 64 + . . . ∆7,B = 7 + ξ6 2 − 23ξ12 64 + . . . ✔ At J = 5, 9, 13 pairs of complex conju- gated dimensions ∆5,± = 5 ± 2iξ3 + 3ξ6 ∓ 31i

4 ξ9 + . . .

∆9,± = 9 ± i

3ξ3 + 7 216ξ6 ± i 223 10368ξ9 + . . .

Remarkable regularity at strong coupling Can be understood using semiclassical solution of the Baxter equation

• p. 15/26

Semiclassical solution of Baxter equation

(∆ − 1)(∆ − 3) 4u2 − iξ3 u3 − 2

• q(u) + q(u + i) + q(u − i) = 0

WKB expansion for ∆ ∼ u ∼ ξ q(u) = exp

• ξ

x dy p(y)dy

• ,

u = xξ Baxter equation becomes a finite gap eq. for a classical system of three noncompact spins cos(p(x)) = i 2x3 − (∆/ξ)2 8x2 + 1 Exact quantization conditions are replaced by Bohr-Sommerfeld relations 1 2πi

• γ

dx p(x) = ℓ + 1 2ξ Two different famililes of states correspond to γ = α and γ = β on the spectral curve α β x1 x2 x3 x4 Semiclassical quantization of finite-gap solutions on AdS3

• p. 16/26

Summary for part 1

Very simple model but very rich properties of the spectrum

✔ Does the model admit dual strigny description? ✔ What the meaning of nonperturbative corrections at strong coupling? ✔ Derive quantization conditions directly from the spin chain picture? ✔ The complete spectrum of states with higher U(1) charge (face 4rd order Baxter equation)?

• p. 17/26

Part 2: Instantons in N = 4 SYM

• p. 18/26

Instantons in gauge theories

Belavin, Polyakov, Schwarz, Tyupkin ’75

✔ The first indication of a nontrivial vacuum structure in nonabelian gauge theories

SE = 1 4g2

• d4xF a

µνF a µν(x) − i

θ 32π2

• d4xF a

µν

F a

µν(x)

✔ Solutions of classical equations of motion in the Euclidean space-time with a finite action

• Fµν = 1

2 ǫµνρλFρλ = ±Fµν

✔ Describe a tunneling trajectory interpolating between the classical minima ✔ Instanton contribution to physical quantities

∼ e−SE = e−8π2/g2+iθ well defined in the semiclassical limit g2 < 1

✔ Fate of instantons in QCD: Semiclassical picture is inapplicable due to growth of the effective

coupling at large distances

[Callan,Dashen,Gross’78]

• p. 19/26

Instantons in N = 4 SYM

Yang-Mills + 6 scalars + 4 gauginos in the adjoint of the SU(N) L = 1 g2 tr

• − 1

16 F 2

αβ − 1

16 F 2

˙ α ˙ β − 1

4 Dα

˙ αφABD ˙ α α ¯

φAB − 2i¯ λ ˙

αAD ˙ αβλA β +

√ 2λαA[¯ φAB, λB

α ]

− √ 2¯ λ ˙

αA[φAB, ¯

λ ˙

α B] + 1

8[φAB, φCD][¯ φAB, ¯ φCD]

• + i θ

8π2 tr 1 16 F 2

αβ − 1

16 F 2

˙ α ˙ β

• ✔ Solution to Euclidean equations of motion with a finite action

✔ Explicit construction for the SU(2) gauge group

[Zumino’77]

Φ(x; ξ, ¯ η) = ei(ξQ)+i(¯

η ¯ S) Φ(0)(x) ,

Φ = {scalar, gaugino and gauge field} A(0)

µ (x − x0) = 2

ηa

µν(x − x0)νT a

(x − x0)2 + ρ2 , φAB,(0) = λA,(0)

α

= ¯ λ ˙

α,(0) A

= 0 ρ and x0 are the size and the location of the instanton; ηa

µν are ’t Hooft symbols

✔ Depends on 16 fermion collective modes ξA

α and ¯

η ˙

αA (α, ˙

α = 1, 2 and A = 1, . . . , 4) φAB = φAB,(2) + φAB,(6) + · · · + φAB,(14) , φ(2),AB = 1 √ 2 ζαAF (0)

αβ ζβB

(ζ = ξ + x¯ η) Φ(n) homogenous polynomial in ξA

α and ¯

η ˙

αA of degree n

• p. 20/26

Why instantons are interesting in N = 4 SYM?

✔ Nonperturbative corrections are saturated by instantons

G(g2, θ) = fpert(g2) +

• K∈Z

e−8|K|π2/g2+iKθ fK(g2) Exponentially small in the planar limit g2 = O(1/N) but

✔ They are needed to restore the S−duality (electro-magnetic)

[Montonen,Olive’77]

g2 4π → g2 4π −1 , θ → θ + 2π The SL(2, Z) transformation of the compexified coupling constant τ = θ 2π + i g2 4π , τ → τS = aτ + b cτ + d , (a , b , c , d ∈ Z)

✔ The S−duality should commute with the (super) conformal symmetry of N = 4 SYM ✔ The spectrum of the scaling dimensions ∆i should be invariant under the S−duality

∆i(τS, ¯ τS) = ∆i(τ, ¯ τ) but operators can transform nontrivially

• p. 21/26

Semiclassical approximation

O1 . . . On =

• DΦ e−SN =4[Φ] O1 . . . On

Contribution from a nontrivial classical solution Φ = Φinstanton + g Φquantum Semiclassical approximation Φquantum = 0 O1 . . . Oninst =

• dµphys e−Sinst O1 . . . On
• Φ→Φinst

×

• 1 + O(g2)
• The SU(2) integration measure for instanton
• dµphys e−Sinst =

g8 234π10 e2πiτ

• d4x0

dρ ρ5

• d8ξ
• d8¯

η , The integrand should soak 16 fermion modes O1 . . . On

• Φ→Φinst

= ξ8¯ η8 × f(x1, . . . , xn; x0, ρ) Correlation function in the semiclassical approximation O1 . . . Oninst = g8 234π10 e2πiτ

• d4x0

dρ ρ5 f(x1, . . . , xn; x0, ρ)

• p. 22/26

Konishi operator

K(x) = 1 g2

6

• I=1

tr(φIφI) , K′(x) = δ2

¯ Qδ2 QK = 1

g4 tr([Z, X][Z, X]) The additional factor of 1/g2 is due to φI(x)φI(0) ∼ g2/x2

✔ Supersymmetric Ward identities

∆K = 2 + γK(τ, ¯ τ) , ∆K′ = 4 + γK(τ, ¯ τ)

✔ Instanton corrections to γK(τ, ¯

τ) γK,inst =

• n≥1

e2πinτ + e−2πin¯

τ ∞

• k=0

γn,k(N)g2k , Series in g2 is due to quantum fluctuations

✔ The calculation of correlation functions of K′ is simpler

K′(1)K′(2)inst = g8 e2πiτ 234π10

• d4x0

dρ ρ5

• d8ξ
• d8¯

η 1 g8 K′(8)(x)K′(8)(0) = O(e2πiτ) Instanton profile K

′(8) = 0

• p. 23/26

Instanton corrections to K′

• Corr. functions involving K′ vanish in the semi-classical approximation

[Bianchi,Kovacs,Rossi,Stanev,’01]

Scaling dimension of the Konishi operator γK′ = e2πiτ ×

• 0 + γ1g2 + γ2g4 + . . .
• γ1 and γ2

are due to quantum corrections x I I I Two possible scenario:

✔ Konishi operator is protected due to a ‘bonus’ symmetry, γ1 = γ2 = · · · = 0

[Intriligator’98]

✗ Incompatible with the expected S−duality of N = 4 SYM ✔ Quantum corrections to the Konishi operator are different from zero ✗ ... but extremely complicated to compute

Little progress over the last decade

• p. 24/26

Instanton corrections to K

K(1)K(2)inst = g8 e2πiτ 234π10

• d4x0

dρ ρ5

• d8ξ
• d8¯

η 1 g4 K(8)(x)K(8)(0) = O(g4 e2πiτ ) The additional factor of g4 as compared with K′ After integration over 16 fermion modes K(1)K(2)inst = 81 16π10 g4 e2πiτ

• d4−2ǫx0
• dρ

ρ5 (x2

12)4ρ12

(ρ2 + x2

10)6(ρ2 + x2 20)6 ,

The integral develops a logarithmic divergence (ρ → 0 , x2

10 → 0)

• r

(ρ → 0 , x2

20 → 0)

Comes from instantons of small size located close to one of the operators K(1)K(2)inst = 36 (4π2)2(x2

12)2

g2 4π2 2 e2πiτ

• − 9

5ǫ (x2

12)−ǫ

• The leading one-instanton correction to the Konishi operator

γSU(2)

K

= − 9 5 g2 4π2 2e2πiτ + e−2πi¯

τ

A complex conjugated term comes from the anti-instanton

• p. 25/26

Unexpected simplicity of instanton corrections

γK′ = e2πiτ ×

• 0 + γ1

g2 4π2 + γ2 g2 4π2 2 + . . .

• ,

γK = − 9 5 g2 4π2 2 e2πiτ +c.c. But N = 4 symmetry dictates γK = γK′ I I I = = γ1 ∼ γ2 ∼ Quantum instanton correction to K′ = Semiclassical result for K

• p. 26/26

Conclusions and open questions

✔ Scaling dimensions in N = 4 SYM do receive instanton corrections but they are suppressed by

powers of the coupling

✔ A very interesting interplay between semi-classical vs quantum instanton effects and the

symmetries of N = 4 SYM

✔ The same phenomenon holds for twist-two operators γS = 2Γcusp(g2) ln S + O(S0)

ΓSU(2)

cusp

= − 4 15 g2 4π2 4 e2πiτ +c.c.

✔ Hint at unexpected simplifications when considering quantum corrections around instanton

backgrounds in N = 4 SYM

✔ Does integrability hold in N = 4 SYM with instanton effects included ? ✔ What are S−duality invariant functions whose first few terms in weak coupling expansion are

given by leading instanton corrections?