\mathcal{N}=1 SCFTs from Brane Monodromy

SCFTs from Brane Monodromy

Abstract

We present evidence for a new class of strongly coupled superconformal field theories (SCFTs) motivated by F-theory GUT constructions. These SCFTs arise from D3-brane probes of tilted seven-branes which undergo monodromy. In the probe theory, this tilting corresponds to an deformation of an SCFT by a matrix of field-dependent masses with non-trivial branch cuts in the eigenvalues. Though these eigenvalues characterize the geometry, we find that they do not uniquely specify the holomorphic data of the physical theory. We also comment on some phenomenological aspects of how these theories can couple to the visible sector. Our construction can be applied to many SCFTs, resulting in a large new class of SCFTs.

SCFTs from Brane Monodromy

Jonathan J. Heckman, Yuji Tachikawa,

[4mm] Cumrun Vafa, and Brian Wecht



School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA


jheckman,yujitach,bwecht@ias.edu


Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA


vafa@physics.harvard.edu

Abstract

September 2010

1 Introduction

The interplay between string theory and geometry provides a rich template for realizing many quantum field theories of theoretical and potentially experimental interest. A common theme is how geometric insights translate to non-trivial field theory statements, and conversely, how statements about the field theory allow us to probe details of the string geometry.

One class of theories which have recently been extensively studied is based on compactifications of F-theory, in part because such constructions combine the flexibility of intersecting D-brane configurations with the more attractive features of GUT models. See [1, 2, 3, 4, 5] and the references in [6] for a partial list of work on F-theory GUTs. E-type geometric singularities play an especially important role in realizing aspects of a GUT model such as the Yukawa coupling. In most cases, the focus of such models has been to realize weakly-coupled field theories which reproduce at least the qualitative features of the Standard Model. These models can also accommodate hidden sectors, which can be added as separate sectors.

D3-branes provide an additional set of ingredients which are present in such constructions. The presence of background fluxes often causes the D3-branes to be attracted to the E-type points of the geometry [7]. An important feature of such points is that the axio-dilaton is of order one, and thus the D3-brane worldvolume theory is strongly coupled. Depending on the details of the geometry, we expect to realize a wide variety of possible strongly coupled quantum field theories. The study of such field theories is of independent interest, but is additionally exciting because the proximity to the visible sector suggests a phenomenologically novel way to extend the Standard Model at higher energy scales.

D3-brane probes of F-theory singularities have been considered in various works, for example [8, 9, 10, 11, 12, 13, 14]. In many cases of interest, the probe theory becomes an interacting superconformal field theory (SCFT). Geometrically, we engineer these SCFTs by considering the D3-brane probe of a parallel stack of seven-branes with gauge symmetry . When the stack is flat, this provides a geometric realization of rank 1 SCFTs with flavor symmetry , where can be . Denoting by the coordinates parallel to the stack, and by the coordinate transverse to the stack, in the probe theory becomes a chiral superfield parameterizing the Coulomb branch, and become a decoupled hypermultiplet . In addition, we have chiral operators in the adjoint representation of parameterizing the Higgs branch. When we have a weakly coupled UV description of the theory, these operators can be written as composites made from quarks, e. g. .

We study deformations of these theories by tilting the seven-branes. This tilting is described by activating a position-dependent vev for an adjoint-valued scalar on the stack. In the probe theory, this tilting corresponds to the superpotential deformation [7]:

(1.1)

The eigenvalues of specify the location of the seven-branes. Geometrically, this tilting process is known as “unfolding a singularity,” and is specified purely in terms of the Casimirs of . A given matrix will satisfy a characteristic equation of the form:

(1.2)

where the ’s depend on the coordinates . The most generic possibility is therefore that the eigenvalues for will have branch cuts, a phenomenon known as “seven-brane monodromy”. Such monodromies are a natural ingredient for F-theory GUT models [15, 16, 17, 18, 19].

Although the unfolding is dictated purely by the Casimirs of , we find that distinct -deformations with the same Casimirs can produce strikingly different behavior in the IR. In other words, the eigenvalues of are not enough to specify the holomorphic data of the physical theory. This freedom opens a new avenue for realizing intersecting seven-brane configurations, which to this point appear to have been relatively unexplored.111Some discussion of the massless open string spectrum for “exotic” intersecting brane configurations based on a nilpotent Higgs field has appeared in [20]. Related issues in the context of F-theory compactifications will be discussed in [21].

In this paper, our aim will be to elucidate these differences from the point of view of a probe D3-brane. Along the way, we will provide evidence for a large class of new deformations of theories.222Though the setup is quite similar to that discussed in [9], here we emphasize the eight-dimensional gauge theory interpretation of these deformations, some aspects of which cannot be seen from the Calabi-Yau fourfold alone. Moreover, we find that the scaling dimensions of operators differ from what was found in [9], a point we shall comment more on in Appendix A. We will provide various consistency checks that these deformations lead to new interacting SCFTs. For example, assuming we realize an SCFT, we can use -maximization [22] to determine the infrared R-symmetry. We can also check that the scaling dimensions of operators remain above the unitarity bound, and that the central charges of the SCFT decrease monotonically after further deformations of the theory. Moreover, in some cases we can argue that a further deformation induces a flow to a well-known interacting SCFT. In such cases, the -deformed theory can be viewed as an intermediate SCFT between the original SCFT and another IR SCFT.

The rest of the paper is organized as follows. In section 2 we introduce the brane setup for realizing the SCFTs of interest. As a first example, in section 3 we discuss the deformation of theory with four flavors, corresponding to a D3-brane probing a singularity. Next, in section 4 we turn to deformations of a broader class of non-Lagrangian theories, determining a general expression for the IR R-symmetry. We also discuss some of the geometric content associated with this class of deformations. Section 5 considers specific examples of deformations. In section 6 we briefly consider further deformations of such theories by superpotential terms fixing the vev of the and , and in section 7 we indicate very briefly how the coupling of the SCFT sector to the visible sector works. Section 8 contains our conclusions. In Appendix A we briefly review some standard tools from field theory.

2 Geometric preliminaries

In this section we review the general setup of D3-branes probing an F-theory singularity. Our aim here is to explain how the geometry of an F-theory compactification filters down to a D3-brane probe theory.

We are interested in F-theory singularities filling . The neighborhood of a singularity is a small patch in parameterized by , and . Over each point is an auxiliary elliptic curve, whose complex structure modulus is . The elliptic curve is given in Weierstrass form by:

(2.1)

where the coefficients and fix .

The locations of the seven-branes are specified by the zeros of the discriminant:

(2.2)

Each irreducible factor of determines a hypersurface in . Throughout this paper we shall be interested in the behavior of a D3-brane probing a seven-brane located originally at . In this convention, the coordinates and denote directions parallel to the seven-brane.

Away from the seven-branes, the worldvolume theory of a D3-brane is given by a gauge theory, with holomorphic gauge coupling whose value is controlled by the position of the D3-brane. From the viewpoint of the open strings, the change in the coupling reflects the renormalization effects from the 3-7 strings. As the D3-brane moves close to a seven-brane, some of these states will become light, and in some cases we expect to recover an interacting SCFT.

When the seven-brane is flat, the probe theory is an SCFT. These theories are described by a D3-brane probing a single parallel stack of seven-branes extending along . The gauge symmetry on the seven-brane translates to a global symmetry of the D3-brane probe theory. The probe theory has a Coulomb branch parameterized by , the position of the D3-brane transverse to the seven-brane. The dimension of is given in Table 1. The Higgs branch of the probe theory corresponds to dissolving the D3-brane into the seven-brane stack as a gauge flux. Examples of such probe theories include with four flavors [8], strongly-coupled theories of the type of Argyres and Douglas [23], and the E-type theories found in [24, 25].

Table 1: The scaling dimension of the Coulomb branch parameter for the SCFTs realized by a D3-brane probing an F-theory singularity at constant dilaton. Here, the for respectively correspond to the Argyres-Douglas theories arising from an gauge theory with flavors.

Our main interest in this paper is to engineer theories by considering more general seven-brane configurations. The positions of the seven-branes are dictated by a complex scalar on the seven-brane stack taking values in the adjoint representation of . Letting depend on corresponds to tilting the original stack of seven-branes. Geometrically, this corresponds to performing a deformation of the original Weierstrass model via:

(2.3)

Terms in and correspond naturally to expressions made from the Casimirs of [26]. In the D3-brane probe theory this shows up as the superpotential deformation:

(2.4)

where and are both in the adjoint representation of , and -invariant information like the positions of the seven-branes is characterized by the Casimirs of . Thus, from a given one can naturally construct and .

This deformation breaks supersymmetry to when , but is admissible as a background field configuration of the seven-brane once gauge fluxes are taken into account [2, 7]. Note that this deformation couples the original theory to the hypermultiplet . The theory then has a moduli space parameterized by and , on a generic point of which the low energy limit is just a theory. The physical coupling of this low-energy vector multiplet is holomorphic in and , and is given by a family of curves [27]. In this F-theoretic setup the required family of curves is exactly the elliptic fibration (2.3). It is worth noting that in contrast to the case, this curve no longer describes a full solution of the low-energy theory because the behavior of the chiral multiplets is no longer controlled by the gauge coupling. The homogeneity of the curve can still be used to fix the relative scaling of further mass deformations and the Coulomb branch parameters. We shall meet examples of this analysis later on.

Although should be holomorphic without branch cuts, its eigenvalues can have branch cuts as we vary . In cases where such structures exist, we say these deformations exhibit “seven-brane monodromy.” A simple example is the matrix:

(2.5)

which has eigenvalues . Closely related to seven-brane monodromy is the generic presence of nilpotent mass deformations. For example, in equation (2.5), the constant contribution is a mass matrix which is upper triangular, and thus nilpotent. This is the source of the monodromy after a further deformation by a lower triangular part proportional to .

Such nilpotent mass deformations are already of independent interest. Indeed, because all of the Casimirs of a nilpotent matrix vanish, the curve of the deformed theory is identical to the original curve. But this apparent invariance of the holomorphic geometry is deceptive. Clearly, we have added a mass term to the theory which breaks supersymmetry in the probe theory and moreover gives a mass to some of the degrees of freedom of the original theory. Thus, we see that the Calabi-Yau fourfold alone does not fully specify the holomorphic data of the compactification. This does not immediately contradict the standard lore in much of the F-theory literature that the Calabi-Yau fourfold and flux data are enough to specify the compactification. The point is that Casimirs of and of the gauge flux are not enough. This is quite exciting from the perspective of F-theory compactifications, because it points to a far greater degree of flexibility in the specification of a compactification, based on more holomorphic data than just the Casimirs of .

The greater freedom in specifying is also connected with the presence of singular fibers in the Calabi-Yau geometry. Indeed, in a compactification in which all singularities of the geometry have been deformed away, our general expectation is that there is no ambiguity in reconstructing a unique choice of . From the perspective of the seven-brane gauge theory this is equivalent to asking whether a given characteristic equation for uniquely determines the physics of the seven-brane configuration. It would be interesting to study whether this natural physical expectation is always met for a general unfolding.

Before going further, let us point out that there is no distinction among , , and from the ten-dimensional point of view. For example, consider the configuration

(2.6)

This can be thought of as either a deformation of an seven-brane at , or as a deformation of an seven-brane at . This suggests that the same theory can be realized by deformations of two different SCFTs. We hope to come back to this question in the future.

3 Probing a Singularity

The cases of E-type flavor symmetry in which we are interested do not have an obvious Lagrangian description. As a warm-up, we start in this section by studying deformations of a D3-brane probing a singularity of F-theory; this setup leads to a Lagrangian theory. The weakly coupled theory of a D3-brane probing a singularity is given by an gauge theory with four quark flavors for [8]. The superpotential is dictated by supersymmetry:

(3.1)

The theory has an flavor symmetry.

The moduli space is characterized in terms of gauge-invariant operators built from the elementary fields. The Coulomb branch of the theory is parameterized by the coordinate:

(3.2)

Next consider the Higgs branch, touching the Coulomb branch at . The Higgs branch is parameterized in terms of composite meson operators quadratic in the quarks. They transform in the adjoint representation of . It is convenient to decompose them in terms of irreducible representations of :

(3.3)

Although at this stage we can write the ’s in terms of the ’s, when we later explore E-type theories, we will have nothing to work with except the analogous operators.

In addition to the degrees of freedom described above, there is a free hypermultiplet representing the position of the D3-brane parallel to the seven-brane. Thus, we initially have two decoupled CFTs. Tilting the seven-branes to some new configuration couples these two CFTs, and generates a non-trivial flow to a new theory.

In F-theory, the geometry of a singularity is given by the Weierstrass equation:

(3.4)

where is a free parameter. The modulus of the torus (3.4), depending on , gives the coupling of the theory.

In the remainder of this section we study in greater detail deformations of the probe theory. In particular, our aim will be to present evidence that these theories realize interacting SCFTs in the IR. For the most part, we focus on nilpotent mass deformations such that takes values in a single Jordan block of . Many of the checks we perform in the following subsections can be viewed as elucidating more details of the interacting SCFT.

3.1 Mass Deformations and the Curve

Mass deformations of the theory correspond to deformations of the form for independent of and . The Casimirs of determine deformations of the original curve:

(3.5)

where the ’s and ’s correspond to degree polynomials in the masses formed from expressions built from the Casimirs of . Using the formulation in terms of the ’s, these deformations can be written as:

(3.6)

Additionally, we can consider field-dependent mass deformations which couple the theory to the free hypermultiplet . From the perspective of the geometry, the only change is that now the and in (3.5) can depend on the coordinates and .

Let us now consider a deformation by the nilpotent mass term:

(3.7)

As the mass terms are nilpotent matrices, all Casimirs built from these operators are trivial, and the curve is identical to the curve.333 In the theory, we have added mass terms for some of the quark flavors. As the theory flows from the UV to the new IR theory, the beta function of the gauge coupling is non-zero. This raises the question: Since we have initiated a flow of the gauge coupling, why does the curve predict that on the Coulomb branch of the deformed theory there is no change to the value of ? To see what is happening, let us note that there are three mass scales of interest. First, there is the mass scale associated with the nilpotent deformation. In addition, there is the scale of the W-bosons of the Higgsed gauge theory. At scales , we have integrated out a quark flavor from the gauge theory, and the coupling increases as the theory flows to the IR. However, below the scale , the W-bosons of the gauge theory are also massive, and this in turn counters the effects of the initial decrease. In particular, we see that as the theory flows to the scale , the value of the gauge coupling has flowed back to its original value. All of this behavior is automatically encoded in the geometry. See [28] for related discussions.

The existence of an curve constrains the relative scaling dimensions of operators in the deformed theory, as in the case. Since the curve is no different from the curve, this implies that the relative scalings of and the mass deformations in the new theory obey the same relations as in the original theory. Let us now check how this works from the viewpoint of the Lagrangian.

To this end, consider the effective superpotential:

(3.8)

after integrating out and .

Assuming we have flowed to an interacting CFT, let us now compute the relative scaling dimensions of the various mass deformations. For simplicity, we restrict attention to the mass parameters transforming in the adjoint representation of . Taking the superpotential of equation (3.8) to be marginal in the IR, we learn that the dimensions of the ’s are related to the dimension of by:

(3.9)

where is the scaling dimension of and . The corresponding mass terms are

(3.10)

where the dimension of the ’s can be easily obtained from the data given above. In order to match these mass deformations to quantities of the curve, we must form invariants under the surviving flavor symmetries. We obtain four invariants with corresponding dimensions:

(3.11)

We now compare these invariants to invariants of the theory. First note that these expressions transform non-trivially under the (now broken) original flavor symmetry. Including appropriate factors of to form flavor invariants of the original theory, we see that these expressions descend from the theory Casimirs:

(3.12)
(3.13)

Note that the same relative scalings are obtained once we set , as appropriate upon treating as a marginal operator in the IR theory. Similar considerations hold for other group theory invariants, and for more general deformations as well.

3.2 -Maximization and Nilpotent Mass Deformations

Nilpotent mass deformations are of particular interest because although they constitute a non-trivial deformation of the theory, they do not alter the geometry of the curve. For simplicity, we again confine our analysis to nilpotent deformations where takes values in . Explicitly, we consider the mass deformations:

(3.14)

Let us first consider in detail the case . Upon integrating out the quarks , we are left with an gauge theory with chiral superfields , , , and with an effective superpotential (3.8). By inspection, one finds

(3.15)

We require that the IR R-symmetry is non-anomalous, and that the two superpotential terms in equation (3.8) are marginal in the IR. We find that and can be expressed in terms of , which can then be determined by -maximization. The calculation is straightforward; we find a local maximum at . As can be checked, all gauge-invariant operators have dimensions above the unitarity bound.

Let us now generalize this to the cases . Upon integrating out the heavy quarks, we are left with the effective superpotential:

(3.16)

Again we impose the conditions that is non-anomalous and that the effective superpotential is marginal in the IR. For both we find one undetermined parameter which we fix by -maximization.

The behavior of the theories is quite similar. The case of presents a new phenomenon, in that here it would appear that there are unitarity bound violations. Indeed, assigning R-charges to the operators and then requires either that both operators saturate the unitarity bound, or that one operator violates this bound.

What are we to make of this case? One possibility is that this theory may not be an interacting conformal theory. This does not appear very plausible, because nothing drastic appears to be happening to the geometry. For example, we can still move onto the Coulomb branch and compute a non-trivial dependence of on the parameters and . In what follows, we shall assume that much as in [29], an accidental symmetry appears which rescues only this individual operator.

We now recompute the dimensions for the quarks in the IR theory under the assumption that only decouples in the IR, with an associated emergent which only acts on . This emergent can be included in -maximization via the procedure described in [30]. The scaling dimensions for the various fields are then given by the values shown in Table 2.

Table 2: Dimensions of the elementary fields obtained from nilpotent deformations of the probe theory. The operators of the CFT are specified by gauge invariant expressions built from these elementary fields. Entries with an “X” indicate fields which have been integrated out of the low energy theory.

Note that although the dimensions of the ’s are less than one, all of the composite operators built from two ’s have dimension above the unitarity bound.

3.3 Large Limit

In the previous section we discussed the specific case of a single D3-brane probing a singularity. In the context of brane constructions, it is natural to consider the theory obtained by D3-branes probing the same configuration. The theory is given by an gauge theory with four quark flavors and an additional hypermultiplet in the two-index antisymmetric representation [10, 12]. The superpotential for this theory is:

(3.17)

This is an SCFT. In addition, there is a free hypermultiplet describing the motion of the center of mass for the configuration.

Let us now deform this theory by nilpotent masses. Assuming that the IR limit is an SCFT without any accidental symmetry, we perform -maximization and expand the result to first order in . The R-charge assignment for and the scaling dimensions for the elementary fields are given in Table 3.

Table 3: Dimensions of the elementary fields for the large limit of the probe theories with a nilpotent mass deformation turned on. Note that although the dimensions of some elementary fields fall far below the unitarity bound, the scaling dimensions of composite operators involving these fields can still remain above this bound.

In this table, indicates any flavors that couple to through , and indicates flavors that couple through , which as in the previous section is present after integrating out the massive quarks. Looking at the table, we see that for , some of the operators quadratic in will fall below the unitarity bound. As before, we can assume that these operators become free fields and decouple from the IR theory and re-do -maximization. However, because in the large limit the number of offending operators is , there is no change at this order to the R-charge assignments, since is .

We still believe that this system flows to an IR SCFT. Consider the related deformation by , given by symmetrizing the original deformation. In the UV theory, this corresponds to a deformation which preserves supersymmetry, and leads to the theory describing D3-branes probing singularities. If we now consider deforming first by , and after some long RG time deforming further by , we then flow to this same theory. Assuming that this further deformation can be added at an arbitrarily late RG time, this strongly suggests that the above description of the theory is legitimate, and that the further deformation by induces a flow from the -deformed theory to the deformed theory. The only concern here would be that somehow the further deformation by is not valid in the deep IR, though this seems rather implausible. Furthermore, nothing dramatic seems to be happening to the F-theory geometry, so it seems reasonable to assume that the CFT is still present.

4  Deformations: Generalities

In the previous section we studied the theory of a D3-brane probing a singularity, relying on a Lagrangian formulation of the theory to analyze the effects of various deformations. In many cases of interest, however, we do not have a weakly coupled Lagrangian description, as when we have a flavor symmetry .

In this section we consider superpotential deformations of the form:

(4.1)

Our main assumption will be that we obtain a CFT in the infrared, and we shall present some consistency checks of this statement. Assuming we do flow to a new CFT, it is important to determine the scaling dimensions of operators, and the values of the various central charges in the infrared. This is of intrinsic interest, but is also of interest in potential model building applications where the degrees of freedom from the D3-brane couple to visible sector degrees of freedom associated with modes localized on seven-branes.

In the case where has no constant terms, the results of [31] establish that this deformation is marginally irrelevant, and so induces a flow back to the original CFT [7]. For this reason, we shall focus on the case where has a non-trivial constant part. Further, since we are interested in the structure of deformations where the geometric singularity is retained at , we also demand that all Casimirs of vanish at . Hence, the constant part of is a nilpotent matrix. This reinforces the point that nilpotent deformations go hand in hand with generic deformations of an F-theory singularity.

We now describe the general procedure for obtaining the R-charge assignments for the matter fields after turning on a combination of relevant and marginal deformations. We first catalogue the symmetries of the UV theory, and then the surviving symmetries compatible with the deformation . The R-symmetry of the theory contains two ’s, which we call and ; see Appendix A for details. The remaining global symmetries are the non-abelian flavor symmetry and the generators which rotate the fields of the free hypermultiplet. Since the infrared R-symmetry is a linear combination of abelian symmetries, it is enough to focus on the Cartan subalgebra of the flavor symmetries , where is the rank of . We shall denote by the corresponding generators, where runs from to . Finally, we denote by the generator under which has charge . Under , and the , the charges of the original operators are given in Table 4.

Table 4: Assignment of UV charges where denotes the dimension of in the UV. The charges refer to the bottom component of each supermultiplet.

The deformation explicitly breaks some of these flavor symmetries. The infrared R-symmetry will then be given by a linear combination of the UV symmetries, and possibly some additional emergent flavor symmetries. In what follows, we shall assume that there are no emergent abelian symmetries in the infrared. Then the R-symmetry is given by

(4.2)

The coefficient of has been chosen for later convenience.

Let us determine which symmetries are left unbroken by the original deformation. At first, it may appear that no solution is available which is compatible with a deformation of the form given by equation (4.1). Indeed, though there are at most flavor symmetries, there will typically be far more independent entries in the matrix . However, in the infrared, our expectation is that some of these deformations will become irrelevant operators. For example, if one of the entries of contains a term of the form , we expect this term to be irrelevant in the infrared. This also matches with geometric expectations. The geometry is well-approximated to leading order by the lowest degree polynomials in the . Higher order polynomials correspond to subleading features of the geometry. Since we are only interested in a small neighborhood of the region where , much of this information is washed out in the infrared. We shall return to this theme later when we discuss the UV and IR behavior of the characteristic polynomial for .

The flow to a new CFT is dominated by the operators of lowest scaling dimension. In the UV, the most relevant terms are the constant matrices. By assumption, the constant matrix is nilpotent, and so by a unitary change of basis, we can present it as an upper triangular matrix. For simplicity, in what follows we assume that the constant part of denoted by decomposes as a collection of blocks, each of which corresponds to an upper triangular matrix:

(4.3)

We assume that the upper triangular matrices are generic in the sense that the first superdiagonal has only nonzero entries.

Associated with each block is an subalgebra of the original flavor symmetry group with generators and in the spin representation, satisfying

(4.4)

In this basis, the generator is:

(4.5)

Most of the data associated with each block of the decomposition in equation (4.3) drops out in the infrared. To see how this comes about, consider the entries of the upper triangular block . Along each superdiagonal of the matrix, the value of the charge is the same. Moving out from the diagonal, the entries of on the first superdiagonal have charge , the second have , and so on until the upper righthand entry which has charge . The operators which pair with in the deformation have respective charges down to . Since all operators on the same superdiagonal have the same charge, we see that the one with the charge of smallest norm will dominate the flow. In other words, the first superdiagonal of dominates the flow. In the following we assume takes the form of a nilpotent Jordan block from the start.

Then the operators have charge . The requirement that is marginal for all in the IR can now be satisfied by the choice

(4.6)

where

(4.7)

is the generator of the diagonal subalgebra. The coefficients are still undetermined. We can now organize the operators into representations of this diagonal . We denote by an operator with spin under this .

To fix the value of the ’s, we need to know which of the remaining operator deformations are most relevant in the IR. In the case of a deformation by a constant , the free hypermultiplet decouples, and we can neglect the ’s. We therefore focus on the additional effects of -dependent deformations. Unitarity dictates that and the have dimensions greater than or equal to one. This means that if two or more ’s multiply an operator , their product will be irrelevant. Hence, it is enough to focus on contributions which are linear in the .

Most of the deformations linear in the will also be irrelevant. Given two operators and which have different charges and , the operator with the larger charge will have lower dimension, and will therefore dominate the flow.

Since the IR behavior is dictated by the operators with the highest values of , it is enough to consider the deformation by just these highest values. Let and denote the operators which respectively multiply and . The parameters are now fixed by requiring that these deformations have R-charge 2 in the IR. In terms of , and , this constraint yields:

(4.8)

We now see that for a given , the only free parameter in is .

4.1 The Curve and Relative Scaling Dimensions

Now let us study to what extent we can read off properties of the deformed theory without determining . As we have already mentioned, on the Coulomb branch of the theory, we can read off the coupling from the curve, which is the F-theory geometry (2.3). Homogeneity of this curve then predicts the relative scaling dimensions of the mass deformations to that of the Coulomb branch parameter.

The form of the infrared R-symmetry (4.6) implies

(4.9)

Therefore the unknown parameter can be eliminated in favor of the ratio , and we find

(4.10)

The dimension of the mass parameter associated with an operator is then:

(4.11)

In particular, when we form flavor invariants out of the mass parameters as in section 3.1, their ratio in the IR is the same in the UV, because the total spin of the flavor invariants is zero.

As a passing comment, let us also note that the value of the IR central charge agrees with the computation in [7]: using (A.5), we easily find

(4.12)

4.2 Characteristic Polynomials in the Infrared

Some aspects of the deformation are irrelevant in the IR. To study the possibilities, we can consider choices for which in the IR induce a flow to the same theory as the original . To indicate the UV and IR behavior we write and .

The matrix is fully characterized by terms which are at most linear in the . Indeed, as the D3-brane only probes a small patch of the geometry, it is insensitive to higher order terms in the geometry, which are effectively gone in the deep infrared. Of course these further effects can still be probed by moving a finite distance onto the Coulomb branch.

Given two different ’s, linearizing in the can produce the same IR behavior for . For example, the characteristic equations

(4.13)
(4.14)

respectively define solvable and unsolvable quintics. For , however, the term linear in drops out in the infrared. Thus, the UV and IR behavior of can be different.

For the more mathematically inclined reader, we note that the “seven-brane monodromy group” corresponds to the Galois group for the characteristic equation for . The monodromy group acts by permuting the roots of the polynomial, and is indicated by the specific branch cut structure present in the eigenvalues of . Here we see that the infrared monodromy groups which can be realized are of quite limited type.

A polynomial of the form:

(4.15)

will generically have maximal Galois group given by , the symmetric group on letters. In particular, we can take the to admit a power series expansion in the . By a general coordinate redefinition of the geometry, and a field redefinition in the CFT, we see that generically, we can take the leading order behavior of the lowest coefficients to be and .

Finding a representative with the corresponding characteristic equation is also straightforward. To illustrate the main points, let us focus on the case of given by a matrix. A representative with characteristic equation as in (4.15) can be taken in the form:

(4.16)

for appropriate . In the infrared, the relevant deformation by is:

(4.17)

for some coefficients and . The characteristic equation for is:

(4.18)

As can be checked, the monodromy group for this degree five polynomial is again .

4.3 Central Charges and -Maximization

We now fix the infrared R-symmetry using -maximization. The trial central charge can be computed using ’t Hooft anomaly matching between the UV and IR theories. Thus depends on , and , as well as the details of the Jordan block structure associated with the deformation .

Plugging (4.6) into (A.3) and rewriting it using (A.7)–(A.9), we obtain the value of the IR central charges as follows:

(4.19)
(4.20)
(4.21)

where in the above, we have introduced the parameter which measures the sizes of the nilpotent block:

(4.22)

We need to find the local maximum of given in (4.19) to find the value . There are two cases of interest, which we analyze separately. The first case corresponds to deformations where is a constant nilpotent matrix. In this case, we formally set . In addition, we must remember that the free hypermultiplet decouples, and in particular does not contribute to the central charges and . The other case corresponds to the more generic geometry in which has some linear dependence in both and . In this case, the contribution from the hypermultiplet must be included in the values of and . These values are tabulated in Appendix A.

Nilpotent Mass Case

First consider the case where is a constant nilpotent matrix. Setting in (4.19), -maximization yields an extremum at:

(4.23)

with as in equation (4.22). Note that for , we recover , corresponding to the correct branch of solutions to the quadratic equation.

Monodromic Case

Next consider the case of position-dependent where a term linear in each appears in the deformation . Applying (4.8) in (4.19) and performing -maximization, we find

(4.24)

where:

(4.25)
(4.26)
(4.27)

The choice of branch cut in equation (4.24) is fixed as in the nilpotent case.

5 Probing an Singularity

Having given a general analysis of the expected IR R-symmetry, we now turn to some examples. In fact the case analyzed in section 3 falls within the analysis presented in the last section. Here we will study the case where a weakly-coupled UV description is not available.

We first consider nilpotent mass deformations of the SCFT. We find a consistent structure of flows between various deformations of this theory. We also study the large limit of such probe theories. After this analysis, we turn to the more generic case of deformations which include a -dependent contribution. In F-theory, allowing a position-dependent profile for the field corresponds to tilting the configuration of the seven-branes. Finally, we consider some particular examples which are of interest for F-theory GUTs.

5.1 Nilpotent Mass Deformations

We now turn to deformations of the theory by valued in .444This decomposition makes the representation content under less manifest, but for our present purposes this is not necessary. The adjoint of decomposes under into the adjoint representation, and a three index antisymmetric tensor as:

(5.1)

Hence, the operators initially transforming in the adjoint representation of will decompose into singlets, and one-, two-, and three-index tensor representations of . Another feature of interest is that this also suggests a natural split between the cases of and . For , the three index representation is already the dual representation of a representation with a smaller number of indices, while for , no such redundancy is present.

For simplicity we confine our analysis to deformations where is given by a single Jordan block. The parameter introduced in (4.22) is then given by

(5.2)

Let us now comment on the representation content of the operators . Under the subalgebra specified by the Jordan block, the fundamental representation becomes a spin irreducible representation of . For the higher tensor index structures, the indices are free to range over the spin irreducible representation, subject to appropriate anti-symmetry or tracelessness conditions. Since the dimension of the operators is specified by its spin content, and thus its tensor structure in , we shall denote by the singlets, an operator in the fundamental of , an operator in the two-index antisymmetric, and so on. We denote by the operator in the fundamental with the lowest scaling dimension, with similar notation for the other ’s. The scaling dimension of is then given by (4.10) for appropriate .

Using equation (4.23) and the expressions for the operator scaling dimensions and the values of the central charges obtained in section 4.3, we find the values for the various parameters given in Table 5.