Introduction to the Spectrum of {\mathcal{N}=4} SYM and the Quantum Spectral Curve

Introduction to the Spectrum of SYM and the Quantum Spectral Curve

Nikolay Gromov
Mathematics Department, King’s College London, The Strand, London WC2R 2LS, UK. & St.Petersburg INP, Gatchina, 188 300, St.Petersburg, Russia
Abstract. This review is based on the lectures given by the author at the Les Houches Summer School 2016. It describes the recently developed Quantum Spectral Curve (QSC) for a non-perturbative planar spectrum of N=4 Super Yang-Mills theory in a pedagogical way starting from the harmonic oscillator and avoiding a long historical path. We give many examples and provide exercises. At the end we give a list of the recent and possible future applications of the QSC.
Dedication. In memory of Ludvig Dmitrievich Faddeev.

Chapter 1 Introduction

The importance of AdS/CFT correspondence in modern theoretical physics and the role of SYM in it is hard to over-appreciate. In these lecture notes we try to give a pedagogical introduction to the Quantum Spectral Curve (QSC) of SYM, a beautiful mathematical structure which describes the non-perturbative spectrum of strings/anomalous dimensions of all single trace operators. The historical development leading to the discovery of the QSC [7, 11] is a very long and interesting story by itself, and there are several reviews trying to cover the main steps on this route [6, 18]. For the purposes of the lectures we took another approach and try to motivate the construction by emphasizing numerous analogies between the QSC construction and basic quantum integrable systems such as the harmonic oscillator, Heisenberg spin chains, and classical sigma-models. In this way the QSC comes out naturally, bypassing extremely complicated and technical stages such as derivation of the S-matrix [19], dressing phase [20], mirror theory [22], Y-system [2], Thermodynamic Bethe Ansatz [3, 24, 25, 23], NLIE [7, 26] and finally derivation of the QSC [7, 11].

We also give examples of analytic solutions of the QSC and in the last chapter describe step-by-step the numerical algorithm allowing us to get the non-perturbative spectrum with almost unlimited precision [13]. We also briefly discuss the analytic continuation of the anomalous dimension to the Regge (BFKL) limit relevant for more realistic QCD.

The structure is the following: in the Chapter 1 we re-introduce the harmonic oscillator and the Heisenberg spin chains in a way suitable for generalization to the QSC. Chapter 2 describes classical integrability of strings in a curved background, which give some important hints about the construction of the QSC. In Chapter 3 we give a clear formulation of the QSC. In Chapter 4 we consider some analytic examples. And in the last Chapter 5 we present the numerical method.

Acknowledgment

I am very grateful to M.Alfimov, A.Cavaglià, S.Leurent, F.Levkovich-Malyuk, G.Sizov, D.Volin, and especially to V.Kazakov and P.Vieira for numerous discussions on closely related topics. I am thankful to D.Grabner, D.Lee and J.111i.e. Julius, who only has a first name for carefully reading the manuscript. The work was supported by the European Research Council (Programme “Ideas” ERC-2012-AdG 320769 AdS-CFT-solvable). We are grateful to Humboldt University (Berlin) for the hospitality and financial support of this work in the framework of the “Kosmos” programe. We wish to thank STFC for support from Consolidated grant number ST/J002798/1. This work has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007-2019/ under REA Grant Agreement No 317089 (GATIS).

Please report typos or send other improvement requests for these lecture notes to nikgromov@gmail.com.

Chapter 2 From Harmonic Oscillator to QQ-Relations

2.1 Inspiration from the Harmonic Oscillator

To motivate the construction of the QSC we first consider the 1D harmonic oscillator and concentrate on the features which, as we will see later, have similarities with the construction for the spectrum of SYM.

The harmonic oscillator is the simplest integrable model which at the same time exhibits nontrivial features surprisingly similar to SYM. Our starting point is the Schrödinger equation

(2.1)

where . Alternatively, it can be written in terms of the quasi-momentum

(2.2)

as

(2.3)

This non-linear equation is completely equivalent to (2.1). Instead of solving this equation directly let us make a simple ansatz for . We see that for large the r.h.s. behaves as implying that at infinity . Furthermore, should have simple poles at the position of zeros of the wave function which we denote . All the residues at these points should be equal to as one can see from . We can accommodate all these basic analytical properties with the following ansatz:

(2.4)

We note that at large the r.h.s. of (2.4) behaves as . Plugging this large approximation of into the exact equation (2.3) we get:

(2.5)

Comparing the coefficients in front of and we get which is the famous formula for the spectrum of the harmonic oscillator. In order to reconstruct the wave function we expand (2.3) near the pole . Namely, we require

(2.6)

obtaining (from the first bracket)

(2.7)

This set of equations determines all in a unique way.

 

Exercise 1.

Verify for and roots that there is a unique up to a permutation solution of the equation (2.7), find the solution.  

Finally, we can integrating (2.2) to obtain

(2.8)

It is here for the first time we see the Q-function, which is the analog of the main building block of the QSC! We will refer to equation (2.7) for zeros of the Q-functions as the Bethe ansatz equation. We will call the Bethe roots.

Let us outline the main features which will be important for what follows:

  • The asymptotic of contains quantum numbers of the state.

  • Zeros of the function can be determined from the condition of cancellation of poles (2.6) (analog of Baxter equation) which can be explicitly written as (2.7) (analog of Bethe equations).

  • The wave function can be completely determined from the Bethe roots or from (by adding a simple universal for all states factor).

  • The Schrödinger equation has a second (non-normalizable) solution which behaves as . Together with the normalizable solution they form a Wronskian

    (2.9)

    which is a constant.

     

    Exercise 2.

    Prove that the Wronskian is a constant for a general Scrödinger equation.  

2.2 -Heisenberg Spin Chain

In this section we discuss how the construction from the previous section generalizes to integrable spin chains – a system with a large number of degrees of freedom. The simplest spin chain is the Heisenberg magnetic which is discussed in great detail in numerous reviews and lectures. We highly recommend Faddeev’s 1982 Les Houches lectures [27] for that. We describe the results most essential for us below.

In short, the Heisenberg spin chain is a chain of spin- particles with a nearest neighbour interaction. The Hamiltonian of the system can be written as

(2.10)

where is an operator which permutes the particles at the position and and is a constant. We introduce twisted boundary conditions by defining

(2.11)
(2.12)

The states can, again, be described by the Baxter function . The Bethe roots have a physical meaning – they represent the momenta of spin down “excitations” moving in a sea of spin ups via (see Fig.2.1). We find the roots from the equation similar to (2.7)111One should assume all to be different like in the harmonic oscillator case.

(2.13)

 

Exercise 3.

Take log and expand for large . You should get exactly the same as (2.7) up to a rescaling and shift of .  

from where one gets a discrete set of solutions for . The energy is then given by

(2.14)

 

Exercise 4.

Take and compute the energy spectrum in two different ways: 1) by directly diagonalizing the Hamiltonian (2.10), which becomes a matrix of the form

Next solve the Bethe equation (2.13) for and compute the energy from the formula (2.14).  

One could ask what the analog of the Schrödinger equation is in this case. The answer is given by the Baxter equation of the form

(2.15)

where is a polynomial which plays the role of the potential, but it is not fixed completely and has to be determined from the self-consistency of (2.15).

 

Exercise 5.

Show that the leading large coefficients of are where .  

In practice we do not even need to know as it is sufficient to require polynomiality from to get (2.13) as a condition of cancellation of the poles.

 

Exercise 6.

For generic polynomial we see that is a rational function with poles at , where . Show that these poles cancel if the Bethe ansatz equation (2.13) is satisfied.  

Figure 2.1: Two equivalent representations of the same state. In the first case we treat spin downs as excitations (magnons) moving with some momenta and all spin ups correspond to the reference (vacuum) state. In the second case we treat spin ups as excitations moving with some momenta .

Notice that given some polynomial there is another polynomial (up to a multiplier to “twist” ) solution to the Baxter equation, just like we had before for the Schrödinger equation. Its asymptotics are where . The roots of also has a physical interpretation – they describe the spin up particles moving in the sea of the spin downs (i.e. opposite to which described the reflected picture where the spin ups played the role of the observers and the spin-downs were considered as particles). The second solution together with the initial one should satisfy the Wronskian relation (in the same way as for the Schödinger equation)222The sign is used to indicate that the equality holds up to a numerical multiplier (which can be easily recovered from large limit).

(2.16)

where satisfies

(2.17)

so we conclude that .

 

Exercise 7.

Show that if and are two linearly independent solutions of (2.15), then (2.17) holds.  

We see that there are strict similarities with the harmonic oscillator. Furthermore, it is possible to invert the above logic and prove the following statement: equation (2.16) plus the polynomiality assumption (up to an exponential prefactor) by itself implies the Bethe equation, from which we departed. This logic is very close to the philosophy of the QSC.

 

Exercise 8.

Show that the Baxter equation is the following “trivial” statement

(2.18)

From that determine in terms of and .  

2.3 Nested Bethe Ansatz and -relations

The symmetry of the Heisenberg spin chain from the previous section is . In order to get closer to (the symmetry of SYM) we now consider a generalization of the Heisenberg spin chain for the symmetry group. For that we just have to assume that there are possible states per chain site instead of , otherwise the construction of the Hamiltonian is very similar.

The spectrum of the spin chain can be found from the “Nested” Bethe ansatz equations [28], which now involve two different unknown (twisted) polynomials and . They can be written as333by the twisted polynomials we mean the functions of the form , for some number .:

(2.19)

and the energy is given by

(2.20)

We denote . We also introduced some very convenient notation

(2.21)

 

Exercise 9.

Show that the Bethe equations reduce to the equations (2.13) and (2.14) when .  

2.3.1 Bosonic duality

From the Heisenberg spin chain we learned that the Baxter polynomial contains as many roots as arrow-downs we have in our state. In particular the trivial polynomial corresponds to the state . One can also check that there is only one solution of the Bethe equations where is a twisted polynomial of degree and it satisfies

(2.22)

 

Exercise 10.

Solve this equation for and and check that also solves the Bethe equations of the spin chain. Compute the corresponding energy.  

As this equation produces a polynomial of degree it must correspond to the maximally “excited” state . It is clear that even though physically these states are very similar our current description in terms of the Bethe ansatz singles out one of them. We will see that there is a “dual” description where the Q-function corresponding to the state is trivial. In the case of the spin chain where we have different states per node of the spin chain, which we can denote , there are equivalent vacuum states , , and , but only one of them corresponds to the trivial solution of the nested Bethe ansatz. Below we concentrate on the case and demonstrate that there are several equivalent sets of Bethe ansatz equations (2.19).

Figure 2.2: Bosonic duality applied to the first node of the BA.

To build a dual set of Bethe equations we first have to pick a -function which we are going to dualise. For example we can build a new set of Bethe equations by replacing , a twisted polynomial of degree , with another twisted polynomial of degree , where is the degree of the polynomial . For that we find a dual -function from

(2.23)

Let’s see that satisfies the same Bethe equation. By evaluating (2.23) at and dividing by the same relation evaluated at we get:

(2.24)

which is exactly the first equation (2.19) with replaced by ! To accomplish our goal we should also exclude from the second equation. For that we notice that at the relation gives

(2.25)

which allows us to rewrite the whole set of equations (2.19) in terms of . We call this transformation a Bosonic duality. Similarly one can apply the dualization procedure to . We determine from

(2.26)

By doing this we will be able to replace by in (2.19). Let us also show that we can use instead of in the expression for the energy (2.14). We recall that , so evaluating (2.26) at we get

(2.27)

We can also differentiate (2.26) in once and then set , so that

(2.28)

Dividing (2.28) by (2.27) we get

(2.29)

which indeed gives

(2.30)
Better notation for -functions

One can combine the above duality transformations and say dualise after dualising and so on. In order to keep track of all possible transformations one should introduce some notation, as otherwise we can end up with multiple tildas. Another question we will try to answer in this part is how many equivalent BA’s we will generate by applying the duality many times to various nodes.

In order to keep track of the dualities we place numbers in between the nodes of the Dynkin diagram. We place the -functions on the nodes of the diagram as in Fig.2.2. Then we interpret the duality as an exchange of the corresponding labels sitting on the links of the diagram, so if before the dualization of we had , after the duality we have to exchange the indexes and obtaining . If instead we first dualised we would obtain . Each duality produces a permutation of the numbers. We also use these numbers to label the -functions. Namely we assign the indexes to the function in accordance with the numbers appearing above the given node. So, in particular, in the new notation

(2.31)

Each order of the indexes naturally corresponds to a particular set of Bethe equations. For instance, the initial set of Bethe equations on correspond to the order and the Bethe ansatz (BA) for correspond to and so on. Now we can answer the question of how many dual BA systems we could have; this is given by the number of permutations of i.e. for the case of we get equivalent systems of BA equations.

Following our prescription we also denote

(2.32)

We note that we should not distinguish Q’s which only differ by the order of indexes. So, for instance, and is the same . We can count the total number of various -functions we could possibly generate with the dualities: different -functions which are

(2.33)

for completeness we also add and so that in total we have . For general we will find different Q-functions. We see that the number of the Q-functions grows rapidly with the rank of the symmetry group. For we get functions, and we should study the relations among them in more detail.

-relations

Let us rewrite the Bosonic duality in the new notation. The relation (2.23) becomes

(2.34)

where we added to the r.h.s. to make both l.h.s and r.h.s be bilinear in . Very similarly (2.26) gives

(2.35)

We see that both identities can be written in one go as

(2.36)

where for general we would have and and represents a set of indexes such that in (2.35) it is an empty set and for the second identity (2.36) contains only one element . Note that no indexes inside are involved with the relations and in the r.h.s. we get indexes and glued together in the new function. We see that proceeding in this way we can build any -function starting from the basic with one index only. For that we can first take and build , then take and build and so on. It is possible to combine these steps together to get explicitly

(2.37)

Whereas the first identity (2.34) is obvious from the definition, the second (2.37) is a simple exercise to prove from (2.36).

 

Exercise 11.

Prove (2.37) using the following Mathematica code

Q[a___] := Signature[{a}] Q @@ Sort[{a}] /; ! OrderedQ[{a}]
(*program bosonic duality*)
Bosonic[J___, i_, j_] := Q[J, i, j][u_] -> (
Q[J, i][u + I/2] Q[J, j][u - I/2] -
Q[J, i][u - I/2] Q[J, j][u + I/2])/Q[J][u];
(*checking the identity*)
Q[1, 2, 3][u] Q[][u + I/2] Q[][u - I/2] /. Bosonic[1, 2, 3] /.
Bosonic[1, 2] /. Bosonic[1, 3] /. Bosonic[2, 3] // Factor

Also derive a similar identity for using the same code.  

From the previous exercise it should be clear that we can generate any as a determinant of the basic Q-functions . In particular the “full-set” Q-function , which is also , can be written as a determinant of basic polynomials . Interestingly this identity by itself is constraining enough to give rise to the full spectrum of the spin chain! Indeed is a polynomial of degree and thus we get nontrivial relations on the coefficients of the (twisted) polynomials , which together contain exactly Bethe roots. This means that this relation alone is equivalent to the whole set of Nested Bethe ansatz equations. So we can put aside a non-unique BA approach, dependent on the choice of the vacuum, and replace it completely by a simple determinant like (2.37). In other words the QQ-relations and the condition of polynomiality is all we need to quantize this quantum integrable model. We will argue that for SYM we only have to replace the polynimiality with another slightly more complicated analyticity condition but otherwise keep the same QQ-relations. We will have to, however, understand what the QQ-relations look like for the case of super-symmetries like , which is described in the next section.

2.3.2 Fermionic duality in

We will see how the discussion in the previous section generalizes to the super-group case. Our starting point will be again the set of nested Bethe ansatz equations, which follow the pattern of the Cartan matrix. Let us discuss the construction of the Bethe ansatz. Below we wrote the Dynkin diagram, Cartan matrix and the Bethe ansatz equations for the super spin chain

(2.38)

The -functions still correspond to the nodes of the Dynkin diagrams and the shift of the argument of the -functions entering the numerators of the Bethe equations simply follow the pattern of the Cartan matrix (with the inverse shifts in numerators). Since the structure of the equations for the bosonic nodes is the same as before, one can still apply the Bosonic duality transformation for instance on and replace it by . However for the fermionic type nodes (normally denoted by a crossed circle), such as , we get a new type of duality transformation

(2.39)

which look similar to the Bosonic one with the difference that we can extract explicitly the dual Baxter polynomial 444Whereas for the Bosonic duality (2.23) the dual Baxter polynomial occur in a complicated way and one had to solve a first order finite difference equation in order to extract it.. Let us show that the middle Bethe equation can be obtained from the duality relation (2.39). Indeed we see again that for both and we get the middle equation

(2.40)

Next we should be able to exclude in the other two equations. For that we set and to get

(2.41)

Dividing one by the other

(2.42)

which allows up to exclude from the second equation of (2.44). This then becomes

(2.43)

As we see this changes the type of the equation from bosonic to fermionic. Thus we also change the type of the Dynkin diagram. This is expected since for super algebras the Dynkin diagram is not unique. Similarly the fourth equation also changes in a similar way. To summarize, after duality we get

(2.44)
Figure 2.3: Fermionic duality
Index notation

Again in order to keep track of all possible combinations of dualities we have to introduce index notation. In the super case we label the links in the Dynkin diagram by two types of indexes (with hat and without). The type of the index changes each time we cross a fermionic node. For instance our initial set of Bethe equations corresponds to the indexes . The fermionic duality transformation again simply exchanges the labels on the links of the Dynkin diagram (see Fig.2.3). So after duality we get , which is consistent with the grading of the resulting Bethe ansatz equations. Finally, we label the -functions by two antisymmetric groups of indexes – with hat and without again simply listing all indexes appearing above the given node of the Dynkin diagram. In particular we get

(2.45)

An alternative notation is to omit hats and separate the two sets of indexes by a vertical line:

(2.46)

 

Exercise 12.

The fermionic duality transformation changes the type of the Dynkin diagram. The simplest way to understand which diagram one gets after the duality is to follow the indexes attached to the links. Each time the type of the index changes (from hatted to non-hatted) you should draw a cross. List all possible Dynkin diagrams corresponding to Lie algebra.  

Fermionic QQ-relations

In index notation (2.39) becomes

(2.47)

For completeness let us write here the bosonic duality relations

(2.48)

In the case of one could derive all functions in terms of functions and . We will demonstrate this in the next section in the example of .

2.4 QQ-relations for Spin Chain

The global symmetry of SYM is . The QQ-relations from the previous section associated with this symmetry constitute an important part of the QSC construction. The symmetry (up to a real form and a projection) is same as . In this section we specialize the QQ-relations from the previous part to this case and derive all the most important relations among Q-functions. In particular we show that all various Q-functions can be derived from just Q-functions with one index

(2.49)

which are traditionally denoted in the literature as

(2.50)

These are the elementary Q-functions.

For us, another important object is . According to the general consideration above it can be obtained from the fermionic duality relation (2.47) with

(2.51)

This is the first order equation on which one should solve; and the formal solution to this equation is555Note that there is a freedom to add a constant to . This freedom is fixed in the twisted case as we should require that has a “pure” asymptotics at large i.e. .

(2.52)

 

Exercise 13.

Find a solution to the equation (2.51) for and also for .  

Once we know we can build any -function explicitly in terms of and . For example using the Bosonic duality we can get

(2.53)

In this way we can build all Q-functions explicitly in terms of and . There is a nice simplification taking place for Q-functions with equal number of indexes:

(2.54)

 

Exercise 14.

Prove (2.54) using the following Mathematica code

Q[a___][b___][u_] := Signature[{a}] Signature[{b}]
Q[Sequence @@ Sort[{a}]][Sequence @@ Sort[{b}]][u]
/; ! (OrderedQ[{a}] && OrderedQ[{b}])
(*program bosonic and fermionic dualities*)
B1[J___, a_, b_][K___] := Q[J, a, b][K][u_] :>
(Q[J, a][K][u + I/2] Q[J, b][K][u - I/2] -
Q[J, a][K][u - I/2] Q[J, b][K][u + I/2])/Q[J][K][u];
B2[K___][J___, i_, j_] :=
Q[K][J, i, j][u_] :> (Q[K][J, i][u + I/2] Q[K][J, j][u - I/2] -
Q[K][J, i][u - I/2] Q[K][J, j][u + I/2])/Q[K][J][u];
F1[K___, a_][J___, i_][u_] := Q[K, a][J, i][u] :>
(Q[K, a][J, i][u - I] Q[K][J][u - I] +
Q[K, a][J][u - I/2] Q[K][J, i][u - I/2])/Q[K][J][u]
F2[K___, a_][J___, i_][u_] := Q[K, a][J, i][u] :>
(Q[K, a][J, i][u + I] Q[K][J][u + I] -
Q[K, a][J][u + I/2] Q[K][J, i][u + I/2])/Q[K][J][u]
(*deriving the identity*)
Q[a, b][i, j][u] /. B1[a, b][i, j] /. B2[a][i, j] /. B2[b][i, j] /.
Flatten[Table[F1[c][k][u + I], {c, {a, b}}, {k, {i, j}}]] /.
Flatten[Table[F2[c][k][u - I], {c, {a, b}}, {k, {i, j}}]] /.
B2[][i, j] // Simplify

Also derive a similar identity for using the same code. The general strategy is to use the bosonic duality to decompose ’s into -functions with fewer indexes. Then use (2.51) to bring all