Pulse Contour Analysis

Ryo Suzuki

Institute for Theoretical Physics and Spinoza Institute,

Abstract

The hybrid NLIE of AdS5 × S5 is applied to a wider class of states.

We Find That The

Konishi state of the orbifold AdS5 × S5/ZS satisfies A1 NLIE with the source terms which are derived from contour deformation trick. For general states, we construct a deformed contour with which the contour deformation trick yields the correct source terms.

Orbifold Konishi State

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

TBA and NLIE in horizontal strips . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

Source terms of A1 NLIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

General Source Terms In The Simplified Tba

. . . . . . . . . . . . . . . . . . . .

3.2

General source terms in A1 NLIE . . . . . . . . . . . . . . . . . . . . . . . . . .

Fourier Transform Method

. . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.2

Contour deformation trick with Konishi’s contour . . . . . . . . . . . . .

3.2.3

Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

Consistent deformed contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

B.1 A1 TQ-relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

B.2 Symmetry in A1 TQ-relations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

B.3 General solution of A1 TQ-relations . . . . . . . . . . . . . . . . . . . . . . . . .

21

C.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

C.2 Transfer matrix for orbifold Konishi . . . . . . . . . . . . . . . . . . . . . . . . .

25

D.1 Derivation of A1 NLIE with source terms . . . . . . . . . . . . . . . . . . . . . .

25

D.2 Contour deformation for A1 NLIE . . . . . . . . . . . . . . . . . . . . . . . . . .

D.2.1

Deformed contour of orbifold Konishi state . . . . . . . . . . . . . . . . .

D.2.2

Various deformed contours . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction And Summary

The primary example of AdS/CFT correspondence is the one between four-dimensional N = 4 super Yang-Mills and AdS5 × S5 string theory . The spectrum of string states on AdS5 × S5 can be computed by the mirror Thermodynamic Bethe Ansatz (TBA) equations [2, 3, 4] based on string hypothesis in the mirror model [5, 6]; or equivalently the extended Y-system on psu(2, 2|4)-hook [7, 8, 9]. It is believed that these methods give the exact answer, because they capture all finite-size corrections [10, 11].

The numerical study of the mirror TBA has made progress [12, 13, 14, 15]. However, it suffers from the problem of critical coupling constants . The analyticity of the unknown variables called Y-functions changes around certain values of ’t Hooft coupling constant, and the explicit form of the TBA equations changes there discontinuously. As a result, it is difficult to solve the equation with high precision around the critical values, and to judge if the exact energy does not show unusual behavior like inflection points around the critical points.

The author has recently applied the method of hybrid nonlinear integral equations (hybrid NLIE) to the mirror TBA for AdS5 × S5 . This method replaces the horizontal part of the mirror TBA equations by A1 NLIE.1 The hybrid NLIE consists of a smaller set of unknown variables than the mirror TBA, and we expect that it suffers less often from the problem of critical coupling constants. We exemplify our expectation in a way similar to .

For this purpose the mirror TBA for the twisted AdS5 × S5 offers a desired playground, because all Y-functions have intricate analytic properties, depending on the twist angle α and

2Π .2

The orbifold Konishi state is the simplest nontrivial example that exhibits critical behavior in the mirror TBA for YM|w. For this state, we find that the hybrid NLIE also exhibits critical behavior; its source terms change discontinuously across certain values of coupling constant.

Orbifold Konishi is a two-particle state in the sl(2) sector of AdS5 × S5/ZS, where the ZS acts on su(2)2 ⊂[su(2|2)2 ∩su(4)]. This is also a special state in the twisted AdS5 × S5, β- or γ-deformed AdS5 × S5 models. The orbifold and γ-deformed models are another important examples of AdS/CFT correspondence, realized in gauge theory [24, 25, 26, 27, 28, 29, 30] and in string theory [31, 32, 33, 34, 35]. Finite-size corrections of deformed theories have been studied in gauge theory [36, 37, 38], in string theory , by L¨uscher formula [40, 41, 42, 43, 44, 45, 46, 47], and by the mirror TBA or Y-system [48, 49, 50, 51, 52]. However, it is not clear if the corresponding sigma model on twisted AdS5 × S5 possesses integrability (see 1This equation is called Kl¨umper-Batchelor-Pearce or Destri-de Vega equation in the literature [19, 20, 21, 22, 23]. We call it A1 NLIE, since it can be derived from A1 TQ-relations and analyticity conditions as shown in .

2In fact, the mirror TBA for YM|w in the untwisted model do not have critical coupling constants asymptot- ically. We checked this claim for several four particle states for g ≲1.

2

for review), though integrable twists exist mathematically. Next, we notice that such discontinuous change of the NLIE for the orbifold Konishi state can be explained by the contour deformation trick. There is a conjecture that the TBA for excited states follows from the TBA for the ground state by analytic continuation of coupling constant [54, 55]. It is expected that such analytic continuation introduces extra singularities of the integrand on the complex rapidity plane, and deforms the integration contour accordingly.

Then, the excited states TBA should be expressed equivalently either as the ground-state TBA integrated over the deformed contour, or as the TBA integrated over the real line with additional source terms. This idea is called contour deformation trick. The contour deformation trick predicts how to correct the TBA when numerical iteration ceases to converge due to the change of analyticity, and is a guideline to study various states in the mirror TBA including boundstates . The A1 NLIE with source terms has been studied in various examples [57, 58, 59, 60, 61, 62, 63, 64, 65], and the contour deformation trick was used in [66, 67].

With successful examples of the contour deformation in mind, we ask what the most general possible source terms are, and if they are obtained by the contour deformation trick. In princi- ple, A1 NLIE can be derived even when the Q-functions are meromorphic, rather than analytic, in the upper or lower half plane. Then the isolated singularities of Q-functions provide extra source terms to A1 NLIE. It is a nontrivial question whether such source terms can be explained by the contour deformation trick, particularly with the same contour as in the orbifold Kon- ishi state. Indeed, mismatch is found between the two results. To reconcile this problem, we construct a deformed contour which is consistent for general states including orbifold Konishi.

The consistent deformed contour picks up only the preferred singularities of the integrand and runs both the lower and upper half planes. The details will be discussed in Section 3. The contour deformation trick illustrates the difference between hybrid NLIE and FiNLIE . In the latter the integrals run over the gap discontinuity of dynamical variables, which is not something to be deformed. In contrast, hybrid NLIE is written in terms of gauge-invariant (but frame-dependent) variables,3 allowing us to handle the equations similar to that of the mirror TBA.

This paper is organized as follows. In Section 2, we study the orbifold Konishi state from the mirror TBA and hybrid NLIE, and clarify the critical behavior in the asymptotic limit. In Section 3, we discuss the source terms of A1 NLIE in view of contour deformation trick. Section 4 is for conclusion. In appendices, we introduce our notation, review the NLIE variables, compute the asymptotic transfer matrix in the form of Wronskian, and derive the results in Section 3.

3See the discussion at the end of Appendix C for the frame dependence.

Tba And Nlie For Twisted Ads5 × S5

We study the critical behavior of hybrid NLIE for the orbifold Konishi state as a specific example. We briefly review the mirror TBA in twisted AdS5 × S5 and their critical behavior.

Orbifold Konishi State

The orbifold Konishi state can be defined in two equivalent ways. The first is to consider the sl(2) Konishi descendant on the orbifold AdS5×S5/ZS, where the ZS action is chosen as follows (see ). We decompose the transverse 8+8 fields of AdS5 × S5 into (2|2) ⊗(2|2) representation of su(2|2)L × su(2|2)R , as

(2.1)

where b, ˙b = 1, 2 refer to the S5 part, and β, ˙β = 3, 4 refer to the AdS5 part of su(2|2)2. The

S

(nL ∈Z).

(2.2)

Similarly, the boundary conditions of y˙b are twisted by αR = 2πnR

. The Orbifold Action (2.2)

affects only the auxiliary part of the asymptotic Bethe Ansatz equations. Thus, if we set the total momentum to zero as in the ordinary Konishi state, the asymptotic Bethe roots remain unchanged before and after orbifolding. This is called orbifold Konishi state.

The second is to introduce integrable twisted boundary conditions to the transfer matrix of AdS5 × S5. To preserve the integrability, the twist operator must commute with the S-matrix. When the twist operator belongs to [su(2|2)2 ∩su(4)] and the twist angle is equal to a multiple of 2π/S, Konishi state of the twisted AdS5 × S5 is equivalent to the orbifold Konishi state.

The second point of view is useful to construct the twisted transfer matrix, as defined by

.

And Similarly For T R

Q,1. The S0i is the S-matrix between the mirror particle and the i-th particle in string theory. We can diagonalize (2.3) by algebraic Bethe Ansatz . In practice, it is easier to twist the generating function for the eigenvalues of transfer matrices [70, 49, 48]. This construction will be discussed in Appendix C, where we also rewrite the transfer matrices in the form of Wronskian. In what follows we set αL = αR ≡α for simplicity.

The mirror TBA for the twisted model is obtained as follows. The twist angle α in string theory corresponds to the insertion of defect operator in mirror theory . In particular, the same mirror string hypothesis is used in both twisted and untwisted models. In the case of

4

orbifold, the defect operator can be identified as an extra chemical potential, and it changes the v →±∞asymptotics of Y-functions . The mirror TBA equations for twisted AdS5 ×S5 are solved by the twisted transfer matrices in the asymptotic limit .

Tba And Nlie In Horizontal Strips

We compare mirror TBA and hybrid NLIE in the horizontal part of the psu(2, 2|4)-hook for the twisted AdS5 × S5. We will consider only the states which are invariant under the interchange (a, s) →(a, −s) of the psu(2, 2|4)-hook.

The simplified TBA equation for Y1|w and YM|w (M ≥2) can be written as

(2.4)

log YM|w = −VM|w + log(1 + YM+1|w) ⋆sK + log(1 + YM−1|w) ⋆sK ,

(2.5)

where VM|w is the source term, which depends on the state and the values of (α, g) under consideration. In the hybrid NLIE, the 1 + YM+1|w on the right hand side is replaced by

(2.7)

where ν = I or II refers to the two sets of Q-functions , and γ (0 < γ < 1) is a regularization parameter, as reviewed in appendix B. We leave s ∈Z≥3 unspecified, though one can substitute s = 3 at any time. The case of ν = I is simpler than ν = II, because the source terms {JI

3}

vanishes in the Konishi state of the untwisted AdS5 × S5 model, at least asymptotically. Below

S , Ai

s only, and omit ν = I. In short, the YM|w functions of the mirror TBA are replaced by three dynamical variables, (a3 , a3 , Y1|w).4 Numerically, the equations (2.6), (2.7) can be checked modulo multiple of πi for the following reason. Since (1+as , 1+as) are complex, their logarithm may choose either of log(−1) = ±πi, which changes the numerical value of the convolution by (2πi) ⋆Kf = πi.

Critical lines and analyticity. The source terms in TBA or NLIE change discontinuously as we vary the parameters (α, g). We divide the (α, g) plane into subregions according to different 4If we consider both left and right horizontal strips of the psu(2, 2|4)-hook, Y (L)

, Y (R)

1|w ).

5

form of the source terms. The boundary of subregions is called critical lines. We denote the

Cr (G) Or G = G(I)

cr (α). The critical lines are different for different integral equations of TBA or NLIE. So the phase space of a given state in the twisted AdS5 × S5 is divided into infinitely many tiny regions as

O

for hybrid NLIE.

(2.9)

The critical lines, or discontinuous changes of source terms, come from the change of the analyticity of unknown variables in a given integral equation. This statement holds true for both simplified TBA and NLIE. The TBA for the orbifold Konishi has already been studied in detail , so we will make this statement more precise for the NLIE.

It should be noted that the critical lines of hybrid NLIE depend on the regularization parameter γ. Also, the critical lines of the mirror TBA change if we pull back the deformed contour to the line R + iδ with δ̸ = 0 instead of the real line.5 Besides its simplicity, there is no particular meaning of setting γ or δ to zero. From the continuity of the equations this implies that physical quantities such as the exact energy should not be singular at g = g(l)

Source Terms Of A1 Nlie

We determine the source terms in A1 NLIE (Js , Js), by taking examples of the twisted ground state and orbifold Konishi state. Source term of twisted ground state.

The Ground State Of The Twisted Ads5 ×S5 Satisfies

the simplified mirror TBA with VM|w = 0 .

Js = +Iα,

Js = −iα.

(2.10)

This result follows immediately from the asymptotic solution discussed in Appendix C. Even for excited states, each term in the A1 NLIE approaches its ground state value in the limit v →±∞, just like TBA. Furthermore, the orbifold Konishi state satisfies the same equation at small α̸ = 0 and small g. For general (α, g) we should add logarithms of S-matrix to the source term.

5It is not practical to solve the mirror TBA using the Y-functions not sitting on the real axis, because the reality of Y-functions is abandoned. 6The author thanks a referee of JHEP for pointing this out.

6

Main strip of hybrid NLIE. Before studying source terms at general (α, g), let us discuss the main strip of the mirror TBA or the hybrid NLIE. The main strip is defined by the region of complex plane in which the respective equation remains valid without modification. It is helpful to identify the main strip in advance, because the critical lines are often related to the movement of extra zeroes going in or out of this strip.

The main strip of the simplified TBA for YM|w (2.4), (2.5) is A−1,1 defined in (A.2). This is because we encounter the singularity of sK along the boundary of A−1,1 . Analytic continuation of the simplified TBA beyond A−1,1 requires us to add an extra term ∼log(1 + Y ±) for some

Y .

The main strip for the hybrid NLIE is smaller than that of the simplified TBA. Consider the holomorphic part of A1 NLIE (2.6), which contains the kernels Kf , K[+2−2γ]

. Since

these kernels are singular at Kf(±2i/g) and sK(±i/g), the main strip of (2.6) is



(0 ≤γ ≤1).

(2.11)

The main strip of the anti-holomorphic part of A1 NLIE (2.7) is the complex conjugate of the above result. Source terms of orbifold Konishi.

We Describe The Source Terms Of Hybrid Nlie For

(as , as) describing the asymptotic orbifold Konishi state at general (α, g). One can check all these results explicitly by using the formulae in Appendix C. The holomorphic part of A1 NLIE (2.6) consists of the dynamical variables (as , as, Ys−2|w), and the variables as , as are related to bs , bs by (B.5).

As Reviewed In Appendix B, These

variables can be expressed by gauge-covariant ones by

T2,S−1 T0,S−1

.

(2.12)

Consider the asymptotic orbifold Konishi state and fix the gauge as given in Appendix C.2. For this state, neither Q- nor L-functions have singularities around the real axis, and all critical behaviors come from the extra zeroes of T-functions, T1,s−1 and T1,s , inside the main strip (2.11).7 Since the location of extra zeroes is determined by the values of (α, g), the critical lines

= 0

at α = αcr(g).

The Solution To The Equations T1,Q(−I

g) = 0 also defines the critical lines of the mirror TBA for the twisted AdS5 × S5, and their asymptotic solutions have been studied in . The first equation of (2.13) has s −1 solutions and the second has s solutions for 0 < α < π and at fixed g with 0 < g ≲1.8 We denote them by αs−1,i(g), αs,i(g, γ) with the ordering

S

< αs,s(g, γ) < π.

(2.14)

It is instructive to keep track of the zeroes of T1,Q in detail, as they behave in an interesting

Way When Α Is Around Nπ

Q for n ∈Z, 1 ≤n ≤Q −1. If α is slightly less than nπ

Q , T1,Q Has

no zeroes around the real axis. Let α grow larger. When α reaches nπ

Q , Then T1,Q Acquires A

pair of real zeroes at ±∞. The pair of zeroes run toward the origin along the real axis as α increases, and collide at the origin. After the collision, they run along the imaginary axis in the

G At Α = Α(I)

cr . There are exceptions at α = 0, π. In the limit α →0, a pair of zeroes of T1,Q run to ±∞along the real axis. Nothing happens around α = π. As for α ∈(π, 2π) the movement of zeroes is symmetric with respect to the flip α →π −α.



.

(2.15)

Whenever α crosses the boundary of the interval Is−1(g) ∪Is(g, γ), the source terms of hybrid

Nlie (Js , Js) Change Discontinuously.9

The (Js , Js) at fixed (α, g) are given explicitly as follows. Start from the source terms for the grounds state (2.10). If α ∈Is−1(g), add (jB , B) to (Js , Js); and then if α ∈Is(g, γ), add (jC , C) to (Js , Js), where jB , B , jC , C are defined

8The Equation T1,Q(−I

g) = 0 has more asymptotic solutions for g ≳1, which are called Type II and Type III critical behaviors in . 9Recall that s = 3 is the minimum choice of hybrid NLIE. In contrast, the phase space (α, g) of the mirror TBA for orbifold Konishi state is classified partially by ∪∞ s=1Is(g), which consists of infinitely many segments

Of The Width ∼Π

s for each s.

8

Figure 1: Asymptotic phase space of the mirror TBA (Left) and hybrid NLIE (Right) in the horizontal part. We set s = 3 and γ = 0 in hybrid NLIE. The lines correspond to α = nπ

The Solutions Of T1,Q(−I

g) = 0 for Q = 2, 3, 4 . . . (TBA) and Q = 2, 3 (NLIE). The phase space of the mirror TBA (Left) should be infinitesimally subdivided if Q is not truncated at Q = 6. where bj , cj are defined as the zeroes of dynamical variables:

= 0,

cj ∈A−1,1 .

(2.19)

All solutions of (2.18), (2.19) must be summed in (2.16), (2.17). The integral equation for these roots can be obtained by analytic continuation of (2.4)-(2.7) as in , noting that Ys−2|w(bj) ∝T1,s(bj) = 0. One can derive the critical lines of (2.13) from these results, by recalling that (1 + as), (1 + as) are related to T1,s , and 1 + Ys−2|w is related to T1,s−1 . It will turn out in Section 3.2.2 that each term of (2.16), (2.17) can be explained by the contour deformation trick of the NLIE (2.6), where the deformed contour runs through the lower half plane. Figure 1 shows the horizontal part of the critical lines in the mirror TBA and hybrid NLIE from the asymptotic analysis.

One remark is needed to evaluate the integrals in TBA and NLIE correctly in a numerical way. Consider the convolutions log(1 + as) ⋆Kf −log(1 + as) ⋆K[+2−2γ]

In (2.6). If (1 + As)

crosses the branch cut of logarithm running the negative real axis, then the integrand changes discontinuously. Suppose there exists vd ∈R such that Im [1 + as(vd)] = 0 with Re [1 + as(vd)] < 0.

(2.20)

Then we need to integrate log(−1) = ±πi over (vd, ∞) or (−∞, vd), which provides extra source terms. As for asymptotic Konishi state, whenever (1 + as) crosses the branch cut of logarithm,

9

then (1 + as) crosses the branch cut at the same point. Thus we get



+ 2πi.

(2.22)

The discontinuity of logarithm can in principle happen for the integral with log(1 + Ys−2|w).

Contour Deformation Trick For Tba And Nlie

In the last section we studied the ground and orbifold Konishi states in the twisted AdS5 × S5, in which the hybrid NLIE acquires source terms. In this section, we turn our attention to the structure of the source term for general states. It is known that the origin of the source term in the simplified TBA for general states can be explained by both integration of Y-system and contour deformation trick. This is no longer trivially so in hybrid NLIE, as we shall see below.

General Source Terms In The Simplified Tba

Take the simplified TBA for Y1|w as an example, and the following discussion applies to other simplified TBA equations as long as the Y-system exists at that node. We will derive the source terms by integration of Y-system and contour deformation trick.

The explanation by integration of Y-system goes as follows.10

Dlf(V) ≡∂

∂v log f(v).

(3.1)

Suppose Y1|w(v) has a set of single zeroes rj inside the strip A−1,1 . If we take the convolution



.

(3.2)

Here all solutions of Y1|w(rj) = 0, rj ∈A−1,1 must be summed. If we integrate both sides with respect to v, we obtain the simplified TBA equation (2.4) with11

(3.3)

10This explanation is also called TBA lemma in the literature.

Y+

ˆ⋆sK owing to Y−(v −i0) = Y+(v + i0) for v ∈(−∞, −2)∪(+2, +∞).

10

where c1|w is an integration constant fixed by the behavior v →±∞, where all Y-functions approach the ground state value. The explanation by contour deformation trick goes as follows. We start from the simplified TBA equation (2.4) for the ground state, V1|w = c1|w . To obtain the TBA equation for excited states, we regard the contour of integration in the right hand side of (2.4) as running somewhere far below in the complex plane. When we pull the deformed contour back to the real axis, we obtain additional terms by picking up the residues as

(3.4)

where C2|w , Cy are the deformed contour for respective convolutions. Let {ρn} be a set of roots Y1|w(ρn) = 0, where ρn ∈An−1,n for n ≥1 and ρn ∈An,n+1 for

N ) = ∞,

n ∈Z̸=0 .

(3.5)

When we straighten the deformed contours of (2.4) running through the lower half plane, the

.

(3.6)

where the contributions from ρ−n (n ≥2) vanish owing to S−S+ = 1.

This Result Agrees

perfectly with (3.3).

Fourier Transform Method

The A1 NLIE was derived from the assumptions that Q[s−2], L[+s] are analytic in the upper

[2−S], L

[−s] are analytic in the lower half plane . This derivation can be generalized to the case where dynamical variables have zeroes or poles in the complex plane:13 T1,s(ts,n) = T1,s(ts,−n) = Q(qn) = Q(qn) = L(ℓn) = L(ℓn) = 0,

∈A−N,−N+1 ,

(n ≥1).

(3.7)

12There can be multiple roots as well as poles inside the same strip of the complex plane. It is straightforward to generalize the whole argument for such cases. 13The Fourier transform of logarithmic derivative diverges if these functions have zeroes on the boundary of Am,n , namely on the line g Im v ∈Z. We should regularize this by shifting the zeroes slightly upward or downward.

11

In general, these functions can have multiple zeroes or poles in the complex plane. The gen- eralization for such case is straightforward; if they have poles, the logarithmic derivative have the residue with the opposite sign. For simplicity we do not discuss poles.

The whole derivation is explained in Appendix D.1. Eventually we obtain the derivative of the source terms Js appearing in the hybrid NLIE (2.6) as

S+1

).

(3.11)

We can neglect the δ-functions, as they just add a constant after integration.

Contour Deformation Trick With Konishi’S Contour

We start from the A1 NLIE for the ground state with constant source terms (Js , Js) = (js , s). Then we apply the contour deformation trick to obtain extra source terms, using the same deformed contour as that of the orbifold Konishi state, depicted in Figure 2. For the NLIE of as , it runs slightly above the line Im v = (1 −s + γ)/g, and run down along the imaginary axis.

Note that the integrands have branch cut discontinuity along the line Im v = (1 −s + γ)/g. We take the limit γ ≪1 in what follows. Again we throw the details of computation in Appendix D.2. After straightening the contour

J

).

12

Figure 2: The deformed contour used in the NLIE for bs for the orbifold Konishi state. (Left) the contour in z-torus, where the vertical and horizontal axes are normalized by the period of the rapidity torus with moduli k = −4g2/Q2, with Q = s−1 for (1+bs), (1+bs) and Q = s−2 for (1 + Ys−2|w). The real line in z-torus corresponds to the real axis of the mirror v-plane, and the line Im z = −1 corresponds to the real axis of the string v-plane. We assumed that there are no singularities like Bethe roots along the string real axis. (Right) the contour in v-plane, where the orange region corresponds to the region surrounded by the deformed contour and the mirror real axis.

Comparison

Let us compare the Fourier transform of the derivative of the source terms (D.19) (Fourier source terms), with the source terms predicted by the contour deformation trick (3.12) (CDT source terms). We can make a similar argument for the NLIE of bs . Since this is complex conjugate to bs , we just have to impose the complex-conjugate constraints in addition.

It turns out that there are mismatches in two results. Let us have a closer look for each of the T, L, Q-functions. T-functions.

The Fourier source terms (3.9) agree with the first line of the CDT source terms (3.12). L-functions.

The Fourier source terms (3.10) partially agree with the second line of the CDT source terms (3.12). The terms with {ℓm} agree with each other if ℓm≥2s−1 lie along the imaginary axis in the lower half plane, so that all of them are picked up by the deformed contour.

Fourier

Q[s−2] , L[+s] are meromorphic in the upper half plane.

Cdt

Q , L[+2] are meromorphic in the upper half plane. Table 1: Analyticity conditions used in the Fourier transformation method and the contour deformation trick.

The complex conjugate conditions for Q, L are also used.

We Make No

assumptions about Q , L[+2] in the lower half plane, Q , L [−2] in the upper half plane. The terms with {ℓm} do not agree, because they have the opposite signs. Moreover, the roots {ℓm} in (3.10) lie in the upper half plane, while those in (3.12) lie in the lower half plane.

Q-functions. Just like the case of L-functions, The Fourier source terms (3.11) partially agree with the third line of the CDT source terms (3.12).

If the deformed contour pick up all {qm}, then the terms with {qm} perfectly agree with each other.

S+N−1} (N ≥2) In (3.11) Lie In The Upper Half

plane, while those in (3.12) lie in the lower half plane. The corresponding source terms have

S

in the Fourier source term (3.11). It lies in the lower half plane, but this term is not present in the CDT source term (3.12). The mismatch between two source terms can be explained by different analyticity conditions used in two methods, as summarized in Table 1. In particular, the extra zeroes of Q(v) at v ∈A0,s−1 and those of L(v) at v ∈A2,s modify only the CDT source terms.

Strictly speaking, the T, L, Q-functions may have singularities which can be simultaneously removed by gauge transformation. We forbid such gauge artifacts, and assume that the roots

{Ts,N , ℓn , Qn , ℓn , Qn} Are Independent.14

In other words, the contour deformation trick with Konishi’s contour works fine as long as one can choose a gauge such that all zeroes and poles can be associated to the T-functions rather than the L- and Q-functions.

Consistent Deformed Contour

In the last subsection we have learned that, for states other than the orbifold Konishi, the contour deformation trick with Konishi’s contour may not yield the correct source terms of A1 NLIE, as given by the Fourier transform method. To remedy this problem, we will look for new deformed contours of A1 NLIE.

For the sake of simplicity let us choose the gauge QI = Q

I = 1. In Other Words, We Will

14The case of boundstates is exceptional, and further analysis is needed to clarify if the contour deformation trick works as in .

14

study the analyticity of gauge-invariant quantities,

L[+S] L

[−s] .

(3.14)

The zeroes of T , L, L can be rephrased in terms of analyticity of bs , bs , Y1,s−1 as,

+

s = 1 + Y1,s−1 = ∞.

(3.15)

As in Section 3.2, we consider only the zeroes of T , L, L and use the notation (D.1).

For

completeness we also introduce L(ℓ−n) = L(ℓ−n) = 0 with ℓ−n ∈A−n,−n+1 , ℓ−n ∈An−1,n for n ≥1. As a warm-up, let us apply the contour deformation trick to A1 NLIE using the contour which encloses all zeroes of T , L, L in the mirror sheet of complex v-plane. Just like the contour deformation trick in TBA, we do not pick up the singularities of the kernels.15 Let ∗↓and ∗↑be the deformed contours which encloses all zeroes in the lower and upper half plane when pulled backed to the real axis, and ∗↕≡∗↓+ ∗↑. We then obtain

#

.

(3.17)

The derivation is discussed in Appendix D.2.2. 15The reason for this prescription is not understood.

15

Let us compare the results with the Fourier source terms. The first line of (3.17) involving the zeroes of T-functions is twice as large as (3.9), and we should apply the principal value prescription to halve this contribution. The second line agrees with (3.10), which implies that the third line should be absent. It is easy to trace the origin of the third line. For example,

K

) come from the zeroes of L[+s] and L[+s+1] in the lower half plane computed in (D.25) and (D.27), respectively. Based on this observation, we can specify a deformed contour which is consistent with the Fourier source terms.16 It turns out that, if we want to apply the contour deformation trick to the consistent deformed contour, we need to study the singularity of integrands first, and classify if they come from T-function or L-functions, following (3.15).

Let us give one example of the consistent contour by modifying the contours ∗↓, ∗↑to ∗d , ∗u . For both ∗d and ∗u , we make the principal value prescription to the zeroes (or poles) of T- functions. As for ∗d , we neglect the zeroes of L[+s] or L[s+1] in the lower half plane, and as for

[−S] Or L

[−s−1] in the upper half plane. We join the two contours as shown in Figure 3, and denote the corresponding convolution by ∗s = ∗d + ∗u . We then obtain

#

.

(3.19)

The derivation is explained again in Appendix D.2.2. This result agrees with (3.9), (3.10). Regarding the anti-holomorphic part of A1 NLIE (2.7), we can construct a consistent deformed contour by taking the complex conjugation.

The source term (3.19) depends on the zeroes (or poles) of T1,s−1 , T1,s in the strip A−1,1 and the zeroes (or poles) of L, L in the upper or lower half planes, {ℓs+1 , ℓs+2 , . . . }, {ℓs+1 , ℓs+2 , . . . }. The latter is related to the poles (or zeroes) of dynamical variables 1 + b−

S , 1 + Y1,S−1

via (3.15). To impose the exact quantization condition on the extra roots lying outside the main strip, we need to analytically continue the NLIE, as mentioned in Section 2.3. This is a noticeable feature of NLIE compared to the mirror TBA.

16The consistent deformed contour is not necessarily unique, so there is no contradiction with our previous claim on the orbifold Konishi state at weak coupling.

16

Figure 3: The deformed contour for bs , bs in v-plane, adjusted to be consistent with the Fourier source term. The symbols ◦, ×, △represent the zeroes or poles of T, L, L, respectively. The orange region corresponds to the region surrounded by the deformed contour and the mirror real axis.

Conclusion

In this paper we generalized the hybrid NLIE of and applied it to a wider class of states. First, we studied the ground and the orbifold Konishi states of twisted AdS5 × S5. In the mirror TBA, the orbifold Konishi states have infinitely many asymptotic critical lines from YM|w nodes. In the hybrid NLIE, the number of critical lines is indeed reduced to a finite number.17 The quantization condition for the extra zeroes is written in terms of NLIE variables (as , as , Y1,s−1).

Second, we derived the source terms of hybrid NLIE for general states in two ways, Fourier transform method and contour deformation trick. We constructed the deformed contour which is consistent with the Fourier transform method.

It is interesting to generalize the gauge-invariant NLIE to An cases. The SU(N) principal chiral models contain boundstate spectrum for N ≥3, and its NLIE has been studied in . We should be able to reproduce their results by A2 NLIE and contour deformation trick.

While this paper is in preparation, hybrid NLIE of AdS5 × S5 made out of A1 and A3 NLIE coupled to the quasi-local formulation of the mirror TBA has appeared in . We expect that the contour deformation trick will also work to obtain this new NLIE for excited states.

17As long as the sl(2) sector is concerned, this conclusion is expected because the exact truncation method of can be applied without modification.

Acknowledgements

The author acknowledges Gleb Arutyunov and Stijn van Tongeren for discussions. This work is supported by the Netherlands Organization for Scientific Research (NWO) under the VICI grant 680-47-602.



.

Together With F [±M] = F(V ± Im

g ) and f(v)± = f(v)[±1]. The complex rapidity plane are divided



.

G )

 .

F S+

f = S1 .

−2

dt F(t) K(v −t).

(A.5)

The logarithmic derivative and its Fourier transform are defined by

Dv Eikv ∂

∂v log X(v).

(A.6)

18This definition is adapted for Fourier transform and different from the usual convolution in the mirror TBA,

E.G. F ⋆K(V) = R ∞

−∞dt F(t) K(t −v). Since the kernels sK(v) is invariant under v →−v, we can still use (A.5) to write down the simplified TBA for YM|w .

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