Intersecting Surface Defects and Instanton Partition Functions
We analyze intersecting surface defects inserted in interacting four-dimensional supersymmetric quantum field theories. We employ the realization of a class of such systems as the infrared fixed points of renormalization group flows from larger theories, triggered by perturbed Seiberg-Witten monopole-like configurations, to compute their partition functions. These results are cast into the form of a partition function of 4d/2d/0d coupled systems. Our computations provide concrete expressions for the instanton partition function in the presence of intersecting defects and we study the corresponding ADHM model.
Intersecting Surface Defects and Instanton Partition Functions
Yiwen Pan and Wolfger Peelaers
July 15, 2019
- 1 Introduction
- 2 Higgsing and codimension two defects
- 3 Intersecting defects in theory of free hypermultiplets
- 4 Intersecting surface defects in interacting theories
- 5 Instanton partition function and intersecting surface defects
- 6 Discussion
- A Special functions
- B The and SQCDA partition function
- C Factorization of instanton partition function
- D Poles and Young diagrams in 3d
- E Poles and Young diagrams in 2d
Half-BPS codimension two defect operators form a rich class of observables in supersymmetric quantum field theories. Their vacuum expectation values, as those of all defect operators, are diagnostic tools to identify the phase of the quantum field theory [1, 2, 3]. Various quantum field theoretic constructions of codimension two defects have been proposed and explored in the literature, see for example the review . First, one can engineer a defect by defining a prescribed singularity for the gauge fields (and additional vector multiplet scalars) along the codimension two surface, as in . Second, a defect operator can be constructed by coupling a quantum field theory supported on its worldvolume to the bulk quantum field theory. The coupling can be achieved by gauging lower-dimensional flavor symmetries with higher-dimensional gauge fields and/or by turning on superpotential couplings. Third, a codimension two defect in a theory can be designed in terms of a renormalization group flow from a larger theory triggered by a position-dependent, vortex-like Higgs branch vacuum expectation value [6, 7].111The gauged perspective of  is equivalent to considering sectors with fixed winding in a ‘Higgs branch localization’ computation. See [8, 9, 10, 11, 12, 13, 14, 15, 16] for such computations in various dimensions. Naturally, some defects can be constructed in multiple ways. Nevertheless, it is of importance to study all constructions separately, as their computational difficulties and conceptual merits vary. Such study is helped tremendously by the fact that when placing the theory on a compact Euclidean manifold, all three descriptions are, in principle, amenable to an exact analysis using localization techniques. See  for a recent comprehensive review on localization techniques.
The M-theory construction of four-dimensional supersymmetric theories of class (of type ) allows one to identify the class of concrete defects of interest to this paper: adding additional stacks of M2-branes ending on the main stack of M5-branes can introduce surface defects in the four-dimensional theory. The thus obtained M2-brane defects are known to be labeled by a representation of . In , the two-dimensional quiver gauge theory residing on the support of the defect and its coupling to the bulk four-dimensional theory were identified in detail. In fact, for the case of defects labeled by symmetric representations two different coupled systems were proposed. For the purposes of this paper, it is important to remark that one of these descriptions can alternatively be obtained from the third construction described in the previous paragraph.222The fact the application of this Higgsins prescription introduces M2-brane defects labeled by symmetric representations was understood in the original paper , see for example also .
Allowing for simultaneous insertions of multiple half-BPS defects, intersecting each other along codimension four loci, while preserving one quarter of the supersymmetry, enlarges the collection of defects considerably and is very well-motivated. Indeed, in  it was conjectured and overwhelming evidence was found in favor of the statement that the squashed four-sphere partition function of theories of class in the presence of intersecting M2-brane defects, wrapping two intersecting two-spheres, is the translation of the insertion of a generic degenerate vertex operator in the corresponding Liouville/Toda conformal field theory correlator through the AGT dictionary [22, 23], extending and completing [19, 24]. Note that such defects are labeled by a pair of representations , which is precisely the defining information of a generic degenerate vertex operator in Liouville/Toda theory.333A generic degenerate momentum reads , in terms of the highest weight vectors of irreducible representations and respectively, and parametrizes the Virasoro central charge.
In , the insertion of intersecting defects was engineered by considering a coupled 4d/2d/0d system. In this description, the defect is engineered by coupling quantum field theories supported on the respective codimension two worldvolumes as well as additional degrees of freedom residing at their intersection to each other and to the bulk quantum field theory. The precise 4d/2d/0d coupled systems describing intersecting M2-brane defects were conjectured. As was also the case for a single defect, intersecting defects labeled by symmetric representations can be described by two different coupled systems.
A localization computation, performed explicitly in , allows one to calculate the squashed four-sphere partition function of such system.444See also  for a localization computation in the presence of a single defect. Let denote the four-dimensional theory and let denote two-dimensional theories residing on the defects wrapping the two-spheres and , which intersect each other at the north pole and south pole. The full partition function then takes the schematic form
where the factors denote the product of the classical action and one-loop determinant of the theory placed on the manifold (in their Coulomb branch localized form). Furthermore, are the one-loop determinants of the degrees of freedom at the two intersection points respectively, and are two copies of the instanton partition function, one for the north pole and one for the south pole, describing instantons in the presence of the intersecting surface defects spanning the local coordinate planes in . In  the focus was on the already very rich dynamics of 4d/2d/0d systems without four-dimensional gauge fields, thus avoiding the intricacies of the instanton partition functions. In this paper we aim at considering intersecting defects in interacting four-dimensional field theories and addressing the problem of instanton counting in the presence of such defects.555By taking one of the intersecting defects to be trivial, one can always simplify our results to the case of a single defect. In  an extensive study was performed of the squashed four-sphere partition function of theories of free hypermultiplets in the presence of a single defect.
Our approach will be, alternative to that in , to construct theories in the presence of intersecting M2-brane defects labeled by symmetric representations using the aforementioned third strategy, i.e., by considering a renormalization group flow from a larger theory triggered by a position-dependent vacuum expectation value with an intersecting vortex-like profile.666To be more precise, the configuration that triggers the renormalization group flow is a solution to the (perturbed) Seiberg-Witten monopole equations , see . When the theory is a Lagrangian theory on , this Higgsing prescription offers a straightforward computational tool to calculate the partition function of in the presence of said intersecting defects. In more detail, it instructs one to consider the residue of a certain pole of the partition function , which can be calculated by considering pinching poles of the integrand of the matrix integral computing . The result involves intricate sums over a restricted set of Young diagrams, which we subsequently cast in the form of a coupled 4d/2d/0d system as in (1.1), by reorganizing the sums over the restricted diagrams into the integrals over gauge equivariant parameters and sums over magnetic fluxes of the partition functions of the two-dimensional theories . This step heavily relies on factorization properties of the summand of instanton partition functions, which we derive in appendix C, when evaluated at special values of their gauge equivariant parameter. More importantly, we obtain concrete expressions for the instanton partition function, computing the equivariant volume of the instanton moduli space in the presence of intersecting codimension two singularities, and their corresponding ADHM matrix model.
The main result of the paper, thus obtained, is the -partition function of a four-dimensional gauge theory with fundamental and antifundamental hypermultiplets,777While there is no distinction between a fundamental and antifundamental hypermultiplet, it is a useful terminology to keep track of the respective quiver gauge theory nodes. We choose to call the right/upper node of each link the fundamental one. i.e., SQCD, in the presence of intersecting M2-brane surface defects, labeled by and -fold symmetric representations respectively. It takes the form (1.1) and can be found explicitly in (4.13). To be more precise, the coupled system we obtain involves chiral multiplets as zero-dimensional degrees of freedom, i.e., it coincides with the one described in conjecture 4 of  with four-dimensional SQCD. The left subfigure in figure 1 depicts the 4d/2d/0d coupled system under consideration.
We derive the instanton partition function in the presence of intersecting planar surface defects and find it to take the form
where we omitted all gauge and flavor equivariant parameters. It is expressed as the usual sum over -tuples of Young diagrams. The summand contains the new fctors , which can be found explicitly in (4.17), capturing the contributions to the instanton counting of the additional zero-modes in the presence of intersecting surface defects, in addition to the standard factors , and describing the contributions from the vector multiplet and hypermultiplets. The coefficient of of the above result can be derived from the ADHM model for -instantons depicted in the right subfigure of figure 1. We have confirmed this ADHM model by analyzing the brane construction of said instantons, see section 5 for all the details. In section 6 we present conjectural generalizations of the instanton counting in the case of generic intersecting M2-brane defects.
The paper is organized as follows. We start in section 2 by briefly recalling the Higgsing prescription to compute squashed sphere partition functions in the presence of (intersecting) M2-brane defects labeled by symmetric representations. We also present its brane realization. In section 3 we implement the prescription for the case where is a four- or five-dimensional theory of free hypermultiplets placed on a squashed sphere. The vacuum expectation value in of intersecting M2-brane defects on the sphere has been computed in  from the point of view of the 4d/2d/0d or 5d/3d/1d coupled system and takes the form (1.1) (without the instanton contributions). For the case of symmetric representations, we reproduce this expression directly, and provide a derivation of a few details that were not addressed in . We notice that the superpotential constraints of the coupled system on the parameters appearing in the partition function are reproduced effortlessly in the Higgsing computation thanks to the fact that they have a common origin in the theory , which in this case is SQCD. These relatively simple examples allow us to show in some detail the interplay of the various ingredients of the Higgsed partition function of theory , and how to cast it in the form (1.1). In section 4 we turn our attention to inserting defects in four-dimensional SQCD. We apply the Higgsing prescription to an gauge theory with bifundamental hypermultiplets and for each gauge group an additional fundamental hypermultiplets, and cast the resulting partition function in the form (1.1). As a result we obtain a sharp prediction for the instanton partition function in the presence of intersecting surface defects. This expression provides concrete support for the ADHM matrix model that we obtain in section 5 from a brane construction. We present our conclusions and some future directions in section 6. Five appendices contain various technical details and computations.
2 Higgsing and codimension two defects
In this section we start by briefly recalling the Higgsing prescription to compute the partition function of a theory in the presence of (intersecting) defects placed on the squashed four/five-sphere [6, 7]. We also consider the brane realization of this prescription, which provides a natural bridge to the description of intersecting surface defects in terms of a 4d/2d/0d (or 5d/3d/1d) coupled system as in .
2.1 The Higgsing prescription
We will be interested in four/five-dimensional quantum field theories with eight supercharges.101010The localization computations we will employ throughout this paper rely on a Lagrangian description, but the Higgsing prescription is applicable outside the realm of Lagrangian theories. We will restrict attention to (Lagrangian) four-dimensional supersymmetric quantum field theories of class and their five-dimensional uplift. Let us for concreteness start by considering four-dimensional supersymmetric theories. Consider a theory whose flavor symmetry contains an factor, and consider the theory of free hypermultiplets, which has flavor symmetry . By gauging the diagonal subgroup of the flavor symmetry factor of the former theory with one of the factors of the latter theory, we obtain a new theory . As compared to , the theory has an extra factor in its flavor symmetry group. We denote the corresponding mass parameter as
The theory can be placed on the squashed four-sphere ,111111We consider defined through the embedding equation in five-dimensional Euclidean space with coordinates in terms of parameters with dimension of length. The squashing parameter is defined as . The isometries of are given by , which act by rotating the and plane respectively. The fixed locus of is a squashed two-spheres: and, similarly, the fixed locus of is . The two-spheres and intersect at their north pole and south pole, i.e., the points with coordinates and . and its partition function can be computed using localization techniques [27, 28]. Let us denote the supercharge used to localize the theory as . Its square is given by
where are generators of the isometries of (see footnote 11), is the Cartan generator and are the Cartan generators of the flavor symmetry algebra. The coefficients are mass parameters rescaled by , where and are two radii of the squashed sphere (see footnote 11), to make them dimensionless. Localization techniques simplify the computation of the partition function to the calculation of one-loop determinants of quadratic fluctuations around the localization locus given by arbitrary constant values for , the imaginary part of the vector multiplet scalar of the total gauge group.121212More precisely, this is the “Coulomb branch localization” locus. Alternatively, one can perform a “Higgs branch localization” computation, see [15, 16]. The final result for the partition function of the theory is then
where denotes the classical action evaluated on the localization locus, is the one-loop determinant and are two copies of the Nekrasov instanton partition function [29, 30], capturing the contribution to the localized path integral of instantons residing at the north and south pole of .
In [6, 7], it was argued, by considering the physics at the infrared fixed point of the renormalization group flow triggered by a position dependent Higgs branch vacuum expectation value for the baryon constructed out of the hypermultiplet scalars, which carries charges and , that the partition function necessarily has a pole when
Moreover, the residue of the pole precisely captures the partition function of the theory in the presence of M2-brane surface defects labeled by -fold and -fold symmetric representations respectively up to the left-over contribution of the hypermultiplet that captures the fluctuations around the Higgs branch vacuum. These defects wrap two intersecting two-spheres the fixed loci of .
The pole at (2.3) of finds its origin in the matrix integral (2.2) because of poles of the integrand pinching the integration contour. To see this, let us separate out the gauge group that gauges the free hypermultiplet to , and split accordingly: where is the vector multiplet scalar of the full gauge group of theory and the vector multiplet scalar. We can then rewrite (2.2) as
The first factor in the second line is the one-loop determinant of the vector multiplet, while the second factor is the contribution of the extra hypermultiplets, organized into fundamental hypermultipets.131313See appendix A for the definition and some useful properties of the various special functions that are used throughout the paper. Here denote the mass parameters associated to the flavor symmetry (with ). The integrand of the -integral has poles (among many others) located at
where denotes a permutation of variables. These poles arise from the one-loop determinant of the extra hypermultiplets. When the mass parameter takes the value of (2.3), they pinch the integration contour if
since we only have independent integration variables. Note that the residue of the pole of at (2.3) is equal to the sum over all partitions of in (2.6) of the residue of the -integrand of at the pole position (2.5) when treating the as independent variables.141414Upon gauging the additional flavor symmetry and turning on a Fayet-Iliopoulos parameter, which coincides with the gauged setup of [6, 7], the residues of precisely these poles were given meaning in the “Higgs branch localization” computation of  in terms of Seiberg-Witten monopoles.
A similar analysis can be performed for five-dimensional theories. The theory can be put on the squashed five-sphere ,151515The squashed five-sphere is given by the locus in satisfying (2.7) Its isometries are which act by rotations on the three complex planes respectively. The fixed locus of is the squashed three-sphere while the fixed locus of is the circle The notation indicates that it appears as the intersection of the three-spheres and A convenient visualization of the five-sphere and its fixed loci under one or two of the isometries is as a -fibration over a solid triangle, where on the edges one of the cycles shrinks and at the corners two cycles shrink simultanously. and its partition function can again be computed using localization techniques [31, 32, 33, 34, 35, 36]. The localizing supercharge squares to
where are the generators of the isometry of the squashed five-sphere (see footnote 15). The localization locus consists of arbitrary constant values for the vector multiplet scalar , hence the partition function reads
One can argue that has a pole at
whose residue computes the partition function of in the presence of codimension two defects labeled by -fold symmetric representations and wrapping the three-spheres obtained as the fixed loci of the isometries (see footnote 15), respectively. These three-spheres intersect each other in pairs along a circle. Again, this pole arises from pinching the integration contour by poles of the one-loop determinant of the hypermultiplets located at
if . The residue of at the pole given in (2.10) equals the sum over partitions of the integers of the residue of the integrand at the pole position (2.11) with the treated as independent variables.161616In , these residues were interpreted as the contribution to the partition function of K-theoretic Seiberg-Witten monopoles.
2.2 Brane realization
To sharpen one’s intuition of the Higgsing prescription outlined in the previous subsection, one may look at its brane realization . Consider a four-dimensional gauge theory described by the linear quiver and corresponding type IIA brane configuration171717The branes in this figure as well as those in figure 2 and the following discussion span the following dimensions: NS5 — — — — — — D4 — — — — — D2 — — — D2 — — — D0 —
Gauging in a theory of hypermultiplets amounts to adding an additional NS5-brane on the right end of the brane array. The Higgsing prescription of the previous subsection is then trivially implemented by pulling away this additional NS5-brane (in the 10-direction of footnote 17), while suspending D2 and D2-branes between the displaced NS5-brane and the right stack of D4-branes, see figure 2.
Various observations should be made. First of all, the brane picture in figure 2 was also considered in  to describe intersecting M2-brane surface defects labeled by and -fold symmetric representations respectively. Its field theory realization is described by a coupled 4d/2d/0d system, described by the quiver in figure 3 (see ).
Note that the two-dimensional theories, residing on the D2 and D2-branes, are in their Higgs phase, with equal Fayet-Iliopoulos parameter proportional to the distance (in the 7-direction) between the displaced NS5-brane and the next right-most NS5-brane. Before Higgsing, this distance was proportional to the inverse square of the gauge coupling of the extra gauge node:
In particular, the Higgsing prescription will produce gauge theory results in the regime where is positive, and where the defect is inserted at the right-most end of the quiver. In this paper we will restrict attention to this regime. Note however that sliding the displaced NS5-brane along the brane array in figure 2 implements hopping dualities [37, 19] (see also [38, 39]), which in the quiver gauge theory description of figure 3 translate to coupling the defect world volume theory to a different pair of neighboring nodes of the four-dimensional quiver, while not changing the resulting partition function.
In , a first-principles localization computation was performed to calculate the partition function of the coupled 4d/2d/0d system when placed on a squashed four-sphere, with the defects wrapping two intersecting two-spheres the fixed loci of , in the case of non-interacting four-dimensional theories. Our aim in the next section will be to reproduce these results from the Higgsing point of view. When the four-dimensional theory contains gauge fields, the localization computation needs as input the Nekrasov instanton partition function in the presence of intersecting planar surface defects, which modify non-trivially the ADHM data. The Higgsing prescription does not require such input, and in section 4 we will apply it to SQCD. This computation will allow us to extract the modified ADHM integral.
The brane realization of figure 2 already provides compelling hints about how the ADHM data should be modified. In this setup, instantons are described by D0-branes stretching between the NS5-branes. Their worldvolume theory is enriched by massless modes (in the Coulomb phase, i.e., when ), if any, arising from open strings stretching between the D0-branes and the D2 and D2-branes. These give rise to the dimensional reduction of a two-dimensional chiral multiplet to zero dimensions, or equivalently, the dimensional reduction of a two-dimensional chiral multiplet and Fermi multiplet. We will provide more details about the instanton counting in the presence of defects in section 5. Our Higgsing computation of section 4 will provide an independent verification of these arguments.
3 Intersecting defects in theory of free hypermultiplets
In this section we work out in some detail the Higgsing computation for the case where is a theory of free hypermultiplets. We will find perfect agreement with the description of intersecting M2-brane defects labeled by symmetric representations in terms of a 4d/2d/0d (or 5d/3d/1d) system. Our computation also provides a derivation of the Jeffrey-Kirwan-like residue prescription used to evaluate the partition function of the coupled 4d/2d/0d (or 5d/3d/1d) system, and of the flavor charges of the degrees of freedom living on the intersection. In the next section we will consider the case of interacting theories .
3.1 Intersecting codimension two defects on
As a first application of the Higgsing prescription of the previous section, we consider the partition function of a theory of free hypermultiplets on in the presence of intersecting codimension two defects wrapping two of the three-spheres fixed by the isometry (see footnote 15, and also figure 4),
say and . Our aim will be to cast the result in the manifest form of the partition function of a 5d/3d/1d coupled system, as in . We consider this case first since the fact that the intersection has a single connected component is a simplifying feature that will be absent in the example of in the next subsection.
3.1.1 partition function of
Our starting point, the theory , is described by the quiver
That is, it is an gauge theory with fundamental and anti-fundamental hypermultiplets, i.e., SQCD.181818Recall our terminology of footnote 7. The -partition function of is computed by the matrix integral (2.9) [40, 41, 31, 32, 33, 34, 35, 36]
The explicit expression for the classical action is given by
while the one-loop determinant is the product of the one-loop determinants of the vector multiplet, the fundamental hypermultiplets and the antifundamental hypermultiplets:
written in terms of the triple sine function. Here we used the notation . Note that we did not explicitly separate the masses for the flavor symmetry, but instead considered masses. Finally, there are three copies of the K-theoretic instanton partition function, capturing contributions of instantons residing at the circles kept fixed by two out of three isometries. Concretely, one has
of a product over the contributions of vector and matter multiplets:
Here we have omitted the explicit dependence on in all factors . The instanton counting parameter is given by and denotes the total number of boxes in the -tuple of Young diagrams. The expression for reads
while those of and are given in (C.2)-(C.3) in appendix C.191919In appendix C we have simultaneously performed manipulations of four-dimensional and five-dimensional instanton partition functions, which is possible after introducing the generalized factorial with respect to a function , defined in appendix A.1, with in four and five dimensions given in (C.1). Note that the masses that enter in (3.7) are slightly shifted (see ):
3.1.2 Implementing the Higgsing prescription
As outlined in the previous section, to introduce intersecting codimension two defects wrapping the three-spheres and and labeled by the -fold and -fold symmetric representation respectively, we should consider the residue at the pole position (2.11) with (and hence for all )202020Recall that we have regrouped the mass for the flavor symmetry and those for the flavor symmetry into masses.
while treating as independent variables, and sum over all partitions of and of . As before, is a permutation of which we take to be, without loss of generality, . At this point let us introduce the notation that “” means evaluating the residue at the pole (3.10) and removing some spurious factors. As we aim to cast the result in the form of a matrix integral describing the coupled 5d/3d/1d system, we try to factorize all contributions accordingly in pieces depending only on information of either three-sphere or . As we will see, the non-factorizable pieces nicely cancel against each other, except for a factor that will ultimately describe the one-dimensional degrees of freedom residing on the intersection.
It is straightforward to work out the residue at the pole position (3.10). The classical action (3.2) and the one-loop determinant (3.3) become, using recursion relations for the triple sine functions (see (A.8)),212121Here we omitted on the right-hand side the left-over hypermultiplet contributions mentioned in the previous section as well as the classical action evaluated on the Higgs branch vacuum at infinity, i.e., on the position-independent Higgs branch vacuum.
Let us unpack this expression a bit. First, is the one-loop determinant of free hypermultiplets, which constitute the infrared theory It reads
Note that the masses of the free hypermultiplets, represented by a two-flavor-node quiver, are . Recall that while . Second, we find the classical action and one-loop determinant of squashed three-sphere partition functions of a three-dimensional supersymmetric gauge theory with fundamental and antifundamental chiral multiplets and one adjoint chiral multiplet, i.e., the quiver gauge theory
We will henceforth call this theory ‘SQCDA.’222222Note that the rank of the gauge group is the rank of one of the symmetric representations labeling the defects supported on the codimension two surfaces, or in other words, it can be inferred from the precise coefficients of the pole of the partition function, see (2.10). These quantities are in their Higgs branch localized form,232323The squashed three-sphere partition function of a theory can be computed using two different localization schemes. The usual “Coulomb branch localization” computes it as a matrix integral of the schematic form [43, 44, 45, 46] while a “Higgs branch localization” computation brings it into the form [10, 11] Here the sum runs over all Higgs vacua HV and the subscript denotes that the quantity is evaluated in the Higgs vacuum HV. Furthermore, one needs to include two copies of the K-theoretic vortex partition function . The two expressions for are related by closing the integration contours in the former and summing over the residues of the enclosed poles. In the main text the theory will always be SQCDA and hence we omit the superscripted label. Note that for SQCDA, the sum over vacua is a sum over partitions of the rank of the gauge group. See appendix B for all the details. hence the additional subscript indicating the Higgs branch vacuum, i.e., the partition . Their explicit expressions can be found in appendix B.2. The Fayet-Iliopoulos parameter , the adjoint mass , and the fundamental and antifundamental masses entering the three-dimensional partition function on are identified with the five-dimensional parameters as follows, with ,
Note that the relation on the mass translates into a relation on the mass of the fundamental chiral multiplets. Finally, both the classical action and the one-loop determinant produce extra factors which cannot be factorized in terms of information depending only on or ,
where captures the non-factorizable factors from the antifundamental one-loop determinant, while captures those from the vector multiplet and fundamental hypermultiplet one-loop determinants, which can be found in (C.21)-(C.22). These factors will cancel against factors produced by the instanton partition functions, which we consider ne