Effective gravitational couplings of Kaluza-Klein gauge theories

We study the effective gravitational couplings of four-dimensional gauge theories with eight supercharges. The class of theories we analyse are arrived at via Kaluza-Klein compactification of five-dimensional gauge theories. We consider both pure SU(N) Yang-Mills theories with Chern-Simons couplings and the conformal gauge theories with 2N fundamental flavours. The resolvent of the gauge theory plays a crucial role in the calculation of these gravitational couplings. The results obtained from the Seiberg-Witten geometry are matched against independent computations using localisation.


Introduction
Gauge theories on four-manifolds have been an active area of investigation for some decades now, and this body of work represents perhaps one of the most striking examples of the richness of physical mathematics.Celebrated early efforts [1,2] at outlining a physical approach to Donaldson invariants have played a significant role, not only in simplifying many proofs, but also shedding light on the dynamics of strongly coupled gauge theories.
A widely studied class of theories in this field are the N = 2 supersymmetric gauge theories, whose low-energy effective dynamics is encoded in the famous Seiberg-Witten geometry [3,4].In its simplest avatar -for the case of pure SU(2) gauge theory -this geometry is an elliptic fibration over the 1-complex dimensional Coulomb moduli space.More generally, when we speak of the Seiberg-Witten geometry, we will mean an algebraic curve (whose moduli space is identified with the quantum moduli space of the supersymmetric gauge theory) and an associated differential.This data is sufficient to fully (i.e.nonperturbatively) solve for the low-energy effective action on the Coulomb branch.
These theories can in fact be defined on arbitrary four-manifolds M using the topological twist [5], and the low-energy effective action includes the gravitational couplings [6] to the Euler character and the signature of the four-manifold: where R is the curvature 2-form and a i are the vevs of the adjoint scalar in the N = 2 vector multiplet.The functions A and B appear as measure factors in the U -plane integral that computes topological invariants.Based on arguments leveraging holomorphy, R-symmetry, and electric-magnetic duality [2,[6][7][8], it is expected that for generic N = 2 gauge theories, these gravitational couplings are specified by their Seiberg-Witten geometry in the following manner: Here, the u i are gauge-invariant coordinates on Coulomb moduli space, and ∆ phys. is proportional to the discriminant of the Seiberg-Witten curve.The constants α and β are not determined at this stage, and are fixed by an independent computation of the same.This is done by studying the theory on a solvable background.In [9], a number of rank-1 gauge theories were studied on the Ω-deformed C 2 , first used in the equivariant localisation computations of [10,11].Since the Euler characteristic and signature of the Ω background are quadratic in the deformation parameters, the gravitational couplings in question can be read off from a small deformation expansion of the partition function of the Ω-deformed gauge theory: In this way, by comparing the results of curve and localisation computations, they were able to determine the dependence of the constants α and β on hypermultiplet masses and the strong-coupling scale.Subsequent work [12] generalised their analysis of mass-deformed N = 4 theories to all higher-rank gauge theories with unitary gauge groups, and [13] studied the gravitational couplings of rank-1 Kaluza-Klein theories using similar techniques.
In this paper we will study the couplings A and B for higher-rank four-dimensional N = 2 theories that are arrived at via Kaluza-Klein compactification of N = 1 theories in five dimensions.Working with higher-rank gauge groups presents certain obvious challenges.For example, computing both gravitational couplined using eq.(1.2) requires knowledge of the relation between the gauge-invariant moduli u i and the classical vacuum expectation values a i , which becomes increasingly complicated as we go to higher orders in the instanton expansion.In order to circumvent this difficulty, we introduce a two-step procedure.
First, we use equivariant localisation to compute 1-point functions of gauge-invariant chiral observables, as in [14][15][16].The generating function of these 1-point functions is the resolvent of the gauge theory, and it is completely determined by the data of the Seiberg-Witten curve [17,18].We show that composing these two operations gives us the sought after u i vs. a j relations as a power series in the instanton counting parameter.This makes it possible to compute the u i (a j ) without having to compute and invert period integrals.
These results are then compared with the couplings A and B read off from a small deformation expansion of the Ω-deformed partition function.Requiring that the two independently computed quantities agree fixes the constants α and β.
We have focused on the case of pure SU(N ) theories and on the case of conformal theories with N f = 2N fundamental hypermultiplets in five dimenions, both Kaluza-Klein compactified on S 1 .In the former case, we find perfect agreement between curve and localisation computations for all admissible Chern-Simons levels once the logarithmic terms arising from the perturbative sector of the theory are accounted for.The latter case is perhaps more interesting in that in addition to perturbative contributions, we must also account for nondynamical discrepancies between the results of equivariant localisation and those coming from the Seiberg-Witten geometry.Since we are interested in gravitational couplings, on dimensional grounds these constants can only be functions of q, the instanton counting parameter.In line with prior work on conformal gauge theories [12], we show that all k-instanton discrepancies can be attributed to the so-called U(1) factor.As we show, this follows as a consequence of a specific choice in the parametrisation of the gauge invariant coordinates on the Coulomb moduli space.
This paper is organised as follows.In Section 2 we review the Seiberg-Witten geometry of Kaluza-Klein compactified SU(N ) gauge theories with eight supercharges.In Section 3, we list our results for the gravitational couplings for low-rank gauge groups.We generalize to the case with N f = 2N fundamental flavours in Section 4 and discuss the mismatch of the curve and localisation results in this case in Section 5. We also analyse the four-dimensional limit of our results in this section.We summarize and discuss our results in Section 6.A brief review of localisation techniques is presented in Appendix A and a discussion of the perturbative contributions can be found in Appendix B.

The Curve and the Differential
The Seiberg-Witten curve for these theories was first studied in the work of [19] (see also [20] where the gauge theory is geometrically engineered) and takes the form: where Here β = 2πR is the circumference of the S1 , the parameters U i are gauge-invariant coordinates on the Coulomb branch of the Kaluza-Klein reduced five-dimensional theory, and κ is the coefficient of the five-dimensional Chern-Simons term, which takes integral values |κ| ≤ N −1 [21].We will henceforth refer to any κ satisfying this condition as an admissible Chern-Simons level.
In the classical limit, the Coulomb moduli reduce to elementary symmetric polynomials in the A u , the classical vevs of the Kaluza-Klein theory: 3) The A u are related to the four-dimensional vevs a u of the adjoint scalar Φ by the exponential map A u = e βau . 1 For the SU(N ) theory, as is evident from eq. (2.2), we have set This ensures that in the classical limit, the quantum gauge polynomial P N (Z) reduces to the classical one: (2.4) The associated Seiberg-Witten differential is given by [20]: where (2.6)

The Resolvent
We have just seen that classically, the U i that appear in the Seiberg-Witten curve are elementary symmetric polynomials of the Coulomb vevs A u .In order to make contact with the gravitational couplings that are the object of our study, we will need to compute instanton corrections to these Coulomb moduli for admissible values of the Chern-Simons coupling.We do this in two steps.
First, we compute the resolvent of the gauge theory, which will provide a precise relationship between Coulomb moduli U i and the one point function of the chiral observables of the gauge theory.Next, we calculate the chiral observables directly using localisation methods.By composing these two steps, we can compute the instanton corrections to the U i at any point on the Coulomb moduli space, order by order in the instanton expansion.We pause to mention that while the resolvent is typically used to study how the trace relations are modified by quantum corrections [17], our goal will be to obtain the U i vs. V ℓ relations for i < N .
Recall that the resolvent of the gauge theory is the generating function of the chiral correlators.For the five-dimensional gauge theory, it is defined to be [18,22]: (2.7) If we expand this for large Z, we obtain where the V ℓ are the one-point functions: The resolvent is directly given in terms of the Seiberg-Witten differential associated to the curve [18] and is given by: Substituting for Ψ(Z) from eq. (2.6), we find that (2.11) By taking the β → 0 limit, one can check that this is the five-dimensional uplift of the resolvent of the four-dimensional pure N = 2 gauge theory that was computed in [17] using the Konishi anomaly.
Expanding this expression for large Z and comparing with eq.(2.8) allows us to read off relations expressing chiral correlators in terms of sums of products of Coulomb moduli, which are easy to invert.At 1-instanton, we have the following relations [18]: (2.12) and so on.We find the universal relation at 1-instanton for all values of n: where the sum over {m p } ⊢ n indicates a sum over partitions of n such that p p m p = n.We emphasise that although the above result is only true at 1-instanton, it is possible to compute k-instanton corrections to the same in a straightforward manner.
We see that for κ > 0, the relation between the U i and the V ℓ are exactly what one would expect from the classical theory; the above relation exchanges power sums for elementary symmetric polynomials.However, for κ < 0, we see that these relations are corrected by instantons.
The next step is to compute chiral correlators using supersymmetric localisation.We refer the reader to Appendix A for details and simply present the 1-instanton result for chiral correlators in the Kaluza-Klein reduced pure SU(N ) gauge theory [18]: (2.14) where we have imposed the SU(N ) condition N u=1 A u = 1.

Proposed Formulas for A and B
As we have reviewed in the Introduction, the proposal for the A and B functions that appear as measure factors in the U -plane integral are Here, ∆ phys. is called the "physical discriminant" and is proportional to the mathematical discriminant of the Seiberg-Witten curve, which in turn is a polynomial in the Coulomb moduli.Given that we have an algorithmic way to compute the U i as a function of the Coulomb vevs, we are now in a position to compute these two functions as a power series in the instanton counting parameter q = (−1) N (βΛ) 2N .
We check these results by explicitly computing these functions in the Ω-background.The instanton contributions are obtained by doing the contour integrals of the (five-dimensional uplift of the) Nekrasov integrand.The integrands, the contours, and the 1-instanton results for all SU(N ) gauge groups are summarised in Appendix A. There is, in addition to this, a leading term that survives the q → 0 limit.This is the perturbative contribution and is computed for the Ω-background in the five-dimensional case in [23].We review this in detail in Appendix B. In the sections that follow, the sum of these two contributions is referred to as the localisation contribution.

SU(2)
The Seiberg-Witten curve for SU(2) theory with Chern-Simons level κ takes the form Here, we have already identified q = (βΛ) 4 with the instanton counting parameter, in order to facilitate the match with the localisation results easier.(We have also set U 1 ≡ U in order to lighten the notation.)The discriminant of the curves for various κ are easily computed to be We then read off the physical discriminant as Recall that the relation between U and V is given via the resolvent as reviewed in Section 2.2.For the SU(2) theory with κ = 0, this relation is simply U (κ) = V .The chiral correlator V , in turn can be computed using localisation techniques.Putting all this together we have (with Given these, one can evaluate log A and log B using the proposed formulae in eq.(2.15).For κ = 0, we obtain Comparing with the localisation results we find a perfect match with the perturbative and instanton contributions provided we set This is our first check of eq.(2.15).2

SU(3)
The Seiberg-Witten curve for the SU(3) gauge theory takes the form In this case we have identified q = (βΛ) 6 with the instanton counting parameter.The resulting expressions are quite lengthy to list out for all cases, so in what follows we present the main results only for the κ = 0 case.The discriminant of the curve in this case is given by (3.9) The physical discriminant is given by ∆ phys.= ∆ 4096q 3 . (3.10) As before, the resolvent tells us how the U k 's are related to the chiral correlators V ℓ and we find Of course, in the above expression, we have We then find We have verified that these results lead to a precise match of the results for the A and B functions coming from localisation at the 2-instanton level for all admissible Chern-Simons levels if we set α = (βΛ) − N (N −1) for N = 3.For higher-rank gauge groups, the above result can be established in perturbation theory by expanding the 1-loop contribution to the Ω-deformed partition function to quadratic order in the deformation parameters and comparing logarithmic terms.

Adding Fundamental Flavours
We now extend our analysis of the previous sections to gauge theories with fundamental flavour.In particular we restrict ourselves to the SU(N ) gauge theory, set the Chern-Simons level κ = 0, and consider the asymptotically conformal case in which we have N f = 2N fundamental flavours.

The Curve and the Differential
The Seiberg-Witten curve of the gauge theory [19] is given by where the relevant functions are given by: Here, the Ũi are proportional to the Coulomb moduli U i up to the addition of some constants on the Coulomb branch. 3Our choice of parametrisation is A more extensive discussion of the rationale behind this prefactor will be given in the subsequent sections.
The S k in eq.(4.2) are the Weyl-invariant symmetric polynomials in the exponentiated masses M ℓ = e βm ℓ of the fundamental flavours, with m ℓ being the four-dimensional masses; the flavour function can be equivalently written as As is clear from eq. ( 4.2), we will restrict ourselves to the case in which the flavour symmetry is SU(2N ), by setting 2N ℓ=1 M ℓ = 1.It is easy to check that in the four dimensional limit β → 0, the Seiberg-Witten curve in eq.(4.1) reduces to the well-known curve for the four-dimensional theory at the β 2N order.
In order to identify the results obtained from the Seiberg-Witten geometry with those obtained via localisation, it is important to relate the dimensionless parameter g 2 with the instanton counting parameter q.This can be achieved by realizing the gauge theory as a system of two parallel NS5 branes transverse to N parallel D4 branes and lifting the brane configuration to M-theory [24][25][26][27].The ratio of the asymptotic positions of the NS5 branes is identified with the Nekrasov counting parameter.Matching the two curves in this way leads to the relation (4.5) The Seiberg-Witten differential associated to the curve takes the form:

The Resolvent and the Proposal Revisited
The next step is to express the U k that appear in the curve in terms of the chiral correlators, which can be computed using localisation methods.The relation between the two is once again given in terms of the resolvent of the gauge theory.As in the case without flavour, the generating function of chiral correlators is obtained from Ψ(Z) by the relation The factor of 1+q 2 in eq.(4.6) ensures that in the large-Z expansion of the resolvent has the correct form Recall that the resolvent has a large-Z expansion in terms of V ℓ = Tr e βℓΦ as shown in eq.(2.8).In order to more easily deal with the large-Z expansion that we need, we rewrite Ψ(Z) as follows: This is essentially the five-dimensional uplift of the analysis that was done for the fourdimensional N = 2 gauge theory with N f = 2N fundamental flavours in [15,28].Substituting this into eq.(4.7), and expanding the resulting expression for large Z, we express the V ℓ in terms of the U k .Inverting these relations, we find the following relations between Coulomb moduli and chiral correlators for the SU(N ) gauge theory with N f = 2N flavours: ) and so on.Once the V ℓ are computed -here, via localisation -we have all the ingredients needed to compute the gravitational couplings from the Seiberg-Witten geometry of the gauge theory.We once again use the relations: While one could, for example, just as well imagine using the Ũi instead of the U i in the above relations, we will argue that the above choice is the most appropriate one in the following sections.To compute log B, one needs to relate the discriminant of the Seiberg-Witten curve and the physical discriminant.We define where 4N is the highest degree of U N −1 that appears in the discriminant of the Seiberg-Witten curve.

Low Rank Results
In this section, we summarise our findings for gauge groups with low ranks.Since the resulting expressions in all cases match perfectly up to purely q-dependent (and therefore non-dynamical) factors, our presentation will largely focus on these discrepancies.

SU(2)
The Seiberg-Witten curve for this case is given by The parameter U is determined in terms of the relation eq.(4.9), with the chiral correlator V , that can be determined from localisation to be Proceeding as before, we first compute U using eq.(4.9), and then from eq. (4.11), we obtain As we have mentioned, on comparing these results -obtained from the Seiberg-Witten geometry -with a localisation computation of the same, one finds a non-dynamical (i.e.independent of Coulomb vevs), purely q-dependent mismatch that has interesting parallels with the results found for other asymptotically conformal models in the four-dimensional cases studied in [9,12].We find that up to 2-instantons, A discrepancy of this kind exists for all conformal gauge theories -we will see another example of it in the following section -and we will discuss this discrepancy in detail in Section 5.

SU(3)
The Seiberg-Witten curve in this case is given by (4.17) The parameters U 1 and U 2 are determined by the relations eq.(4.9) and eq.(4.10), with the chiral correlators V 1 and V 2 determined from localisation to be Given these, the functions log A and log B can be calculated from the curve.The expressions are not particularly illuminating, so we will not reproduce them here.For our purposes, however, it suffices to observe that just as in the SU(2) case, the functions log A and log B computed via (i) the Seiberg-Witten geometry, and (ii) via localisation match exactly up to purely q-dependent terms.We find that log .

The Four Dimensional Limit and the U(1) Factor
For the SU(N ) gauge theory with N f = 2N flavours, we observed a mismatch (see eq. (4.16) for N = 2 and eq.( 4. 19) for N = 3) in the result for log A and log B computed from the Seiberg-Witten curve and that computed from localisation.It is important to emphasise that all the non-trivial dynamical dependence on the Coulomb vevs and the masses matches, and the mismatch between the two can be parametrized by a constant (on the Coulomb branch) function of the instanton counting parameter or, equivalently, the gauge coupling.

The Four Dimensional Limit
In this section we attempt to give a uniform characterization (for any rank) of the mismatch between the results of the curve and localisation results for log A and log B. However the calculations in the conformal five-dimensional theory turn out to be progressively difficult from a computational point of view for the higher-rank cases.We therefore take the β → 0 limit and study the four-dimensional SU(N ) gauge theory with N f = 2N fundamental matter.The mismatch between the results from the curve and localisation is identical; this is not surprising as the mismatch is purely q-dependent and independent of the circle radius.
The ingredients needed to carry out this program are once again the Seiberg-Witten geometry and the resolvent of the four-dimensional gauge theory, apart from the expectation values for the 1-point functions of single trace operators on the Coulomb branch.All of these are already available in the literature, so we begin with a brief review of the relevant results.The Seiberg-Witten curve is given by where As was the case with the curve of the five-dimensional theory, we define ũk = 1 − q 1 + q u k . (5. 3) The u k are the gauge-invariant coordinates on the four-dimensional Coulomb moduli space, and the s k are elementary symmetric polynomials in the fundamental masses.Note that we have imposed SU(2N ) flavour symmetry at the level of the curve, which is the condition s 1 = 0.As before, the key ingredient in the calculation is the resolvent of the gauge theory, which is given by [15,28] Tr log (5.4) By doing a large-z expansion and comparing coefficients, we read off the u vs. v relations, where the v's refer to the single trace operators ⟨Tr Φ ℓ ⟩.We refer the reader to [28] for details and present only the first couple of relations: ) We can now repeat what was done in the five-dimensional case, and compute the log A and log B functions from localisation, along with the vevs of the single trace operators needed to compute the u k on the Coulomb branch.The localisation calculations are done for a U(N ) gauge theory with N f = 2N fundamental flavours, on which the constraints i a i = 0 and ℓ m ℓ = 0 are subsequently imposed.
For the results from the curve, we calculate where, for the four dimensional theory we define the physical discriminant to be where 4N − 2 is the highest degree of u N in the discriminant of the four dimensional curve.
We once again isolate the discrepancies between curve-and localisation-based computations of the effective gravitational couplings log A and log B, which we will call log δ A and log δ B respectively.These results are compiled for low-rank gauge groups in Table 1.
Note that the 1-and 2-instanton contributions in the first two rows are identical to the results obtained in the five-dimensional case in the previous section.Also, these results can just as easily be obtained in the massless theory, since at this order in the expansion in terms of the Ω deformation parameters, on dimensional grounds there can be no massdependent terms.The 4-instanton result for the SU(2) N f = 4 theory and all the results The differences between curve-and localisation-based computations of effective gravitational couplings for gauge groups of low rank.These discrepancies are computed in the fourdimensional theory with gauge group SU(N ) and N f = 2N fundamental flavours.
for the N = 4, 5, 6 cases were obtained in the limit in which we take the flavours to be massless.
Based on our study of these discrepancies for gauge groups of low rank and up to a few instantons, we conjecture the following: Although we do not have a proof of the above conjectures from first principles, these formulas agree on all test cases we studied and exactly capture the discrepancies as a function of the gauge coupling to all orders and the rank of the gauge group.

The U(1) Factor
Earlier efforts to characterise such mismatches (between curve-and localisation-based computations) of A and B for rank-1 theories with flavour [9] and the higher-rank N = 2 ⋆ theories [12] successfully attributed it to the contribution of a U(1) factor, which has its origins in the AGT correspondence [29].In that context, factoring out the contribution of the U(1) factor was crucial in order to match Liouville conformal blocks with the instanton partition function of conformal quiver gauge theories with SU(2) gauge groups.Subsequent work [30] found that a similar factorization was required to match the instanton partition function of higher-rank gauge theories with the Toda conformal blocks.
The origin of the U(1) factor can be understood by recalling that the Nekrasov integrand is naturally defined for U(N ) theories, while the curve we have worked with is for the SU(N ) theories.Therefore, in order to make comparisons between the results of curveand localisation-based computations, one must take into account the global U(1) factor.In terms of partition functions, we have the decomposition (5.10) It is easily checked that the discrepancies in eq.(5.9) can be accounted for by considering a U(1) factor of the form log where the • • • indicate other terms in a small deformation expansion.Based on our study of just the effective gravitational couplings, this is as much as we can say about the U(1) factor with absolute certainty.It is, however, a small step to conjecture the following U(1) factor for the four dimensional SU(N ) gauge theory with 2N flavours with masses m i in the fundamental representation: (5.12) Firstly, this reduces to the factor proposed in [9] for the case of N = 2. Secondly, at linear order in the ϵ i -expansion, the contribution to H is proportional to the sum of all the masses of the fundamental flavours, which vanishes due to the SU(2N ) flavour constraint.At second order in the ϵ i -expansion, we recover the factor derived in eq. ( 5.11) for all N .Finally, at leading order, this introduces a quadratic term in the masses that breaks the Weyl symmetry acting on the masses.One can proceed, as in [9], to define a Weylinvariant prepotential (and associated Coulomb vev) by subtracting out a constant (moduli independent) term.
Let us now return to the Kaluza-Klein theories.As we have seen, in the ϵ i → 0 limit, the discrepancy in the Kaluza-Klein compactified theory is identical to that of the fourdimensional theory.This is essentially because ϵ i → 0 is identical to the β → 0 limit.It is therefore a simple matter to lift the above formula for the U(1) partition function to five dimensions.In terms of the exponentiated Ω deformation parameters E i = e βϵ i , and mass parameters M i = e βm i , we have: (5.13) Taking into consideration the contribution of the above U(1) factor and the perturbative (1-loop) contribution (see Appendix B for details) we find that with the following choices: we find perfect agreement between curve and localisation results.As in the case of the pure gauge theory, the factors α and β are functions of the dimensionless constant βΛ, which ensures that the five-dimensional results smoothly go over into the four-dimensional results in the limit of vanishing circle radius.

Summary and Discussion
Our investigations in this paper have focused on the case of four-dimensional theories arrived at via Kaluza-Klein reduction of five-dimensional N = 1 supersymmetric gauge theories.We studied both the pure SU(N ) gauge theories at admissible Chern-Simons level, and the conformal gauge theories with N f = 2N fundamental hypermultiplets.In studying the case of non-zero Chern-Simons levels and the case of higher-rank gauge groups, we have extended earlier work [9,13] on the study of these effective gravitational couplings.Our analysis crucially used the resolvent of the gauge theory, which in turn allowed us to compute quantum corrections to the Coulomb moduli order by order in the instanton expansion.In all the theories we studied, we found that the the effective gravitational couplings independently determined via equivariant localisation match the expectations in eq.(2.15) arising from considerations of holomorphy, R-symmetry, etc. up to the constants of the proportionality.
The case of the conformal gauge theories was particularly interesting, as we found a nondynamical, purely q-dependent discrepancy.Such discrepancies are not new in the study of conformal gauge theories -equivariant localisation in these cases reproduces the results derived from the Seiberg-Witten curve, but only up to constants on the Coulomb branch.We required that these discrepancies in the effective gravitational couplings be absorbed by an appropriate U(1) factor.Significantly, our rescaling of the Coulomb moduli in the curve was crucial for this to work.While many alternative parametrisations of the Coulomb moduli space are permissible [31], each will leave its imprint in the choices of constants of proportionality forced upon us to match the results of curve-and localisation-based computations.We have presented in this paper the unique choice attributes all k-instanton discrepancies between curve-and localisation-based computations to the U(1) factor, in line with other conformal gauge theories.It would be an interesting task to derive the full five-dimensional U(1) factor, perhaps along the lines of [32][33][34].
Another line of investigation that naturally presents itself is the question of resummation.
As is well-known, the constraints from S-duality on superconformal N = 2 gauge theories takes the form of a modular anomaly equation, which can be used to reconstruct the prepotential [35][36][37][38].It is natural to ask if the effective gravitational couplings presented in this paper are similarly constrained, and whether they can be resummed into modular forms of the relevant S-duality group.We hope to return to some of these questions in the near future.

A.1 The Instanton Partition Function
The partition function of an N = 1 SU(N ) gauge theory with N f fundamental hypermultiplets on R 4 × S 1 and computed via supersymmetric localisation is given by [10]: where the contribution of the vector and hypermultiplets is captured by the integrands In the above expressions, g(x) = 2 sinh βx 2 , β is the circumference of the S 1 , and κ is the five-dimensional Chern-Simons level.The Coulomb branch of the theory has been parametrized by the vev a u with u = 1, . . ., N of the adjoint scalar field Φ in the vector multiplet.They satisfy the SU(N ) condition N u=1 a u = 0.
We have studied two models in the main text.
• For the pure gauge theory in which we omit the contribution due to fundamental matter, the instanton counting parameter q is related to the complexified strong coupling scale Λ and the radius of the S 1 by the relation: • For the case with conformal matter, we set N f = 2N , and the instanton counting parameter q is related to g 2 , the parameter that appears in the Seiberg-Witten curve, by eq.(4.5).
The integral in eq.(A.1) is a (closed) contour integral over the complex χ σ -planes.We briefly review the contour prescription now.We take the a u to be real and assign an imaginary part to the Ω deformation parameters, such that With this choice, the poles in the integrand lie either in the upper-half plane or the lowerhalf plane of χ σ .As is well known [10,39], the poles are in a one-to-one correspondence with Young tableaux, such that the total number of boxes is equal to k, the instanton number.We observe that the locations of the poles is completely independent of the value of the five-dimensional Chern-Simons coupling.If (i, j) label the row and column of the Young tableau, the poles are located at We restrict ourselves to just the fundamental domain, with n = 0, and we choose the convention in which we select the poles in the upper-half planes.This, and the choice of contours at 1-instanton has been discussed in great detail in [18], to which we refer Similar expressions can be obtained, both for the partition function and for the chiral correlators in the case with fundamental flavour.But the expressions are not very illuminating and we present just the results for the low rank cases in the main text.

B The Perturbative Contribution
The one loop or perturbative contribution to the partition function of an Ω-deformed gauge theory of the Kaluza-Klein type that we have been studying has been obtained in the classic work [23].The main ingredient here is the five-dimensional lift of the special function γ ϵ 1 ,ϵ 2 (x|Λ) that already appeared in the earlier works [11,40].In what follows we shall briefly review the definitions and then provide the particular combinations of the special functions of interest to the gauge theories studied in the main text.
Following [23] we first of all define We list the first few coefficients that will be of relevance in our calculations: Then, the relevant function in terms of which the perturbative contribution will be written is given by It is shown in [23] that this function has the expected four dimensional behaviour in the limit that β → 0. Our present goal is to find linear combinations that give the correct logarithmic terms in log A and log B for the pure gauge theory and in the case with flavour in the limit q → 0. For this purpose we consider the small-ϵ i expansion of the function in eq.(B.4), which is valid for x > 0: The linear combination shown below proves to be useful to write down the perturbative contribution for the pure gauge theory; we present the small-ϵ i expansion: This combination ensures that H = 0, and that at second order in the expansion, the contribution to both the log A and log B terms is identical as far as the log-terms are concerned.The contribution of the linear term cancels out once we sum over all roots of the Lie algebra.
The following shifted function in turn allows one to write down the perturbative contribution from the matter fields in the fundamental representation: with the constraints N i=1 a i = 0 and N f ℓ=1 m ℓ = 0 imposed on the Coulomb v.e.vs and mass parameters (this also ensures the vanishing of the linear terms in the combination of special functions that contribute to the one loop contribution for the fundamental representation).We note that in the four-dimensional limit, the same combinations of the special functions with shifted arguments occur in the 1-loop contribution proposed in [9].