Strings of Minimal 6d SCFTs
We study strings associated with minimal 6d SCFTs, which by definition have only one string charge and no Higgs branch. These theories are labelled by a number with or . Quiver theories have previously been proposed which describe strings of SCFTs for . For the strings interact with the bulk gauge symmetry. In this paper we find a quiver description for the string using Sen’s limit of F-theory and calculate its elliptic genus with localization techniques. This result is checked using the duality of F-theory with M-theory and topological string theory whose refined BPS partition function captures the elliptic genus of the SCFT strings. We use the topological string theory to gain insight into the elliptic genus for other values of .
Six-dimensional superconformal theories have light strings as their basic building blocks. One approach to a better understanding of these theories involves unlocking the mysteries associated with these strings. In particular one would like to describe the free propagation of such strings and the degrees of freedom on their worldsheet. Recently, many advances have been made in our understanding of 6d SCFTs Heckman:2013pva (); Gaiotto:2014lca (); DelZotto:2014hpa () including in many cases an effective description of their strings’ worldsheet QFT Haghighat:2013gba (); Haghighat:2013tka (); Kim:2014dza (). The goal of this paper is to study the strings associated with minimal 6d SCFTs. These are SCFTs in six dimensions which have only one string charge (i.e. a one dimensional tensor branch), and are non-Higgsable. They are labelled by an integer excluding , and are realized within F-theory as elliptic fibrations over a base Morrison:2012np (). It is also natural to include the case here although strictly speaking it is not part of the non-Higgsable family. The cases also arise in the F-theory context as simple orbifolds of Witten:1996qb () where we rotate each plane of by an -th root of unity and compensate by rotating the elliptic fiber by .
For the and cases, quivers have been found which describe the worldsheet dynamics of the corresponding strings Haghighat:2013gba (); Haghighat:2013tka (); Kim:2014dza (). The case corresponds to the exceptional CFT with global symmetry describing an M5 brane near the M9 boundary wall Witten:1995gx (); Ganor:1996mu (); Klemm:1996hh (); Seiberg:1996vs (); the case, on the other hand, corresponds to the SCFT of type . In this paper we extend this list by finding the quiver for the case. This is one of the orbifold cases for which the elliptic fiber can have arbitrary complex modulus , as the only symmetry required in the fiber is , which does not fix the modulus of the torus. To find the quiver describing the strings of this theory, we use Sen’s limit of F-theory Sen:1997gv (), which corresponds to taking the modulus of the torus . Following this approach we are able in particular to compute the elliptic genus of these strings, which we do explicitly for the first few string numbers.
If we compactify the theory on a circle, the elliptic genus computes the BPS degeneracies of the wrapped strings. Following the duality between F-theory and M-theory and the relation between M-theory, BPS counting in five dimensions, and topological strings, we find that the elliptic genera are encoded in the topological string partition function defined on the corresponding elliptically-fibered Calabi-Yau, similar to the observation for the case in Klemm:1996hh (). Within topological string theory the genus zero BPS invariants can be easily calculated using mirror symmetry even for high degree in the base, which corresponds to times wrapped strings. However, since the boundary conditions are only known to some extent Huang:2013yta (), the higher genus theory cannot be completely solved with the generalized holomorphic anomaly equations; on the other hand, the elliptic genus computation provides the all genus answer. In particular we can use this relation to successfully check our answer for the elliptic genus of the strings.
For the other values of , no worldsheet description of the associated strings is known. For these cases we employ topological string techniques to obtain BPS invariants of the corresponding geometry, which can be related to an expansion of the elliptic genus of small numbers of strings for specific values of fugacities.
The organization of this paper is as follows: In Section 2 we review the classification of minimal 6d SCFTs and their F-theory realization. We also review the quivers describing the worldsheet dynamics of the strings of the and models. In Section 3 we derive the quiver for the strings of the theory by exploiting its orbifold realization. Furthermore, using the quiver we obtain an integral expression for the elliptic genus of strings which we evaluate explicitly for the cases and . We then discuss how one can extract from this the BPS degeneracies associated to the strings. In Section 4 we construct explicitly the elliptic Calabi-Yau manifolds corresponding to as hypersurfaces in toric ambient spaces, solve the topological string theory and calculate the genus zero BPS invariants associated to these Calabi-Yau manifolds, from which one can obtain BPS degeneracies associated to the strings. In Appendix A we give a description of the local mirror geometry for some of these elliptic Calabi-Yau threefolds in terms of non-compact Landau-Ginzburg models.
2 Minimal SCFTs in six-dimensions
Six-dimensional SCFTs can be classified in the context of F-theory by considering compactifications on an elliptically fibered Calabi-Yau threefold with non-compact base . In the case where all fiber components are blown down the fibration can be described in terms of the Weierstrass form
where and are sections of the line bundles and . The discriminant locus, along which the elliptic fibers are singular, is a section of and has the following form:
The discriminant locus corresponds to the location of seven-branes in the system. More precisely, each component of the discriminant locus is identified with a seven-brane wrapping a divisor . Each seven-brane supports a gauge algebra which is determined by the singularity type of the elliptic fiber along Morrison:1996na (); Morrison:1996pp ().
In the maximally Higgsed phase (that is, when all hypermultiplet vevs that can be set to non-zero value are turned on) one can classify the resulting models in terms of the base geometry only Morrison:2012np (). Non-Higgsability requires that the divisor be rigid. This implies that must be a curve with self-intersection for a positive number (in the following we will refer to this as a curve), and the local geometry is the bundle . Furthermore, it can be shown that is only allowed to take the values or Morrison:1996na (); Morrison:1996pp (); Morrison:2012np (). In the case, corresponding to the E-string SCFT Witten:1995gx (); Ganor:1996mu (); Seiberg:1996vs (), the discriminant vanishes along the non-compact fiber over isolated points on the . In this case instead of a gauge symmetry one finds an global symmetry. In the case the fiber is everywhere non-singular, and one finds the SCFT which corresponds to the world-volume theory of M5 branes in flat space. For , the seven-branes wrap the compact , and therefore the 6d SCFT has non-trivial gauge symmetry. In the non-Higgsable case this gauge symmetry is completely determined by the integer . We summarize the list of possibilities in the following table:
In the case, one finds that in addition to gauge symmetry the 6d theory also contains a half-hypermultiplet. The cases , and lead to gauge symmetry but additionally contain “small instantons”; these cases can be reduced to chains of the more fundamental geometries summarized in the table, as discussed in Heckman:2013pva ().
These geometries (excluding the cases ) can equivalently be realized as orbifolds of the form , . Here, acts on the , coordinates of and as
with . This construction will be in particular useful when we study the SCFT, as it will enable us to find a weak coupling description for the corresponding model.
2.1 Strings of the and models
Let us next discuss the strings that appear on the tensor branch of 6d SCFTs. From the point of view of F-theory these strings arise from D3 branes which wrap the curve in the base in the limit of small size. Let us first review the ‘M-strings’ that arise in the case. Since in this case the orbifold acts trivially on the torus, its modulus can be taken to be arbitrary, and in particular one can take the weak coupling limit and study this system from the point of view of Type IIB string theory compactified on . It turns out Haghighat:2013gba (); Haghighat:2013tka () that the dynamics of M-strings are captured by the two-dimensional quiver gauge theory depicted in Figure 1. For strings, this quiver describes a two-dimensional theory with gauge group and the following field content: and are chiral multiplets in the fundamental representation of , while and are fundamental Fermi multiplets. Furthermore, the Fermi multiplet and vector multiplet combine into a vector multiplet, and the adjoint chiral multiplets combine into a hypermultiplet. One intuitive way to see how this comes about is to look at the configuration of curves which captures the local geometry DelZotto:2014hpa () and is pictured in Figure 2.
The left and right curves are non-compact, whereas the curve in the middle is a compact . Choosing the elliptic fiber to be trivial would lead upon circle compactification to gauge theory; it is in fact possible to deform this theory to by letting the elliptic fiber degenerate over each curve to an singularity (that is, by wrapping a D7 brane over each curve). D3 branes wrapping the compact curve give rise to the strings of the resulting 6d SCFT, and upon circle compactification their BPS degeneracies then capture the BPS particle content of the theory. It is easy to understand how the field content of the quiver in Figure 1 arises from strings that end on the D3 branes: D3-D3 strings give rise to a vector multiplet in the adjoint of consisting of the (2, 0) multiplets ; strings stretching from the D3 branes to the D7 brane wrapping the same compact give rise to the chiral multiplets and ; finally, strings stretching between the D3 branes and the D7 branes that wrap the non-compact curves give rise to the Fermi multiplets and . Whether D3-D7 strings give rise to chiral or Fermi multiplets is determined by the number of dimensions that are not shared by the D3 and D7 branes (four for the D3-D7 strings leading to , eight for the ones leading to ).
Recently, a quiver gauge theory was also found that describes the dynamics of E-strings, corresponding to the case Kim:2014dza (). In terms of multiplets, the theory of E-strings was found to have the following field content: a vector multiplet and a Fermi multiplet in the adjoint representation of , two chiral multiplets in the symmetric representation of , and a Fermi multiplet in the bifundamental representation of and of a flavor group, which enhances to at the superconformal point. The relevant quiver is shown in Figure 3.
2.2 From strings of 6d SCFTs to topological strings
In cases where a quiver gauge theory description is available for the strings of minimal six-dimensional SCFTs, one can use the methods of Gadde:2013dda (); Benini:2013nda (); Benini:2013xpa () to compute the elliptic genus for an arbitrary number of strings. The elliptic genus will depend on the complex structure of the torus as well as a number of fugacities corresponding to various symmetries enjoyed by the two-dimensional quiver theory. In particular, it will always depend on two parameters that correspond to rotating the transverse to the strings’ worldsheet in the six-dimensional worldvolume of the SCFT. In addition to this, the elliptic genus will depend on a number of fugacities parametrizing the Cartan of the flavor symmetry group of the worldsheet theory. In the F-theory picture these fugacities correspond to Kähler parameters of the resolved elliptic fiber of the Calabi-Yau.
The elliptic genus encodes detailed information about the spectrum of the strings. Being able to reproduce this information with an alternative method is therefore an important check of the validity of the quiver theory. This can be achieved by exploiting duality between F-theory and M-theory Vafa:1996xn (); Morrison:1996na (), and in particular the relation between D3 branes on one side and M2 branes on the other Klemm:1996hh (). This duality relates F-theory on (where is an elliptically fibered Calabi-Yau threefold) to M-theory on ; under this duality the complex structure of the gets mapped to the Kähler parameter of the elliptic fiber of on the M-theory side. D3 branes wrapping the base as well as correspond to strings wrapped on the torus. It turns out Klemm:1996hh () that the BPS states of a configuration of strings with units of momentum along a circle get mapped to BPS M2 branes wrapping the base times and the elliptic fiber times; furthermore, if a string BPS state has nonzero flavor symmetry charges, the corresponding BPS M2 brane will also wrap additional curves in .
The precise relation between the counting of BPS states on the two sides turns out to be Haghighat:2013gba ():
where is the topological string partition function that counts BPS configurations of M2 branes on the M-theory side (or, equivalently, 5d BPS states of the theory arising from M-theory compactification on ), and is the elliptic genus of strings of the six-dimensional SCFT. Furthermore is the Kähler class of the base and is proportional to 111The proportionality factor will be a combination of Kähler classes in the resolution of the elliptic fiber and its exact form can be determined by requiring to be invariant under the monodromy associated to , as in Candelas:1994hw ().. In other words, the topological string partition function is given by a sum over elliptic genera of the six-dimensional strings, except for a simple piece which captures contributions coming from vector multiplets and can be obtained straightforwardly.
In the next section we will discuss the case of the theory and determine the quiver describing its strings. Furthermore, we will find an integral expression for the elliptic genera of these strings; we will evaluate these integrals explicitly for one and two strings and present an answer in a form from which BPS degeneracies may be readily extracted. In Section 4.2.3 we will compute the topological string partition function of the corresponding Calabi-Yau geometry and extract BPS invariants which can be shown to agree with the elliptic genus computations.
3 Quiver for the model
We now turn to the strings of the (1,0) SCFT in 6d and construct a quiver theory that describes their dynamics. Recall that the six-dimensional theory is obtained by compactifying F-theory on the following orbifold geometry:
where the orbifold action on the complex coordinates of is given by:
and . To obtain a F-theory construction in terms of a non-compact elliptic Calabi-Yau one has to first blow up the singularity at . The resulting space is described by the bundle
with the singular elliptic fiber over the base. The resolution of this fiber leads to the fiber in the Kodaira classification of elliptic fibrations. In fact, one can obtain an infinite family of six-dimensional theories by taking the singular fiber to be of type , with . Lowering corresponds in physical terms to Higgsing. This geometry can be equivalently viewed in the weak coupling limit as a type IIB orientifold of the singularity Sen:1997gv (). In this limit the singular elliptic fiber over can be interpreted as the presence of D7-branes wrapping the together with an orientifold 7-plane. This gives rise to a gauge theory in the six non-compact directions parallel to the branes. Furthermore, D3-branes wrapping the give rise to strings in the six-dimensional theory.
In the following we study the worldsheet theory of these strings and obtain a quiver gauge theory description for it. The particular orientifold we are interested in has been studied in some detail in Uranga:1999mb () and we shall describe it here briefly. The theory we want to study is type IIB theory on , modded out by , where is world-sheet parity and acts as
with , parametrizing the two complex planes in . The D7-branes wrapping the can, in the singular limit,be thought of as D5-branes probing together with an orientifold 5-plane at . Similarly, D3-branes become D1-branes whose worldvolume theory we wish to determine. We start by describing the brane system probing and successively add the orbifold and orientifold actions. Before the orbifolding, the theory living on the D1-branes is a gauge theory with one adjoint and fundamental hypermultiplets, where denotes the number of D5-branes Haghighat:2013tka (). To summarize, we have the following massless field content on the worldvolume:
where the indices represent the fundamental representations of the two groups rotating the directions (the directions orthogonal to D1 but parallel to D5) while are indices for the ’s rotating (the directions orthogonal both to D1 and D5). Next, we embed the orbifold action generated by
into and hence obtain the following action on fields with -index
The resulting theory has supersymmetry and its field content can equally well be described in terms of chiral superfields , , , , , , a gauge superfield , and Fermi superfields . The decomposition of (4,0) fields in terms of components is as follows:
Following Douglas:1996sw (), the theory one obtains after the orbifold (3.11) is a quiver gauge theory whose gauge nodes correspond to the nodes of the affine quiver (for more general orbifolds one would obtain the affine quiver). The fields that do not carry a index are localized at each node, while those with a index connect adjacent nodes Haghighat:2013tka (). In order to turn this D1 quiver into a D3 quiver one needs furthermore to turn off D1 brane charge and instead introduce D3 branes wrapped around blow-up cycles of the resolved singularity. This transformation corresponds to removing the last node of the inner quiver as well as all links ending on it. Correspondingly, in the case of the singularity which is of interest here, the single remaining gauge node contains an vector multiplet ) and an adjoint hypermultiplet .
Next, we come to the orientifolding. Orientifolds of orbifolds were discussed in Uranga:1999mb (). There it was found that for orbifolds the gauge group in the -twisted () D-brane sector is of -type if is even and of -type if is odd. This implies for our case that we have an orthogonal gauge group on the D5-branes in the untwisted sector and a symplectic one on the D5-branes of the -twisted sector. Furthermore, anomaly cancellation in six dimensions fixes the ranks of the gauge groups such that the allowed configurations are Uranga:1999mb (). This corresponds to having D5-branes together with an O5-plane at the Orbifold singularity. Uplifting this to F-theory one finds that the case is obtained from the fiber while the cases come from fiber types.
In fact, in the F-theory setup the six-dimensional theory has gauge group and two flavor nodes. The situation here is analogous to the case: the two flavor nodes correspond to non-compact D7 branes intersecting the compact curve as shown in Figure 4 (see for example DelZotto:2014hpa () for more details about the geometry).
From the point of view of the two-dimensional theory living on the strings the gauge node and the flavor nodes descend to flavor nodes. Furthermore, orientifolding implies that transform in the symmetric (that is, adjoint) representation of 222Orientifolding amounts to projecting the gauge group from to ., while transform in the antisymmetric representation Gimon:1996rq (). It is interesting to note that the introduction of two nodes is also necessary from gauge anomaly cancellation in two dimensions which will be reviewed later. The resulting two-dimensional quiver is the one depicted in Figure 5.
The various fields in the quiver have different charges with respect to the two that rotates the directions. We denote the fugacities by , as they are the same parameters that appear in the Nekrasov partition function. For completeness we also present the charges of the fields of the quiver under the different ’s and gauge groups; these charges are obtained directly by the orbifolding construction, as in Haghighat:2013tka ().
We have arrived at the conclusion that the theory for strings is an gauge theory with a (2,0) vector multiplet and a Fermi multiplet in the adjoint (i.e. symmetric) representation, two chiral multiplets in the antisymmetric representation, and two chiral multiplets , each in the bifundamental representation of . If one also has Fermi multiplets in the bifundamental of . One can pick a basis of the weight lattice of in which the fundamental representation has weights . In this basis, the symmetric representation has weights , while the antisymmetric representation has weights . We also pick to be the Cartan parameters dual to the weights of the fundamental representation of , and and to be the Cartan parameters for the two flavor groups.
Let us next comment on gauge anomaly cancellation in two dimensions. The contribution of chiral fermions running in the loop to the anomaly is proportional to the index of their representation defined as:
Furthermore, left-moving fermions contribute with a positive sign to the anomaly while right-moving ones contribute with a negative sign. Thus, for our particular quiver we obtain the following result:
where use has been made of the identities
and the fact that the fundamental fields transform in real representations and therefore only have half the number of degrees of freedom.
3.1 Localization computation
Having written down the field content of the two-dimensional theory of strings, it is straightforward to compute its elliptic genus, following the localization computation of Benini:2013xpa (); Benini:2013nda (). The elliptic genus is given by a contour integral of a one-loop determinant:
where is a -form on the complex-dimensional space of flat connections on , which is a complex torus parametrized by variables , and the contour of integration is determined by the Jeffrey-Kirwan prescription JeffreyKirwan (). The one-loop determinant is obtained by multiplying together the contributions of all multiplets and takes the following form:
where333We use the following definitions for the Dedekind eta function and Jacobi theta function : (3.18) where .
and , , , , . The integral itself is then obtained by computing a sum over Jeffrey-Kirwan residues of the one-loop determinant:
where labels poles of and the role of will be clarified shortly. In the following sections we will compute the residue sum for one and two strings, in which case the evaluation of Jeffrey-Kirwan residues turns out to be straightforward and we do not need to resort to the full-fledged formalism.
For a single string, the one-loop determinant is given by a one-form:
One first needs to identify the singular loci of the integrand. Each of the theta functions in the second line of (3.25) determines a (0-dimensional) singular hyperplane within the one complex dimensional space spanned by , for a total of distinct singular points at
To determine which poles contribute to the residue sum, one needs to consider the normal vectors to the singular hyperplanes. In this case, the normal vector is simply , where the sign is the one multiplying in (3.26). The data that enters the Jeffrey-Kirwan residue computation corresponds of two quantities: the position of the pole in the plane and a choice of a vector . In this case, we can choose either ; let us pick . For two-dimensional theories, it can be argued that once the sum over residues is performed the answer is independent of the choice of . Next, one picks the poles satisfying the property that lies within the one-dimensional cone spanned by the vector normal to the corresponding hyperplane. In this trivial example one finds that only the following poles contribute to the integral:
Evaluating the Jeffrey-Kirwan residues in this situation corresponds to summing over the ordinary residues at these poles. Summing over the eight residues and dividing by leads to the following answer:
Note some features of this expression: The existence of theta functions in the denominator which depend on fugacities suggests that the continues to be carried by some bosonic degrees of freedom in the IR. Also, the fact that the expressions include a mixture of (captured by ) and suggests a non-trivial structure for the theory which makes it unlikely to correspond to a free theory in the IR. It would be interesting to identify the non-trivial CFT whose elliptic genus is given by the above expression. Perhaps ideas similar to the ones employed in Gadde:2014ppa () can be used to do this.
The computation for two strings proceeds analogously; first, one should identify the hyperplanes in the two-dimensional space along which the denominator of vanishes. There are such hyperplanes:
where . For concreteness, let us focus from now on to the case where , keeping in mind that the computation for arbitrary proceeds analogously. We display the vectors normal to the hyperplanes, as well as our choice of , in Figure 6.
The next step is to identify the points at which hyperplanes intersect. The computation of Jeffrey-Kirwan residues is simplified by the fact that for generic values of at most two hyperplanes intersect at the same time. The poles whose residues contribute to the elliptic genus are those for which lies within the cone spanned by the vectors normal to the corresponding hyperplanes. For example, since lies in the cone spanned by and , but not in the one spanned by and , the residue evaluated at
will contribute, while the one at
will not. Following this prescription, one arrives at the following list of poles whose residues contribute to the computation:
The prescription outlined above also picks up some additional poles, but they do not contribute to the elliptic genus since the numerator of turns out to vanish for them. Therefore, the elliptic genus of two strings is obtained by summing over the residues that correspond the 112 poles listed in Equations (3.34)–(3.38). In practice, one can exploit Weyl symmetry to show that the residues of poles and are identical to the ones of . For the same reason, one can set in (3.38) and multiply the corresponding 24 residues by a factor of .
After these considerations, we are ready to write down the elliptic genus of two strings:
where we have divided by an overall factor of 8 , and the residues have the following explicit form: