Exact partition function in $U(2)\times U(2)$ ABJM theory deformed by mass and Fayet-Iliopoulos terms

We exactly compute the partition function for $U(2)_k\times U(2)_{-k}$ ABJM theory on $\mathbb S^3$ deformed by mass $m$ and Fayet-Iliopoulos parameter $\zeta $. For $k=1,2$, the partition function has an infinite number of Lee-Yang zeros. For general $k$, in the decompactification limit the theory exhibits a quantum (first-order) phase transition at $m=2\zeta $.


Introduction
The dynamics of two coincident M2 branes on the orbifold R 8 /Z k is described by ABJM theory, three-dimensional U (2) k × U (2) −k supersymmetric Chern-Simons theory with bifundamental matter [1]. For this particular gauge group, the ABJM theory has N = 8 superconformal symmetry and is in fact equivalent to Gustavsson-Bagger-Lambert theory [2,3]. The partition function for the theory on S 3 can be computed by supersymmetric localization [4,5]. This theory can be deformed, preserving N = 4 supersymmetry, by adding mass and Fayet-Iliopoulos (FI) parameters m, ζ, and the localization technique then reduces the full supersymmetric functional integral to the matrix integral [5] Z = 1 4 where i, j = 1, 2. The parameter ζ represents a Fayet-Iliopoulos parameter for the diagonal U (1) subgroup, whereas m corresponds to a mass for the chiral multiplets. The partition function should be understood as a function Z(2ζ, m; k), but for ease of presentation we will omit its arguments unless needed. For k = 1, the theory is mirror dual to N = 4 supersymmetric super Yang-Mills theory with gauge group U (2) coupled to a single fundamental hypermultiplet and a single adjoint hypermultiplet [5]. By shifting the integration variables, x ≡ µ − ζ, y ≡ ν + ζ, the partition function becomes where m 1 , m 2 are Note that ζ has dimension of mass. We are using units where the radius R of the threesphere is R = 1. The purpose of this note is to explicitly carry out the integration in (2). In the m = ζ = 0 case, the integral was computed in [6] (a discussion of the partition function in the more general ABJ case can be found in [7]). On the other hand, the m, ζ-deformed ABJM theory was studied in [8] using the Fermi-gas formulation [9] and at at large N for the U (N ) k × U (N ) −k gauge group in [10] (with ζ = 0) and in [11] (with general m, ζ = 0), where phase transitions in the complex parameter space generated by m 1 , m 2 and g = 2πi/k were investigated. Our explicit formula will uncover some interesting physical properties of the mass-deformed system with gauge group U (2) k × U (2) −k .
For the k = 1 case, this symmetry already appeared in [5], where it was also explained by the fact that the corresponding brane configuration is self-mirror. The symmetry implies, in particular, that a FI-deformation ζ on the massless theory is equivalent to a mass-deformation m = 2ζ in the theory with vanishing FI-parameter. The case m = 2ζ -representing a fixed point of this symmetry-is special, as we shall shortly see. In the dual N = 4 supersymmetric super Yang-Mills theory, m 2 = 0 corresponds to coupling the theory to a massless adjoint hypermultiplet.

Residue integration
The partition function for the m, ζ-deformed ABJM theory with U (N ) k × U (N ) −k gauge group can be written in the following form [5,11] where the sum goes over permutations. The derivation uses a trigonometric identity, Fourier integrations and only holds for opposite Chern-Simons levels (see sect. 2 in [11] for details). For N = 2, the formula (5) then leads to the following expression with and Using the identity and the formula for the Fourier transform [11] du e −ikm 2 u cosh πk we obtain In the limit m 2 → 0, the partition function becomes In the following, we compute the integrals (11), (12) by residue integration.
To compute (11) we follow the ideas in [6], where the partition function was computed in the case m = ζ = 0. Thus we start by writing the integrand as the product of two even functions f, g with Under the shift u → u + ik these functions transform as These properties imply that the integral in (13) along the curve u = x+ik with x ∈ R will differ from the integration along the real axis by the factor (−) k cosh(m 2 k). Therefore, the rectangular contour composed by the real axis, two vertical segments and the displaced real axis u = x + ik becomes appropriate for residue computation in the case m 2 = 0 (see Fig.1 The residues encircled by the contour comprise the ones arising from the poles of f (z) located at z = in with n = 1, . . . , k and those of g(z) located at z ± = ± m 1 k 2π + i k 2 . The pole located at z = ik does not contribute due to a double zero in the numerator of g(z). Calling C the closed rectangular contour described above and 1 It is easily seen that the vertical contours do not contribute when we push them to infinity. where Case m 2 = 0, k odd: it is evident from (16) that the m 2 → 0 limit of (13) is smooth, the result is Case m 2 = 0, k even: the factor multiplying the bracket in (16) prevents taking m 2 → 0 in the even k case. To compute the integral in (12) we consider with g(u) as in (14) andf Upon integration, the odd piece inf vanishes against g(u) and therefore the partition function (12) can be written as The shift u → u + ik inf (u) gives As discussed below (15), this property makes the rectangular contour in Fig.1 appropriate for computing I by residues.
For the residues analysis we should now consider the pole inf (z) at the origin z = 0 but a zero in g(z) eliminates it; along the same lines the residue from z = ik/2 is absent since a zero appears forf . CallingF(z) =f (z)g(z) one finds The n = k 2 term in the sum vanishes as expected. The final result is

Summary of results and limits
Thus we have obtained where and we can finally put the partition function in the form In the formulas (27)-(28), the symmetry m 1 ↔ m 2 -which is manifest in the integral form (2)-is hidden. Interestingly, this symmetry is only recovered upon summation over n.
On the other hand, the symmetry m 2 → −m 2 is manifest. Note that Z is real. While this is expected in a unitary theory, it is not generally the case in Chern-Simons theories (for a discussion, see [12]). In the present case, it is related to the fact the theory is a combination of two Chern-Simons theory with opposite levels. 2 Consider, as particular examples, the important cases k = 1, 2. The partition functions take the form Now the symmetry m 1 ↔ m 2 has become manifest. Note that the partition functions for k = 1, 2 have zeros. Restoring the R dependence, the zeros are located at They represent Lee-Yang zeros (see, for example, [13]). In the infinite volume, R → ∞, the zeros condense in a certain line, and a phase transition should emerge. The fact that the partition function has zeros seems to be related to the fact that the coupling, g = 2πi/k, is imaginary for real k. Indeed, from the general expressions (24)-(25) we see that the arguments of the sine and cosine functions in (29), (30) contain a factor π/k. If the coupling g is (unphysically) continued to the real line by taking k → ik, the partition function zeros would then lie on the imaginary g-axis, in accordance with the Lee-Yang theorem (see [11] for a related discussion).
For the undeformed ABJM theory, the k = 1 case is of special interest, since it is conjectured to describe the dynamics of two M2 branes in eleven-dimensional Minkowski spacetime. An interesting question is what is the origin of these Lee-Yang singularities in the brane realization.
The partition function Z(2ζ, m; k) does not have any zeros for k > 2. For higher values of k, the partition function becomes more involved, below we quote explicitly the k = 3 and k = 4 cases Note that the symmetry under the exchange m 1 ↔ m 2 is manifest.

Asymptotic formulas
Let us consider the limit of a large sphere, mR 1, at fixed k. Assuming m 1 > 0, m 2 > 0 and restoring the R dependence, we find The general asymptotic formula with arbitrary sign for m 2 and m 2 = 0, is obtained by replacing m 2 by |m 2 |. The absolute value implies a discontinuity in the first derivative of F = − ln Z. This indicates a first-order phase transition in the parameter m 2 at m 2 = 0, i.e., when the two mass scales m, 2ζ cross. Explicitly, at large R, we have Hence For k = 1 the discontinuity in the first derivative of ∆F is equal to 3R, as can be seen from (34).
For the general theory with gauge group U (N ) k × U (N ) −k , large N phase transitions in the complex parameter N g = 2πiN/k were studied in [10,11]. These phase transitions require taking infinite volume and, at the same time, a strong coupling limit with fixed kR -a limit that already appeared in the context of supersymmetric U (N ) Chern-Simons theory with massive fundamental matter in [14,15]. It should be noted that such decompactification limit is different from the present (more physical) limit of large R at fixed k.
Another interesting aspect of (36) is that it is in a form suitable for a weak coupling expansion in powers of 1/k: The perturbative expansion has an infinite radius of convergence. However, the original theory on the three-sphere of finite radius R has an asymptotic perturbative expansion, with 2n! asymptotic behavior for the 1/k 2n term. This can be seen by using the integral form (11) and generalizing the study of [16,17] on the resurgence properties of the perturbation series of ABJM theory. Now, expanding the integrand in (11), one finds a series with finite radius of convergence determined by the poles of sech(πu/k ± m 1 /2) in the complex u-plane. The integral over u then adds an extra (2n)!, leading to an asymptotic (but Borel summable) perturbation series.
4 The special case m 2 = 0 The m 2 = 0 case is special and must be considered separately. In particular, it represents the critical point in the phase transitions that arise in the decompactification limit. In section 2 we have obtained the following formulas: Odd k: .
Note that the partition function does not have zeros in this case.
Asymptotic formulas m 2 = 0 Consider again the limit of a large sphere, mR 1, at fixed k, but now with m 2 = 0. We find Note that these formulas differ from the asymptotic formulas (34)-(36) given above for Z(m 1 , m 2 ) at m 2 = 0. This is expected, since the latter were obtained by assuming |m 1 R|, |m 2 R| → ∞. Unlike the m 2 = 0 case, the perturbation series for this flat-theory limit has now finite radius of convergence |π/k| < π/2, therefore perturbation series is convergent for all k > 2, where the formula applies. On the other hand, just like the general m 2 = 0 case, the theory on a finite-radius S 3 has an asymptotic perturbation series with 2n! asymptotic behavior.
Finally, it would be interesting to study supersymmetric Wilson loops in the present mass/FI deformed theory, along the lines of [18].