Classical Lie Bialgebras for AdSCFT Integrability by Contraction and Reduction

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arxiv:2210.11150

Classical Lie Bialgebras

for AdS/CFT Integrability

by Contraction and Reduction

Niklas Beisert, Egor Im

Institut f¨ur Theoretische Physik,

Eidgen¨ossische Technische Hochschule Z¨urich,

Wolfgang-Pauli-Strasse 27, 8093 Z¨urich, Switzerland

{nbeisert,egorim}@itp.phys.ethz.ch

Abstract

Integrability of the one-dimensional Hubbard model and of the fac-

torised scattering problem encountered on the worldsheet of AdS strings

can be expressed in terms of a peculiar quantum algebra. In this arti-

cle, we derive the classical limit of these algebraic integrable structures

based on established results for the exceptional simple Lie superalgebra

d(2,1; ) along with standard sl(2) which form supersymmetric isome-

tries on 3D AdS space. The two major steps in this construction consist

in the contraction to a 3D Poincar´e superalgebra and a certain reduc-

tion to a deformation of the u(2|2) superalgebra. We apply these steps

to the integrable structure and obtain the desired Lie bialgebras with

suitable classical r-matrices of rational and trigonometric kind. We il-

lustrate our ﬁndings in terms of representations for on-shell ﬁelds on

AdS and ﬂat space.

arXiv:2210.11150v2 [hep-th] 14 Feb 2023

Contents

Contents 2

1 Introduction 3

2 Contraction 5

2.1 Algebra ..................................... 5

2.2 Geometry .................................... 7

2.3 Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 IrrepContraction................................ 13

3 Loop Algebras and r-Matrices 16

3.1 SimpleRationalCase.............................. 17

3.2 Contraction ................................... 19

4 Reduction 21

5 Trigonometric Case 25

5.1 Contraction ................................... 25

5.2 Reduction .................................... 27

6 Supersymmetric Extension 29

6.1 Superalgebras.................................. 30

6.2 Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.3 Reduction .................................... 35

6.4 r-Matrix..................................... 36

6.5 DoubleSupersymmetry............................. 39

7 Conclusions and Outlook 40

References 42

1 Introduction

Integrable systems play an important role in theoretical physics, especially for models of

condensed matter physics as well as of high energy physics. Integrability of such models

is often attributed to the existence of particular extended types of symmetry algebra. In

the case of many integrable quantum models, the relevant algebras are known as quantum

groups and quantum algebras [1, 2]. These describe not only the extended symmetries of

a quantum model, but also allow one to formulate the integrable structure purely in the

algebraic language. Quantum algebras thus give us a useful tool to describe, evaluate and

study integrable systems.

Two important, yet elaborate examples of integrable models are given by the one-

dimensional Hubbard model, and by the planar limit of N= 4 supersymmetric gauge

theory, which is AdS/CFT dual to strings on AdS4,1×S5. In fact, these two particular

examples are not unrelated. The algebraic structures underlying integrability in the 1D

Hubbard model and factorised worldsheet scattering in the AdS/CFT models are given

by one and the same algebra. Many features of this quantum algebra have already been

worked out, importantly, that it is based on certain extensions of the Lie superalgebra

u(2|2) and that it is of some exceptional kind. The latter means that established stan-

dard constructions in quantum algebra based on simple or semi-simple Lie algebras and

superalgebras are not suﬃcient to describe it. Let us elaborate on these achievements and

on open questions.

The quantum R-matrix for the integrable structure of the 1D Hubbard model [3],

see [4], has been proposed by Shastry [5]. It is the result of an elaborate combination of

two six-vertex models at the free fermion point using elliptic functions and it was shown

to satisfy the quantum Yang–Baxter equation. A signiﬁcant feature of this R-matrix is

that it is not of a so-called diﬀerence form, a feature that most of the known solutions to

the Yang–Baxter share. Much later, the R-matrix was reproduced [6] by a construction

of the AdS/CFT worldsheet scattering matrix [7], see [8], which was based on a central

extension of the Lie superalgebra psu(2|2) in combination with dynamics of excitations

on the worldsheet. Quantum algebra structures were established for this system in [9],

and the algebra was extended to an inﬁnite-dimensional Yangian algebra in [10].

Equipped with these algebraic tools, scattering matrices for some higher representa-

tions [11, 6, 12] have been constructed [13]. Importantly, also the overall phase for the

scattering matrix could be pinned down by consistency considerations of the quantum

algebra together with considerations of the underlying physical system [14]. All of this

calls for the formulation of a universal R-matrix which (in principle) could be evaluated in

arbitrary representations in order to yield the corresponding scattering matrix along with

a suitable overall phase. Yet, our understanding of the quantum algebra is not complete,

nor is the original Drinfel’d presentation well suited towards the construction of a uni-

versal R-matrix. Alternative presentations of the Yangian algebra have been formulated

in [15] with the aim of providing a complete formulation of this quantum algebra from

which all relevant properties of the algebra can be derived using established methods.

Progress towards a complete formulation of the quantum algebra is compromised by

an elevated complexity of the structures for this case. In general, quantum algebras are

highly non-linear objects, but certain functions and structures have been established to

formulate their objects in more convenient terms. Among others, these are so-called q-

deformations of group actions, exponential functions, logarithms, dilogarithms, factorials,

Gamma functions and Pochhammer symbols. Unfortunately, in the present case, it is

not yet known precisely how to compose these to formulate, e.g., the universal R-matrix.

Here, the classical limit comes along very handy, where expressions are reduced to their

leading order terms. Consequently, the quantum algebra reduces to a Lie bialgebra where

most relations are linearised. The classical limit for the R-matrix has been introduced

in [16–18], and a formulation as a Lie bialgebra was completed in [19]. The underlying

algebra turned out to be a novel deformation of the u(2|2) loop superalgebra. A corollary

of this result was the discovery of an additional u(1) derivation to extend the psu(2|2)

symmetry algebra [20].

The article [19] made an auxiliary proposition for the derivation of the deformed u(2|2)

loop superalgebra as a curious reduction of a maximally extended sl(2) npsu(2|2) n R2,1

loop superalgebra. The latter is a non-simple Lie superalgebra, yet its bialgebra structures

take a standard form in the rational case.

A further clue in this direction was provided in the article [21], where the above

maximally extended sl(2) npsu(2|2) n R2,1superalgebra was shown to be an algebraic

contraction of the semi-simple algebra d(2,1; )×sl(2) involving the exceptional Lie su-

peralgebra d(2,1; ).1This contraction follows along the lines of the contraction of the 3D

AdS algebra so(2,2) = sl(2) ×sl(2) to the 3D Poincar´e algebra iso(2,1) = sl(2) n R2,1.

The latter contraction can be supersymmetrised by the replacement of one (or two) factors

of sl(2) by d(2,1; ), and by the introduction of one (or two) factors of the superalgebra

psu(2|2) into the Poincar´e algebra sl(2) n R2,1.

The combination of the latter two insights opens up a path towards a complete al-

gebraic formulation of the classical integrable structures purely in terms of established

elements of simple loop superalgebras and their Lie bialgebra structures. In the present

article, we carry out the full procedure in order to obtain the complete Lie bialgebra

with its classical r-matrix. In particular, we will explore two concepts, contraction and

reduction, that are essential in avoiding the complications mentioned above. Moreover, we

will resort to the well-established representation theory of sl(2) together with analogous

representations of d(2,1; ) to express a relevant class of representations for the resulting

algebra. This will allow us to express the classical r-matrix as the classical limit of a

particle scattering matrix, and it will generally illustrate some of the abstract results in

more applied terms.

In this article we ﬁll some of the missing steps of the above construction in the classical

limit. In Sec. 2 we start with the reduced case of the contraction of the algebra so(2,2)

of isometries of AdS2,1to the 3D Poincar´e algebra iso(2,1). In particular, we describe

on-shell ﬁeld representations on AdS2,1and on ﬂat R2,1, and we show how to perform the

contraction between the two. Then we promote the discussion to loop algebras in Sec. 3

and establish the contraction of the r-matrix of rational type. In Sec. 4 we discuss a par-

ticular reduction of the algebras, their r-matrices and representations. We then extend

the receding construction from the rational to the trigonometric case in Sec. 5. Finally,

the supersymmetric extension of the above constructions involving the exceptional super-

algebra d(2,1; ) is discussed in Sec. 6. Eventually, our construction yields the classical

r-matrix which describes the classical limit of the 1D Hubbard model and of the AdS/CFT

worldsheet S-matrix.

1The idea to involve the exceptional Lie superalgebra d(2,1; ) in the limit →0 appeared earlier

in [22].

algebra generators indices sl(2) form a∈ {0,±}

so(3) Jk1,2,3sl(2) (generic) Ja

so(2,2) Mαβ x, y;u, v sl(2) ×sl(2) Ma

1,Ma

iso(2,1) Lµν ,Pµx, y;tsl(2) n R2,1La,Pa

Table 1: Generators of spacetime symmetries

2 Contraction

Our aim is to understand integrable structures of a physical model with (an extension

as well as a reduction of) Poincar´e symmetry on ﬂat Minkowski space R2,1. A diﬃculty

is that the Lie algebra iso(2,1) = sl(2) n R2,1incorporating Poincar´e symmetry is non-

simple, whereas structures of integrability are best developed for simple and semi-simple

algebras, see [23]. Moreover, unitary representation of this non-compact algebra are

necessarily inﬁnite-dimensional, which complicates constructions in terms of physically

relevant representations. Our resolution to these problems is to resort to the fact that

the Poincar´e algebra is a contraction of the Lie algebra so(2,2) = sl(2) ×sl(2) which

incorporates the isometries of anti-de Sitter space AdS2,1. The algebra factors sl(2) are

simple, and their representation theory is well-understood and easy to handle. We will

then move, step by step, towards the originally intended situation in the subsequent

sections.

In order to set the stage for some more elaborate constructions in this work, we will

review the algebra contraction,

so(2,2) = sl(2) ×sl(2) −→ iso(2,1) = sl(2) n R2,1,(2.1)

ﬁrst at the level of the algebra, then in terms of geometry and ﬁnally at the level of

inﬁnite-dimensional representations. This will also introduce the notation and relate the

abstract mathematical considerations to physical ﬁelds on the symmetric space AdS2,1

and on ﬂat Minkowski space R2,1.

2.1 Algebra

We start by introducing the above Lie algebras in terms of their generators, see the

summary in Tab. 1, and by describing the contraction that relates the two.

Spacetime Algebras. Here we present the generators of the relevant spacetime Lie

algebras so(2,2) and iso(2,1) along with their Lie brackets and invariant quadratic forms.

The AdS algebra so(2,2) is spanned by a set of generators which we shall denote by

Mαβ =−Mβα with the indices α, β ∈ {u, v, x, y}. Their Lie brackets are given by2

[Mαβ,Mγδ] = −˚ıηβγMαδ +˚ıηαγMβδ +˚ıηβδMαγ −˚ıηαδ Mβγ ,(2.2)

where ηdenotes a metric tensor of signature (−,−,+,+) corresponding to the directions

(u, v, x, y). The algebra has two independent invariant quadratic forms

+:= −1

2ηαγηβδ Mαβ ⊗Mγδ,M2

−:= 1

4εαβγδ Mαβ ⊗Mγδ,(2.3)

2Here and below, we follow the physics convention that the generators for the real form of a Lie algebra

are typically assumed to be purely imaginary.

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arxiv:2210.11150ClassicalLieBialgebrasforAdS/CFTIntegrabilitybyContractionandReductionNiklasBeisert,EgorImInstitutfurTheoretischePhysik,EidgenossischeTechnischeHochschuleZurich,Wolfgang-Pauli-Strasse27,8093Zurich,Switzerland{nbeisert,egorim}@itp.phys.ethz.chAbstractIntegrabilityoftheone-dimensio...

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