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The arguments suggested above allow to solve several two-dimensional gauge theories with one space dimension and one time dimension. Many such models are relativistic and have massive particles4.
On the other hand, the known particle physics (described by the standard model), is a four-dimensional relativistic gauge theory, which is asymptotically free. This means that the interactions which occur above an energy scale are well described by sums of “Feynman diagrams” which correspond to diﬀerent possible interaction processes. For instance, the simplest way for two electrons to interact is by exchanging one photon. But they could as well exchange two photons, or more. Or an electron could emit a photon which transforms afterwards into an electron-positron pair which annihilates into photons that are finally absorbed by the other electron. All the processes which can happen are described by “Feynman diagrams”, and in general one should sum an infinite series of these processes. Asymptotic freedom (which occurs for instance in the standard model) means that above a given energy scale, the more complicated5 a diagram is, the less it contributes to the sum. This allows to show that, in order to compute the properties of an interaction to a given accuracy, it is suﬃcient to keep a finite number of terms.
We see that in general, for these gauge theories, what we can do is to write an infinite series (which is an asymptotic expansion), which is called a perturbative expansion. This captures important physical properties of the interactions, but it cannot be used below a given energy scale. As a consequence, there are some aspects of these gauge theories that are not captured by this approach. For instance, one of these non-perturbative aspects is the confinement of quarks inside hadrons (such as the neutrons and the protons), which explains that we cannot observe an isolated quark, but only some particles made of multiple quarks.
These questions arise for lots of diﬀerent gauge theories and we will see that there exists at least one four-dimensional gauge theory, the so-called super Yang-Mills field theory with four supersymmetries (N = 4 SYM) for which some exact computations can be done (as opposed to the perturbative expansion mentioned above). This the-ory is a conformal field theory (CFT), which means that it is invariant under several transformations including dilatations (see below).
Conformal invariance and dimension of operators Every field theory describes some fields which are functions of the positions (these functions are operator-valued for quantum field theories). Conformal field theories are invariant under the transforma-tions of positions which preserves angles (i.e. these transformations locally look like compositions of translations, rotations, and dilatations).
Integrability in the AdS/CFT duality Interestingly enough the integrable proper-ties of the super Yang-Mills field theory are best understood and tested in the framework of the “AdS/CFT correspondence”. This conjectured duality [Mal98, GKP98, Wit98] says that several quantities (such as correlation functions) that we wish to compute on one side of the duality, for instance in super Yang-Mills, can be obtained by computing other quantities in the other side of the duality, for instance AdS. This “AdS” denotes a string theory (i.e. a quantum theory of gravity) on a 10-dimensional space-time having the ge-ometry AdS5 S5, where AdS5 denotes the 5-dimensional anti de Sitter space, a curved manifold which is roughly speaking a multi-dimensional hyperboloid. Given a quantity that we want to compute (for instance) in super Yang-Mills, it is not easy to understand what computation in the AdS string theory is associated to this quantity. Nevertheless, this duality is very interesting because it turns out to relate the perturbative domain of one model to the non-perturbative domain of the other model. It means that for instance, classical string theory is related to deeply non-perturbative gauge theory.
We will not enter deeply into the details of this duality, but we will simply work with one of the predictions of this duality: the fact that the energy spectrum of the super-strings is equal to the spectrum of the conformal dimensions of the super Yang-Mills operators.
On the super Yang-Mills side, integrability was first noticed in [Lip94], and in the planar limit (a limit when the rank of the gauge group is large), a mapping was noticed [FK95, MZ03] between the study of the spectrum of the conformal dimensions in super Yang-Mills and an SL(2) spin chain (see for instance [Min12]8). More precisely, it was shown that in this planar limit the only relevant operators are linear combinations of operators of the form tr(O1; O2; ; OL) where O1, O2, , OL are arbitrary fields. In general, the operator n1; ;nL tr(On1 ; ; OnL ) is not a primary operator, which transforms as in (I.31). Instead it is a linear combination of primary operators, and the operators of this form transform as 0A(x0) = ZAB B(x) where the “mixing matrix” ZAB has eigenvalues (x) i where i denotes the dimensions of primary operators. In particular there is a sector called SU(2) sector, where only two elementary operators can appear (i.e. each Oi is either equal to X or Y ). Then the operator tr(XXY X) can be mapped to the state j » »# »i of an SU(2) spin chain of size L = 4. Then it was shown that for “long operators”, i.e. for composite operators made of the trace of the product of many elementary operators, this mapping transforms the mixing matrix into an integrable spin chain Hamiltonian, and some Bethe equations arise that allow to find the spectrum of super Yang-Mills’ long operators.
The equations obtained for these “long operators” can then be continued to short op-erators. This gives a Y-system which was conjectured in [GKV09a] and then understood in terms of the thermodynamic Bethe ansatz approach [BFT09, GKKV10, AF09]. This Y-system was successfully tested in the weak coupling regime, by comparison with per-turbative expansion in super Yang-Mills [JŁ07, HJŁ08, BJ09, FSSZ08, Vel09, MOSS11, AFS10, BH10], but also in the strong coupling regime [Gro10].
In this manuscript, we will take these equations as the starting point for the chapter IV, and derive a simpler set of equations [11GKLV]. This work, performed during this PhD, is similar in spirit to the analysis of the principal chiral model in the chapter III, but we will see that the analyticity conditions are much richer. In particular we will find a new symmetry of the Y-system (which we call “quantum-Z4 symmetry”, and which we interpret from the string theory on AdS5 S5), and show how to recast the infinite set of equations arising from the thermodynamic Bethe ansatz into a finite set of equations, where the analyticity of several functions is much better understood than in previous analyses.
In this chapter we will prove the integrability of a class of spin chains which generalize the Heisenberg spin chain studied in chapter I.1.
To do this, we will first construct a family of commuting operators (and which also commute with the Hamiltonian H), called the T-operators [BR90]. We will then show, following the paper [KV08], that these T-operators obey some fusion relations governed by the “Cherednik-Bazhanov-Reshetikhin” determinant formula (II.80) [Che86, BR90], which can also be recast into the bilinear form of the Hirota equation [KP92, KN92, KLWZ97, Tsu97]. The proof of this relies on combinatorial identities introduced in the appendix B.1, and it will conclude the section II.1. In the next sections, we will show how to diagonalize the T-operators and the Hamiltonian, by writing a “Bäcklund flow”. More precisely we will start by motivating the introduction of the Bäcklund flow in section II.2, where we will show (as in [KLWZ97, Zab96, KSZ08, Zab08]) that if this flow exists and is polynomial, then some strong constraints arise that allow to diagonalize the T-operators. Finally new results of this PhD will be presented in the section II.3, where this Bäcklund flow is constructed explicitly at the operatorial level, and an original construction of the so-called “Q-operators” is presented for the GL(KjM) spin chain.
Starting from the introduction of “Q-operators” in [Bax72] for the eight-vertex lattice model, some Q-operators have been constructed for a large variety of integrable systems1 The construction given in this thesis for the GL(KjM) spin chain is quite diﬀerent from these constructions, and allows to define Q-operators directly as operators and to show their polynomiality. In particular, this gives Wronskian determinant expressions for the T-operators in terms of Q-operators. These Wronskian expressions are given in section II.3.2.3, and can also be found in the literature [BLZ97a, KLWZ97, BT08, 11GKLT, Tsu10]. These Wronskian expressions are known solutions of the Hirota equation, and we will prove that these expression apply to the T-operators of these spin chains2.
Interestingly, this construction turns out to have very deep connections [11AKL+] with the “classical integrability”, as explained in the section II.4.
Spin chains and T-operators
Spin chains are particularly simple examples of integrable systems. In this section, we will see how to construct the family of conserved charges which will allow to diagonalize the Hamiltonian. We will also see that these charges can be expressed through the action of a “co-derivative”, and we will see that it allows to prove the “Cherednik-Bazhanov-Reshetikhin” determinant formula (II.80).
Construction of the T-operators
Heisenberg spin chain
The “Heisenberg XXX1=2 spin chain” is the simplest spin chain, and corresponds to a quantum version of the Ising model. As we already saw in the introductory section I.1, its Hilbert space is H = i=1L Hi = (C2) L. In this spin chain, the interactions are only between nearest neighbors, and are governed by the Hamiltonian (I.1), or by the (more N physical) expression below (we will show that the two expressions coincide)
Yang-Baxter equation and the construction of conserved charges
Rewriting the Hamiltonian (II.1) in terms of permutation operators will be useful due to the simple algebra satisfied by these permutation operators: for instance, the relation Pi;jPj;k = Pj;kPi;k allows to check the following equality Ri;j(u v)Ri;k(u)Rj;k(v) = Rj;k(v)Ri;k(u)Ri;j(u v) where Rm;n(u) = u + Pm;n .
The equality (II.14) will be crucial in what follows, and it is called the “Yang-Baxter” identity. More precisely, we will say that the R-matrix defined by (II.15) does satisfy the “Yang-Baxter” identity (II.14).
Now, we will see how this identity allows to define a family of operators which com-mute with each other and with the Hamiltonian. To this end, we need to introduce a larger Hilbert space H H a 1 H a 2 H a n , where each H a k = C K is an “auxiliary” L K N i=1LHi = CK , space. Here, the symbol H denotes the original Hilbert space, H = whereas the smaller symbol H denotes the smaller spaces (isomorphic to CK) which appear in the tensor products. Each K-dimensional space Hi = C corresponds to one spin in a superposition of K states.
We will then show by recurrence that Ra1;a2 (u v)L(1)(u)L(2)(v) =L(2)(v)L(1)(u)Ra1;a2 (u v) (II.16)
where L(k)(u) = RL;ak (u)RL 1;ak (u)R1;ak (u) : (II.17)
In the definition (II.17), the “monodromy matrix” L(k )(u) is an operator acting on H Hak .
By contrast the operators in (II.16) are acting on H H a 1 H a 2 , and in this equation it is implicit that for instance L (u) rigorously denotes the operator L
Proof of the relation (II.16). The proof relies on the following recurrence relation: Ra1;a2 (u v)L(1)(u)L(2)(v) = LL(2);i+1(v)LL(1);i+1(u)Ra1;a2 (u v)Li(1);1(u)Li(2);1(v) ; (II.18)
where Li(;kj)(u) = Ri;ak (u)Ri 1;ak (u) Rj;ak (u) ; (II.19)
where we see that L(k )(u) = L(Lk;1)(u). The initialization of the recurrence for i = L is trivial, whereas the case i = 0 is the statement (II.16) that we want to prove. As explained graphically in the figure II.1, going from i to i 1 in (II.18) is done by using the Yang-Baxter relation (II.14), together with the commutation relation L Ri;a1 (u) ; Rj;a2 (v) = 0 if i 6= j (II.20) where A ; B A B B A : (II.21)
Indeed, the R -operators have the form R I , and the commutation relation (II.20) is nothing but the statement that A I ; I B =0 : (II.22) diagrammatically the structure of this iterative proof, It can be interesting to write L M
which is done in figure II.1. In this figure, the vertical lines stand for the Hilbert spaces H1, H2, : : :, whereas the horizontal lines stand for the auxiliary spaces Ha1 et Ha2 . The dots (colored in the online version) stand for the R-operators and the order of multiplications is given by the arrows. These arrows allow an ambiguity in the order of operators, which is fully consistent with (II.20), and crucial for the proof. Graphically, we see that the iterative structure of the proof simply consists in moving vertical lines from the right side of the blue dot to its left side, and the relation (II.14) exactly allows to move lines this way across each other.
Hirota equation and Cherednik-Bazhanov-Reshetikhin de-terminant formula
In this section, we will introduce the “fusion relations” between the T-operators corre-sponding to various representations. We will introduce the results obtained in [KV08] (in particular the derivation of the CBR determinant formula (II.80) found in [Che86, BR90]), and briefly introduce the proof of these results.
on characters (see appendix A.3.4), where (s) denotes the character associated to the representation [1;s]. It allows to express an arbitrary T-operator in terms of the operators T1;s(u). That will allow to prove the commutation relation (II.52), but also to show the Hirota equation (II.101), found in [KP92, KN92, KLWZ97, Tsu97].
Though this determinant relation was proven in [KV08], it will be helpful to recall this proof, which extensively relies on the “co-derivative” formalism. The proof is done in two main steps: the first step shows that the right-hand-side of (II.80) is a polynomial (by showing the vanishing of specific minors (II.83) of the determinant), and the second step checks that it really coincides with T( )(u).
Proof of the CBR formula : Part one
Let us start by proving that the right-hand-side of (II.80) is polynomial:
By definition, all the T1;s(v) inside the determinant are polynomial functions of the variable v. Moreover, the denominator is a polynomial, which can be written explicitly due to the relation T0;0(u) ui = (u i) : (II.82) i=1 i=1
This relation is nothing but the definition (II.63), where the representation associated to an empty Young diagram [0;0] = [a;0] = [0;s] has the character [1;0] (g) = 1, as it can be seen from the relation (A.55) in the appendix A.3.2.
Hence in order to prove that the right-hand-side of (II.80) is polynomial, it is suﬃcient to prove that it has no pole, i.e. that the determinant is zero5 when u = i+k for arbitrary i 2 J1; LK; k 2 J1; j j 1K. To do this, we can expand the determinant with respect to the two successive lines at position k and k + 1. That gives a sum of terms of the form 5 Rigorously, the argument works under the condition that the denominator only has simple zeroes i.e. under the condition that the i j’s are not integer. We will explain later in the text how to deal with this constraint.
T1;s1 ( i + 1)T1;s2 ( i) T1;s2 1( i + 1)T1;s1+1( i) D(k; k + 1; j0; j1), where D(k; k0; l; l0) denotes the (j j 2) (j j 2) minor obtained by removing the lines k and k0 and the columns l and l0 from the determinant, and where we introduced s1 = j0 + k j0 and s2 = j1 + k + 1 j1.
Table of contents :
I Integrability and Bethe ansatz
I.1 Coordinate Bethe ansatz for the Heisenberg spin chain
I.2 Generalization to other integrable models
I.3 AdS/CFT duality
II Q operators for spin chains
II.1 Spin chains and T-operators
II.1.1 Construction of the T-operators
II.1.2 Differential expression of the T-operators
II.1.3 Generalization to super-groups
II.1.4 Hirota equation and CBR determinant formula
II.1.5 Jacobi identity and bilinear equations
II.1.6 Conservation of the number of particles
II.2 Bäcklund transform and Bethe equations
II.2.1 Introduction of the Bäcklund flow
II.2.2 Bethe equations and energy spectrum
II.3 Differential expression of Q-operators
II.3.1 Derivation of the simplest Q-operators, when L = 1
II.3.2 General expression of the Bäcklund flow
II.3.3 Degree of the T-operators
II.4 Relation to the classical integrability
II.4.1 The MKP hierarchy and the CBR formula
II.4.2 The rational solution of the MKP hierarchy
III Thermodynamic Bethe Ansätze and Y-systems
III.1 Example of the principal chiral model
III.1.1 The asymptotic Bethe ansatz
III.1.2 Thermodynamic Bethe ansatz
III.2 General solution of Hirota equation
III.2.1 Examples of (a; s) lattice
III.2.2 Equivalence of Hirota equation and Y-system equation
III.2.3 Typical solution of Hirota equation
III.2.4 Gauge conditions and Wronskian expressions of the T-functions
III.2.5 Writing FiNLIEs
III.3 FiNLIE for the principal chiral model
III.3.1 Y-system equation
III.3.2 Asymptotic limit
III.3.3 Parameterization of the q-functions
III.3.4 Set of equations
III.3.5 Computation of the energy
IV FiNLIE for the AdS/CFT Duality
IV.1 The Y-system for AdS/CFT
IV.2 The asymptotic limit
IV.3 Parameterization of the T- and q-functions
IV.3.1 Analyticity strips
IV.3.2 Parameterization of the q-functions
IV.4 Set of equations
IV.4.1 The “Physical Gauge”
IV.4.2 The Z4 symmetry
IV.4.3 Set of equations and iterative algorithm
IV.5 Numerical results
Conclusion and outlook
A Introduction to representations of matrix groups
A.1 Notations and tensor product
A.2 Representations of SU(2)
A.3 Young diagrams and representations of GL(K)
A.3.1 Young diagrams
A.3.2 Decomposition of the tensor representation of GL(K)
A.3.3 Finite rank groups
A.4 Generalization to the super-group GL(KjM)
B Properties of co-derivatives
B.1 Diagrammatic expressions for co-derivatives
B.2 Identities involving co-derivatives
B.2.1 Yang-Baxter Equation
B.2.2 Bilinear identities
B.3 Co-derivatives and eigenvalues
Index and notations