Quantum periods and TBA equations for N = 2 SU ( 2 ) N f = 2 SQCD with flavor symmetry

We apply the exact WKB analysis to the quantum Seiberg-Witten curve for 4-dimensional N = 2 SU(2) Nf = 2 SQCD with the flavor symmetry. The discontinuity and the asymptotic behavior of the quantum periods define a Riemann-Hilbert problem. We derive the thermodynamic Bethe ansatz (TBA) equations as a solution to this problem. We also compute the effective central charge of the underlying CFT, which is shown to be proportional to the one-loop beta function of the SQCD. E-mail: k.imaizumi@th.phys.titech.ac.jp

It is important to generalize the SU(2) SYM to the SQCD, which has the moduli space with higher dimensions. The quantum Seiberg-Witten curves for 4d N = 2 SU(2) SQCD with N f ≤ 4 can be obtained from the quantization of the Seiberg-Witten curves [14]. The quantum curves are also obtained from the AGT correspondence with the mass decoupling limits, which decouple the different hypermultiplets considered in [21]. In this paper, we derive the TBA equations for N f = 2 SQCD with the flavor symmetry based on the curve in [14]. This paper is organized as follows. In section 2, we apply the exact WKB analysis to the quantum Seiberg-Witten curve and define a Riemann-Hilbert problem. In section 3, we derive the TBA equations as a solution to the Riemann-Hilbert problem. We then study some special limits of the TBA equations, the massless limit, the decoupling limit and the Argyres-Douglas limit. We also compute the effective central charge of the underlying CFT, which is found to be proportional to the coefficient of the one-loop beta function of the SQCD.

Exact WKB analysis and Quantum SW curve
The quantum Seiberg-Witten curve for 4-dimensional N = 2 SU(2) SQCD with two fundamental hypermultiplets take the form of the Schrödinger type differential equation [14], where u is the Coulomb moduli parameter, Λ 2 is the dynamically generated scale, m 1 , m 2 are the bare masses of the hypermultiplets, ǫ is the deformation parameter in the Nekrasov-Shatashvili limit of the Ω-background and q is a complex variable.
The standard WKB method produces an asymptotic expansion in ǫ of the solution to (2.1), By substituting the solution (2.2) into (2.1), we can obtain p 2n (q) recursively. Especially we find p 0 (q) = u − V (q).
P even (q)dq in (2.2) can be regarded as a one-form on the Riemann surface Σ defined by the following algebraic curve, We will call Σ WKB curve. The one-cycles γ ∈ H 1 (Σ) generate the periods of P even (q)dq, which we will call quantum periods. The quantum periods determine the low-energy effective dynamics of the SQCD. As P even (q) goes, the quantum periods are even power series in ǫ, We can compute the higher order corrections to the quantum periods by using the differential operator technique [14]: where O n is a differential operator with respect to the moduli parameters on the WKB curve.
The quantum periods are asymptotic series, which converge only at |ǫ| = 0, and therefore need to be properly resumed. In the exact WKB analysis, we take Borel resummation technique. First, we define Borel transformation of a quantum period as follows: where ξ is a complex variable. With the help of the factor 1/(2n)!, the Borel transformation has a finite convergence of radius. Therefore the Borel transformation can be analytically continued on the whole of ξ-plane. The Borel resummation of the quantum period is then defined by the laplace integral of the Borel transformation, where ϕ is the phase of ǫ (ǫ = |ǫ|e iϕ ). The Borel resummation of the quantum period s ϕ (Π γ ) (ǫ) is an analytic function and has the quantum period Π γ as the asymptotic expansion in ǫ → 0.
In general, the analytic continuations of the Borel transformations have singularity, which are typically poles and blanch cuts, on the ξ-plane. If the Borel transformation of a quantum period Π γ has singularities on the ray along a direction in the ξ-plane, then the integral (2.9) cannot be defined in this direction. In that case, instead of (2.9), we use the integrals which avoid the singularities to the left or right, (2.10) The discontinuity of a quantum period Π γ are then given as the difference of (2.10), Let us compute the discontinuity of the quantum periods for (2.1). We take a special region on the moduli space in which all the solutions to u = V (q * ) become real and different.
This region can be realized as follows: We consider the case that the hypermultiplets have Restricting the parameters to − Λ 2 2 < m < Λ 2 2 and Λ 2 |m| − , the solutions to u = V (q * ) become real and different (see Fig.2.1) 1 . In the context of the gauge theory, the restriction of u corresponds to the strong coupling region.
One can then finds four independent cycles on the WKB curve, α,α cycles, encircling the classically allowed intervals, and β,β cycles, encircling the classically forbidden intervals ( Fig.2.2). We choose the orientations of the cycles so that are real and positive.
Under the change of variable e iq = z, p 0 (q)dq becomes the root of the quadratic differential for 4d N = 2 SU(2) N f = 2 gauge theory considered in [31]. The quadratic differential has two irregular singularities at z = 0, ∞ and theirs residues lead to the following relation, (2.14) Then one can show that the higher order coefficients are equivalent respectively, because the differential operator O n can be expressed by only using u-derivative [14].
The discontinuity of the Borel resummations of the quantum periods can be captured by the Delabaere-Dillinger-Pham formula (theorem 2.5.1 of [32], and theorem 3.4 of [33]).
For ϕ = 0, the formula says that the periods for the classically allowed intervals have the discontinuity, and it is encoded in the periods for the classically forbidden intervals, where γ ca is a classically allowed cycle, γ cf is the classically forbidden cycles and γ ca , γ cf is the intersection number of them. By using this formula, the discontinuities of Π α , Πα are given as follows, Π α and Πα also have the discontinuity for the direction ϕ = π because the quantum periods are even power series in ǫ. Similarly, Π β and Πβ have the discontinuity for the direction ϕ = ± π 2 because, in this direction, the classically allowed intervals and classically forbidden intervals are switched.
The asymptotic behavior and the discontinuities for the Borel resummations of the quantum periods define a Riemann-Hilbert problem for themselves [25]. In the next section, we derive the TBA equations as a solution to this problem.
3 Quantum periods and TBA equations 3

.1 TBA equations
The discontinuities obtained in the previous section can be put into a uniform description by introducing functions ǫ i (θ) as The discontinuities are then put together into a simpler one, and we define L 0 = L 4 , L 5 = L 1 .
The functions ǫ i (θ) have the following asymptotic behavior,  [22]. For the precent case, the solution is given by the following TBA system, (3.6) The identification (2.14) leads to ǫ 2 (θ) = ǫ 4 (θ) and the TBA equations can be collapsed to three, (3.7) So far we only consider the special parameter region, but we can also derive the TBA equations for a pure imaginary region (m, Λ 2 ∈ iR) and an anti-same mass region (m 1 = −m 2 ) in the same way. For general region, we need to analytically continue the TBA equations [34]. Let us consider the analytic continuation for (3.6). Taking u, m, Λ 2 as complex values, Z i also becomes complex, Then we obtain the following TBA equations, The TBA equations (3.9) are only valid for the region |φ i − φ j | < π 2 because the integrands have the pole at |φ i − φ j | = π 2 . For |φ i − φ j | > π 2 , the residue of the pole deforms the TBA equations and typically we find an infinite number of the integral equations.
The number of the TBA equations is equivalent to the number of the stable BPS states in the gauge theory [31,35]. For the 4d N = 2 SU(2) N f = 2 gauge theory, there are four stable BPS states in the strong coupling region and an infinite number of the states in the weak coupling region. Therefore we conclude that the TBA equations (3.9) is valid for the strong coupling region.
In principle, we can also derive the TBA equations for m 1 = m 2 by solving the Riemann-Hilbert problem. But in this case the potential V (q) is not a real-valued function on the ℜ(q) axis and there is no way to analytically determine the directions of the discontinuity as the same-mass case. This difficulty also exists for 4d N = 2 SU(2) N f = 1, 3, 4 gauge theories. However, the number of the stable BPS states for the 4d N = 2 SU(2) N f = 2 gauge theory with m 1 = m 2 is the same for the same-mass case. Therefore we conjecture that the TBA equations (3.9) is also valid for the m 1 = m 2 case. Now we discuss the TBA equations at some special points in the moduli space. In the massless case m = 0, (2.14) leads to Π α = Πα or equivalently ǫ 1 (θ) = ǫ 3 (θ) and the TBA equations can be collapsed to two, (3.11) (3.11) agrees with the TBA equations for the Mathieu equation [26,27]. This agreement is compatible with that the quantum SW curve (2.1) becomes the Mathieu equation in the massless case. We also obtain the TBA equations (3.11) in the decoupling limit (m → ∞ and Λ 2 → 0 while mΛ 2 being fixes), which turns the N f = 2 theory into the pure gauge theory, because in this limit the quantum SW curve (2.1) becomes the one for the pure SU(2) theory.
The point m = Λ 2 2 , u = 3 8 Λ 2 is the supercomformal or Argyres-Douglas point where mutually nonlocal BPS states become massless [36]. In the limit m → Λ 2 2 , u → 3 8 Λ 2 with keeping the theory in the strong coupling region,α-cycle shrinks and Πα goes to zero. If we neglectα-cycle, we obtain the following TBA equations, (3.12) This TBA system agrees with the TBA equations for the quartic potential derived in [22], which is the quantum SW curve for (A 1 , A 3 ) AD theory. Moreover, under the universality of the AD theory [16], the quantum SW curve for (A 1 , A 3 ) AD theory is equivalent to the SU(2) N f = 2 AD theory.
The large θ expansion of the TBA equations provides the all-older asymptotic expansion of the epsilon functions. For example, for (3.6), can be replaced with the coefficients of the quantum periods as follows: Moreover, (3.14) indicates the following identifications: These are agree with (2.15).
We compare the calculation of the quantum periods by using the TBA equations and the differential operator (2.7). In the same mass case, the first and second orders can be calculated by using the following differential operators: Note that there are at least first order u-derivative in each terms. Therefore we can evaluate the higher order collections to the quantum periods by using the u-derivative of Π where q 1 , q 2 are the solution to u = V (q) (see Fig.2

Effective central charge and one-loop beta function
By using the TBA equations, we can calculate the effective central charge c eff = c − 24∆ min of the underlying CFT , where c is the central charge of the Virasoro algebra and ∆ min is the minimum eigenvalue of the Virasoro operator L 0 . For (3.6), c eff is given by where In θ → −∞ limit, The TBA equations (3.6) lead (3.23) and therefore (3.24) In fact, there are no mathematically rigorous solutions to (3.24) [27,37,38]. But we can formally consider that ǫ ⋆ i → −∞ are the solutions. Then and Therefore we obtain c eff = 4. (3.27) This result agrees with the numerical calculation.
The large θ expansion of the TBA equations (3.14), (3.15) leads to a relational expression between c eff and the quantum periods, β can be expressed only Π (0) γ by using the differential operator O 1 (3.17). After some transpositions, we finally get the following relation, (3.29) In the massless case m = 0, the second, third and forth terms in the r.h.s become zero.
This relation is also derived in the context of N = 2 gauge theories [39,40]. In these papers, it is pointed out that the constant term in the r.h.s is proportional to the oneloop beta functions of the N = 2 gauge theory. Therefore (3.29) indicates that c eff for 2-dimensional CFT is proportional to the one-loop beta function for the SQCD.

Summary and discussions
In this paper, we have investigated the exact WKB analysis for the quantum Seiberg-Witten As future works, we want to derive the TBA equations for other gauge theories whose quantum Seiberg-Witten curves form the Schrödinger type differential equations (e.g. SU (2) with N f ≤ 4 [14,21], N * = 2 theory [41][42][43]). One of the possible way to derive the TBA equations is the ODE/IM correspondence proposed in [44]. For example, the TBA equations for pure SU(2) case have already derived in [45] by using the ODE/IM correspondence.
Another possible way is using the integral equations proposed by Gaiotto, Moore and Neitzke in [35]. The conformal limit of these integral equations becomes the TBA equations [37] and the author argued that these TBA equations calculate the quantum periods. This argument was numerically demonstrated for pure SU(2) case [26] and showed in [46]. It is also interesting to study the relation to [47], which studied the same Riemann-Hilbert problem we considered but used different methods. More ambitious generalization is the N = 2 supersymmetric gauge theories with higher rank gauge group, whose quantum Seiberg-Witten curves form higher-order differential equations (for pure SU(N) case [12] and with matters [13]). A good starting point is the A n -type ODE studied in [48], which relates to the quantum Seiberg-Witten curve for the (A n , A m )-type Argyres-Douglas theories [49].
It is also interesting to apply the TBA equations to study black hole physics. In [50], the authors claimed that the quasinormal mode frequencies for black holes are determined by the Bohr-Sommerfeld quantization condition for the quantum periods of 4-dimensional N = 2 SU(2) gauge theory with N f = 2, 3 and explicitly demonstrated at some lower levels.
The more we included the higher order collections of the quantum periods, the more the spectrum obtained from the Bohr-Sommerfeld quantization condition matched to the true value. Therefore we expect that the Borel resummations of the quantum periods provide more precision.