#### Moduli spaces of SO(8) instantons on smooth ALE spaces as Higgs branches of 4d \( \mathcal{N} \) = 2 supersymmetric theories

Yuji Tachikawa
0
0
Department of Physics, Faculty of Science, University of Tokyo
, Bunkyo-ku,
Tokyo 133-0022
,
Japan Institute for the Physics and Mathematics of the Universe, University of Tokyo
, Kashiwa, Chiba 277-8583,
Japan
The worldvolume theory of D3-branes probing four D7-branes and an O7-plane on C2/Z2 is given by a supersymmetric USp USp gauge theory. We demonstrate that, at least for a particular choice of the holonomy at infinity, we can go to a dual description of the gauge theory, where we can add a Fayet-Iliopoulos term describing the blowing-up of the orbifold to the smooth ALE space. This allows us to express the moduli space of SO(8) instantons on the smooth ALE space as a hyperkahler quotient of a flat space times the Higgs branch of a class S theory. We also discuss a generalization to C2/Z2n, and speculate how to extend the analysis to bigger SO groups and to ALE spaces of other types.
1 Introduction 2 3 4
SO(8) instantons on C2g/Z2
2.1 Basic mathematical facts
2.2 Analysis using string dualities
2.2.1 T-duality to the D4-D6-O6 system
2.2.2 Re-interpretation as a class S construction
2.2.3 Identification of the new contribution
2.3 Summary of the procedure
2.4 A more field-theoretical approach
2.4.1 The infrared dual of the USp(2k + 2) theory
2.4.2 The infrared dual of the total system
SO(8) instantons on C2g/Z2n
3.1 String-theoretic analysis
3.2 Field-theoretic analysis Conclusions and speculations
The objective of this paper is to revisit the problem of the gauge theory description of
D3branes probing a D7-O7 system on smooth asymptotically-locally-Euclidean (ALE) spaces.
Let us first recall what the difficulty was.
It is by now well-known that the open-string description of k Dp-branes probing flat
D(p + 4)-branes, with or without O(p + 4)-plane, realizes the ADHM construction of the
moduli space of instantons. When the (p + 4)-branes are put on an orbifold C2/Z2, the
world-volume theory of Dp-branes becomes a quiver gauge theory. Without any O(p +
4)plane in place, the gauge group is of the form U(k) U(k0), and the blow-up parameters of
the orbifold are given by the FI terms of the theory. This reproduces Kronheimer-Nakajima
construction [1, 2] of unitary instantons on the ALE space C^2/Z2, as first shown in [3].
There is a problem with an O(p + 4)-plane, however. The gauge group of the system,
which can be found by quantizing open strings on the orbifold, is now USp(2k) USp(2k0),
for which we cannot add any FI terms. Still, it is clear geometrically that we can still
blow up the orbifold. We would like to understand this process better in the gauge theory
language and to find a way to describe the moduli space of orthogonal instantons on smooth
ALE spaces.
U(k)U(k) gauge theory
U(k)U(k) gauge theory + FI term
USp(2k)USp(2k) gauge theory
When p = 2, the gauge theory is three dimensional with N = 4 supersymmetry. FI
terms cannot be added directly to the USp USp gauge theory, but we can use the 3d
mirror description (see e.g. [46] for recent discussions), where the blow-up parameters are
visible as hypermultiplet mass terms. The moduli space of orthogonal instantons on a
smooth ALE space is then given as the quantum-corrected Coulomb branch of this mirror
theory. In general, describing the Coulomb branch of 3d N = 4 theories is a difficult
problem, and therefore this construction does not yet tell us much about the moduli space
of orthogonal instantons on the smooth ALE spaces.
When p = 5, the gauge theory is six dimensional with N = (1, 0) supersymmetry.
Here, as noticed first in [7], the blow-up parameters are a part of hypermultiplets involved
in the transition between the tensor branch and the Higgs branch, about which not much
is understood yet either.
In this paper, we take p = 3, so that the gauge theory is four dimensional with
N = 2 supersymmetry. In the last few years, a significant progress has been made in the
understanding of the duality of such systems. We will see that, at least for SO(8) instantons
with a particular holonomy at infinity, we can go to a dual description of the original gauge
theory, where we can add appropriate FI terms. This method will give a description of the
moduli space of such instantons as a hyperkahler quotient of a flat space times the Higgs
branch of a particular class S theory. When the instanton number is sufficiently small, the
moduli space reduces to a hyperkahler quotient of a flat space times a nilpotent orbit.
In the next section, we study SO(8) instantons on the blown-up ALE space C^2/Z2.
We provide two complimentary approaches leading to the same conclusion: one uses the
embedding to string theory directly, and the other uses a field-theoretical infrared duality
recently discussed in [8, 9]. In section 3, a generalization to C^2/Z2n will be described. We
conclude with a discussion in section 4, where we speculate how we can extend the analysis
to larger SO groups and to ALE spaces of other types.
SO(8) instantons on C^2/Z2
Basic mathematical facts
Let us first recall the quiver description of the moduli space of SO(N ) instantons on the
orbifold C2/Z2, with the holonomy at infinity given by
diag(+ + + )
|N+{tzimes} |N{tzimes}
where N = N+ + N. Note that N is even.1
The moduli space is given by the Higgs branch of a USp(2k+)USp(2k) gauge theory,
with N+, N fundamental half-hypermultiplets for the first and the second gauge factors,
and a bifundamental hypermultiplet of the two USp factors.2 The holonomy at the origin
can be computed by a method explained e.g. in appendix B of [11], and is given by
diag(+ + + ),
|N+0{tzimes} |N0{tzimes}
N +0 = N+ 4(k+ k),
N 0 = N 4(k k+).
K = k + N8
where k is an integer, and the dimension3 of the moduli space is
(N 2)K N+8N (2.5)
where the second term is the contribution from the invariant at the asymptotic boundary.
The second Stiefel-Whitney class of the bundle is determined by the holonomy at infinity,
and therefore does not give additional topological data. For more explanations of the facts
in this paragraph, see e.g. section 4 of [12].
When N is a multiple of four, we see that the dimensions of the moduli spaces on
the orbifold and the smooth ALE space agree when we take k+ = k, k = k + N/4.
For unitary instantons on the ALE space, the difference k+ k controls the first Chern
1A readable account of the moduli space of SU(N ) instantons on the orbifold C2/ can be found in [2, 10].
2Our convention is that USp(2) = SU(2).
3We always refer to quaternionic dimensions in this paper.
class of the bundle, given by 2(k+ k) + N, on the smooth space [1]. For orthogonal
instantons, on the contrary, the difference k+ k does not correspond to any data of the
gauge configuration on the smooth ALE space.
Analysis using string dualities
T-duality to the D4-D6-O6 system
We are going to study this system using D3-branes probing N D7-branes and an O7-plane
on the orbifold or the smooth ALE space. For a techincal reason, we choose N and k so
that the resulting four-dimensional supersymmetric gauge theory is conformal or slightly
asymptotically free. This is so that we can apply the field theoretical dualities found in the
last few years, starting in [13]. This choice also facilitates the analysis using branes, since
the bending of NS5-branes will be (almost) absent. Concretely, we choose N+ = N = 4,
and set
(k+, k) = (k, k), or (k+, k) = (k, k + 1)
Note that N+ + N = 8 corresponds to the familiar choice where the dilation tadpole of
the O7-plane is canceled by that of the D7-branes.
We first deform the ALE space to a two-centered Taub-NUT space, around whose S1
fibers we perform the T-duality. The resulting configuration is shown in figure 2. The
spacetime is of the form R3,1 (C S1)/Z2 R3, where Z2 is the orientifolding action.
Every brane fills R3,1. In addition, the NS5-branes, the D4-branes, and the D6-branes
extend along C, S1, and R3, respectively. Each of the two O6-planes has four D6-branes
on top, corresponding to the choice N+ = N = 4.
Recall that the relative distance along R3 between the NS5-brane and its orientifold
image is the blow-up parameter of the ALE space. This can be nonzero only when the
NS5brane is on top of the O6-plane, where it meets its mirror image under the orientifolding
action, as shown in the same figure. In terms of the gauge theory describing the dynamics
of the D4-branes, this means that the blow-up parameter can only be introduced when one
of the gauge couplings is extremely strong.
Re-interpretation as a class S construction
Now, let us lift the set-up to M-theory. When k+ = k = k, this can be done very
easily, since there is no bending of the NS5-brane. The result is shown in figure 3. The
Figure 4. The change in the ultraviolet curve under the process. The symbols , are for the
puncture of type [k2] and for the simple puncture, respectively. The symbol ? is for a new type of
puncture introducing FI-like deformation.
spacetime is now of the form R3,1 (C T 2)/Z2 R3; two O6-planes become four Z2
singularities. When the vertical M5-brane (the one not wrapping the M-theory circle) is
on top of a Z2 singularity, we can separate it into two, and each piece can be moved along
R3 independently. Their relative distance is the blow-up parameter.
Let us study this process from the point of view of the class S-theory. We have 2k
M5-branes wrapping S2 ' T 2/Z2, intersected by a vertical M5-brane. Using the standard
rules [1315], we know that a Z2 singularity is a puncture of type [k2] and an intersection
with a vertical M5-brane is a simple puncture. Then, when the vertical M5-brane comes
very close to the Z2 singularity, we can go to a dual frame, as shown by a white arrow in
figure 4.
We now have a weakly-coupled dual SU(2) gauge group from a long neck. The
threepunctured sphere on the right hand side in the figure, containing a simple puncture and a
puncture of type [k2] corresponds to an empty matter content.4 Putting the vertical
M5brane on top of the Z2 singularity is to make the coupling of the dual SU(2) gauge group to
be exactly zero. This is not a continuous process in the field theory language. We therefore
represent the process of separating the M5-brane and its mirror image on the Z2 singularity
by gluing in a different sphere on the right hand side, with a new puncture representing the
separated M5-branes on the Z2 singularity. We showed this procedure by a black arrow in
figure 4. Let us call the new contribution on the right hand side of the neck as an FI fixture.
4This is true only when k > 1. When k = 1 a slight modification of the analysis is necessary, since the
three-punctured sphere on the right hand side also gives a trifundamental.
2.2.3 Identification of the new contribution
To understand what the FI fixture does, we can reduce the system close to the Z2 singularity
to Type IIA theory with a different choice of M-theory circle, such that the Z2 singularity
sits at the tip of the cigar. The result is shown in figure 5. When the vertical M5 is not
exactly on top of the singularity, the outcome of the reduction is an NS5-brane, intersecting
2k D4-branes ending on two D6-branes in equal numbers. When the vertical M5 is exactly
on top of the singularity and separated along R3, the outcome of the reduction is essentially
given by the Hanany-Witten effect: now k 1 D4-branes end on each D6-brane, and there
are in addition two semi-infinite D4-branes whose boundary condition is given by the
separation along R3. To visualize the very-weak SU(2) group, we can artificially cut the
two D4-branes by introducing four D6-branes, remembering that we need to couple the
two resulting SU(2) flavor symmetries by a gauge symmetry. With this process, we clearly
see the brane realization of the FI fixture.
Luckily, this brane set-up realizing the FI fixture was already studied in [16]. Field
theoretically, it is given by a U(1) gauge theory with two flavors with a FI term, and its
Higgs branch is just C^2/Z2. Coming back to figure 4, we replaced the empty matter content
on the right hand side with a one-dimensional Higgs branch. Therefore, the process shown
by the black arrow there adds one dimension to the Higgs branch.
Summary of the procedure
Let us summarize the process described so far in a field theoretical language, see the first
three rows of figure 6.
We start from a USp(2k) USp(2k) gauge theory, with two fundamental
hypermultiplets for each USp group and a bifundamental hypermultiplet. The dimension of
the Higgs branch is 6k. We want to make the right USp(2k) very strongly coupled.
We go to an S-dual frame on the right USp(2k) gauge group: this involves a
threepunctured sphere with a full puncture, a puncture of type [k2], and another puncture
of type [(k 1)212]. The last puncture has a flavor symmetry SU(2) associated to
the parts 12 of the last puncture, to which the SU(2) gauge multiplet couples weakly.
We take a U(1) gauge theory with two flavors with a FI term , and couple its
SU(2) flavor symmetry to the SU(2) gauge multiplet we already have. Note that the
SU(2) gauge multiplet is now infrared free. The dimension of the Higgs branch is
6k + 1. This is the correct dimension of the moduli space of SO(8) instantons with
the holonomy at infinity diag(+, +, +, +, , , , ).
The Higgs branch M k, of this final system, from our chain of string dualities, should
give (a component of) the moduli space of SO(8) instantons on the smooth ALE space
C^2/Z2 with the prescribed holonomy at infinity. Here k is the instanton number and is
the blow-up parameter of the smooth ALE space.
For general k, it is a hyperkahler quotient of a flat space times the Higgs branch of the
class S theory on a three-punctured sphere. This is unfortunately not very explicit yet.5
For k = 1 and k = 2, the construction becomes completely explicit. For k = 1, we need
to make a small modification as was mentioned in the last footnote: the rightmost SU(2)
is coupled to another three-punctured sphere. Then the theory is an SU(2) SU(2)
U(1) gauge theory, with bifundamental hypermultiplets for consecutive gauge groups, and
5Ginzburg and Kazhdan have an unpublished manuscript in which the Higgs branch of these theories
are constructed as holomorphic symplectic varieties [17].
additional two flavors for each SU(2) factor. The Higgs branch is a hyperkahler quotient of
a vector space. For k = 2, the Higgs branch of the three-puncture sphere with punctures
[14], [14] and [22] is the minimal nilpotent orbit of E7. Then the Higgs branch of the total
system is the hyperkahler quotient of the minimal nilpotent orbit of E7 times a vector
space by USp(4) SU(2) U(1). In both cases, the blow-up parameter of the ALE space
is given by the value of the moment map for U(1).
A more field-theoretical approach
This final gauge theory we arrived at after a lengthy analysis using string theory dualities
can also be directly obtained field-theoretically, starting from the choice (k+, k) = (k, k+1)
in (2.6). The gauge theory which we use as the new starting point is shown in the fourth
row of figure 6. Note that the USp(2k) gauge multiplet is infrared free with 2k + 4 flavors,
while USp(2k + 2) gauge multiplet is asymptotically free with 2k + 2 flavors.
The infrared dual of the USp(2k + 2) theory
Let us first focus on the asymptotically free part. The strongly-coupled dynamics of
USp(2k + 2) with 2k + 2 flavors was analyzed in [9], as an extension of the work [8].
Here we quote the results of [9], with additional comments on the Higgs branch.
The case k = 0 is the familiar SU(2) theory with Nf = 2. Classically, the Higgs branch
has two components, each of which is C2/Z2, joined at the origin. Quantum mechanically,
we have two singular points on the Coulomb branch, at which a Higgs branch component
emanates. The two components of the Higgs branch, together with the two singular points
on the Coulomb branch where they touch, are exchanged under the parity of the flavor
symmetry O(4).
For general k, the Higgs branch classically is the nilpotent orbit of O(4k + 4) of type
[22k+2]. (For a discussion of the nilpotent orbits of orthogonal groups, see e.g. [18].) This
is a very even orbit, and consists of two components, exchanged by the parity of O(4k + 4).
Quantum mechanically, on the Coulomb branch, we have two most singular points, at
which each of the two components of the Higgs branch touches. There, the infrared limit
is captured by the following system:
First, take a class S theory of type SU(2k) on a sphere with three punctures, of type
[12k], [k2] and [(k 1)212], respectively. Let us call this the matter sector A. This
has an SU(2) flavor symmetry associated to the parts 12 of the last puncture.
Second, take two free hypermultiplets and couple them to U(1) gauge multiplet with
zero FI term. This has SU(2) flavor symmetry. Let us call this the matter sector B.
We then couple the matter sectors A and B by an SU(2) gauge multiplet.
At this stage, we realized a very-even nilpotent orbit [22k+2] of SO(4k + 4) in terms of
a hyperkahler quotient by SU(2) U(1). Note that we can introduce the FI term for the
U(1) gauge multiplet; this should deform the nilpotent orbit to a nearby coadjoint orbit of
the same dimension.
The infrared dual of the total system
We can then carry over this result to analyze the strongly-coupled physics of the USp(2k)
USp(2k + 2) gauge theory we are interested in, by just adding the USp(2k) gauge multiplet
together with two flavors for it. The result is, again, given by the theory shown in the third
row of figure 6, albeit with zero FI term for the U(1) gauge multiplet. We denote the Higgs
branch of the theory on the third row by M k,, where is the FI parameter. The discussion
above shows that the Higgs branch of the USp(2k) USp(2k + 2) theory, i.e. the moduli
space of SO(8) instantons on C2/Z2 with a holonomy at infinity diag(+, +, +, +, , , , )
and with a trivial holonomy at the origin, is given by two copies of Mk,0.
At present, the author does not have a good argument purely within this second
approach why the FI term for the U(1) gauge group in the infrared dual description can
be identified with the blow-up parameter of the ALE space. Once such an explanation
is given, the approach here would give a much quicker way to derive the moduli space of
SO(8) instantons on the smooth ALE space.
SO(8) instantons on C^2/Z2n
In the string-theoretic approach taken in section 2.2, we can generalize fairly easily the
analysis in the last section to SO(8) instantons on the smooth ALE space C^2/Z2n with the
holonomy at infinity diag(+, +, +, +, , , , ). The result, when viewed from a
fieldtheoretical approach in section 2.4, gives a slightly new class of infrared dual description
of supersymmetric gauge theories. This section is mainly meant to describe this latter
field-theoretical phenomenon.
String-theoretic analysis
We start by considering the gauge theory with gauge group USp(2k)U(2k)2n2USp(2k),
bifundamental matter fields between two consecutive gauge factors, and two additional
fundamentals for each of USp(2k) gauge factors. This is the gauge theory describing k
D3-branes probing four D7-branes and one O7-plane on C2/Z2n, with the holonomies at
infinity and at zero both given by diag(+, +, +, +, , , , ).
We follow the same steps as we did in section 2. First, we take the T-dual, and we
lift the resulting configuration to M-theory. The system is described by a class S theory of
type U(2k), put on a sphere with four punctures of type [k2] and n simple punctures. The
process of putting all n vertical M5-branes on top of the Z2 singularity can be decomposed
into two steps. Namely, we first bring all n 1 simple punctures to one puncture of type
[k2], and then replace it with a FI fixture, see figure 7. We assume k > n for simplicity.
The nature of the FI fixture can be found by reducing the system to a Type IIA setup
around the Z2 singularity, as shown in figure 8. One finds that the FI fixture can be thought
of as a theory with gauge group U(2n 1) U(2n 2) U(1), with bifundamental
hypermultiplets between two consecutive gauge groups, and with additional 2n flavors for
the first U(2n1) group. This theory has 2n1 Fayet-Iliopoulos parameters ~. When they
are all zero, the Higgs branch is the nilpotent orbit NA2n1 of sl(2n), and when they are
send to
turned on, the Higgs branch is a semisimple orbit OA2n1,~ of sl(2n). The 2n 1 complex
FI parameters control the conjugacy class of the orbit, and therefore there is an action of
the Weyl group of SU(2n) on the 2n 1 FI parameters. This matches the number of the
blow-up parameters for C2/Z2n, and the action of the Weyl group of SU(2n) on them.
Let us summarize the process described so far in a field theoretical language, see the
first three rows of figure 9.
We start from a USp(2k) U(2k)n1 USp(2k) gauge theory, with two fundamental
hypermultiplets for each USp group and a bifundamental hypermultiplet for each
consecutive gauge groups. The dimension of the Higgs branch is 6k.
We go to an S-dual frame. Using the standard techniques of the class S analysis, we
find that the resulting theory consists of
USp(2k) group coupled to two fundamentals,
which is coupled further to a class S theory of type SU(2k) on a sphere with
three punctures, of type [12k], [k2], and [(k n)212n],
whose SU(2n) symmetry associated to the parts 12n of the last punctureis
coupled via an SU(2n) gauge multiplet,
to another gauge theory with gauge group U(2n 2) U(2n 4) U(2),
with bifundamental hypermultiplets between two consecutive gauge groups, and
with additional 2n flavors for the first U(2n 2) group.
We now replace the last item, namely the U(2n2) U(2) gauge theory, with the
gauge theory representing the FI fixture. This is, as explained above, given by a gauge
USp(2k) [12k] [(k-n)212n]
theory with U(2n1)U(2n2) U(1), with bifundamental hypermultiplets
between two consecutive gauge groups, and with additional 2n flavors for the first group.
Field-theoretic analysis
Let us instead consider the quiver gauge theory with gauge group USp(2k) U(2k + 2)
U(2k + 4) U(2k + 2n 2) USp(2k + 2n), with bifundamental hypermultiplets
between consecutive gauge groups and additional two flavors for each of USp groups. This
is shown in the last row of figure 9.
The Higgs branch is the moduli space of SO(8) instantons on C2/Z2n, with the
holonomy at infinity given by diag(+, +, +, +, , , , ), and a trivial holonomy at the
origin. The dimension of the Higgs branch is 6k + n, which agrees with that of the moduli
space of SO(8) instantons on the smooth ALE space C^2/Z2n with the same holonomy at
infinity. Then it is likely that the gauge theory we obtained in the stringy approach will
arise as an infrared description close to the most singular points on the Coulomb branch
of this fourth theory, as in section 2.4.
Here we show it is indeed the case. The coupling of the leftmost USp(2k) gauge factor
is infrared free, so we just neglect them and consider the U(2k + 2) U(2k + 2n 2)
USp(2k + 2n) gauge theory, with bifundamental hypermultiplets between two consecutive
gauge groups and additional 2k, 2 flavors for U(2k + 2) and USp(2k + 2n), respectively.
Its Seiberg-Witten curve is known with all mass parameters turned on [19], but the
form is somewhat unwieldy. When the SO(4) mass parameters are zero, the curve can
be embedded into an orbifold of the (v, t) space under the action (v, t) (v, 1/t). The
equation of the curve is given by
with the standard Seiberg-Witten differential = vdt/t. Here, Pj (v) is a polynomial of
degree 2k + 2j whose highest coefficient is one, and Pn(v) = Pn(v). The coefficients of
Pn(v) are the Coulomb branch parameters of USp(2k + 2n) gauge multiplet, and those of
Pj (v) for j = 1, . . . , n 1 are the Coulomb branch parameters of U(2k + 2j), and finally
those of P0(v) are the mass parameters for the U(2k) flavor symmetry. The coefficients c1
to cn encode the gauge coupling parameters.
By tuning all the Coulomb branch parameters, we can make the Seiberg-Witten curve
singular at the orbifold fixed point t = 1. Let us choose t = 1 for concreteness. Say
Pn(v) = v2k+2n + U v2k+2n2 + where U is the dimension-2 Coulomb branch parameter
of USp(2k + 2n) theory. Then, in (3.1), the coefficient of the v2k+2n2 is c1(t + t1) + U ,
and the choice U = 2c1 makes the curve more singular. We can continue this process,
and make the curve very singular there.
Expanding t 1+s with very small s, the local form of the curve close to (v, s) = (0, 0)
is given as
where the summation is over the following pairs (i, j) of non-negative integers:
i + j 2Z, i + j < 2k + 2n,
i 2n.
We perform the identification (s, v) ' (s, v). The differential is vds. In (3.2), the
coefficients c0i encode (a remnant of) the original gauge couplings. Among ui,j , those with
i = 2n and i = 2n 1 are the mass parameters for U(2k), and the rest are the Coulomb
branch parameters. They are displayed in figure 10, for k = 4, n = 2.
Now we look for an appropriate way to scale the parameters, as in [8, 9], so as to keep
the mass parameters for the non-Abelian flavor symmetry to have canonical dimensions,
and to keep as many terms as possible. Given a very small number , a consistent way is
to take
ui,j
( 2kj
if j i 2k 2n,
k+n(i+j)/2 if j i 2k 2n.
Figure 10. Terms in the Seiberg-Witten curves in the singular limit, for k = 4 and n = 2. The
symbols , , represent the couplings, the Coulomb branch parameters, and the mass terms,
respectively. Powers of in (3.4) is also given at each points. The three regions shaded or enclosed
are for j i > 2k 2n, j i = 2k 2n, j i < 2k 2n. The symbols [s] and [v] are the scaling
dimensions of s and v.
In the region s 1, v , only the terms in (3.2) with j i 2k 2n survive.
As the differential is sdv, we can assign scaling dimension 1. The same
curve arises when we study the strongly-coupled limit of the superconformal
USp(2k) U(2k)n1 USp(2k) theory we treated earlier. We can thus identify this
theory as the class S theory of type SU(2k), on a sphere with three punctures of type
[12k], [k2], and [(k n)212n]. The parameters u2nl,2kl are the mass parameters for
the flavor symmetry U(2n) associated to the parts 12n of the last puncture. Let us
call this the matter sector A.
1/2, v 1/2 , only the terms in (3.2) with j i 2k 2n survive.
has scaling dimension 1. The resulting curve has the form
Q2n(z) + yQ2n1(z) + + y2n1Q1(z) + y2n = 0
in terms of the invariant coordinates x = s2, y = v2 and z = sv, with the differential
given by = zdy/y. Here, Qj (z) is a polynomial of degree (at most) j. From our
construction, we see that the coefficient of zj of Qj (z) encodes a coupling. This coefficient
is zero when j is odd. The coefficients of Q2n(z) is the mass parameter for SU(2n)
flavor symmetry, and the coefficients of other Qj (z) are Coulomb branch parameters.
This is the standard Seiberg-Witten curve of a quiver gauge theory with gauge group
U(2n 1) U(2n 2) U(1), with bifundamental hypermultiplets between two
consecutive gauge groups, and 2n additional flavors for the U(2n 1) group, with a
special choice of the coupling constants. Let us call this the matter sector B.
1, only the coefficients u2nl,2kl survive. The curve is
z2n + u2n1,2k1z2n1 + + u0,2k2n = 0
with the differential = zds/s. This tube generates an SU(2n) gauge multiplet,
connecting the SU(2n) flavor symmetries of the two sectors A, B given above.
From this, we find that the physics at the singularity is given by an infrared free SU(2n)
gauge theory coupled to the matter sector A and B.
We can also turn on the SO(4) mass parameters in the analysis. They are the mass
terms for the flavor symmetry SU(2) SU(2) for the parts k2 and (k n)2 of the two
punctures of the matter sector A. We can check that the SO(4) mass parameters do
not modify the matter sector B. One of the two mass parameters deform the orbifold
singularity at t = 1, which clearly does not affect the sector B. The other mass parameter
deform the singularity at t = 1, and modify the relations between the variables x, y,
z introduced above to xy = z2 + . The curve (3.5) of the sector B is written purely in
terms of y and z, and the differential is still = zdy/y. Therefore the mass parameter
does not affect the sector B either.
By coupling USp(2k) gauge multiplet and two additional fundamental hypermultiplets
to the matter sector A via the flavor symmetry SU(2k) associated to the puncture [12k], we
realize the theory shown in the third row of figure 9. This is what we wanted to demonstrate.
Conclusions and speculations
In this paper, we considered k D3-branes probing four D7-branes and an O7-plane on the
orbifold C2/Z2n and on the smooth ALE space C^2/Z2n. For technical reasons, we chose
the holonomy at infinity to be diag(+, +, +, +, , , , ).
On the orbifold, the worldvolume theory on the D3-branes is a 4d N = 2
supersymmetric theory with gauge group USp(2k+) Qin=11 U(ki) USp(2k) with bifundamental
hypermultiplets between two consecutive gauge groups, and two fundamental
hypermultiplets for each of the two USp groups. We chose in particular the case
USp(2k) U(2k) U(2k) USp(2k)
USp(2k) U(2k + 2) U(2k + 2n 2) USp(2k + 2n)
which corresponds to the holonomy at the origin diag(+, +, +, +, +, +, +, +).
In the former case (4.1), we analyzed the system using string duality, and found that
the blow-up parameters can be introduced only in a strongly-coupled limit. There, we
have weakly-coupled dual gauge multiplets of the form U(2n) U(2n 2) U(2).
We argued that giving non-zero blow-up parameters requires that we replace this chain of
gauge multiplets with another chain, U(2n) U(2n 1) U(1), and that the blow-up
parameters are the FI parameters for these gauge multiplets. The result is shown in the
third row of figure 9.
In the latter case (4.2), we analyzed the system field theoretically, along the line
of [8, 9]. One of the gauge group, USp(2k + 2n), is asymptotically free, and at one of two
most singular points on the Coulomb branch, we can find an infrared dual description,
which is again the theory shown in the third row of figure 9. Here the author does not
currently have a direct argument to show that the FI terms of the unitary gauge multiplets
correspond to the blow-up parameters.
As a result, we have a description of the moduli space of SO(8) instantons on the
smooth ALE space C^2/Z2n with the holonomy at infinity being diag(+, +, +, +, , , , )
in terms of a hyperkahler quotient of a flat space times the Higgs branch of a class S theory.
As a holomorphic symplectic manifold, this construction is mathematically completely
explicit, assuming the result in an unpublished work [17]. When k is sufficiently small, we
can give an explicit description even without assuming the content of [17].
Let us discuss how we might extend our analysis to larger SO groups. Our first method
which used the string dualities is not very adequate, as our argument relied on the fact
that the gauge theory is superconformal and there is no bending of the NS5-branes. Our
second method which used the field-theoretical duality should be applicable, although we
do not have a direct way to show that the FI terms in the infrared dual description are
the blow-up parameters. The field-theoretical duality employed is the one studied in [8, 9],
and is not currently developed sufficiently enough to allow us to analyze this general case.
Hopefully this will change in the near future.
A natural question is whether class-S technique can be used to study instanton moduli
spaces of groups other than SO groups on smooth ALE spaces. The unitary groups might
look easier than the orthogonal groups, for example. The standard quiver gauge theories
describing unitary instantons on the ALE spaces have all the FI parameters corresponding
to the blow-up parameters of the ALE space; but one can still ask if the class S technique
would shed new light on the system. Unfortunately, these quiver gauge theories often
have gauge nodes that are very infrared-free, and class S constructions are at present
not immediately applicable here, as they are developed thus far mainly for systems that
are conformal or slightly ultraviolet-free. We need to wait until the class S technique is
extended to infrared free systems.
Finally, let us speculate how we might study the moduli space of orthogonal instantons
on ALE spaces of type D and E. Note that at least for SO(8) and with the holonomy at
infinity diag(+, +, +, +, , , , ), we found the following structure:
A2n1, with the blow-up parameter ~, XA2n1,k is a certain fixed hyperkahler
manifold, and OA2n1,~ is the semisimple orbit of SU(2n) with the parameter ~, and ///
denotes the hyperkahler quotient construction. The dimension of OA2n1,~ for generic is
(dim SU(2n)rank SU(2n))/2. Therefore, XAn1,k has (dim SU(2n)+rank SU(2n))/2 more
quaternionic dimensions than M A2n1,k,~.
where MA2n1,k is the moduli space of instantons on the orbifold C2/Z2n with a trivial
holonomy at the origin, and (VA2n1,k, GA2n1,k) is a known pair of a vector space and a
group realizing this moduli space as a hyperkahler quotient of a flat space.
It is noticeable that the objects involved, namely OA2n1,~, NA2n1 , SU(2n) are all
naturally associated to the type of the ALE space. So, for other ALE spaces of type
= Dn and En, the author would speculate that we might have the same structure, where
the semisimple orbits and the nilpotent orbits involved are replaced with those of the Lie
algebra of type :
The author thanks A. Dymarsky and J. Heckman for old discussions in 2011, and D. Gaiotto
for reminding me of the reference [7], which rekindled his interest on this subject. He also
would like to thank N. Mekareeya for the discussion on [20], and H. Nakajima for helpful
explanations on the moduli space of orthogonal instantons on the ALE spaces. S. Cherkis
and S. Giacomelli kindly read the prepreprint carefully, pointed out many typos, and
suggested various improvements of the draft. It is a great pleasure for the author to thank
them. The author is supported in part by JSPS Grant-in-Aid for Scientific Research No.
25870159, and in part by WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo.
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