On the Quantization of Seiberg-Witten Geometry

We propose a double quantization of four-dimensional ${\cal N}=2$ Seiberg-Witten geometry, for all classical gauge groups and a wide variety of matter content. This can be understood as a set of certain non-perturbative Schwinger-Dyson identities, following the program initiated in arXiv:1512.05388. The construction relies on the computation of the instanton partition function of the gauge theory on the so-called $\Omega$-background on $\mathbb{R}^4$, in the presence of half-BPS codimension 4 defects. The two quantization parameters are identified as the two parameters of this background. The Seiberg-Witten curve of each theory is recovered in the flat space limit. Whenever possible, we derive these results from type IIA string theory.

1 Introduction N = 2 supersymmetric Yang-Mills holds a special place in the realm of quantum field theories. Non-perturbative effects play a crucial role in defining the effective action, yet these effects are tractable. In particular, the low energy physics can be determined with the help of a holomorphic function known as the prepotential. Indeed, this function encodes both the perturbative one loop physics, as well as the non-perturbative instanton corrections to the effective action. In the seminal work [2], Seiberg and Witten put forward a formidable proposal to solve the pure SU (2) gauge theory. In short, they showed that the prepotential can be determined via the symplectic geometry of an algebraic curve, referred to nowadays as the Seiberg-Witten curve. In this way, a large class of examples was solved by means of a hyperelliptic curve. As the gauge group and matter content is varied, it is also not uncommon to encounter non-hyperelliptic curves, of finite or infinite order.
The present paper deals with the quantization of such geometries, a notion that comes about in the following way: first, consider a (conformal or asymptotically free) four-dimensional quiver gauge theory T 4d with N = 2 supersymmetry, where the gauge group is a product of special unitary groups, resulting in a quiver whose graph Γ is a Dynkin diagram of type g = A, D, E,. It was shown in [3] that there exists a relation between the Seiberg-Witten geometry of T 4d and the representation theory of the quiver itself. In particular, one can express the Seiberg-Witten curve of the gauge theory in terms of the characters of the fundamental representations of g. where (M, y) ∈ C × C * , Λ is the QCD dynamical scale, and T (M ; {e i }) is a polynomial of degree N in M whose roots define the vacuum of the theory, and depend implicitly on the classical Coulomb vevs of T 4d . In the quiver denomination, T 4d consists of a single gauge node, so the associated algebra is simply g = A 1 . There is a unique fundamental character of A 1 , and it is given by y + 1/y (the character of the spin 1/2 representation of SU (2)). The second term can be constructed from the first term simply by acting on it with a Weyl reflection, y → 1/y. This is not quite what we want; instead, we should slightly modify the definition of the Weyl reflection to account for the scale Λ, meaning y → Λ 2N /y. This results in a twisted fundamental character of A 1 , which now reads y + Λ 2N /y ≡ χ A 1 1 (y). We immediately recognize this expression as the left-hand side of the Seiberg-Witten curve: More generally, it is always possible to deduce the Seiberg-Witten curve of a g-type quiver gauge theory made up of special unitary groups as was done above, in terms of twisted g-characters. In order to reach this result, a powerful method was first developed to directly evaluate the prepotential of T 4d [4,5]. Crucially, the construction relies on the fact that the four-dimensional spacetime admits a two-parameter deformation known as the Ω-background, which can be thought of as a weak N = 2 supergravity background. We denote it as C 2 1 , 2 , where i rotates the i-th C-line, for i = 1, 2. In this background, the prepotential, plus an infinite number of corrections which vanish in the flat space limit, are equal to the logarithm of a sum of instanton integrals. By exploiting equivariant localization, the computation simplifies drastically to become a mere multiple contour integral. It can then be shown [3] that the Seiberg-Witten curve is nothing but the saddle point equation of such an instanton integral 2 . The saddle point analysis can be carried out by simply taking the limit 1 , 2 → 0. In this picture, the complex variable y which appears on the left-hand side of the Seiberg-Witten curve (1.1) is realized as the vev of an operator, This Y -operator is a generating function of the chiral ring of T 4d . So-called "i-Weyl" reflections are defined to act on the operator Y as Y → Λ 2N Y −1 . Starting with a "highest weight" Y and acting on it with such a reflection, we generate a representation whose character is once again the left-hand side of (1.1), as we saw before. These i-Weyl reflections can be defined for any 4d N = 2 quiver gauge theory.
In fact, sending both 1 and 2 to zero is overkill to carry out a saddle point analysis, as only one parameter really needs vanish, say 2 → 0, while the other parameter 1 can be kept arbitrary 3 . Therefore, a natural question to ask is what type of saddle point equations one obtains with 1 still present. This problem was addressed in [6]. The authors showed that the saddle point equations can be understood as a q-deformation of the Seiberg-Witten curve of T 4d , where the role of Planck's constant is played by the surviving Ω-background parameter 1 ≡ . The saddle point equation no longer defines an algebraic curve, but is now instead a difference equation 4 . In our SU (N ) example, the saddle-point equation reads: The function T (M ; {e i }) is still a polynomial of degree N in the variable M , whose roots {e i } define the vacuum of the theory. We made the dependence of the roots on explicit by using a superscript notation. By analogy with the previous construction of the curve (1.2), the left-hand side of (1.4) is sometimes called the fundamental q-character of A 1 . More precisely, it is a (twisted) Yangian q-character of the fundamental representation of Y (A 1 ) 5 .
It turns out to be possible to obtain a similar equation in the full Ω-background, with both 1 and 2 present. In particular, because none of the parameters can be sent to zero, a saddle point analysis of instanton integrals is no longer possible, so a different strategy is needed. This is the program carried out in [1]. Namely, one first defines a half-BPS point-defect in T 4d and an associated operator Y (M ), with M ∈ C a coordinate on an auxiliary complex line, Second, one computes the instanton partition function of T 4d on the Ω-background, in the presence of the defect 6 . This defect partition function can be expressed in two ways: First, it is a sum of Y -operator correlators 7 : . Though it is a priori far from obvious, the above equation has a beautiful interpretation as a non-perturbative Schwinger-Dyson identity for the theory T 4d [1]. Roughly, the idea is as follows: Given a correlator defined by a path integral in quantum field theory, the Schwinger-Dyson equations can be understood as constraints that must be satisfied by such a correlator. This comes about from demanding that the path integral remain invariant under a slight shift of the contour (provided that the measure is left invariant by such a shift). In particular, we could ask about a contour modification that takes us from a given topological sector of T 4d to another distinct topological sector, related to the first by a large gauge transformation. In other words, are there certain symmetries of T 4d that manifest themselves as we change the instanton number? The above Y -operator is a natural observable to answer this question: as a codimension 4 defect operator in T 4d , it can mediate the change in instanton number of the theory. The equation (1.7) can then be understood as a regularity condition on the correlator Y (M ) . Namely, the correlator has poles in the fugacity M , but the Schwinger-Dyson equation tells us that there exists a precise combination of Y -operator vevs (the left-hand side of the equation) which is pole-free in M . In this fashion, non-perturbative Schwinger-Dyson identities can be derived for SU (N ) gauge theories with or without fundamental and adjoint matter, and for quiver gauge theories made up of such special unitary groups 9 . These identities are what we will refer to as "quantized Seiberg-Witten geometry" in this paper.
A natural question is whether it is possible to extend this analysis to SO(N ) and Sp(N ) gauge groups, as well as SU (N ) gauge theories with different matter content, such as symmetric and antisymmetric matter. The Seiberg-Witten geometry for these models has been known since the 90's (see [8] for a summary). However, no attempt has been made to quantize it. In this paper, we address this point, by constructing non-perturbative Schwinger-Dyson identities for all the above models. Whenever possible, we provide a string theory derivation of our results, based on a generalization of the original SU (N ) brane setup considered in [1]. For example, the quantized geometry of the SO(2N ) and Sp(N ) theories will be defined using orientifold planes, with a D4/D4 /O8 brane system. In some cases, such as SO(2N + 1), our construction will rely on field theoretic arguments only, with no underlying brane setup.
It will turn out that all the non-perturbative Schwinger-Dyson identities we derive take the form . . will be an infinite Laurent series in the point defect operator vev Y (M ) , organized as an instanton expansion. The series is finite only in the case of a pure SU (N ) gauge theory, as in the left-hand side of (1.7), or with fundamental matter. We give an explicit algorithm to write this Laurent series to arbitrary high order in the instanton counting parameter. We then conjecture the highly nontrivial fact that the right-hand side of the equation, T (M, {e 1 , 2 The paper is organized as follows: in Section 2, we describe the general method to construct non-perturbative Schwinger-Dyson identities for gauge theories with a classical gauge group. The theories will be five-dimensional on a circle, and four-dimensional results will be recovered when shrinking the circle size to zero. We also describe how to extract the Seiberg-Witten geometry of the models. In section 3, we test our methods on a wide variety of examples, with a type IIA string theory picture whenever we can.

Constructing Non-Perturbative Schwinger-Dyson Identities
Though our discussion so far has been purely phrased in a four-dimensional context, it will be worthwhile for us to start our analysis in five dimensions. We claim that the construction of quantized Seiberg-Witten geometry is intimately related to the following problem: consider 5d SYM with gauge group G, defined on the manifold S 1 (R) × R 4 . How does a supersymmetric Wilson loop wrapping S 1 (R) interact with the instantons living on R 4 ? Our goal in this section is to make the connection between these two ideas explicit.

The 5d Gauge theory and Instantons
Consider 5d maximal SYM theory, with classical gauge group G = SU (N ), SO(N ), Sp(N ), defined on S 1 (R) × R 4 , where R is the radius of the circle S 1 (R). The theory has SO(1, 4) Lorentz symmetry and SO(5) R-symmetry. We give a non-zero vev to a vector multiplet scalar, Φ = 0. This forces the theory to go on the Coulomb branch, where the gauge symmetry G breaks to a maximal abelian subgroup. Correspondingly, the SO(5) R-symmetry is broken to SO(4) R = SU (2) R + × SU (2) R − .
In this paper, we will be interested in counting instantons, which are solutions of the self-dual Yang-Mills equations on R 4 . In five dimensions, these instantons are massive BPS particles. As such, they preserve the little group of SO(1, 4), which we write as SO(4) L = SU (2) + × SU (2) − .
We now want to ask what happens when we introduce a Wilson loop in the 5d theory.

Wilson Loops and Instantons
Recall that a Wilson loop is formulated as the trace of a holonomy matrix, where a quark is parallel-transported along a closed curve in spacetime, and the trace is evaluated in some irreducible representation of the gauge group G. The vev of such a loop is the phase shift of the quark wavefunction. Going back to the 5d maximal SYM theory, we introduce a supersymmetric Wilson loop wrapping the S 1 (R) and sitting at the origin of R 4 . This can be done with the help of a one-dimensional fermion field χ, transforming in the fundamental representation of G and in the fundamental representation of another background gauge group G (we will refer to it as the defect gauge group), coupled to the 5d gauge field in the bulk as [17] Above, A t and Φ are the pullback of the 5d gauge field and the adjoint scalar of the vector multiplet, respectively. A t is the (nondynamical) gauge field the 1d fermions couple to. i and j are indices for the fundamental representation of G, while ρ and σ are indices for the fundamental representation of G . The variable t is periodic, with period R/(2π). The eigenvalues M ρ of the background gauge field A t are (large) masses for the fermions. Those parameters set the energy scale for the excitation of the fermions. Such a loop is indeed 1/2-BPS, as it preserves the supercharges Q αa and Qα a . Then, evaluating the path integral of the coupled 5d/1d system amounts to computing Here, ψ denotes collectively all the fields of the bulk 5d theory, written as S 5d , while S 1d denotes the coupling term (2.1).
Since instantons are particles in 5 dimensions, counting them amounts to computing the partition function of their quantum mechanics, which is essentially a Witten index. Crucially, one needs to treat carefully the contribution of coincident zero-size instantons, as the moduli space is singular there. In general, the so-called ADHM [12] construction is a powerful way to resolve such singularities. In our case, the instantons are also coincident with the Wilson loop, so regularizing their contribution must be done with extra care. Naively, one could try to simply localize the Wilson loop at ADHM solutions in the absence of a loop, but that is not the way to proceed. Instead, one should generalize the ADHM construction altogether, so that the resulting instanton partition function correctly captures the contribution of zero-size instantons near the Wilson loop [13]. The instantons are then described by a N = (0, 4) gauged quantum mechanics 11 , where the preserved supercharges are Qα a .
When the gauge group G is SU (N ), this gauged quantum mechanics has a description given in string theory . First, maximally SYM is realized as the low energy effective field theory of N D4 branes in type IIA. Second, a string realization of Wilson loops was proposed in [14,15], in the context of holography; in particular, a Wilson loop in the first fundamental representation of SU (N ) can be described as a fundamental string whose worldsheet ends at the loop, located at the boundary of an AdS geometry. Later, a realization in terms of branes in flat space was given instead [16][17][18], allowing for loops in more general representations. More precisely, a 1/2-BPS Wilson loop can be realized as N D4 branes that are codimension 4 with respect to the original N D4 branes; the representation in which the Wilson loop transforms is then determined by the integer N . Lastly, k instantons can be introduced as k D0 branes nested inside the D4 branes [19]. It follows that the ADHM mechanics we are after is the quantum mechanics of D0 branes in a background of D4 and D4 branes. In particular, after integrating out the (heavy) fermionic D4-D4' string modes, the Wilson loop is recovered.
One of the main goals of this paper will be to generalize the above type IIA picture to other classical gauge groups, using orientifold planes. However, this will not always be possible, and in some cases we will only be able to provide a field theory construction.

The Partition Function as a Witten Index
Let T 5d be a 5d N = 1 gauge theory with classical gauge group G and flavor symmetry group K, on the manifold S 1 (R) × R 4 . In the rest of this paper, the 5d bare Chern-Simons term (when it exists) is set to zero. If we view R 4 as C × C, then we can denote the coordinate on the first C-line as z 1 , and the coordinate on the second C-line as z 2 . We introduce an Ω-background by viewing the 5d spacetime as a C 2 bundle over S 1 (R), where as we go around the circle, we make the identification We write the partition function of T 5d with Wilson loop as the Witten index of the following N = (0, 2) ADHM gauged quantum mechanics: 4) H QM is the Hilbert space of the five-dimensional field theory on R 4 . The trace is taken over all BPS states of the ADHM quantum mechanics. These are the BPS states annihilated by the supercharges Q ≡ Q 11 and Q † ≡ Q 22 , with Hamiltonian {Q † , Q}; it therefore counts states in Q−cohomology.
F is the fermion number. k is the topological U (1) charge, conjugate to the instanton counting fugacity q ≡ exp −8π 2 R/g 2 5d , with R the radius of S 1 . We have also defined J + , J − , and J R + as the Cartan generators of SU (2) + , SU (2) − , and SU (2) R + , respectively. Π i , Λ ρ and L d are Cartan generators of the 5d gauge group G, the Wilson loop defect group G , and the additional 5d flavor symmetry group K, respectively. They are all flavor symmetry groups from the one-dimensional perspective. As far as the conjugate variables are concerned, we have introduced the 5d Coulomb parameters {a i }, the Wilson loop defect fugacities {M ρ }, additional masses as {m d }, and redefined the Ω-background parameters as The index is the grand canonical ensemble of instanton BPS states. It is a product of a perturbative factor involving the classical and 1-loop contributions, and of a factor capturing the instanton corrections. The perturbative part plays no role in our story, so we will safely ignore it in the rest of this work. Meanwhile, we organize the instanton part as a sum over all instanton sectors k.
We can evaluate the gauge theory index in the weak coupling regime of the UV quantum mechanics, where it reduces to Gaussian integrals around saddle points. These saddle points are parameterized by φ = Rϕ (QM ) + iRA the gauge field and ϕ QM the scalar in the vector multiplet of the quantum mechanics. We denote the gauge group of this quantum mechanics as G, and the (complexified) eigenvalues of φ as φ 1 , . . . , φ k . Performing the Gaussian integrals over massive fluctuations, the index reduces to a zero mode integral of various 1-loop determinants, which we write schematically for now in five-dimensional language as: The five-dimensional gauge coupling is implicitly written in the instanton counting fugacity q ≡ exp −8π 2 R/g 2 5d . |W ( G)| is the Weyl group order of G. If G = U (N ), then G = U (k).
The factor Z (k) vec ({φ I }, {a i }, 1 , 2 ) contains all the physics of the 5d vector multiplet. We have made explicit the dependence on the N Coulomb parameters {a i } of G, the k integral variables φ I of G, and the Ω-background parameters 1 and 2 .
The factor Z  Since the partition function is a multi-dimensional integral in the 1d Coulomb moduli φ I 's, we need a robust method to compute the residues. We adopt the so-called Jeffrey-Kirwan (JK) residue prescription [20]. The prescription was first introduced in our context in the works [21,22], and has been very popular since, so we will be brief in reviewing it. Each factor Z (k) vec , Z (k) matter , and Z (k) def ect has the following general form: where ρ is a k-tuple vector, which consists exclusively of entries in the set {0, ±1, ± 1 2 }. n 1 and n 2 are positive integers specified by the details of the ADHM quantum mechanics. Finally, ". . ." stands for a linear function of the spacetime fugacities 1 , 2 , as well as 1d flavor fugacities. Since sinh(iπ) = sinh(0) = 0, there can be many poles in (2.7) at a specific value of φ = φ 0 . The JK prescription instructs us to first pick a k-vector η; in this paper, we find the choice (1, 3, 5, . . . , 2k − 1) convenient. Ultimately, the physics will not depend on that choice. Then, we are to choose k hyperplanes from the arguments of sinh functions in the denominator of (2.7). Those hyperplanes will take the following form: (2.8) The contours of the partition function are then chosen to enclose poles which are solutions of the above linear system of equation, but only if the vector η also happens to lie in the cone spanned by the vectors ρ l . One practical way to test this condition is to construct a k × k matrix Q = Q ji = (ρ j ) i , where ρ j = ((ρ j ) 1 , . . . , (ρ j ) k ), and test if all the components of ηQ −1 are positive.
Sometimes, it can happen that a solution of the system of equations (2.8) yields additional zeroes in the denominator of (2.7). This typically results in non-simple poles, and we discuss them in the appendix.

The Wilson Loop Defect Operator Y
Before we can make contact with Seiberg-Witten geometry, we need to discuss in some detail the contours of the partition function.
First, we specialize the Wilson loop factor Z Second, we organize the poles in the partition function into two sets: for a given instanton number k, let M k be the set of poles selected by the JK-residue prescription in the partition function (2.6) (and should therefore all be enclosed by the contours). Meanwhile, let M k be the set poles selected by the JK-residue prescription in the pure partition function, in the absence of the factor Z (k) def ect in the integrand. We denote the instanton partition function in the absence of defects as 1 .
For a given instanton number k, the sets M k and M k have finite, but different sizes: as we will see explicitly in the next section, Z (k) def ect always contains at least one pole depending on the defect fugacity M 14 , so for a given instanton number k, the former set is always strictly larger than the latter set: |M k | > |M k |. This simple observation has important consequences, as we can derive non-perturbative Schwinger-Dyson equations for Wilson loop vevs from it. We first define the expectation value of a defect operator: . (2.9) At first sight, it may seem like Y (M ) is just the partition function (2.6), but we still need to specify the contours of the above integral. Namely, we choose the contours in (2.9) to only enclose poles in the set M k , meaning no enclosed pole will depend on M . Put differently, we ignore all the poles originating from Z Our first main result in this paper will be to show that given a classical gauge group G, the partition function of T 5d can be written as where F k,j is a rational function of the defect Y -operator vev, the spacetime fugacities 1 , 2 , and the various masses. In using the notation Z(M ), we made the dependence of the partition function on the defect fugacity M explicit, while all the other fugacities are kept implicit. For instance, the dependence on the 5d Coulomb parameters is implicitly encoded in the vev Y (M ) and the function F .
The meaning of this expression is as follows: as we have noted, the partition function has the same integrand as the defect operator vev, the first term Y (M ) on the right-hand side. However, the contours on the left-hand side enclose more poles than those of Y (M ) , since |M k | > |M k |, for each instanton number k. The sum is there to make up for that deficit of M -dependent poles. Each term in the sum on the right-hand side stands for a residue of the integral (2.6), evaluated at a pole in M k \ M k , that is to say a pole in the set M k not present in the set M k . Quite nontrivially, we find that the residues turn out to be rational functions of Y -operator vevs. After summing over all the terms in the right-hand side, the partition function is recovered. def ect in the integrand. On the left, we show a possible contour for the computation of the 5d partition function at k = 1, say for a SU (3) theory. Note that by the JK prescription, we must in particular enclose the new pole in red. Remarkably, it is equivalent to trade this contour for the one on the right, which now only encloses the poles in the set M 1 , but with a modified integrand; for the contour on the right, the integrand will now contain insertions of new Y -operators, with an instanton shift of one unit to account for the missing pole.
We emphasize here that at no point in the discussion did we need to know how to compute the instanton partition function of the gauge theory in the absence of the defect, that is to say the content of the set of poles M k . What is instead relevant here is the set of poles M k \ M k , due entirely to the insertion of the defect.
We normalize the partition function, Here, 1 is the pure partition function we defined earlier, (2.6) without the factor Z }) is therefore possible. When G is one of the other classical gauge groups, however, an explicit description of the poles is lacking, so we leave the general proof of the statement to future work. For now, we will content ourselves with a check of the conjecture following the JK-residue algorithm, which can be performed to arbitrarily high instanton number.
We therefore conclude that (2.13) together with the requirement that T (M ; {e 1 , 2 i }) be finite in e M , can be thought of as non-perturbative Schwinger-Dyson equations for the theory. The defect Y -operator vevs we construct as (2.9) are explicit solutions of these equations. In five dimensions, the Yoperator is understood as a loop defect. Since instantons in 5d are particles, the Y -operator can correctly mediate a change in instanton number k, as long as the particles wrap the loop. This changes the topological sector of T 5d , and the left-hand side of (2.13) encodes a corresponding symmetry.
Before we end this section, we point out that the above discussion straightforwardly generalizes to an arbitrary number of fermions, N > 1. In that case, we still obtain Schwinger-Dyson equations similar to the ones above, but involving correlation functions of a higher number of Y -operators.

The Doubly-Quantized Seiberg-Witten Geometry
We would like to propose that the above Schwinger-Dyson (SD) identities can be understood as a double quantization of Seiberg-Witten (SW) geometry, where the two quantum parameters are the spacetime Ω-background parameters 1 and 2 . In this work, we test the idea by taking the flat space limit 1 , 2 → 0 in (2.13), and we argue that we recover the SW curve of the theory. The SD equations involve correlators of Wilson loop operators, which should now be thought of as a complex coordinate on an auxiliary cylinder [3], (2.14) We argue that the resulting equation This result has highly nontrivial implications; for instance, the left-hand side of (2.13) typically is an infinite sum of correlators, since the set M k \ M k becomes increasingly bigger as k grows. However, when T 5d is a pure G-theory, we conjecture that all but a finite number of terms disappear in the flat space limit. More precisely, in those cases, we will argue that when the instanton number k is sufficiently high 15 in the sum (2.13), all terms will cancel against each other at each order, resulting in a finite sum that reproduces the known SW curve of the model.
Perhaps the presentation of the curves is more familiar in a four-dimensional context, so we deem it useful to rewrite our non-perturbative SD equations in four dimensions first. There are a priori many different ways to take a 4d limit, so we should be specific: here, we want T 5d to become a four-dimensional gauge theory T 4d , with the same gauge group G and flavor content as the higher-dimensional theory. Recall that T 5d is defined on the manifold S 1 (R) × R 4 , so we will conveniently reduce the theory on the circle S 1 (R) by sending R → 0. At this point, it is necessary to reintroduce the explicit dependence of the various 5d fugacities on the radius R. We first rescale the gauge coupling as and keep q fixed as we send R to zero. h(G) is the dual Coxeter number of G and k(R) the quadratic Casimir of the representation R. Furthermore, we require the Ω-background parameters 1 , 2 , the Coulomb moduli {a i }, the masses {m d } and the defect fugacity M to be kept fixed as we reduce the circle size. In practice, we simply rescale these fugacities by R, and then take the limit R → 0. The 4d gauge coupling of T 4d is related to the 5d gauge coupling of T 5d by . (2.17) It follows that (2.16) can be understood as an RG flow equation, which describes the running of the coupling g 4d as the UV scale R −1 is varied. An associated dynamical scale Λ can therefore be defined: We can now safely take the limit R → 0 in the SD equations (2.13), and obtain nonperturbative SD equations for T 4d , which have the same functional form as before: (2.19) Note that this time,the function T 4d (M ; {e 1 , 2 i }) on the right-hand side is a finite polynomial in M , with a finite number of roots {e 1 , 2 i } which characterize the vacuum. On the left-hand side, the Y -operator vev Y 4d is now a point defect sitting at the origin of spacetime, since it is the reduction of the Wilson loop that used to wrap the circle S 1 (R). Finally, we can proceed to take the flat space limit 1 , 2 → 0 just as we did in 5d, to obtain We will argue in the next section that in all examples we present, the above is nothing but the SW curve of T 4d , with coordinates (M, y) ∈ C × C * .
Let us end this section with a remark on representation theory. In the 5d picture and for a pure SU (N ) gauge theory, the left-hand side of the SD equation (2.13) can be interpreted as a deformation of the q-character of the fundamental representation of the quantum affine algebra U ( A 1 ) 16 ; see also [1,26]. Now, in four dimensions, the left-hand side is understood instead as a deformation of the q-character of the fundamental representation of the Yangian Y (A 1 ) 17 . In that context, the terms on the left-hand side are built from an i-Weyl group action acting on the highest weight Y (M ) . When G = SU (N ), the relation to the representation theory of Yangians (in 4d) or quantum affine algebras (in 5d) is lacking, and we will see that the i-Weyl group is no longer finite.

Existence of the Theories
Gauge theories in five dimensions are non-renormalizable. As such, T 5d is not useful as a microscopic theory. Still, there is by now a large amount of evidence that T 5d can be understood as a relevant deformation of an interacting 5d superconformal field theory in the UV [27]. This is the point of view we have taken in this work so far, and we have implicitly assumed that T 5d has a UV completion 18 . After taking the 4d limit, we should worry whether or not the resulting gauge theory T 4d makes sense in the first place, as a conformal or asymptotically free theory. Therefore, we henceforth impose the condition 2h(G) − k(R) ≥ 0. In particular, for a fixed gauge group rank, this translates to restrictions on the allowed flavor symmetry of the theory. For instance, as far as fundamental matter is concerned, if G = SU (N )/SO(N )/Sp(N ), the allowed flavor content must now satisfy There is a subtlety, however. Indeed, consider a 4d conformal or asymptotically free theory T 4d , and its 5d uplift T 5d . In the ADHM quantum mechanics description of T 5d , it is possible a continuum opens up in the 1d Coulomb branch, which manifests itself in 5d as extra decoupled states in the bulk. In practice, such a continuum arises when there are new poles at ∞ 19 for the 1d vector multiplet scalar in the integrand (2.6). For example, consider the case G = SO(N ). The conformality/asymptotic freedom condition on fundamental matter is N f ≤ N − 2, but there are poles at infinity in the ADHM quantum mechanics of Our analysis in this paper is only strictly valid when N f ≤ N − 5. For definiteness, then, we will only limit ourselves in the rest of this paper to the cases where the ADHM quantum mechanics does not develop such continua in the 1d Coulomb branch. We leave the detailed study of the remaining cases and their 4d limits to future work. 16 This deformed character was first constructed in a completely different context almost two decades earlier [23][24][25], in the study of W -algebras and two-dimensional conformal field theories. This is yet again a manifestation of the BPS/CFT correspondence. 17 Note this is consistent with the fact that the 5d instanton partition function is a K-theoretic version of the 4d construction in equivariant cohomology. 18 In almost all the examples we will study, the UV completion will be given by string theory. 19 This continuum can be probed, for instance, by changing the sign of the 1d F.I. parameter.

Explicit Derivation of the Defect Partition Function
We now come to the explicit derivation of the quantized Seiberg-Witten geometry, for all classical gauge groups and a wide variety of matter contents.

The SU (N ) Gauge Theory
Recently, the quantized SW geometry of SU (N ) gauge theories has been an active topic of investigation. Before we review it in our formalism, let us give a brief overview of what has been done in the literature. The 5d SU (N ) Wilson loop defects we study here were first considered in [13], in the same type IIA string background as us. Our presentation will follow closely the discussion in [29], where the instanton partition function of 5d SU (N ) SYM with a single Wilson loop insertion was computed as the Witten index of a (0, 4) quantum mechanics. The derivation of non-perturbative SD identities was carried out in [1]. A type IIB realization is given, which is one T-duality away from ours; this is consistent with the fact that the focus in that work is on a point defect in 4d N = 2 SU (N ) SYM. Our presentation is simply the K-theoretic version of the cohomological results obtained there, as can be seen after reducing our 5d instanton partition on the circle wrapped by the Wilson loop. [1] and [29] both compute the defect partition function for pure SU (N ) SYM, as well as in the presence of fundamental and adjoint matter. The S-duality of the SU (N ) Wilson loop in 5d was studied with fundamental matter in [31]. The S-duality of the Wilson loop in the presence of adjoint matter was the focus of [32]. Yet another T-dual brane configuration was given in [33], with the aim of giving a UV description to the holographic dual of AdS 3 × S 3 × S 3 × S 1 . The quantization of the various open strings in this last setup was done in [34]. Finally, a q-CFT perspective was given in [26] in the context of so-called deformed Walgebras; the non-perturbative SD equations arise there as a q-deformed Ward identities in two dimensions. Quiver gauge theories were studied in this context [1,9,10]. The case of a single SU (N ) gauge group corresponds to the case g = A 1 in the quiver denomination.
Our starting point is type IIA string theory in flat space. We compactify the x 0 -direction and introduce the following branes: The effective theory on the D4 branes is a 5d U (N ) gauge theory on S 1 ( R) × R 4 , called T 5d . Separating the D4 branes along the x 9 -direction correponds to giving a non-zero vector multiplet vev Φ 9 = 0, which describes the Coulomb branch of the theory. The D4 brane realizes a 1/2-BPS Wilson loop wrapping the x 0 -circle and which sits at the origin of R 4 . Pulling the D4 brane a distance M away in the x 9 -direction, there are now open strings with nonzero tension between the D4 and D4 branes. These are heavy fermion probes, with mass proportional to the distance M . The gauge group on the D0 branes is G = U (k). We put the theory on the 5d Ω-background S 1 (R) × R 4 1 , 2 . Finally, we find it convenient to introduce the following fugacities, where 1 , 2 , 3 , and 4 are the chemical potentials associated to the rotations of the R 2 12 , R 2 34 , R 2 56 , and R 2 78 planes, respectively. The instanton partition function is the index of the D0 brane quantum mechanics. In our context, it reads 20 : For the sake of brevity, we only made the dependence on the defect fugacity M explicit in writing the partition function Z(M ). |W | is the order of the Weyl group of G, and the various factors are given by the 1-loop determinants of the quantum mechanical fields, coming from the quantization of the strings stretching between the various branes.
strings Multiplets In this paper, we set N = 1 throughout. t-hyper denotes a (0, 4) twisted hypermultiplet. All multiplets in the second column are (0, 4) multiplets, with the exception of the last row, which is an honest (0, 2) Fermi multiplet; still, it can be made compatible with (0, 4) supersymmetry [33], as is the case here.
The 1-loop determinants for the D0/D0 strings give .
In order to deduce the contribution of D0/D4 strings, we can make use of a symmetry of the brane configuration Table 1. Namely, under the exchange of the coordinates x 1,2,3,4 ↔ x 5,6,7,8 , D4 and D4' branes get swapped, while the D0 branes are unaffected. Hence, we can write the D0/D4 contribution from the D0/D4 one, after simply exchanging 1 , 2 with 3 , 4 , meaning We obtain The quantization of the D4/D4 string was first carried out in our context in [17] 21 . The corresponding multiplet is Fermi, in a bifundamental representation of G × G , and we obtain In the above, we defined sh(x) ≡ 2 sinh(x/2) and products over all signs inside an argument should be considered; for example, sh For completeness, let us also write the index in terms of field theory quantities, as Z This identification has a potential caveat, though. The D0 brane quantum mechanical index typically counts states that are present in the UV complete string theory but not necessarily part of the low energy QFT, which was pointed out in [21]. This manifests itself as extra spurious contributions to the index, which we denote collectively as Z extra . We will need to normalize the partition function by this factor when it is nontrivial to get sensible results. We will be more explicit in what follows.
--Pure case --Let us fist study the case of a pure SU (N ) gauge theory. This can be done by decoupling the adjoint mass m → ∞. In the present case, integration commutes with the limit (as long as we properly rescale the instanton counting parameter with m), so we can simply take the limit inside the integrand, which amounts to setting Z (k) adjoint → 1: In order to derive non-perturbative SD equations, we apply our program and study the pole structure of (3.12) in the defect fugacity M , as we argued in detail in section 2.4. Namely, for a fixed instanton number k, let M k be the set of poles selected by the JK-residue prescription in the defect partition function Z(M ). Meanwhile, let M k be the set poles selected by the JK-residue prescription in the partition function 1 , that is to say in the absence of the factor Z (k) def ect . The set M k happens to contain exactly one element depending on M , for all k. To see this, first note that M k contains at least one element depending on M , since one of the k contours, say the I-th one, is required to pick up exactly one pole coming from the defect factor Z Crucially, we take the contour prescription in the definition of the Y -operator vev to only enclose poles in the set M k , meaning it ignores the pole at φ I = M + + . Then, we find that the partition function can be expressed in terms of these Y -operators, as a sum of exactly two terms: .
The meaning of this expression is as follows: The first term on the right-hand side encloses 22 Though this fact is not needed in our analysis, we mention here in passing that the set of poles M k which do not depend on M have a famous classification in terms of N -colored Young diagrams − → µ = {µ 1 , µ 2 , . . . , µ N } such that − → µ = k. In practice, this assigns one Young tableau per U (1) Coulomb modulus. Explicitly, the k integration variables φ1, . . . , φ k are chosen such that See [4] and [5] for details. almost all the "correct" poles in the partition function integrand, but we are missing exactly one: the extra pole at φ I − M − + = 0. The second term on the right-hand side makes up for this missing pole, and relies on a key observation: one can trade a contour enclosing this extra pole for a contour which does not enclose it, at the expense of inserting the operator Y (M + 2 + ) −1 inside the vev. This result is derived at once from the integral expression (3.12), and the Y -operator integral definition (3.15). Finally, note the presence of the parameter q in the second term; it counts exactly one instanton, to make up for the missing M -pole. After normalizing the partition function and expanding it in the (exponentiated) defect fugacity e M , we find To argue that this is the case, one can simply show that the left-hand side is pole-free in M ; for details, see for instance [29]. Furthermore, the asymptotics of Z(M ) at M → ±∞ tell us that the number of terms in the e M -series is N + 1. Therefore, (3.17) can be thought of as non-perturbative SD equations for the SU (N ) gauge theory, solved explicitly by the Y -operator vev (3.15). Taking the flat space limit 1 , 2 → 0, we recover the SW curve of 5d pure SU (N ) super Yang-Mills.
Let us study the more familiar four-dimensional limit in detail. Namely, we reintroduce the radius R explicitly in the partition function, and take the limit R → 0, leaving all fugacities fixed in the process (essentially, all sinh(x) functions simply become x, inside (3.15) and (3.12)). The SD equations become where The roots {e 1 , 2 i } characterize the vacuum, and are determined straight from the expansion of the left-hand side of (3.18) in the fugacity M . As an example, here is T 4d when N = 2, meaning G = SU (2), with Coulomb parameter a ≡ a 1 = −a 2 : Taking the flat space limit 1 , 2 → 0 of the SD equations, we obtain at once The parameters {e 0,0 i } are simply the 1 , 2 → 0 limit of the roots (3.19). Rewriting the instanton counting parameter as the dynamical scale q ≡ Λ 2N , we recover the familiar SW curve of 4d pure SU (N ): We therefore need to modify the index to account for these new Fermi multiplets in the quantum mechanics: As we explained in section 2.6, we will only restrict ourselves to a number of hypermultiplets where the 4d limit is a conformal or asymptotically free theory, and where Z Then, the matter factor does not contribute new poles, and we compute the partition function to be . (3.27) Notice the argument of the matter factor Q(M + + ) is precisely the locus of the M -dependent pole we did not consider in the first term. After normalizing the partition function and expanding it in the (exponentiated) defect fugacity e M , we find the SD equation solved by the Y -operators: Let us study the more familiar four-dimensional limit in detail. Namely, we reintroduce the radius R explicitly in the partition function, and take the limit R → 0, leaving all fugacities fixed. The SD equations become (3.30) As an example, we write T 4d when N = 2 and N f = 2, meaning G = SU (2) with 2 fundamental hypermultiplets. If the Coulomb parameter is denoted as a ≡ a 1 = −a 2 and the two masses as m 1 and m 2 , we get Taking the flat space limit 1 , 2 → 0 and reintroducing the dynamical scale Λ, we recover the familiar SW curve of 4d SU (N ) with N f fundamental hypermultiplets: We come back to our initial quantum mechanics index, (3.8), which we rewrite here for convenience: Then, we compute the partition function to be The above ". . ." stands for an infinite series in the instanton counting parameter q, because there are now an infinite number of new poles depending on the defect mass M . Namely, each term corresponds to an element of the set M k \ M k . In particular, the k = 1 term stands for the residue at the pole φ I = M + + , the only element of M 1 \ M 1 . It turns out that the infinite sum can be written combinatorially as a sum over partitions, but we will not need this fact here, and refer instead to [1,29,32] for details. The coefficient c 1 , 2 (1) is a function of the adjoint mass m, the defect fugacity M , and the Ω-background parameters only. We will give precise expressions in four-dimensional variables momentarily.
After normalizing the partition function and expanding it in the (exponentiated) defect fugacity e M , we find the SD equation solved by the Y -operators: D0/D4 is the usual partition function in the absence of defect, and D0/D4 . Taking the flat space limit 1 , 2 → 0, the above equation describes the SW geometry of 5d SU (N ) super Yang-Mills with adjoint matter.
Let us study the more familiar four-dimensional limit in detail. Namely, we reintroduce the radius R explicitly in the partition function, and take the limit R → 0, leaving all fugacities fixed. The SD equations become Taking the flat space limit 1 , 2 → 0, we recover the SW geometry of 4d SU (N ) with adjoint matter (see for instance section 5 of [36]): Having reviewed what has been studied in the literature, we will now apply our techniques to study new matter content and gauge groups.
--Symmetric matter -- Here, we provide a definition of the Witten index that does not originate from a D0 brane quantum mechanics in type IIA. Namely, we introduce symmetric matter following the field theory analysis performed in [37]. We propose the following index: Then, we compute the partition function and find The above ". . ." is an infinite series in the instanton counting parameter q. As usual, each term represents an element of the set M k \ M k . Following the JK residue prescription, here are the first poles: The k = 1 term is the residue at φ I = M + + . The (2, 1) term is the residue at This is a priori puzzling because the SW curve of SU (N ) with symmetric matter is a cubic curve. As we will see momentarily, the resolution of this paradox is that an infinite number of terms will elegantly cancel out against each other once we turn off the Ω-background. Let us study the more familiar four-dimensional limit in detail. Namely, we reintroduce the radius R explicitly in the partition function, and take the limit R → 0, leaving all fugacities fixed. The SD equations become The coefficients in (3.45) become and the right-hand side becomes (3.49) Taking the flat space limit 1 , 2 → 0, we see some remarkable simplifications: All terms at order 2 cancel out. We further find that six terms at order 3 survive the limit. Those are the ones corresponding to the following JK-poles: The sum of the corresponding coefficients at order 3 gives Looking at instanton corrections beyond order 3, we found numerically that all terms cancel in a similar fashion to the terms at order 2. We conjecture that this is a generic feature of all higher instantons contributions. We therefore conjecture the following flat space limit for the SD equations (3.47): (3.54) After introducing the dynamical scale q ≡ Λ N −2 , multiplying both sides by y(M ) 2 and rescaling, this is precisely the SW curve of T 4d (see for instance [38,39] or more recently [40]): --Symmetric and Fundamental matter --As another example, we consider a SU (N ) gauge theory with both a symmetric and N f fundamental hypermultiplets together. Deriving the quantized SW geometry simply amounts to performing the JK-residue prescription on the following integral: Evaluating the integrals, we find once again an infinite q-series in defect Y -operator vevs: The various coefficients c 1 , 2 (i) are the same as we found before, in the case with symmetric matter only. We will not repeat here the full analysis, as it is identical to that case. We will however deduce the SW geometry directly from the observation that the fundamental matter factors do not contribute new poles. Therefore, after evaluation of the residues, the argument of these factors will simply record the various M -dependent poles we enclosed. For instance, note the presence of the fundamental matter factor Q(M + + ) at k = 1, signaling that the k = 1 term has a pole at φ I = M + + . Then, to deduce the SW curve of the theory, we simply have to keep track of the poles picked up by JK at order 1 and at order 3, and take the flat space limit. The k = 3 poles were recorded above (3.52). Every set of pole there is of the form φ I = M + . . ., φ J = M + . . ., and φ K = −M − m + . . .. Remarkably, this is exactly the required pole structure to produce the SW curve of the corresponding four-dimensional theory: --Antisymmetric matter --As a final example, We briefly mention here how to proceed with antisymmetric matter. The field theory ADHM analysis in the absence of defects was carried out in [37]. We will consider the 5d uplift in the partition function integrand: The partition function is again an infinite q-series when expressed in terms of defect Y -operator vevs: The k = 1 term originates from the pole at φ I = M + + . In the 4d limit, the corresponding coefficient is given by . (3.62) Taking the flat space limit, this is indeed the one-instanton correction to SU (N ) with antisymmetric matter as obtained from usual SW theory [41]. In order to obtain the quantized geometry, we simply expand (3.61) to arbitrarily high order, following the JKresidue prescription. We then take the four-dimensional and flat space limits. An infinite number of terms should disappear in the limit, to yield a cubic SW curve in the end. We will skip the details here as the analysis is similar to the previous example.

The SO(2N ) Gauge Theory
We now construct the quantized SW geometry of gauge theories with other classical groups. The ADHM D0-brane quantum mechanics of such theories has been worked out in [21] (see also the field theory perspective of [42], and [43] in a four-dimensional setup). The challenge is twofold: first, is it possible to incorporate a Wilson loop using branes? Second, how can we make sense of the D0 brane quantum mechanics in field theory terms? When G = SO(2N ) or G = Sp(N ), we claim that we are able to construct such a defect using O8 orientifold planes, by making use of a particular symmetry of the brane setup. We further claim that we will be able to derive non-perturbative SD equations for the low energy gauge theories, and further write down the SW geometry after taking the flat space limit.
Our starting point is the same setup we considered in the SU (N ) case, but with the addition of an O8 + plane, as follows:  (2), wrapping the x 0 -circle and which sits at the origin of R 4 . Pulling the D4 brane a distance M away in the x 9 -direction, there are now open strings with nonzero tension between the D4 and D4 branes. These are heavy fermion probes, with mass proportional to the distance M . The gauge group on the D0 branes is G = Sp(k). We put the theory on the 5d Ω-background S 1 (R) × R 4 1 , 2 . Correspondingly, we introduce the same fugacities as in the SU (N ) case: where 1 , 2 , 3 , and 4 are the chemical potentials associated to the rotations of the R 2 12 , R 2 34 , R 2 56 , and R 2 78 planes, respectively.
Before we go further, it is important to point out that the O8 + plane has introduced a Romans mass in type IIA. This will affect the D4 brane theory by introducing an effective Chern-Simons term interaction. As a result, the U (1) topological charge, aka the instanton number k, receives an anomalous contribution and gets shifted. Thankfully, the quantum mechanics index is not sensitive to this shift, and we can safely proceed.
The instanton partition function is the index of the D0 brane quantum mechanics. In our context, it reads For the sake of brevity, we only made the dependence on the defect fugacity M explicit in writing the partition function Z(M ). |W | is the order of the Weyl group of G = Sp(k), and the various factors are given by the 1-loop determinants of the quantum mechanical fields, coming from the quantization of the strings stretching between the various branes.
The field content of the quantum mechanics on the D0 branes is Of interest to us is the case of a single D4 brane, N = 1. t-hyper denotes a (0, 4) twisted hypermultiplet.
The 1-loop determinants for the D0/D0 strings give . (3.65) The 1-loop determinants for the D0/D4 strings give The hard part is to find the contribution of D0/D4 strings, but thanks to a symmetry of our brane configuration and we obtain The quantization of the D4/D4 string is responsible for a Fermi multiplet, in a bifundamental representation of SO(2N ) × SO (2), which results in sh(M ± a i ) . (3.69) In the above, we used the same notation as in the SU (N ) case: we defined sh(x) ≡ 2 sinh(x/2), and products over all signs inside an argument have to be considered; for The various factors in the integrand can be written in gauge theory terms, with the caveat of factoring out possible spurious contributions Z extra present in the UV. We identify Z def ect . Then the index can also be written as --Pure case --Let us first study the case of a pure SO(2N ) gauge theory. This can be done by decoupling the adjoint mass m → ∞. As long as we properly rescale the instanton counting parameter with m, we can take the limit inside the integrand, and study the following integral: In order to derive non-perturbative SD equations, we need to know the content of the set M k \ M k , that is the set of "new" poles due solely to Z (k) def ect (M ). We find that the set grows unbounded as k increases, which tells us that Z(M ) is an infinite q-series in defect Y -operator vevs. We present the first few terms here: The set M 1 \ M 1 has two elements, which means we expect two terms at k = 1. The (1, 1) term is the residue at φ I = −M + + . The (1, 2) term is the residue at φ I = M + + . Meanwhile, the set M 2 \ M 2 has four elements, so we expect four terms at k = 2. The (2, 1) term is the residue at Let us study the more familiar four-dimensional limit in detail. Namely, we reintroduce the radius R explicitly in the partition function, and take the limit R → 0, leaving all fugacities fixed. The SD equations become We compute the coefficients in (3.75) to be and we find for the right-hand side As an example, here is T 4d when N = 2, meaning G = SO(4), with Coulomb parameters a 1 and a 2 : where from usual root/coefficient relations for a polynomial.
Taking the flat space limit 1 , 2 → 0, the partition function simplifies drastically: Remarkably, the terms at order k = 1 add up, while the ones at k = 2 cancel out. There are 42 nonzero residues at k = 3. We find that they also do not survive the flat space limit. We were able to show this phenomenon at k = 4 as well. We conjecture that this pattern continues at every instanton order, and that only the first instanton contribution k = 1 survives the flat space limit 23 . In the end, the four-dimensional SD equations (3.77) take the following form: After introducing the dynamical scale q ≡ Λ 4N −4 and rescaling, this is nothing but the SW curve of the pure 4d SO(2N ) theory (see for instance [44] or [45]): We cannot add D8 branes in the presence of an O8 + plane. Then, we will abandon the string theory picture and introduce N f fundamental hypermultiplets following a purely field theoretic route. The inclusion of fundamental matter will introduce new fermionic zero modes arising from Fermi multiplets in the bifundamental representation of Sp(k) × Sp(N f ).
We therefore need to modify the index to account for these new Fermi multiplets: The set M k \ M k is the same whether we consider a theory with or without fundamental matter (as long as it does not introduce new poles at infinity, which was our assumption to begin with). The matter function Q simply keep track of the various JK-poles in its argument. In our case, we compute at once: The coefficients c 1 , 2 (i,j) are the same as in the pure case, and the derivation of non-perturbative SD equations is identical. Taking the four-dimensional and flat space limits, we find once again that all terms at k ≥ 2 cancel out against each other, for every k we tested. Notice that the function Q(M ) is even. It follows immediately that at order k = 1, the two terms above combine into one. After introducing the dynamical scale q ≡ Λ 4N −4−2N f , we therefore obtain the SW curve of SO(2N ) with N f fundamental flavors: (M ± e 0,0 i ) . We come back to our initial quantum mechanics index, (3.70), which we rewrite here for convenience: We compute the partition function Let us study the more familiar four-dimensional limit in detail. Namely, we reintroduce the radius R explicitly in the partition function, and take the limit R → 0, leaving all fugacities fixed. The SD equations become where (3.96) Taking the flat space limit 1 , 2 → 0, we conjecture that we recover the SW geometry of 4d SO(2N ) with adjoint matter. In particular, the k = 1 term (3.91) we presented here is in perfect agreement with the first instanton correction computed in [8]. The higher corrections are identified exactly as we did above, following the JK prescription.

The SO(2N + 1) Gauge Theory
Engineering a SO(2N + 1) gauge theory from string theory can usually be done with the help of a different type of orientifold, sometimes called Op + plane. For p ≤ 5, such orientiolds can be interpreted as an Op orientifold plane with a Dp brane stuck on it, and their construction relies on the existence of a Z 2 discrete torsion associated with the RR fields of the theory. However, when p ≥ 6, no such Z 2 torsion exists, and the existence of Op planes becomes a subtle question. It turns out in particular that O8 planes do not exist [46]. Therefore, we will not rely on a stringy construction here, and provide only a field theory definition of the defect partition function. The pure 5d SO(2N + 1) instanton partition function was constructed in [21,42]. The field content of the quantum mechanics is the same as that of the SO(2N ) case, but there is a change in the 1-loop determinants. To simplify notations, let us denote by Z (k,2N ) the SO(2N ) 1-loop determinants written down in the previous section, and let us denote by Z (k,2N +1) the SO(2N + 1) 1-loop determinants.
Without an underlying brane construction, it is a priori unclear how to construct Z We propose that the defect group should be G = SO(2N ). In particular, this implies that the set of poles M k \ M k depending on the defect fugacity M will be exactly the same as in the previous section 24 . We will see that this choice enables us to make contact with SO(2N + 1) SW geometry. Even though the defect group G will be the same as we previously encountered, the 1-loop determinants are not exactly identical: we must remember that there is a classical contribution (what we previously called D4/D4 strings) resulting in a Fermi multiplet in the bifundamental representation of G × G . In our case, we can readily write the 1-loop determinant for such a SO(2N + 1) × SO(2N ) Fermi multiplet: sh(M ± a i ) . (3.97) All in all, we therefore propose the following defect partition function sh(±φ I + m) , We decouple the adjoint mass by taking the limit m → ∞, which once again commutes with integration. Then, consider the integral The partition function organizes itself as an infinite q-series of defect Y -operators, as follows: The JK residue prescription tells us that the set M 1 \ M 1 has two elements, which means we expect two terms at k = 1. We wrote them explicitly above. The (1, 1) term is the residue at φ I = −M + + . The (1,2) term is the residue at φ I = M + + .
After normalizing the partition function and expanding it in the (exponentiated) defect fugacity e M , we derive the following SD equation: Let us study the more familiar four-dimensional limit in detail. Namely, we reintroduce the radius R explicitly in the partition function, and take the limit R → 0, leaving all fugacities fixed. The SD equations become We compute the coefficients in (3.102) to be and the right-hand side comes out to be As an example, here is T 4d when N = 2, meaning G = SO (5), with Coulomb parameters a 1 and a 2 : from usual root/coefficient relations for a polynomial.
Taking the flat space limit 1 , 2 → 0, the partition function simplifies drastically: We find once again that k = 1 terms add up, while the terms at k ≥ 2 cancel out among each other (we checked this up to k = 4). We conjecture that this pattern continues at every instanton order, and that only the first instanton contribution k = 1 survives the flat space limit. In the end, the four-dimensional SD equations (3.104) take the following form: After introducing the dynamical scale q ≡ Λ 4N −2 /16 and rescaling the Y -operators by M , this is nothing but the SW curve of the pure 4d SO(2N + 1) theory (see for instance [44] or [45]): --Fundamental matter --Introducing N f fundamental hypermultiplets amounts to considering new fermionic zero modes. These will arise from Fermi multiplets in the bifundamental representation of Sp(k) × Sp(N f ). We therefore need to modify the index accordingly: The set M k \ M k is the same whether we consider a theory with or without fundamental matter (as long as it does not introduce new poles at infinity, which was our assumption to begin with). The matter function Q simply keeps track of the various JK-poles in its argument. In our case, we compute at once: The coefficients c 1 , 2 (i,j) are the same as in the pure case, and the derivation of non-perturbative SD equations is identical.
Taking the four-dimensional and flat space limits, we find once again that all terms at k ≥ 2 cancel out against each other, for every k we tested. The function Q(M ) is even. It follows immediately that at order k = 1, the two terms above combine into one. After introducing the dynamical scale q ≡ Λ 4N −2−2N f and rescaling, we land on the SW curve of SO(2N + 1) with N f fundamental flavors: We now consider the addition of adjoint matter: Following the JK-residue prescription, we compute the partition function Let us study the more familiar four-dimensional limit in detail. Namely, we reintroduce the radius R explicitly in the partition function, and take the limit R → 0, leaving all fugacities fixed. The SD equations become where (3.123) Taking the flat space limit 1 , 2 → 0, we conjecture that we recover the SW geometry of 4d SO(2N ) with adjoint matter. In particular, the k = 1 term (3.91) we presented here is in perfect agreement with the first instanton correction computed in [8]. The higher corrections are identified exactly as we did above, following the JK prescription.

The Sp(N ) Gauge Theory
We now move on to our last class of examples, the G = Sp(N ) gauge theory. We once again engineer the defect partition function as the index of a D0 brane quantum mechanics, this time in the presence of an O8 − orientifold plane 25 : The effective theory on the D4 branes is a 5d Sp(N ) gauge theory on S 1 ( R) × R 4 , called T 5d . The D4 brane realizes a 1/2-BPS Wilson loop with symmetry group Sp(1), wrapping the x 0 -circle and which sits at the origin of R 4 . Pulling the D4 brane a distance M away in the x 9 -direction, there are now open strings with nonzero tension between the D4 and D4 branes. These are heavy fermion probes, with mass proportional to the distance M . The gauge group on the D0 branes is G = O(k). We put the theory on the 5d Ω-background S 1 (R) × R 4 1 , 2 . We will rely on the same fugacity notation we have used throughout this paper:  1, 1, 1, 1, 2N, 2N ) Table 6: The field content of the quantum mechanics. O(k) = G is the D0 brane group, Sp(N ) = G is the D4 brane group and Sp(N ) = G is the D4 brane group.
Of interest to us is the case of a single D4 brane, N = 1. t-hyper denotes a (0, 4) twisted hypermultiplet.
The instanton partition function is the index of the D0 brane quantum mechanics. In our context, it reads with In the above, for the sake of brevity, we only made the dependence on the defect fugacity M explicit in writing the partition function Z(M ). |O(k) + | is the order of the Weyl group of SO(k), and |O(k) − | is the order of the Weyl group of O(k) − . Throughout the rest of this section, we will make use of the following convenient notation for the instanton number, We collected all the 1-loop determinants in the appendix (see also the work [28]).
We can write the index in terms of field theory quantities, as Z (k,±) An important comment is in order. Note that in the absence of D8 branes, we do not cancel the charge of the O8 plane, which means the dilaton runs in the direction x 9 . In fact, this will be true even when we consider fundamental matter, as we are only interested in theories that have a well-defined four-dimensional limit (conformal or asymptotically free), implying N f ≤ 4; the number of D8 branes is still too low to cancel the orientifold charge. As we already mentioned, the U (1) instanton charge will then receive an anomalous contribution, which causes a (fractional) shift to k. The quantum mechanics index we compute is not sensitive to this shift 26 , so we will safely proceed.
--Antisymmetric matter --We will start with the case of a Sp(N ) gauge theory with one antisymmetric matter multiplet, of corresponding mass m. We find it easier to study the pure gauge later, by sending the mass m → ∞ only at the very end 27 .
First, we note a new feature of this index: the k-th term in the integral (3.129) only requires k/2 integrals to be performed. For instance, the first instanton correction k = 1 is just the integrand itself. So far in this paper, every term in the Schwinger-Dyson identities stood for a residue at some pole in the set M k \ M k . However, we are claiming here that the sets M 1,± and M 1,± , for example, are empty to begin with. Then, we need to slightly modify our prescription to write down our partition function as a Laurent series in Y -operator vevs. We think that the odd-k instanton corrections should contribute a constant term to the SD equations. The question of how we should precisely normalize these terms is subtle, and we leave it to future work.
The first integral we have to perform is found at k = 2. The JK-residue prescription tells us that M 2,+ \ M 2,+ has exactly two elements; they are the pole at φ I = −M + + and the pole at φ I = M + + , respectively. Meanwhile, M 2,− and M 2,− are the empty set, so the O(2) − sector only contributes at most a constant q 2 to the SD equations. Explicit formulas get involved very quickly, so let us focus on the four-dimensional limit R → 0. Note that Z (k,+) and Z (k,−) scale differently in the limit, and only the O(k) + sector contribution survives. We therefore identify the k = 2 term in the defect partition function to be We used superscripts on the Y -operators to make explicit which O(k) sector they are defined with respect to. After normalizing the partition function and expanding it in the defect fugacity M , we derive the following four-dimensional SD equation: 132) 27 We suspect there may be some spurious contributions Zextra that do not decouple properly if we take the limit inside the integrand form the start, as we did for the other gauge groups.
where we checked up to k = 4 that D0/D4 is crucial here in obtaining this result, as it decouples contributions that are not part of the QFT and makes T 4d a finite polynomial in M .
Taking the flat space limit 1 , 2 → 0, we conjecture that we recover the SW geometry of 4d Sp(N ) with adjoint matter. In particular, the k = 2 term (3.130) in that limit is in perfect agreement with the literature [8]. The higher corrections are similarly computed as we did above, following the JK prescription.
--Pure case --The pure Sp(N ) quantized geometry can now easily be deduced from the above results, by taking the limit m → ∞ in the evaluated integrals. The q 2 term derived above becomes The right-hand side of the SD equation, T 4d (M ; {e 1 , 2 i }), is again a finite polynomial in M , which we checked up to k = 4. As an example, here is T 4d when N = 1, meaning G = Sp(1), with Coulomb parameter a: Note this is exactly what we had found for G = SU (2), see (3.21).
We now take the flat space limit 1 , 2 → 0. Following encouraging computer experiments, we conjecture that only the q and q 2 terms in the SD equation survive this limit. As we argued in the previous example, the q term should simply be a constant in the SD equations, since there are no poles associated to it. Meanwhile, the q 2 term (3.134) greatly simplifies in the limit. After introducing the dynamical scale q ≡ Λ 2N +2 , we multiply the equation by M 2 and rescale the Y -operators. All in all, we recover the SW curve of the pure Sp(N ) gauge theory (see [44] or [45]): --Fundamental matter --Finally, we consider adding N f fundamental hypermultiplets. In our brane setup, this can be done by adding N f D8 branes. As usual, we limit ourselves to a number of D8 branes such that in the four-dimensional limit, the resulting low energy gauge theory T 4d is conformal or asymptotically free. In the presence of antisymmetric matter, this translates to N f ≤ 4, while in the absence of antisymmetric matter, this means N f ≤ N + 2. In fact, we impose the stricter condition that the amount of fundamental matter should not introduce new poles at ∞ in the φ I integrals.
In the appendix, we wrote down the various D0/D8 1-loop determinants. In particular, in the O(k) + sector, Then, it is not hard to see that for k even, the argument of the matter factor is the locus of the various JK poles, while a single factor is picked up for k odd. In the context of our previous discussion, the k = 2 term (3.134) (in 4d) is then modified as Meanwhile, the k = 1 term picks up a factor N f d=1 sh(m d ). All higher k corrections follow this pattern. Taking the flat space limit, we conjecture (and checked up to k = 4) that only the k = 1 and k = 2 terms survive in the SD equation. After introducing the dynamical scale q ≡ Λ 2N +2−N f and rescaling the Y -operators, we recognize the SW curve of Sp(N ) with N f fundamental hypermultiplets (see [44] or [45]): (M ± e 0,0 i ) .
(3.141) The case with both antisymmetric and fundamental matter can be treated the same way, by including the D0/D8 1-loop we just discussed determinants in the integrand (3.125).

On the Uniqueness of the Defects
We end this paper with an open question. When G = SO(N ), Sp(N ), there is some evidence that the codimension 4 defect we studied may not be unique. Indeed, already string theory suggests at least two different UV completions of the theories: instead of using O8 planes as we did, a natural construction is to rely on the use of O4 planes, aligned in the same direction as the D4 branes; an immediate advantage is that the case G = SO(2N + 1) would now be realized in a stringy picture, since O4 planes do exist. However, quantizing the various strings may prove subtle; as we saw in the examples, the use of O8 planes enabled us to exploit a particular symmetry of the brane system, from which we easily deduced the new D0/D4 1-loop determinants due to the Wilson loop. With an O4 plane instead, such a symmetry is broken, since the D4 branes are now orthogonal to the O4 plane, while the D4 branes sit on top of it. Therefore, one would need to do more work to quantize the D0/D4 strings.
In the O8 plane setup, the number of Dirichlet-Neumann directions was equal to 4 for the D4 /O8 configuration, the same as for the D4/O8 configuration. In conclusion, we saw an enhancement of both the gauge and defect groups to the same classical group type: the groups G and G were either both orthogonal or both symplectic. We can expect a similar enhancement when using an O4 plane: now, the number of Dirichlet-Neumann directions is equal to 8 for the D4 /O4 configuration, while it is 0 for the D4/O4 configuration. The implication is once again that G and G would both see a symmetry enhancement to either an orthogonal group, or both to a symplectic group.
Even though we were able to derive the SW geometries in the O8 construction, it would then be interesting to further define (and compute, if possible) the index of the quantum mechanics on D0 branes using O4 planes, and find out if the resulting quantum geometry is distinct from ours or not.
Luigi Tizzano for useful discussions and comments at various stages of this project. JO is grateful to Chi-Ming Chang and Ori Ganor, who developed together a JK integral package in Mathematica, which was crucial for high instanton computations in this project. The research of JO is supported in part by Kwanjeong Educational Foundation and in part by the Berkeley Center of Theoretical Physics. The research of NH is supported by the Simons Center for Geometry and Physics.
A 1-loop determinants for G = Sp(N ) and G = O(k) We use the notation sh(x) ≡ 2 sinh(x/2) and ch(x) ≡ 2 cosh(x/2). Products over all signs inside an argument have to be considered, and χ = 0, 1. When d = 0, dealing with such poles is fact benign 28 . As an example, suppose N = 2, with g(φ 1 , φ 2 ) = (φ 1 − a)(φ 1 − φ 2 − b), and suppose f (φ 1 , φ 2 ) is regular at φ 1 = a and φ 1 − φ 2 = b. Then In other words, there is a pair-wise cancellation of residues. Such a phenomenon is a generic feature of d = 0 non-simple poles. Therefore, they do not contribute to the integral, even 28 See also the page 36 of [21] when singled out by the JK residue prescription, and vanish in the final instanton partition function formula. When G = SU (N ), non-simple poles with degree d > 0 can appear at high instanton number. We do not make any claims on how to deal with them in full generality, and we treated them on a case-by-case basis when we encountered them at high instanton number in this work.