6d String Chains
We consider bound states of strings which arise in 6d (1,0) SCFTs that are realized in F-theory in terms of linear chains of spheres with negative self-intersections 1,2, and 4. These include the strings associated to small instantons, as well as the ones associated to M5 branes probing A and D type singularities in M-theory or D5 branes probing ADE singularities in Type IIB string theory. We find that these bound states of strings admit (0,4) supersymmetric quiver descriptions and show how one can compute their elliptic genera.
Recently considerable progress has been made in classifying and understanding basic properties of six-dimensional superconformal field theories Heckman:2013pva (); Ohmori:2014pca (); Ohmori:2014kda (); DelZotto:2014hpa (); Heckman:2014qba (); Intriligator:2014eaa (); DelZotto:2014fia (); Heckman:2015bfa (); Bhardwaj:2015xxa (); DelZotto:2015isa (); Miao:2015iba (); Ohmori:2015pua (); Haghighat:2013gba (); Haghighat:2013tka (); Haghighat:2014pva (); Kim:2014dza (); Haghighat:2014vxa (). Formulating these theories in a way which leads to a computational scheme of physical amplitudes is still a major challenge. One idea along these lines is to note that on their tensor branch these theories have light strings as basic ingredients, which naturally suggests that these strings should play a key role in any computational scheme. For example, it is already known that their elliptic genus can be used to compute the superconformal index of these 6d theories Lockhart:2012vp (); Kim:2012qf (); Haghighat:2013gba (). The main aim of this paper is to expand the known dictionary Haghighat:2013gba (); Haghighat:2013tka (); Kim:2014dza (); Haghighat:2014vxa () for describing the degrees of freedom living on these strings.
The six-dimensional SCFTs that can be obtained via F-theory compactification were classified in Heckman:2013pva (); DelZotto:2014hpa (); Heckman:2015bfa () (see also Bhardwaj:2015xxa ()). This includes all presently known 6d SCFTs, and it may well be the case that all consistent 6d SCFTs can be realized this way. In this context, six-dimensional SCFTs are realized by putting F-theory on elliptic Calabi-Yau threefold with a non-compact base with the property that all compact curves can be simultaneously contracted to zero size. These curves are in fact two-spheres that have negative self-intersection ranging from 1 to 12 (excluding 9,10,11) and intersect one another at a point, and are arranged into trees according to specific rules. In particular, one can identify some basic building blocks (the curves with negative self-intersection and 12, and a few combinations of two or three curves Morrison:2012np ()) that can be glued together by joining them with -curves. Each curve gives rise to a tensor multiplet in the 6d theory. D3 branes wrapping a curve correspond to strings coupled to the corresponding tensor multiplet. Furthermore, gauge multiplets arise whenever the elliptic fiber degenerates over a curve, the nature of the resulting gauge group being determined by the type of degeneration. In particular, non-Higgsable clusters necessarily support a nontrivial gauge group, whose rank can be made larger by increasing the degree of the singularity.
Under favorable circumstances, bound states of strings admit a description in terms of a two-dimensional (0,4) quiver gauge theory. In this paper we extend the list of theories for which such a description is available to three additional classes of 6d SCFTs:
The theory of M5 branes probing a singularity of A or D type. The first case was studied in Haghighat:2013tka () and corresponds to a linear chain of curves and leads to a 6d theory with SU-type gauge group; the second case is novel and corresponds to a linear chain of alternating and curves which support respectively gauge group and .
The theory of small instantons, or equivalently M5 branes probing the M9 plane of Hořava-Witten theory Seiberg:1996vs (); Ganor:1996mu (). This corresponds to a single curve linked to a chain of curves of self-intersection . Upon circle compactification, this theory admits deformation by a parameter corresponding to the mass of a 5d anti-symmetric hypermultiplet. We focus on the case where this parameter is turned off.
The theory of D5 branes probing an ADE singularity. This corresponds in F-theory to an ADE configuration of curves supporting gauge groups of type.
Once the quiver gauge theory corresponding to a given configuration of strings is specified, the elliptic genus can be computed by means of localization Gadde:2013dda (); Benini:2013xpa (). We do this for certain specific bound states of strings for the first two classes of theories, and for arbitrary bound states of strings for the third class.
The organization of this paper is as follows: In Section 2 we review some of the building blocks for the (0,4) supersymmetric 2d quiver gauge theories which will be needed for the description of the worldsheet degrees of freedom of the tensionless strings. In Section 3 we discuss the 2d quiver for the strings of the theory of M5 branes probing A- or D- type singularities. In Section 4 we study the quiver for the strings of the theory of small instantons. In Section 5 we discuss the quiver for strings of the theory of D5 branes probing an ADE singularity. In Appendix A we discuss a candidate 2d quiver for the strings of the theory of small instantons with mass deformation.
2 Chains of Strings
We are interested in computing the elliptic genera of the strings that arise on tensor branches of 6d SCFTs with several tensor multiplets, along the lines of Haghighat:2014vxa (), and are wrapped around a torus of complex modulus . In this paper we aim to obtain 2d quiver gauge theories for a variety of 6d SCFTs that arise within M- and F-theory. These will generally consist of quiver gauge theories with gauge group , where is the gauge group associated to strings of the th type, and will capture the dynamics of a bound state of such a collection of strings. The gauge groups arising in the theories discussed in this paper are either unitary, symplectic, or orthogonal.
Let us now discuss global symmetries of the 2d theory. A number of the global symmetries have a geometric origin, since they arise from rotations of an along the worldvolume of the six-dimensional theory but transverse to the string’s worldsheet, or from rotations of an perpendicular to the six-dimensional worldvolume. Overall, an group acting as rotations of :
The right-moving supercharges transform under the R-symmetry given by
Let us identify the Cartan of as follows:
where we identify with the R-symmetry group of the the theory when viewed as a theory. When computing elliptic genera, we will turn on fugacities and for the remaining factors in the Cartan.
Below we summarize our notation for the following sections. In drawing quiver diagrams, we will denote Fermi multiplets by dashed lines and hypermultiplets by a solid line. The various fields of the quiver gauge theory can be organized in terms of four combinations of (0,2) multiplets:
To each gauge node corresponds the following field content valued in representations of (corresponding to strings of the th kind): a vector multiplet ; a Fermi multiplet ; and two chiral multiplets .
Symbol (0,2) field content (vector) 0 0 0 0 adj. (Fermi) 0 1 adj. (chiral) 1 0 0 0 R (chiral) 0 1 0 0 R
The representation R is the adjoint representation whenever the gauge group is unitary, symmetric whenever the gauge node is orthogonal, and anti-symmetric if the gauge group is symplectic.
Between each pair of nodes such that the corresponding element of the adjacency matrix of the underlying quiver is non-zero one has the following bifundamental fields of : two Fermi multiplets , and chiral multiplets forming a twisted hypermultiplet.
Symbol (0,2) field content (Fermi) 1/2 1 (Fermi) 1/2 1 (chiral) 1 (chiral) -1 1
Between each gauge node and the corresponding global symmetry node one has a link corresponding to two chiral multiplets, , charged under , where is the global symmetry group at that node, which we depict by a square in the quiver.
Symbol (0,2) field content (chiral) 1/2 1/2 0 0 (chiral) 1/2 1/2 0 0
Between each gauge node and any successive node one has a Fermi multiplet ; between the same gauge node and any preceding node one has a Fermi multiplet .
Symbol (0,2) field content (chiral) 0 0 1 (chiral) 0 0 1
3 Partition functions of M5 branes probing ADE Singularities
In this section we consider 6d SCFTs which arise from M5 branes probing singularities of type A and D, and obtain the 2d quiver gauge theory describing the self-dual strings that arise on the tensor branch of the corresponding 6d theory.
3.1 M5 branes probing an singularity
Consider a setup where parallel M5 branes span directions and are separated along the direction in 11d spacetime. Taking the transverse space of the M5 branes to be and blowing up the singular locus gives rise to a 6d SCFT on the tensor branch which enjoys a flavor symmetry. The singularity can be thought of as a limit of Taub-NUT space with charge ; this space has a canonical circle fibration over , and compactifying M-theory along this circle one arrives at a system of parallel D6-branes stretched between NS5 branes. The dynamics of the strings that arise in this system are captured by the two-dimensional quiver theory of Figure 1:
The quiver corresponds to a 2d theory obtained from a orbifold of a supersymmetric Yang-Mills theory. As such, each gauge node contains a vector multiplet together with an adjoint hypermultiplet and the bifundamental fields between the gauge nodes consist of Fermi and twisted hypermultiplets. Furthermore, between each gauge node and the adjacent flavor nodes one has Fermi multiplets in the fundamental representation of the gauge group and between each gauge node and the corresponding flavor node a fundamental hypermultiplet. The exact field content is described in Haghighat:2013tka (). Following the rules of Benini:2013xpa () and the charge tables of Section 2 one can straightforwardly write down an expression for the elliptic genus for any configuration of strings, corresponding to different choices of the ranks of the gauge groups in the 2d quiver. We will perform the computation in Section 5.
We can also relax the condition that all nodes should have the same flavor symmetry. In particular, we can consider the situation where the th flavour node has symmetry together with the convexity condition
which ensures that the parent 6d theory is not anomalous. In case of equality, all are ordered along a linear function and gauge anomaly cancellation is automatically satisfied. However, if is strictly greater than the net number of right moving fermions is greater than the number of left-moving ones and the theory will be anomalous. To cure this, we introduce for each gauge node a fourth flavor node with left-moving fermions to compensate for the excess of the right-moving ones. The corresponding quiver is the one depicted in Figure 2.
Again the elliptic genus can be computed straightforwardly using the charge table of Section 2, however one has to be careful with the charge assignments of the new vertical Fermi multiplets: these are not charged under or . The origin of the different flavor groups can be explained from the brane construction that corresponds to this theory: one has D6 branes separated by NS5 branes. The difference between the number of D6 branes on the two sides of an NS5 brane must equal the negative of the cosmological constant in that region Hanany:1997sa (). So, for instance, if we have an NS5 brane with D6 branes on the left and on the right we must have cosmological constant there. At the next NS5 brane, however, we must have cosmological constant . This is achieved by placing D8 branes between the two NS5 branes Hanany:1997gh (), which has the effect of changing the cosmological constant as required. This leads to the two-dimensional quiver considered above.
3.2 M5 branes probing a singularity
A singularity gives rise to 7d SYM theory with gauge group for . We can place M5 branes at the singularity and separate them along the remaining direction in seven dimensions. Each M5 brane actually splits into two fractional branes, which gives rise to parallel domain walls in the 7d theory DelZotto:2014hpa (). Reducing along this direction leads to a 6d SCFT with global symmetry. Following DelZotto:2014hpa () we can obtain a Type IIA description by replacing the singularity with the corresponding ALF space and taking the circle fiber to be the M-theory circle. This results in a stack of parallel D6 branes on top of an plane, together with their mirrors. Furthermore, one has fractional NS5 branes Hanany:2000fq () which are of codimension 1 with respect to the orientifold plane. Whenever an plane meets an NS5-brane, it turns into an plane. A system of D6-branes parallel to an plane gives rise to an gauge theory, and therefore one obtains alternating and gauge groups in 6d. On top of this, the NS5 branes contribute a total of tensor multiplets.
Furthermore, M2 branes suspended between M5 branes in the M-theory picture become D2 branes suspended between NS5 branes in Type IIA. The brane setup we have arrived at is pictured in Figure 3. Upon reduction along the direction, the D2 branes give rise to the self-dual strings that arise on the tensor branch of the 6d SCFT. The resulting two-dimensional quiver theory is depicted in Figure 4. One can easily check that gauge anomalies correctly cancel out for this theory. However, one finds that is anomalous. The reason for this anomaly can be traced back to geometry: the -type singularity transverse to the M5 branes has only symmetry which is the which commutes with the action of the binary extension of the dihedral group. This is the R-symmetry group of 6d SCFT. The situation has to be contrasted with the -type singularity where the surviving isometry of the space is . Therefore, we see that is not present for the -type theory and hence the elliptic genus of it’s strings should not be refined with respect to it.
Having an explicit description of the two-dimensional quiver theory makes it possible to compute the corresponding elliptic genus. In the simplest case of a single tensor multiplet corresponding to a curve, this corresponds to the E-string elliptic genus which was studied in detail in Kim:2014dza () (although in the present setup one must identify the fugacities associated to the two subgroups of the flavor symmetry group). For the sake of illustration, let us also consider the non-Higgsable theory with three tensor multiplets, gauge group and flavor group . Let us denote by , the fugacities associated to and by () the ones associated to (). From the previous discussion, one can write down the elliptic genus for any bound state of the strings associated to this theory. For instance, if one considers the bound state of one string coupled to the first tensor multiplet and one string coupled to the multiplet, one finds:
The contour integral can be performed by using the Jeffrey-Kirwan prescription for computing residues Benini:2013xpa (). Similarly, one can compute the elliptic genus for other bound states of strings.
4 Multiple M5 branes probing an M9 wall
In this section we study the six-dimensional theory of small instantons Ganor:1996mu (); Seiberg:1996vs (); upon moving to the tensor branch, this becomes the theory of parallel M5 branes in the proximity of the M9 boundary wall of M-theory. The strings originate from M2 branes that are suspended between neighboring M5 branes or between the M5 branes and the M9 plane (see Figure 5). Upon circle reduction to five dimensions with an background Wilson line
(which breaks global symmetry to ), the theory of small instantons reduces to
the theory with fundamental and antisymmetric hypermultiplets Seiberg:1996vs (). The instanton calculus for this five-dimensional theory provides a way to check elliptic genus computations and will be exploited in Section 4.2.
4.1 Two-dimensional quiver
In order to derive a quiver description for the theory of the strings, it again proves useful to switch to an equivalent brane configuration within string theory. Let us begin by discussing theory of a single small instanton, whose associated two-dimensional quiver gauge theory has been worked out in Kim:2014dza (). The quiver was derived from a Type I’ brane configuration, which arises as follows: upon reduction of M-theory on a circle, the M9 plane is replaced by eight D8 branes on top of an O orientifold plane (which has D8 brane charge ); the M5 brane, on the other hand, becomes an NS5 brane. Furthermore, M2 branes are replaced by D2 branes that stretch between the NS5 brane and the D8-O8 system. By studying the two-dimensional reduction of the worldvolume theory of the D2 branes in the limit of small separation between the NS5 and eight-branes, one arrives at the two-dimensional quiver gauge theory of Kim:2014dza (). The (0,4) quiver gauge theory for strings has gauge group and the following field content: a vector multiplet in the adjoint (anti-symmetric) representation of , a hypermultiplet in the symmetric representation, and eight Fermi multiplets in the bifundamental representation of . Elliptic genera for this theory have been computed in Kim:2014dza () for up to four strings and shown to agree with results from the instanton calculus for the five-dimensional theory with eight fundamental hypermultiplets.
The generalization of the Type I’ brane setup to the case of small instantons is illustrated in Figure 6 and is again obtained by reducing the above M9-M5 setup on a circle. The brane setup is a combination of the E-string and M-string brane configurations without the D6 branes which are usually present for the M-string system. As we will see this becomes crucial when we look at the quiver-gauge theory governing the dynamics of the strings to which we now turn.
The brane setup implies a simple quiver gauge theory governing the dynamics of the strings. The first D2 branes ending on the D8-O8 system correspond to a gauge node; from the D2-D8 strings one finds eight bifundamental Fermi multiplets charged under . Furthermore, there is a symmetric hyper at the node as already observed in Kim:2014dza (). All other gauge nodes corresponding to the the D2 branes suspended between NS5 branes have unitary gauge groups with bi-fundamental matter between them familiar from the orbifolds of M-strings Haghighat:2013tka (). Finally, one also obtains bifundamentals from strings ending on the and D2 branes. These bifundamental fields consist of a (0,4) hyper and a (0,4) Fermi multiplet, as is the case for M-strings. The resulting quiver is illustrated in Figure 7.
We comment on the global symmetries of this quiver gauge theory, and compare them with the symmetries that we expect for the infrared CFT on these strings. Our gauge theory has R-symmetry. The first is part of the symmetry which rotates along the worldvolume of NS5-branes, transverse to the strings. The second rotates the space transverse to the NS5-branes and D2-branes. The infrared (or equivalently strong coupling) limit of the 2d gauge theory is realized by going to the M-theory regime of the type I’ theory. Then the space transverse to NS5-D2 is replaced by , including the M-theory circle, and becomes in the strong coupling limit. So in the IR, we expect the symmetry to enhance to . Any analysis from our gauge theory, such as the elliptic genus calculus below, will be missing the extra Cartan charges of the enhanced . Let us denote by the chemical potentials for the rotations on , as in the previous sections. Apart from rotating along the 5-brane, there will be an extra rotation on transverse to the M5-brane, with . Let us denote by the chemical potential for the missing Cartan of the enhanced IR symmetry. Then the part in the type I’ setting is rotated by , while the rotation by is invisible on . Thus, our UV gauge theory will be computing the elliptic genus only at 111The parameter is the one appearing in the instanton calculus of the Nekrasov partition function and should not be confused with the fugacity of in the 2d gauge theory. With respect to the 2d fugacity it is shifted by ..
At , it is known that the 6d SCFT engineered by a single M5 and M9 brane does not see the extra Cartan of (conjugate to ) at all. In other words, all the states in the Hilbert space of the 6d SCFT are completely neutral under this charge (while the full M-theory would see the charged states decoupled from the 6d SCFT). One way to see this is from the 5 dimensional gauge theory obtained after circle compactification. Namely, the parameter above corresponds to the mass parameter for the antisymmetric hypermultiplet in the resulting 5d theory. At , the antisymmetric representation is neutral in and the corresponding hypermultiplet decouples. This implies that the extra for decouples from the 6d CFT at , and this has been tested from the instanton partition function in Hwang:2014uwa (). This is the reason why the 2d gauge theory above provided maximally refined elliptic genera at in Kim:2014dza (), since the restriction loses no information about the 6d SCFT. However, the parameter appears in the 6d CFT spectrum for , which was checked from the instanton calculus Hwang:2014uwa ().
Below we present sample computations for the elliptic genera corresponding to the lowest charge sectors, namely , , and for the quiver.
Combining the one-loop determinants, the zero-mode integral is given by
where . Repeated signs in the arguments mean that both factors are multiplied: . The contour integral given by the JK-Res is done with . Then the only nonzero JK-Res comes from the pole . The result is
The zero-mode integral is given by
If we choose in which , nonzero JK-Res can only come from the following poles.
Actually evaluating the residues, it turns out that all these poles yield vanishing residues, so that
Now the gauge theory comes with gauge group. In the elliptic genus calculus, there are seven disconnected sectors of flat connections Kim:2014dza (). In six sectors, the flat connections are discrete, while in one sector it comes with one complex parameter.
In the first sector with continuous parameter, which we label by superscript , one has to do the following rank contour integral for the elliptic genus:
If we choose in which , nonzero JK-res can only appear from the following poles.
. Its residue is zero.
Collecting all the residues, is given by
On the second line, we used the following identity
which we checked in an expansion in , for the first terms up to order.
The contributions from the other six sectors are given by
where we take the discrete holonomies , , , , , for , respectively. JK-res with can be nonzero only at the pole or , yielding the following result:
which exhibits a factorization structure.
In the next subsection, we will show that all the results above are in complete agreement with the 5 dimensional instanton calculus of Hwang:2014uwa (). Before that, let us first try to interpret these rather simple results that we have found at .
The strings made of and D2-branes in Fig. 6, winding a circle, contribute to the elliptic genus as both multi-particle states, and also through various threshold bound states with lower particle numbers. There could be various kinds of bound states. Generally, of the strings and of the strings may form bounds. One can first deduce that the index is zero at in the sector which contains bound states with charges . This is because the bounds are basically M-strings in a maximally supersymmetric theory. Note that the M-strings are half-BPS states of the 6d theory, so will see broken supercharges as Goldstone fermions. This is in contrast to the strings in 6d QFTs preserving SUSY only. The extra fermionic zero modes for M-strings provide the factor
to the elliptic genus Kim:2011mv (). Thus, M-strings which are unbound to E-strings (i.e. at ) will contribute a factor to the elliptic genus at .
With this understood, let us start by considering the sector with . At , there is no contribution from the two particle states due to the above reasoning. So one should only obtain a single particle contribution in the sector. This is consistent with our finding . A slightly surprising fact from our finding is that the single particle bound with charges behaves exactly the same as a single E-string with charge , at least at . Although the bound look like one long E-string suspended between the M9-plane and the second M5-brane, it penetrates through the first M5-brane so in principle there could be extra contributions to the BPS degeneracies from the intersection. For instance, in the case of M-strings, it is known that the charge M-string and the single particle bound part of the M-string exhibit different spectra (at general chemical potentials, with ) Kim:2011mv (). So we interpret that implies some simplification of the elliptic genus at .
Other results also have nontrivial implications on the E-/M-string bound state elliptic genera at . For , implies that there are no bound states captured by the elliptic genus at , since we know that , , or multi-particles cannot contribute to the elliptic genus at . As we will consider from the 5d instanton calculus, this feature generalizes to higher string numbers: the bounds with do not contribute to the elliptic genus at .
Finally, can also be understood with the above observations. Namely, with a particle yielding a factor of zero in the elliptic genus, the nonzero contribution can come from two particle states. But since we already know that these contributions give equal elliptic genera namely that of a single E-string, we can naturally understand this relation. (So our finding implies that the bound does not contribute to the index at .)
Based on the above observations, we propose that
Namely, at , the string elliptic genus factorizes to two E-string elliptic genera. Although we have shown this result for only a few charges from the 2d gauge theories, we shall confirm such factorizations to a much higher order in from the 5d instanton calculus below.
4.2 Five dimensional instanton calculus
In this subsection, we shall consider the circle compactification of the theory on M5 and one M9, and consider the string spectra from the instanton calculus of the resulting 5d gauge theory.
Let us consider the six-dimensional conformal field theory living on two M5 branes probing the M9 plane. The space transverse to the two M5-branes is , where the latter is obtained by the action of M9. This space has symmetry. The first is the superconformal R-symmetry, and the second is a flavor symmetry. The full flavor symmetry is thus .
We compactify this system on a circle, with an Wilson line which breaks into . Then at low energy, one obtains a 5 dimensional gauge theory with fundamental and one anti-symmetric hypermultiplet. The masses for the fundamental hypermultiplets and the mass for the antisymmetric hypermultiplet are in 1-1 correspondence to the chemical potentials of the flavor symmetries. The precise relations that we use are given in Hwang:2014uwa (); Kim:2014dza (). is simply uplifting to the flavor chemical potential, while the masses are related to the chemical potentials by Kim:2014dza ()
The chemical potentials , for the electric charges are related to those for the string winding numbers by
We chose the convention for in a way that they correspond to the distances from the M5-branes to the M9-plane.
The elliptic genera of the previous subsection are related to the instanton partition function for this 5d