$T$, $Q$ and periods in $SU(3)$ ${\cal N}=2$ SYM

We consider the third order differential equation derived from the deformed Seiberg-Witten differential for pure ${\cal N}=2$ SYM with gauge group $SU(3)$ in Nekrasov-Shatashvili limit of $\Omega$-background. We show that this is the same differential equation that emerges in the context of Ordinary Differential Equation/Integrable Models (ODI/IM) correspondence for $2d$ $A_2$ Toda CFT with central charge $c=98$. We derive the corresponding $QQ$ and related $TQ$ functional relations and establish the asymptotic behaviour of $Q$ and $T$ functions at small instanton parameter $q \rightarrow 0$. Moreover, numerical integration of the Floquet monodromy matrix of the differential equation leads to evaluation of the $A$-cycles $a_{1,2,3}$ at any point of the moduli space of vacua parametrised by the vector multiplet scalar VEVs $\langle \textbf{tr}\,\phi^2\rangle$ and $\langle \textbf{tr}\,\phi^3\rangle$ even for large values of $q$ which are well beyond the reach of instanton calculus. The numerical results at small $q$ are in excellent agreement with instanton calculation. We conjecture a very simple relation between Baxter's $T$-function and $A$-cycle periods $a_{1,2,3}$, which is an extension of Alexei Zamolodchikov's conjecture about Mathieu equation.


Introduction
Ever since 1994 when Seiberg and Witten derived exact low-energy Wilsonian effective action of (pure) SU (2) N = 2 SYM [1], the interest in this kind of theories has been remaining extremely high. The reason is their remarkably rich physical and mathematical content. In fact, these theories provide a framework to address in a precise manner such problems as strong coupling, non-perturbative effects and confinement in non-Abelian gauge theory (so relevant, for instance, in the Standard Model). The impact of Seiberg-Witten theory in pure mathematics is also very substantial. Likely, the most famous applications are in algebraic geometry and topology of four-dimensional differentiable manifolds where e.g. the notions of Seiberg-Witten and Gromov-Witten invariants are of primary importance.
The effective action of N = 2 SYM is given in terms of prepotential: a holomorphic function of the vacuum expectation values (VEV) of the vector multiplet scalar field. Large VEV expansion of the prepotential reveals its structure as sum of classical, one-loop and instanton contributions. Many researchers tried to restore the instanton contributions directly from the microscopic theory but succeeded only in the case of first few instantons [2]. The progress has been achieved when the idea of using equivariant localization techniques in moduli space of instantons [3], [4], especially in combination with introduction of so called Ω background (see [5] and further developments [6], [7], [8]). Considering theory in Ω-background effectively embeds the system in a finite volume ∼ 1 1 2 , where the parameters 1 , 2 are sort of angular velocities on orthogonal planes of (Euclidean) 4d space-time. This makes the partition function a finite, well defined quantity (commonly referred as Nekrasov partition function). Then the corresponding free energy coincides with (generalized) prepotential. The usual SW prepotential is recovered simply by sending the parameters 1,2 → 0. Another, crucial consequence of introducing this background is the fact that instanton moduli integrals are localized at finitely many points. This property eventually leads to an elegant combinatorial formula for instanton contributions [6].
Later developments are even more surprising. It appears that introduction of Ω background is not merely a regularization trick. Thus, keeping 1,2 finite a deep relation between conformal blocks of 2d CFT and Nekrasov partition function [9] emerges, so that the Virasoro central charge is related to these parameters, the masses of hypermultiplets specify inserted primary fields, while VEVs identify the states of the intermediate channel.
The special case 1 = − 2 bridges the theory with topological string, -expansion of Nekrasov partition function coinciding with topological (string) genus expansion.
Another special case of great interest is the Nekrasov-Shatashvili (NS) limit [10] when one of parameters, say 1 is kept finite while 2 = 0. From AGT point of view this case corresponds to semiclassical CFT when the central charge c → ∞. Besides this, another interesting link to quantum integrable system emerges, now the remaining nonzero parameter 1 being related to Plank's constant. In NS limit many quantities familiar from original Seiberg-Witten theory become deformed or quantised in rather simple manner.
More recently, moving from Gaiotto's idea of looking at these equations as quantum versions of the (suitable power of the) SW differential [26] 1 , it has been proposed to investigate their monodromies (quantum periods over cycles) through the connection (Stokes) multipliers appearing in the ODE/IM correspondence [27,28]: [31] describes the general idea by exemplifying it in the simple case of pure SU (2) gauge theory 2 and in particular the link between the a-period and the Baxter transfer matrix T function. In this perspective, Thermodynamic Bethe Ansatz (TBA)-like considerations about pure SU (2) gauge theory were initiated in [32] at zero modulus (of the Coulomb branch) for the dual period a D , and then more recently pursued by [33].
In fact, in this paper we show how to compute the gauge A-periods of the pure SU (3) theory (without any matter hypermultiplet) as Floquet monodromy coefficients of the aforementioned differential equation (in the complex domain). Then, we propose a connexion between them and the integrable Baxter's T function which extends non trivially what happens in the SU (2) case and shows that the latter is not an accident. More in details, we obtain a third order linear differential equation with some similarities (and differences 3 ) with the third order oper of ODE/IM correspondence in [29], [30]. As the latter correspond to some 'minimal' case M > 1/2, we may conjecture, along the lines of [34] and [35], that we are describing A 2 -Toda CFT with central charge c = 98.
In a very interesting unfinished paper [35] Alexei Zamolodchikov has proposed ODE/IM for the Liouville CFT TBA. Special attention has been payed to the selfdual case c = 25, when the related ODE becomes the modified Mathieu equation and a elegant relationship between Floquet exponent and Baxter's T function has been suggested. As written, the implication of this conjecture for the period on the A-cycle of (effective) SU (2) gauge theory has been highlighted and used by [31]. But it was not clear from there if and how it is possible to generalise this beautiful connection between transfer matrix and periods for higher rank groups.
In the details of this paper we derive QQ and T Q functional relations (see eqs. (4.6), (4.8)) and extend Zamolodchikov's conjecture for the case of gauge group SU (3) (see eq. (5.13)). We show that numerical integration of the differential equation leads to evaluation of the 'quantum' A-cycle periods a 1,2,3 at any point of the moduli space of vacua parametrised by the vector multiplet scalar VEV's u 2 = tr φ 2 and u 3 = tr φ 3 even for large values of q at which the instanton series diverges. We have checked that the numerical results at small q are in excellent agreement with instanton calculation. Thus the main message of this paper is that the differential equation provides an excellent tool for investigation of deformed SW theory in its entire range from weak to strong coupling.
The paper is organized as follows: Section 1 is a short review on instanton calculus for SU (N ) SW theory without hypers in Ω-background. Here one can find explicit expressions as a sum over (multiple) Young diagrams for Nekrasov partition function and VEV's tr φ J . Section 2 is a brief introduction to deformed SW theory. We present the main results of [11] in a form convenient for our present purposes. Starting from section 3 we consider the case of SU (3) theory. The main tool of our investigation, a third order linear ODE is derived and its asymptotic solutions are found.
In section 4 we identify a unique solution χ(x) which rapidly vanishes for large negative values of the argument x → −∞. The three quantities Q 1,2,2 are defined as coefficients of expansion of χ(x) in terms of three independent solutions U 1,2,3 (x) defined in asymptotic region x 0. Investigating symmetries of the differential equation we find a system of difference equations for Q k and their analogsQ k obtained by flipping the sign of parameter u 3 → −u 3 . Based on this QQ system we introduce Baxter's T function and write down corresponding T Q relations.
In section 5 we show how numerical integration of the differential equation along imaginary direction with standard boundary conditions allows one to find the monodromy matrix and corresponding Floquet exponents, which in the context of gauge theory, coincide with the A-cycle periods a 1,2,3 . We have convincingly demonstrated the correctness of this identities trough comparison with instanton computation. But the main value of this method is that it makes accessible also the region of large coupling constants, which is beyond the reach of instanton calculus. Eventually, we close this section by suggesting a simple relation between Baxter's T -function and A-cycle periods a 1,2,3 of SU (3) theory, which can be thought of as a natural extension of Alexei Zamolodchikov's conjecture relating Floquet exponent of Mathieu equation to Baxter's T function in c = 25 Liouville CFT.
Finally appendix A contains few technical details for derivation of the T Q relation.

Nekrasov partition function and the VEVs tr φ J
Consider pure SU (N ) theory without hypers in Ω-background. The instanton part of partition function is given by [5] where sum runs over all N -tuples of Young diagrams Y = (Y 1 , · · · , Y N ) , | Y | is the total number all boxes, a = (a 1 , a 2 , · · · , a N ) are VEV's of adjoint scalar from N = 2 vector multiplet, 1 , 2 , as already mentioned, parametrize the Ω-background and the instanton counting parameter q = exp 2πiτ , with τ = i g 2 + θ 2π being the (complexified) coupling constant. The coefficients Z Y are factorized as where the factors P (λ, a|µ, b) for arbitrary pair of Young diagrams λ, µ and associated VEV parameters a, b, are given explicitly by the formula [6] If one specifies location of a box s by its horizontal and vertical coordinates (i, j), so that (1, 1) corresponds to the corner box, its leg length L λ (s) and arm length A λ (s) with respect to the diagram λ (s does not necessarily belong to λ) are defined as where λ i (λ j ) is i-th column (j-th row) of diagram λ with convention that when i exceeds the number of columns (j exceeds the number of rows) of λ, one simply sets λ i = 0 (λ j = 0). The instanton part of (deformed) prepotential is given by [5] Instanton calculus allows one to obtain also the VEV's tr φ J , φ being the adjoint scalar of vector multiplet: where Z Y is already defined by (1.2), (1.3), and [36,37] 2 A Baxter difference equation

Bethe ansatz equation for NS limit
It was shown in [11] that in NS limit 2 → 0, the sum (1.1) is dominated by a single term corresponding to a unique array of Young diagrams Y (cr) specified by properties (the i-th column length of a diagram Y u will be denoted as Y u,i ): • Though the total number of boxes → ∞ in 2 → 0 limit the rescaled column • The rescaled column lengths at small q behave as ξ u,i ∼ O(q i ). This means in particular, that in order to achieve accuracy up to q L , it is consistent to consider restricted Young diagrams with number of columns ≤ L.
• Up to arbitrary order q L the quantities The system of equations (2.1) together with the property ξ u,i ∼ O(q i ) uniquely fixes the quantities x u,i up to order q L . Of course, calculations become more cumbersome if one increases L. Examples of explicit computations for first few values of L can be found in [11].

Baxter's difference equation and deformed Seiberg-Witten 'curve'
The BA equations can be transformed into a difference equation [11] Y (z where Y (z) is an entire function with zeros located at z = x u,i : and is the logarithmic derivative of Gauss' gamma-function. Finally P N (z) is an N -th order polynomial which parametrizes the Coulomb branch of the theory. Explicit expressions of coefficients of this polynomial in terms of VEVs will be presented later for the case of our current interest N = 3. For more general cases one can refer to [11]. Now, let us briefly recall how the difference equation (2.2) is related to the Seiberg-Witten curve. Introducing the function one can rewrite (2.2) as At large z the function y(z) behaves as Notice that setting 1 = 0 in (2.5) one obtains an equation of hyperelliptic curve, which is just the Seiberg-Witten curve. When 1 = 0, everything goes surprisingly similar to the original Seiberg-Witten theory. For example the rôle of Seiberg-Witten differential is played anew by the quantity and, as in the undeformed theory, the expectation values (2.4) are given by the contour integrals where C is a large contour, enclosing all zeros and poles of y(z).

Details on SU (3) theory
Without any essential loss of generality, from now on we will assume that Representing y(z) as a power series in 1/z and inserting in eq. (2.6) one easily finds the relations Now, consistency of (2.8), (2.9) and (2.5) immediately specifies the polynomial P 3 (z) (we omit the subscript 3, since only the case N = 3 will be considered later on) (2.10) 3 The differential equation and its asymptotic solutions 3.1 Derivation of the differential equation To keep expressions simple, from now on we will set 1 = 1. In fact, at any stage the 1 dependence can be easily restored on dimensional grounds. Taking the results of previous subsection, the difference equation for N = 3 case (2.2) can be rewritten as By means of inverse Fourier transform, following [12,19,22], from (3.1) we can derive a third order linear differential equation for the function At least when |q| is sufficiently small, it is expected that the series is convergent for finite x, provided a takes one of the three possible values a 1 , a 2 or a 3 . Taking into account the difference relation (3.1), one can easily check that the function (3.2) solves the differential equation and shifting the variable x → x − log Λ 3 the differential equation (3.3) may be cast into a more symmetric form

Solutions at x → ±∞
It is convenient to introduce parameters p 1 , p 2 , p 3 satisfying p 1 + p 2 + p 3 = 0 such that In the Λ → 0 limit the parameters p i and a i coincide. At large positive values x 3 ln Λ the term Λ 3 e −x in (3.4) can be neglected. In this region the differential equation can be solved in terms of hypergeometric function 0 F 2 (a, b; z) defined by the power series is the Pochhammer symbol. Three linearly independent solutions can be chosen as where by definition Λ ≡ exp θ (3.9) and the indices (i, j, k) are cyclic permutations of (1, 2, 3). We used the symbol ≈ in (3.8) to mean that the approximations of the solutions hold, striktly speaking, only for x 3θ (at leading order). In the end, we must verify that the Wronskian of the three solutions (3.8) (below and later on, for brevity, we use the notation is not zero provided the parameters p i are pairwise different. Thus, (3.10) confirms that generically the U i (x) are linearly independent and constitute a basis in the space of all solutions. Similarly in region x −3θ the term Λ 3 e x of (3.4) becomes negligible and one can write down the three linear independent solutions In fact, we obtain the same result for the Wronskian The functional relations

The QQ relations
All three solutions V i (x) grow very fast at x → −∞, but there is a special linear combination (unique, up to a common constant factor) which vanishes in this limit. If it is the fastest one (as we suspect), this solution is usually referred to as subdominant.
Using formulae for asymptotics of 0 F 2 , which can be found e.g. in [38], we are able to establish that the correct combination is Its asymptotic expansion at x → −∞ is given by and u 2 , u 3 are defined in terms of p i in (3.5).
Since U i (x) constitute a complete set of solutions one can represent χ(x) as a linear combination where the important quantities Q n (θ), based on general theory of linear differential equations, are expected to be entire functions of θ (and also of parameters p dependence on which will be displayed explicitly only if necessary). The following, easy to check property plays an essential role in further discussion. Namely the Wronskian of any two solutions f (x), g(x) of the differential equation (3.4) satisfies the adjoint equation, i.e. the one obtained by reversing the signs p → −p and Λ 3 → −Λ 3 . Taking inspiration from this property, it is then possible to show exactly that whereχ(θ) = χ(θ, −p). In fact, the property entails that both sides of (4.4) satisfy the differential equation (3.4) with substitution p → −p. Besides, by using the identity 4 it is not difficult to show the match of the x → −∞ asymptotics of both sides. Of course, the combination of these two statements implies the equality (4.4) everywhere. Let us investigate the x → ∞ limit of (4.4). Taking into account (4.3) and using the identity (4.5) (with x substituted by −x), we obtain the functional relations where again, the bar on Q n indicates the sign change p → −p Q n (θ, p) ≡ Q n (θ, −p) and (n, j, k) is a permutations of (1, 2, 3). At the end of this section let us establish the θ → −∞ asymptotics of Q k (θ) and Q k (θ). Obviously, in this case both (3.8) and (3.11) are approximate solutions of (3.4) at x ∼ 0. Thus, comparison of (4.1) with (4.3) ensures that for θ 0 It is easy to see that above asymptotic behavior is fully consistent with functional relations (4.6).

SU (3) version of Baxter's T Q relation
The functional relations (4.6) suggest the following SU (3) analog of Baxter's T Q equations: for j, k ∈ {1, 2, 3} with j = k. To uncover the essence of this construction, notice that for a fixed pair of indices (i, j) (4.8) can be thought as definition of function T (θ) in terms of Q's. Then the nontrivial question is "do other choices of (j, k) lead to the same T ?" Fortunately, elementary algebraic manipulations with the help of (4.6) ensure that the answer is positive. As mentioned earlier, Q i (θ) are entire functions. A thorough analysis shows that due to (4.6) all potential poles of T (θ) have zero residue. Thus T (θ) is an entire function too. Details on proofs of above two statements can be found in appendix A. The Bethe ansatz equations can be represented as (see equality (A.4)) where θ are the zeroes of Q j (θ). Functional relations similar to (4.6) and (4.8) emerge also in the context of ODE/IM for 'minimal' 2d CFT with extra spin 3 current (W 3 symmetry) [29], [30]. From there we can extrapolate that our case might correspond to the special choice of Virasoro central charge c = 98 for Toda CFT. In fact, this value of the central charge lies outside the region discussed in above references. Nevertheless, it should be possible to derive the corresponding TBA equations: we leave this task for future publication.
Since the functions f n (x + 2πi) are solutions too, we can define the monodromy matrix M k,n as The solutions (3.2) with a ∈ {a 1 , a 2 , a 3 } have diagonal monodromies and can be represented as certain linear combinations of f n (x). In other words the eigenvalues of the monodromy matrix M k,n must be identified with exp(2πia k ), with k = 1, 2, 3: For any fixed values of parameters Λ, p, it is easy to integrate numerically the differential equation (3.4) with boundary conditions (5.1), find the matrix M k,n and then its eigenvalues exp(2πia n ). Taking into account Matone relation [39], valid also in the presence of Ω-background [37], we can access the deformed prepotential for any value of the coupling constant.

Comparison of the instanton counting against numerical results
Using formulae of section 1 it is straightforward to calculate tr φ 2 or tr φ 3 as a power series in q. Here are the 3-instanton results (it is assumed that a 1 + a 2 + a 3 = 0 and by definition a jk ≡ a j − a k ) where and P 2,2 = 36(220 − 1027h 2 + 1659h 2 2 − 698h 3 2 − 958h 4 2 + 1257h 5 2 − 521h 6 2 (5.8) By means of numerical integration of the differential equation  In table 1 we present some characteristic excerpt from the resulting data: Inserting the values of a k , Λ in (5.5), (5.6) we can calculate tr φ 2 and tr φ 3 . The consistency requires that at small values of q one should always obtain tr φ 2 = p 2 1 + p 2 2 + p 2 3 = 0.1274 and tr φ 3 = p 3 1 + p 3 2 + p 3 3 = −0.017748. Table 2 displays the results of actual computations. One expects an essential deviation from the instanton series starting from the value of Λ at which the polynomial   Table 2. The values tr φ 2 , tr φ 3 obtained by inserting the values of a 1 , a 2 from Table 1 into (5.5), (5.6). To be compared with (by definition) tr φ 2 = p 2 1 + p 2 2 + p 2 3 = 0.1274 and tr φ 3 = p 3 1 + p 3 2 + p 3 3 = −0.017748.

Extension of Zamolodcikov's conjecture to SU (3)
The simpler case of the gauge group SU (2) has been analyzed recently in [31]. In this case one has to deal with the Mathieu equation. Corresponding T Q relation was investigated in [35], where Al. Zamolodchikov conjectured (and demonstrated numerically) an elegant relationship between T -function and Floquet exponent ν of Mathieu equation: T = cosh(2πν) . Notice, that at θ 0 the asymptotic (4.7) leads to which is consistent with (5.13), since for θ 0 instanton corrections disappear and a k coincides with p k .

Few perspectives
It would be very interesting to have a TBA for our case and check our conjecture (5.13) as it was done by Al. Zamolodchikov in [35]. Actually, even relevant would be a Ysystem and a gauge TBA that may shed light on the dual B-cycle periods a D along the route presented in [31] for the SU (2) case (see also the presentation of [32] and [33]).
Of course, it is very plausible that the imaginable generalisations of our results, and in particular of (5.13), might hold for arbitrary SU (N ) gauge groups.