Q-Operators, Yangians, and all that

Matthias Staudacher

Humboldt University Berlin & AEI Potsdam & IAS Jerusalem

N = 4 Super Yang-Mills Theory, 35 Years After

California Institute of Technology, 30 March 2012

1

My Collaborators

• Vladimir Bazhanov

• Rouven Frassek

• Tomasz !Lukowski

• Carlo Meneghelli

arXiv: 1005.3261, 1010.3699, 1012.6021, 1112.3600, and to appear.

2

2O5

Zur Theor ie der MetaUe. I. Eigenwerte und Eigenfunktionen der Hnearen Atomkette.

Von H. Bethe in Rom.

(Eingegangen am 17. Juni 1931.)

Es wird eine Methode angegeben, um die Eigenfunktionen nullter und Eigen- wer~e erster N~iherung (im Sinne des Approximationsverfahrens von London und t te i t le r ) for ein ,,eindimensionales Metall" zu berechnen, bestehend aus einer linearen Kette yon sehr vielen Atomen, yon denen jedes auBer ab- geschlossenen Sehalen ein s-Elektron mit Spin besitz~. Neben den ,,Spinwellen" yon Bloch treten Eigenfunktionen auf, bei denen die naeh einer Rich~ung weisenden Spins mSglichst an dicht benachbaxten Atomen zu sitzen suchen; diese dorf~en fOr die Theorie des Ferromagnetismus *con Wichtigkeit sein.

w 1. In der Theorie der l~ietalle hat man sich bis vor einiger Zeit darauf beschr~nkt, die Bewegung der einzelnen Leitungselektronen im Potential- feld der Metallatome zu untersuchen (Sommer fe ld , Bloch). Von der Wechselwirkung der Elektronen un~ereinander wurde abgesehen, wenigstens soweit sie nicht summarisch in dem auf die Elektronen wirkenden Potential untergebracht werden ka~m. Dieses Veffahren war flit die Probleme der metallischen Leitf~higkeit (mit Ansnahme der Supraleitung) sehr fruchtbar, liel~ aber ein tieferes Eindringen etwa in das Problem des Ferromagnetismus nich~ zu 1) und machte z. B. die Berechnung der Koh~isionskr~fte im Metall zu einem ganz hoffnungslosen Unternehmen: Die flit die St6rungsenergie erster N~herung mal~gebenden Austanschkr~fte zwischen den Leitungs- elektronen sind yon gleicher Gr5~enordnung wie die Nullpunktsenergie des Elektronengases (Energie nullter Naherung), und dementsprechend kann man ~bsch~tzen, dal3 die zweite N~herung wieder yon derselben GrSBen- ordnung wird usw. Dieser Umstand stlmm~ etwas skeptisch gegen die ganze Approximation, bei der die Bewegung des einzelnen ]~lektrons (kine~i- sche Nullpunktsenergie) als iiberragencI v~_:chtig gegenfiber der Wechsel- wirkung (Austauschenergie) angesehen wird.

Deshalb ist in letzter Zeit yon S la te r 2) und Bloch s) versucht worden~ das Problem yon der anderen Seite her zu approximieren, d.h. die Atome als fest vorgebildet anzunehmen und ihre Wechselwirkung als StSrung zu betrachten, wie es der London- t t e i t l e r s chen Approximation flit Molekiile

1) F. B loch , ZS. f. Phys. 57, 545, 1929 zeigte, da~ tinter gewissen Voraussetzungen auch ,,freie Elektronen" Fe~romagnetismus zeigen kSnnen.

2) j . C. Slater , Phys. Rev. 35, 509, 1930. a) F. Bloch, ZS. f. Phys. 61, 206, 1930 (ira folgenden mit 1. o. zitler~): 3

Bethes Ansatz of 1931

(

uk + i2

uk −

i2

)L

=

K∏

j=1j !=k

uk − uj + i

uk − uj − i,

−−−−− −− −−−−−−−−

E =

K∑

j=1

2

u2j + 1

4

.

4

Nuclear Physics B121 (1977) 77-92 © North-Holland Publishing Company

S U P E R S Y M M E T R I C Y A N G - M I L L S T H E O R I E S *

Lars BRINK ** and John H. SCHWARZ California Institute o f Technology, Pasadena, California 91125

J. SCHERK Laboratoire de Physique Thkorique de l'Ecole Normale Supbrieure, 24 rue Lhomond, 75231 Paris, France

Received 22 December 1976

Yang-Mills theories with simple supersymmetry are constructed in 2, 4, 6, and 10 dimen- sions, and it is argued that these are essentially the only cases possible. The method of di- mensional reduction is then applied to obtain various Yang-Mills theories with extended supersymmetry in two and four dimensions. It is found that all possible four-dimensional Yang-Mills theories with extended supersymmetry are obtained in this way.

1. Introduct ion

Three types of supersymmetric Yang-Mills theories in four dimensions are known. In the first one that was found [1] the infinitesimal parameter of the supersymmetry transformation is a Majorana spinor ("simple" supersymmetry). In the second one [2] it is a Dirac spinor ("complex" supersymmetry). In the third case it consists of four Majorana (or Weyl) spinors [3]. This last model was obtained recently by applying the method of dimensional reduction to a supersymmetric Yang-Mills theory in ten- dimensional space-time.

The goal of this paper is to classify all the possible supersymmetric Yang-Mills theories in both two and four dimensions. The interest in four dimensions is obvious, of course, as one of these schemes may be part of a correct theory. The two-dimen- sional cases are also emphasized because of the possibility of coupling such Yang- Mills multiplets to a corresponding two-dimensional supergravity theory [4] in order to get a modified string theory. Our technique consists of two stages. In the first stage Yang-Mills theories with simple supersymmetry are constructed for all space- time dimensions in which it is possible. Then in the second stage each of the higher-

'~ Work supported in part by the US Energy Research and Development Administration under contract E(11-1)-68, and by the Swedish Atomic Research Council under contract 0310-026. On leave of absence from Institute of Theoretical Physics, GiSteborg, Sweden.

77 5

1 =

K2Y

j=1j !=k

u2,k − u2,j − i

u2,k − u2,j + i

K3Y

j=1

u2,k − u3,j + i2

u2,k − u3,j − i2

,

1 =

K2Y

j=1

u3,k − u2,j + i2

u3,k − u2,j − i2

K4Y

j=1

x3,k − x+4,j

x3,k − x−4,j

,

0

@

x+4,k

x−4,k

1

A

L

=

K4Y

j=1j !=k

u4,k − u4,j + i

u4,k − u4,j − iσ2(x4,k, x4,j)

! K3Y

j=1

x−4,k

− x3,j

x+4,k

− x3,j

K5Y

j=1

x−4,k

− x5,j

x+4,k

− x5,j

,

1 =

K6Y

j=1

u5,k − u6,j + i2

u5,k − u6,j − i2

K4Y

j=1

x5,k − x+4,j

x5,k − x−4,j

,

1 =

K6Y

j=1j !=k

u6,k − u6,j − i

u6,k − u6,j + i

K5Y

j=1

u6,k − u5,j + i2

u6,k − u5,j − i2

,

− − − −− − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −

E(g) = 2

K4X

j=1

0

@

i

x+4,j

−i

x−4,j

1

A =1

g2

K4X

j=1

r

1 + 16 g2 sinpj

2− 1

!

, ∆ = ∆0 + g2

E(g) , K4 = K .

1 =

K4Y

j=1

0

@

x+4,j

x−4,j

1

A =

K4Y

j=1

eipj , uk = xk +

g2

xk, uk ± i

2 = x±k

+g2

x±k

.

The Asymptotic All-Loop AdS/CFT Bethe Equations

[ Beisert, Staudacher ‘05,‘06 ]

6

The Birthday Child

N = 4 SYM, unique action: [ Brink, Schwarz, Scherk ‘76; Gliozzi, Scherk, Olive ‘77 ]

S =N

λ

∫

d4x 2Tr

(

14 F

µνFµν + 12 D

µΦm DµΦm − 14[Φ

m,Φn][Φm,Φn]

+ Ψaασαβ

µ DµΨβa − i2 Ψαaσ

abm εαβ[Φm,Ψβb] − i

2 Ψaασm

abεαβ[Φm, Ψb

β]

)

+θYM

16π2

∫

d4x 2TrFµνFµν

Free parameters: λ = Ng2YM and N and theta angle θYM.

Non-trivial CFT4. [ Sohnius, West ‘81; Brink, Lindgren, Nilsson ‘83; Mandelstam ‘83; Howe, Stelle, Townsend ‘84 ]

su(N) gauge symmetry, Olive-Montonen symmetry, and PSU(2, 2|4).Hidden string description (AdS/CFT). [ Maldacena ‘97; Gubser, Klebanov, Polyakov ‘98; Witten ‘98 ]

7

Exact Solvability of N = 4 Yang-Mills Theory

Mathematically arguably the “most beautiful Yang-Mills theory”!There are striking hints, that it is exactly solvable.

What is the final secret symmetry of this theory?

+ ?

8

1 =

K2Y

j=1j !=k

u2,k − u2,j − i

u2,k − u2,j + i

K3Y

j=1

u2,k − u3,j + i2

u2,k − u3,j − i2

,

1 =

K2Y

j=1

u3,k − u2,j + i2

u3,k − u2,j − i2

K4Y

j=1

x3,k − x+4,j

x3,k − x−4,j

,

0

@

x+4,k

x−4,k

1

A

L

=

K4Y

j=1j !=k

u4,k − u4,j + i

u4,k − u4,j − iσ2(x4,k, x4,j)

! K3Y

j=1

x−4,k

− x3,j

x+4,k

− x3,j

K5Y

j=1

x−4,k

− x5,j

x+4,k

− x5,j

,

1 =

K6Y

j=1

u5,k − u6,j + i2

u5,k − u6,j − i2

K4Y

j=1

x5,k − x+4,j

x5,k − x−4,j

,

1 =

K6Y

j=1j !=k

u6,k − u6,j − i

u6,k − u6,j + i

K5Y

j=1

u6,k − u5,j + i2

u6,k − u5,j − i2

,

− − − −− − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −

E(g) = 2

K4X

j=1

0

@

i

x+4,j

−i

x−4,j

1

A =1

g2

K4X

j=1

r

1 + 16 g2 sinpj

2− 1

!

, ∆ = ∆0 + g2

E(g) , K4 = K .

1 =

K4Y

j=1

0

@

x+4,j

x−4,j

1

A =

K4Y

j=1

eipj , uk = xk +

g2

xk, uk ± i

2 = x±k

+g2

x±k

.

The Asymptotic All-Loop AdS/CFT Bethe Equations

[ Beisert, Staudacher ‘05,‘06 ]

9

Luscher Corrections and Thermodynamic Bethe Ansatz

[ Luscher ‘86; A.B. Zamolodchikov ‘90 ]

[ Ambjørn, Janik, Kristjansen ‘06; Arutyunov, Frolov, ‘07, ‘08; Bajnok, Janik ‘08 ]

In the TBA, one “turns around” the world sheet cylinder of the stringσ-model, and considers scattering in the cross channel.The virtual field-theoretic corrections solve the wrapping problem.

10

All-loop TBA equations, Y-System

Ground State TBA: [ Bombardelli, Fioravanti, Tateo ‘09, Arutyunov, Frolov ‘09, Gromov, Kazakov, Kozak, Vieira ‘09 ]

Claim: Exact spectrum of planar N = 4. [ Gromov, Kazakov, Kozak, Vieira ‘09 ]

Infinite system of integral equations for “Y-functions” living on a lattice:

How to derive (as opposed to guess) the Y-system?What is the physics of it?

11

Predictions from the All-Loop TBA and the Y-System

[ Gromov, Kazakov, Vieira ‘09 ]

• A numerical plot for Konishi was obtained and confirmed in [ Frolov ‘10 ].

• It fits very well weak coupling. Analytical results up to five loops.

• It also fits well the known strong coupling behavior 2λ14 + 2λ−1

4 + . . . .

• Analytic Structure of the expansion not quite clear though.

12

Weak Coupling Challenges for AdS/CFT Integrability

• It is very important to explore the predictions of integrability for shortoperators in N = 4 gauge theory to much higher orders.

• Analytic 5-loop Konishi [ Bajnok, Hegedus, Janik, !Lukowski ‘09 ] confirmed in [ Eden et. al. ‘12 ]

γ = 12 g2 − 48 g4 + 336 g6 − 96 (26 − 6 ζ(3) + 15 ζ(5)) g8 + +96“

158 + 72 ζ(3) − 54 ζ(3)2 − 90 ζ(5) + 315 ζ(7)”

+ . . .

• The dots are not boring!

• Results from mathematical Feynman graph theory and algebraic geo-metry indicate that at some order multiple zeta functions are expected.6 loops in φ4-theory. [ Bloch, Broadhurst, Brown, Kreimer ] 8 loops? Double-Wrapping?

• A non-Tate graph appears in planar φ4-theory at 9 loops. [ Brown, Schnetz ‘10 ]

Integrals not expressible as zeta functions! 12 loops? Triple-Wrapping?

13

Solvable Structures in the Planar AdS/CFT System

• Spectral Problem

• Gluon Amplitudes

• Wilson Loops

• High Energy Scattering (BFKL)

These are all related, and all show (unproven) hints of integrability!

• Recently much progress with N = 4 amplitudes and Wilson loops.

• Exciting hints at Yangian structures in planar N = 4 amplitudes.

[ Drummond, Henn, Plefka ‘09; Drummond, Ferro ‘10; Beisert, Henn, McLoughlin, Plefka ‘10 ]

14

Strategy

• Strategy for dealing with the finite size corrections:

TBA↓

Y = TTTT ↑ Y-system ↓

easy ↓ hardT = det Q ↑ T-system ↓

↑ ↓ ↓↑ Q-system ↓

↓FINLIE

• Analyticity of T- and Y-functions is inherited from Q-functions!

• How to construct the Q-system of AdS/CFT? Start from spin chains ...

15

Strategy

• Strategy for dealing with the finite size corrections:

TBA↓

Y = TTTT ↑ Y-system ↓

easy ↓ hardT = det Q ↑ T-system ↓

↑ ↓ ↓↑ Q-system ↓

↓FINLIE

• Analyticity of T- and Y-functions is inherited from Q-functions!

• How to construct the Q-system of AdS/CFT? Start from spin chains ...

16

Motivation for this Talk

• Learn how to construct Q-operators for quantum integrable systems.

• This had not yet been properly done even for spin chains such as theones appearing at one-loop in N = 4.

• Deeper exploration of algebraic structure of quantum integrability.

• Yangian algebras for the user.

17

Quantum Integrability

• Symmetry: [JA, JB] = fCAB JC

• Fundamental R-matrix: [R12(z), J1 + J2] = 0

• Yang-Baxter equation (YBE):

R12(z1 − z2)R13(z1)R23(z2) = R23(z2)R13(z1)R12(z1 − z2)

18

Yangian*

gl(n) R-matrix: R12(z) : Cn ⊗ Cn → Cn ⊗ Cn, R12(z) = z I12 + P12 .

More interesting solutions of the YBE: L(z) : V ⊗ Cn → V ⊗ Cn

R12(z1 − z2)L1(z1)L2(z2) = L2(z2)L1(z1)R12(z1 − z2)

Ansatz:

L(z) =n

∑

i,j=1

∞∑

r=0

eij ⊗ !(r)ji z−r

RLL = LLR defines a quadratic algebra for !(r)ji which we call Yangian*.

19

Truncation, Yangian and Yangian*

Let us restrict to !(r)ij := 0 for r > 1. Then

Lij(z) =1

z(z !

(0)ij + !

(1)ij )

The YBE says that !(0)ij are central; they are structure constants for !

(1)ij :

[!(1)ij , !(1)kl ] = !

(0)kj !

(1)il − !

(0)il !

(1)kj

gl(n)-variance of R(z) says that L = FLG also satisfies the YBE. Hence

!(0)ij = !

(0)i δij , !

(0)i =

{

1, i ∈ I

0, i #∈ I

Yangian: I = {1, . . . , n} while Yangian*: I ⊂ {1, . . . , n}.

20

Solution

Put w.l.o.g. I = {1, . . . , p}. The quadratic algebra then yields

LI(z) =

(

z + jrs + arrars ars

−ars δrs

)

,

{

r, s = 1, . . . , p ,

r, s = p + 1, . . . , n

gl(p) subalgebra: [jrs, jkl] = δksjrl − δrljks

H(I,I) Oscillator algebras: [ars, akl] = δksδrl with I = {p + 1, . . . , n}.

All other LI′ with |I ′| = |I| are related by Weyl symmetry.

These solutions are evaluation homomorphisms

Y ∗(gl(n)) → AI := gl(I) ⊗H(I,I)

where AI defines a representation of the Yangian* algebra.

21

Standard Representations and Minimal Representations

We thus found a generalization of pure gl(n) representations where |I| = n

with Jrs → (jrs, ars, ars). For |I| = n no oscs are left, we are back to

L(z) = z + ers ⊗ Jsr .

Conversely, if jrs = 0 (trivial rep for gl(p)) the matrix elements of LI(z)only contain oscillators, yielding Lax-operators for Q-operators.

22

Intertwining Relations

Now intertwine any two Yangian* reps: A daunting, unsolved problem.

Special case: Intertwine pure gl(n) rep and pure osc rep AI∗ := H(I,I) :

L(z1 − z2)LI(z1)RI(z2) = RI(z2)LI(z1)L(z1 − z2)

Linear equation for the intertwiner RI(z). Solution:

RI(z) =

23

New Solutions of the YBE

Solution through shifted weight operators A Ik of the subalgebra gl(n−p) :

RI(z) = ears Jsr

n−p∏

k=1

Γ(z − A Ik )e−ars Jsr

They are related to the Casimir operators of the gl(n − p) subalgebra

cn = J i1in

J i2i1

. . . J inin−1

, i1, . . . , in ∈ I

by

cn =∑

k

∏

l "=k

(

1 +1

A Ik − A I

l

)

(A Ik )n .

24

Analytic Structure of the New Solutions

Compact representations: Since the spectra of all A are bounded, one cannormalize by ρI(z) such that all RI(z) are polynomials

RI(z) = ρI(z) ears Jsr

n−p∏

k=1

Γ(z − A Ik )e−ars Jsr

Non-compact highest/lowest weight reps: Some RI(z) can be normalizedto be polynomials, some are meromorphic functions:

Hasse diagram: su(1, 2)

25

Supersymmetry

Modest changes in the supersymmetric case gl(n|m).

Now the solution is in terms of shifted weight ops A Ik and B I

l of thebosonic subalgebras gl(n−p) and gl(m− q), using superoscs [ξ, ξ} = δδ :

RI(z) = ρI(z) eξrs Jsr

n−p∏

k=1

m−q∏

l=1

Γ(z − A Ik )

Γ(z − B Ik )

e−ξrs Jsr .

RI(z) again polynomial (compact) or meromorphic (non-compact).

Hasse diagram: gl(2|1)

26

Q-Operator Construction

Now apply the standard QISM procedure:

Quantum space: Λ ⊗ . . . ⊗ Λ︸ ︷︷ ︸

L

∣∣∣

∣∣∣

∣∣∣

∣∣∣

∣∣∣

∣∣∣

∣∣∣

∣∣∣

∣∣∣

∣∣∣

∣∣∣

∣∣∣

︸ ︷︷ ︸

L

However, we need a convergence factor D (Aharonov-Bohm fluxes):

QI(z) = TrAI∗[DRI(z) ⊗ . . . ⊗RI(z)]

The analytic structure of the Q-operators is inherited from RI(z).

27

T-Operator Construction

Compare to the standard QISM construction of transfer matrices TΛ′ :

If the Lax-operators satisfy the YBE, we have

[TΛ′(z′), TΛ′′(z′′)] = 0 .

In fact, “everything commutes with everything”:

[QI(z), QI′(z′)] = 0 , [QI(z), TΛ′(z′)] = 0 .

28

Fusion

Now one may algebraically derive a host of operatorial fusion relations.There are three basic types:

QQ-relations:

QI∪a∪b(z)QI(z) = QI∪a(z + 12)QI∪b(z − 1

2) − QI∪a(z − 12)QI∪b(z + 1

2)

Determinant formulas for Transfer matrices (“partonic” Q-operators):

TΛ(z) = deta,b

Qa(zb) , zb+1 − zb − 1 = λb

(For Y-systems, at the heart of reducing infinitely TBA eqs to FINLIE!)

There is also a hierarchy of TT-relations (not shown).

29

Solution

The eigenvalues of Q-operators are the Q-functions, which encode theBethe roots. Polynomial case:

Q(z) =∏

k

(z − zk).

The nested Bethe equations follow immediately from the QQ-relations.Example:

No Bethe ansatz required!

30

Hamiltonians, I

Λ′ = Λ is used for deriving nearest neighbor spin chain Hamiltonians.

To derive it, intertwine two gl(n) reps Λ :

R(z1 − z2)L(z1)L(z2) = L(z2)L(z1)R(z1 − z2)

Linear equation for the intertwiner R(z). Solution:

R(z) = with R(0) =

31

Hamiltonians, II

For rectangular representations of gl(n) algebras with a rows we found

R(z) = ρ(z)a

∏

k=1

Γ

(

z +Ak − Ak + 1

2

)

Γ

(

z −Ak − Ak + 1

2+ 1

)

,

where k = 2a − k + 1 and Ai are two-site operatorial shifted weights.

Hamiltonian density H = ddz logR(z)

∣∣∣z=0

:

−2a

∑

k=1

[

ψ

(

Ak − Ak + 1

2

)

− ψ

(

Ak + Ak + 1

2+ 2a − k

)]

Generalization of the famous gl(2) formula (same for simple gl(4|4)reps!)

Hsu(2) = 2(ψ(J + 1) − ψ(1)) , J(J + 1) = (J1 + J2)2 .

32

Conclusions

• Systematic procedure for the construction of Q-operators proposed.

• Explicit results for a very general class of nearest neighbor spin chains.

• Discovery of novel reps of Yangian algebras, termed Yangian*.

• Yangian algebras for the user: Leads to solution of the systems studied.

33

Open Problems

• Construct Q-operators for the AdS/CFT Y-system.

• What is the Yangian algebra of N = 4 at finite ‘t Hooft coupling λ?

• Is there a meaning to Yangian* for N = 4 amplitudes?

34

Happy Birthday!

N = 4 SY M

35