Dualities of Deformed \mathcal{N}=2 SCFTs from Link Monodromy on D3-brane States

# Dualities of Deformed N=2 SCFTs from Link Monodromy on D3-brane States

Antonella Grassi\emailgrassi@sas.upenn.edu, James Halverson\emailj.halverson@neu.edu, Fabian Ruehle\emailfabian.ruehle@physics.ox.ac.uk and Julius L. Shaneson\emailshaneson@math.upenn.edu
###### Abstract

We study D3-brane theories that are dually described as deformations of two different superconformal theories with massless monopoles and dyons. These arise at the self-intersection of a seven-brane in F-theory, which cuts out a link on a small three-sphere surrounding the self-intersection. The spectrum is studied by taking small loops in the three-sphere, yielding a link-induced monodromy action on string junction D3-brane states, and subsequently quotienting by the monodromy. This reduces the differing flavor algebras of the theories to the same flavor algebra, as required by duality, and projects out charged states, yielding an superconformal theory on the D3-brane. In one, a deformation of a rank one Argyres-Douglas theory retains its flavor symmetry and exhibits a charge neutral flavor triplet that is comprised of electron, dyon, and monopole string junctions. From duality we argue that the monodromy projection should also be imposed away from the conformal point, in which case the D3-brane field theory appears to exhibit confinement of electrons, dyons, and monopoles. We will address the mathematical counterparts in a companion paper.

\affiliation

Department of Mathematics, University of Pennsylvania,
David Rittenhouse Laboratory, 209 S 33rd Street, Philadelphia, PA 19104, USA
Department of Physics, Northeastern University, Boston, MA 02115, USA
Rudolf Peierls Centre for Theoretical Physics, Oxford University, 1 Keble Road, Oxford, OX1 3NP, UK

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Introduction

Historically, the study of D3-branes has led to a rich array of physical phenomena in supersymmetric quantum and conformal field theories. For example, at orbifold singularities D3-branes give rise to rich quiver gauge theories [1]; in F-theory [2] compactifications they realize [3, 4] a variety of Seiberg-Witten theories [5, 6] and superconformal field theories (SCFTs) of Argyres-Douglas [7, 8] and Minahan-Nemeschansky [9, 10], and also other theories [11, 12, 13, 14, 15]; finally, most famously D3-branes give rise to gravity [16, 17] in the large limit. Many of the most interesting results exist at strong coupling, but are tractable due to invariance.

F-theory itself has a plethora of strongly coupled phenomena and SCFTs beyond its D3-brane sectors. For example, its seven-brane configurations may realize exceptional gauge symmetry and seven-brane structures, which is central to certain phenomenological aspects of F-theory GUTs [18, 19, 20]; there is growing evidence that non-trivial seven-brane structures, so-called non-Higgsable clusters [21, 22], are generic [22, 23, 24, 25, 26, 27] in F-theory; and in recent years there has been a resurgence of interest in [28, 29] and SCFTs [15, 30] that arise from F-theory and in SCFTs in general [31, 32, 33]. All of these typically involve strongly coupled physics.

In this paper we initiate the study of string junctions on D3-brane theories that probe non-trivial seven-brane configurations in lower (than eight) dimensional compactifications of F-theory. Specifically, we will develop a mathematical and physical formalism for studying the spectrum of D3-brane theories at certain isolated seven-brane singularities (non-trivial self-intersections of an -locus) that should be extendable to broader classes of singularities.

One physical aspect we will study is how duality arises geometrically from deforming rather different SCFTs. Specifically, the D3-brane theory we study in this paper, which we call Theory for brevity, is a deformation of two different SCFTs realized on D3-branes in simpler F-theory backgrounds. We will call the latter two Theory and Theory , and denote their flavor symmetries as and , which can take values and with in general. Schematically, D3-brane positions relative to the seven-brane configurations in these theories appears as

where the D3-brane theory with flavor group in the non-trivial seven-brane configuration at the bottom may be obtained either from a deformation of the vertical or horizontal seven-branes of theories and . The deformed theories are necessarily dual since the D3-brane theory in the non-trivial background can be obtained from either deformation. Said differently, the coordinates that parameterize the Coulomb branches of the theories are on equal footing as spatial coordinates from a ten-dimensional perspective, and in the deformed theory the seven-branes that the D3-brane probes spread out in both directions.

The seven-brane backgrounds that we study are easily described in F-theory. We will study specific backgrounds, but our techniques should be generalizable to others as well. They are described by an elliptic fibration over with coordinates and with fiber coordinates , which in Weierstrass form are given by

 Xh:  y2=x3−zax+tb, (1)

where , ; see work [34] of Grassi, Guralnik, and Ovrut for the case. A seven-brane is localized on the locus where . Note that for the elliptic threefold defined by (1) has an isolated singularity at ; we will address the role of the singularity in this context in a sequel paper [35]. The worldvolume theory of the D3-brane at is Theory , and the SCFTs Theory and Theory are obtained by turning off the terms and in (1):

 Xf:  y2=x3+tb,Xg:  y2=x3−zax. (2)

Theories and have different flavor symmetries, which must be reduced to a common one by the deformation to Theory . The “paradox” can be seen directly in the background (1), since Theory may be obtained by taking a D3-brane to via coming in along the locus or the locus . These processes naively look like turning off mass deformations of Theory and Theory , respectively, but this cannot be the full story since then the flavor symmetries would disagree. This (incorrect) conclusion is obtained by looking too locally in the geometry, and by looking more globally the issue is resolved. Specifically, torus knots or links on which seven-branes are localized arise naturally in the geometry, and we will use them to reconcile the naive flavor symmetry discrepancy between theories , , and .

Throughout this work, our focus will be on the implications of the geometry for the D3-brane spectrum, but there are many interesting questions for future work.

The sketch of our results are as follows. As is well-known, D3-brane probes of seven-brane backgrounds in eight-dimensional F-theory compactifications have string junctions stretching between the D3-brane and the seven-brane. These describe a rich spectrum of states in non-trivial flavor representations that are generally charged both electrically and magnetically under the of the D3-brane. Mathematically, these string junctions are topologically described by elements of relative homology; they are two-cycles in an elliptic fibration over a disc relative a chosen fiber above a point , which means they are two-chains that may have boundary in . Thus, topologically a junction is . Here the elliptic fiber is the elliptic fiber over the D3-brane, so the “asymptotic charge” gives the electromagnetic charge of the junction ending on the D3-brane. There is a pairing to the integers on that is the intersection pairing on closed classes, i.e. those with . Finally, following [36, 37, 38], the set

 R:={J∈H2(X,Ep)|(J,J)=−2, a(J)=0} (3)

has the structure of an ADE root lattice. In particular, we can use the intersection pairing to compute the Cartan matrix

 Aij:=2(αi,αj)(αi,αi),i,j=1,…,rk(G), (4)

where the are those junctions that form simple roots of an underlying ADE algebra. We will label the sets with subscripts , , to denote the relevant objects in theories , , , and in particular , , define the flavor algebras , , . Non-trivial flavor representations and BPS states of () can be constructed [39, 27] from string junctions111 and are particular elliptic surfaces and elliptic fibers in those elliptic surfaces; they will be defined in Sections 0.2 and 0.3. () with , i.e. they are charged under the of the D3-brane.

What changes geometrically for the D3-brane in this paper is that the lower-dimensional F-theory background that it probes has seven-branes extending in multiple directions. The seven-brane wraps the divisor defined by

 −4z3a+27t2b=0, (5)

and locally cuts out a knot or a link on a three-sphere near the singularity . String junctions with one end on the D3-brane then have their other end on the link, and as the D3-brane traverses the link and eventually comes back to its initial position there is an associated monodromy action on the string junction states. The knot, or link, associated to equation (5) has two canonical braids representations, the -braid with strands and and the -braid with strands. These braids define two solid tubes, which we call respectively the -tube and the -tube. A transverse section of the -tube, for example, is a disc, which we call , parameterized by the angle and centered at . A transverse section gives a natural string junction interpretation of the singularity of , we then study the associated action on states. Mathematically, these are monodromies

 Mf:H2(Xf,Ef)→H2(Xf,Ef)Mg:H2(Xg,Eg)→H2(Xg,Eg), (6)

obtained from studying two one-parameter families of elliptic fibrations, and we will compute them explicitly. See Figure 1 for a pictorial representation of the -tube with , its relation to the -braid with strands, and the monodromy induced by identifying the various strands of the braid upon traversing the torus.

Though the flavor symmetries of the theories generally differ, as captured by the fact that generally , one of our main results is that

 H2(Xf,Ef)Mf≅H2(Xg,Eg)Mg. (7)

That is, the string junctions that are invariant under the link-monodromy, and thus may exist as massless states on the D3-brane theory at the isolated singularity, generate the same lattice regardless of whether one takes the or perspective. Specifically, the link-monodromy associated with the deformations222For the sake of brevity, we will from now on implicitly talk about these deformed fibers/theories without mentioning it explicitly every time. reduces the flavor algebras and to a common algebra . This leaves us mathematically with two Lie algebras at each point, which share a common reduction. Interestingly, though is sometimes non-trivial, no charged string junctions are monodromy-invariant.

In summary, the theories we study are dual deformations of two different SCFTs and the geometry shows that the deformations sometimes break the flavor symmetry of the theories, but always break the gauge symmetry as deduced by the absence of charged string junctions. This deformation yields an SCFT for the D3-brane at . One such theory, which is a deformation of the rank one Argyres-Douglas theory , exhibits a charge neutral flavor triplet that is comprised of electron, dyon, and monopole string junctions, even though none of those charged junctions survive the monodromy projection themselves.

We argue that duality also requires imposing the monodromy projection for theories away from , in which case the D3-brane theories are related to deformations of massive field theories, or deformation of one massive field theory and one SCFT. Then the geometry implies that the D3-brane theory can exhibit massive charge-neutral monodromy-invariant string junctions in non-trivial flavor representations that are comprised of electron, monopole, and dyon string junctions. The presence of this massive state, together with the absence of charged states, suggests an interpretation as confinement of an electron, monopole, and dyon.

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Review of Seven-branes and String Junctions There is a rich literature on string junctions, and we review some aspects of them here.

String junctions have been introduced [40, 41, 42, 43] as a generalization of ordinary open strings stretching between D-branes in Type II theories. They occur as non-perturbative objects in these theories and are hence closely related to F-theory, as first pointed out by Sen in [44, 45]. One introduces -strings that carry units of NS-charge and units of Ramond-charge. In this notation a fundamental Type II string corresponds to a string. Alternatively, in the context of Seiberg-Witten theory [5, 6] one can think of them as states carrying units of electric charge and units of magnetic charge. Via an action a string can be turned into a string [46]. The seven-branes are then defined as seven-branes on which strings can end. Note that, since D3-branes are invariant, any string can end on them and we need not attach a label to them. In the worldsheet description of the D3-branes, the -branes act as flavor branes. String junctions arise if several strings join at a common vertex. Since the overall charge needs to be conserved at each vertex, this means that the sum of the incoming charges is zero.

The mathematics of string junctions has been worked out in [43, 39, 36] and in [37, 23, 38]. We will review the latter description since it makes direct contact with F-theory geometries, as will be useful for describing the seven-brane backgrounds utilized in this paper. This description can be related to the former if paths from the base point to seven-branes can be chosen so as to reproduced the labels of [39], for example.

We describe an elliptically fibered Calabi-Yau -fold via a Weierstrass model, i.e. we start with the anti-canonical hypersurface in with homogeneous coordinates ,

 E:y2−(x3+fxw4+gw6)=0. (8)

To describe a fibration over some -dimensional base with canonical bundle such that the whole space is CY, and are sections of and , respectively. Models of this type always have a holomorphic section, the so-called zero section, at . The elliptic fiber becomes singular if , which means that the zero section is non-singular. We thus set from now on when we wish to study the singularities. In this case becomes singular if the discriminant vanishes, i.e. on the locus .

For such that is a Kodaira type fiber (as will be the case when string junctions are utilized), the singular fiber is an elliptic fiber where a one-cycle has vanished. In this way a vanishing cycle is associated with a zero of . In more detail, this association works as follows: Consider a discriminant with vanishing loci with Kodaira fibers. We fix a base point of and a basis of the first homology of the fiber above and choose a path starting at and ending at . Upon reaching , a cycle vanishes, and if a basis on is chosen this can be written as . This corresponds to a 7-brane along in F-Theory, cf. Figure 2.

The inverse image of the path , , is a Lefschetz thimble, which looks like the surface of a cigar and is commonly referred to as a “prong” in the junction literature. This prong is a string and has boundary , which is called the asymptotic charge in the literature. A multi-pronged string junction is then

 J=∑JiΓi, (9)

with . The absolute value corresponds to the number of prongs ending on the 7-brane and the sign specifies their orientation. The asymptotic charge of a general junction is given as . Note that string junctions with asymptotic charge are two-spheres and can be thought of as strings (perhaps passing through a D3-brane at ), whereas junctions with are strings, or perhaps a part of a larger junction, the remainder of which ends on a seven-brane.

The picture is simplified for elliptic surfaces, which can be useful for higher dimensional fibrations since, when restricted to a patch, a local model for an elliptic fibration can be thought of as a family of elliptic surfaces. Consider the case of a disc , a neighborhood in one of the bases of those elliptic surfaces that is centered at , where intersects at a point and the paths are chosen to be straight lines from to , which gives rise to an ordered set of vanishing cycles [38]. Let be the elliptic surface and . Then the form a basis, the “junction basis” on the relative homology . There is a pairing on that becomes the intersection pairing on closed classes in , i.e. those elements of that are also in . In certain cases, such as the being obtained from the deformation of a Kodaira singular fiber [47] with associated ADE group , there is a distinguished set of interesting junctions

 R:={J∈H2(X,Ep)|a(J)=0,(J,J)=−2} (10)

that furnish the non-zero weights of the adjoint representation of from the collection of seven-branes. This is the gauge symmetry on the seven-brane of the singular (undeformed limit) in which the collide, or alternatively the flavor symmetry on the D3-brane probing the seven-brane.

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D3-branes near Seven-brane Self-intersections: Traversing Links

In this paper we are interested in D3-brane theories located at certain isolated singularities in non-trivial seven-brane backgrounds; the isolated singularity is located at the self-intersections of the seven-brane. To study these theories we will first consider D3-brane theories near these singularities, and the effect on the spectrum of moving them around loops in the geometry. We will study the implications for the D3 brane at the singularity in Section 0.5.

### 0.1 The Seven-brane Background, Links, and Braids

We will take the F-theory description of the seven-brane background, utilizing a Weierstrass model as discussed. If the base of the Weierstrass model is comprised of multiple patches, then the associated global Weierstrass model across the entirety of may be restricted to a patch, giving a local Weierstrass model, which suffices here since the D3-brane sits at a point in the elliptic fibration and is affected only by local geometry.

We study a D3-brane in a particular collection of self-intersecting seven-brane backgrounds defined by the local Weierstrass model

 y2=x3−zax+tb,Δ=−4z3a+27t2b, (11)

and the integers , . The seven-branes are localized on and the D3-brane will move around near the origin , where the seven-brane self-intersects (technically, where it is singular in the base). In Section 0.5 we will study the D3-brane theory at .

We wish to study the local structure of this codimension two singularity by surrounding it with a three-sphere and moving the D3-brane around on the three-sphere. The knot, or link, associated to equation (5) has two canonical braid representations, the -braid with strands and the -braid with strands. These braids define two solid tubes, which we will call the -tube or -tube. Writing and , the three-sphere of radius is and the discriminant locus is . On the discriminant modulo . Intersecting the discriminant locus with the three-sphere gives a link

 LΔ:={S3}∩{Δ=0}, (12)

which is a torus link (torus knot if and are coprime); that is the seven-branes intersect the three-sphere at a torus link. It can be described by either of the equations

 4(R2−r2t)3a/2e3iaθz=27r2bte2ibθt,4r3aze3iaθz=27(R2−r2z)be2ibθt. (13)

Consider a one-parameter family of discs, , centered at with parameter . The first equation intersects each member of the family at a collection of points, and as is varied in the positive direction from to the intersection points encircle the origin, creating a spiral that could be thought of as sitting on a tube. For a pictorial representation see e.g. Figure 7. Call this the -tube. Alternatively, there is also a one-parameter family of discs ; the second equation intersects a member of this family at some points, and the whole family at a spiral that sits on the -tube, see e.g. Figure LABEL:fig:IIITorusLinkTube.

Formally, associated to the -tube and -tube, respectively, are periodic one real parameter families of elliptic surfaces

 Xθzπθz−−→Dt(θz),Xθtπθt−→Dz(θt). (14)

We will be interested in studying the string junctions in the members of these families and , and also the monodromy action on string junctions associated with taking a loop in the family. For consistency of notation, we will will also define and .

### 0.2 General analysis of the g-tube

We now study string junctions emanating from the seven-brane link and ending on a D3-brane sitting on the three-sphere, as well as the seven-brane action on the D3-spectrum associated with traversing the -tube. We must specify the initial location of the D3-brane. We choose this point to be , , , which sits on the three-sphere and at the origin of the disc in the -tube at . Mathematically, the selection of this point selects a distinguished fiber in the elliptic fibration from which to build the relative homology associated with string junctions. We will be more precise about this definition in a moment.

We must study , define a basis of cycles there, and determine the action on this basis of cycles as varies from to , i.e. as the D3-brane travels down the -tube. At , the Weierstrass model simplifies to

 y2=x(x+za2)(x−za2)=x(x+Ra2eiθza2)(x−Ra2eiθza2) (15)

and defines a one-parameter family of elliptic curves depending on . At , where , and the elliptic curve is a double cover of the -plane with four branch points a , , and . We will study the first three points, which sit on the real axis. If , note that as passes from to the points at swap via a counterclockwise rotation. In general we find that the root at becomes the root at . This determines some monodromy that can be computed explicitly, and to do so it is convenient to choose a basis of one-cycles.

We will consider two different bases and ensure that they give the same theory of string junctions. The elliptic curve is a double cover of the -plane with branch points at , and . Let () be a straight line connecting to on the -plane. Some details of the analysis can be easily understood in a small neighborhood of and its inverse image in the double cover. Let be a local coordinate on and a local coordinate on , and and the inverse images of and in the double cover. sits along the negative Re-axis and along the positive Re axis; therefore sits along the entire Im-axis and sits along the entire Re-axis. is an , and there is an orientation according to whether the circle is traversed coming out from along the positive or negative Im axis. Note that this distinction is lost in , since coming out from along the positive or negative axis corresponds to the same path of exit from due to the double cover.

The two bases of that we study are defined as follows. In basis one, abusing notation, define to be with the orientation associated with departing along the positive Im-axis, and to be with the orientation associated with departing along the positive Re-axis. In basis two, is the same as in basis one, but is instead defined to be with the opposite orientation, i.e. departing along the negative Im-axis. In the usual complex structure on , defined by the phase of going counter-clockwise rather than clockwise333Equivalently, versus ., determine a positive basis on the tangent space in basis one (two), and therefore in basis one (two). The intersection product of arbitrary one-cycles in the usual complex structure is , where and are one-cycles in some basis. To use this intersection product, we can choose and in basis one and and in basis two. Using the notation of [23], we define

 π1=(10),  π2=(−1−1),  π3=(01),π1+π2+π3=0. (16)

with the usual intersection product, and we will map onto this language later. The cycle will be used in the next section.

Traveling down the -tube via passing from to rotates counter-clockwise in by and counter-clockwise in by . The latter gives an action on the bases

 Basis One: (W,Z)↦(Z,−W) for a=1,(W,Z)↦(−W,−Z) for a=2, (W,Z)↦(−Z,W) for a=3, Basis Two: (W,Z)↦(−Z,W) for a=1,(W,Z)↦(−W,−Z) for a=2, (W,Z)↦(Z,−W) for a=3.

All of these can be seen by direct inspection of Figure 3. The associated monodromy matrices are () with

 M1=(0−110),M2=(01−10). (17)

Having determined the bases on , let us determine the vanishing cycles. We do this on the three-sphere at , where we read off the vanishing cycles by following straight line paths from the D3-brane at to the seven-branes. The seven-branes intersect the three-sphere at a link, and at this determines a set of points in a disc centered at that are the solutions to the equation

 4(R2−r2t)3a2=27r2bte2ibθt, (18)

which requires

 θt=πkb,k∈{0,1,…,2b−1}. (19)

The -dependent part of (18) is satisfied for some , and therefore the seven-branes intersect the disc at the points , and each of the vanishing cycles may be read off by following a straight line path from to . Let us determine the vanishing cycles explicitly using a simple analysis from calculus. The Weierstrass model over is

 y2=x3−(R2−r2t)a2x+rbteibθt (20)

which on any straight line path from to simplifies to

 y2=x3−(R2−r2t)a2x+rbteiπk=:vk(x). (21)

At (that is, at ), the cubic has three real roots, and it is positive for real and negative for real . Letting vary from to , all of the roots remain real, but two of them collide at . To determine which two roots collide, note that

 ∂vk∂rt=art(R2−r2t)a2−1x+(−1)kbrb−1t (22)

so that is positive for even and negative for odd. Then the center and right root collapse for even as goes from to , and the center and left root collapse for odd. That is, if is even (odd) the vanishing cycle is (). Since we choose to index our seven-branes starting from , the ordered set of vanishing cycles is

 {W,Z,W,Z,…} (23)

where the , pair repeats times, for a total of vanishing cycles. Note that this set applies to both bases discussed above since vanishing cycles do not have a sign, but the basis choice must carefully be taken into account when studying monodromy associated with taking closed paths in the geometry (as we will see).

Finally, before studying examples we briefly discuss the map on seven-branes that is induced by the braid upon traveling down the -tube. At the seven-brane is at an angle in the -plane given by , due to points on the discriminant satisfying . Upon traveling down the -tube, varies from to and the term in the discriminant picks up a phase so the associated phase condition on discriminant points becomes

 2bθt=2πk+6πa=2π~k, (24)

where mod . So a seven-brane that starts with index spirals down the -tube and becomes the seven-brane with index modulo . This seven-brane mapping, together with undoing the action on associated with traveling down the -tube will induce a map on string junctions, allowing for the comparison of closed cycles representing simple roots and the determination of whether or not traveling down the -tube gives an outer automorphism on string junctions.

### 0.3 General analysis of the f-tube

In the discussion of the -tube we can proceed similarly to the analysis of the previous Section. This time we choose the point of the D3-brane to be at , , and study the string junctions with respect to the elliptic fiber . The corresponding one-parameter family of Weierstrass models at read

 y2=x3+tb=(x+Rb3e2πi3bθt2π)(x+Rb3e2πi3(1+bθt2π))(x+Rb3e2πi3(2+bθt2π)). (25)

At , we have and the curve is a double cover of the -plane, this time with the four branch points; three at , , and one at . We focus on the first three. Upon traveling from to , we find that these points are permuted according to modulo three. Consequently, it is convenient to phrase the following discussion in terms of segments that connected the three branch points. Let connect the branch points, connect the branch points, and connect the branch points. Let be local neighborhoods of the branch point , and be the inverse image of these neighborhoods in the double cover.

Let us look at the neighborhood and its double cover in more detail. We choose local coordinates such that is oriented along the positive Re-axis. The angle of is () in (). Note that the cycle is not visible in this local neighborhood. By a similar analysis as in the -tube, and we take and .

Upon traversing the -tube from to , we find a counter-clockwise rotation by in the -plane. When , this rotates the segment to the segment on the left hand side of Figure 4, and similarly the () segment to the () segment. This mapping of segments determines the mapping of each associated cycle up to a sign. Let us determine the signs, writing , , with . Encircling the origin three times via going from to we have , , . This corresponds to a rotation in , and therefore a rotation by in , which reverses the orientation of the cycle, requiring . At this point there are two possibilities: all negative, or one negative. However, preserving the intersection of the cycles under the mapping, or alternatively symmetry considerations, requires . Thus, for one rotation in gives , , and . For general , the braid acts as

 (W,U,Z) ↦(−1)b(U,Z,W) for b=1,4, (W,U,Z)↦(−1)b(Z,W,U) for b=2,5, (W,U,Z) ↦−(W,U,Z) for b=3.

Recalling , from the previous paragraph and also the matrix

 M=(1−110), (26)

encodes the monodromy, which is given by .

In order to determine the vanishing cycles we proceed similarly to the previous Section. The discriminant of the Weierstrass model at , intersected with the three-sphere, yields

 4r3aze3iaθz=27(R2−r2z)b⇒θz=2πk3a,k∈{0,1,…,3a−1}. (27)

We study again the solutions of the Weierstrass equation along straight line paths from to , which reads

 y2=x3−raze2πik3x+(R2−r2z)b2=:vk(x). (28)

As varies from 0 to , two roots collide and we determine which ones by studying the imaginary part. Starting from , we find the ordered set of vanishing cycles

 {U,W,Z,U,W,Z,…} (29)

such that the vanishing cycles are given by repeating the vanishing cycles a total of times. Finally, we find that upon traveling down the -tube by varying from to , the term in the discriminant is rotated by a phase , such that

 3aθz=2πk+4πb=2π~k,~k=(k+2b) mod 3a (30)

which means that the braid induces a permutation which sends the seven-brane with index to the seven-brane with index modulo .

### 0.4 Braid action on intersection form

In the previous Sections we studied an elliptic fibration over a disc with the inverse image of the origin of the disc being a smooth elliptic curve. For unified notation in this Section, we take the elliptic fibration to be with . String junctions are elements of two-cycles relative , i.e. .

Following [38], we can define a self-intersection pairing for a given junction written in terms of a basis (the junction basis) on , where the boundary of the junction is . Then the pairing is

 ⟨J,J⟩0=−N−1∑k>l≥1JkJl γk⋅γl−N−1∑l=0J2l. (31)

Note that the first index is skipped in the first sum. This is the case since the pairing is a pairing on relative classes which depends on choosing a second base point nearby the first base point . The rays that connect the base point to the points on the discriminant locus divide the plane into cones. In writing (31), we have arbitrarily put this second base point into the cone between and . Since the intersection pairing depends on which cone the second base point lies in, we indicate the cone used in the pairing on relative homology with a subscript . While this choice is irrelevant for the intersection pairing on classes with vanishing asymptotic charge, it becomes relevant for junctions whose asymptotic charge is non-zero, which correspond to matter states. Consequently, if we want to compare these junctions at and , we need to track the motion of the second base point upon traveling down the - or -tube. If the braid induces a permutation where the indices are to be read mod , the cone is in also moves counter-clockwise by , so ends up in the cone, cf. Figure 5. The new intersection form then reads

 ⟨J,J⟩c−1=−N−1+c∑k>l≥cJkJl γk⋅γl−N−1∑l=0J2l, (32)

where the indices in the first sum are to be read modulo .

### 0.5 Monodromy Action on Bases of String Junctions

Having performed a general analysis of the -tube and -tube, we are ready to state the associated action on relative homology, which will be utilized in examples to perform a map on simple roots, studying associated Lie algebraic structure in the quotient.

Let us begin with the -tube. The results of [38] shows that the thimbles or prongs with form a basis for the relative homology , that is, a basis of string junctions. Each thimble has an associated vanishing cycle , and the results of the previous Section show that

 ∂Γi=Wfor i even,∂Γi=Zfor i odd.

The combined action of the braid map and monodromy on induces the following map in basis 1:

 Γi↦(−1)⌊a2⌋Γ(i+3a) mod 2b  for i even,Γi↦(−1)⌊a2+12⌋Γ(i+3a) mod 2b  for % i odd, (33)

where is the floor function, for , where is the largest integer satisfying .

We now turn to the -tube. Now there are prongs with that form a basis on relative homology . The results of the previous Section show that

 ∂Γi =Ufor i mod 3=0,∂Γi=Wfor i mod 3=1, ∂Γi =Zfor i mod 3=2. (34)

The braid map and the monodromy on induce a map on the basis

 Γi↦(−1)bΓ(i+2b) mod 3a∀i,

which is simpler than that of the -tube.

Summarizing, when the D3-brane traverses the -tube or -tube it is taking a small, closed loop in the geometry near the seven-brane self-intersection at . The seven-branes spiral around the D3-brane as it traverses the tube; since string junctions end on the spiraling seven-branes and the D3-brane, this induces a monodromy on string junction states. Mathematically, in the -tube and -tube we have computed the monodromy maps

 Mf:H2(Xf,Ef)→H2(Xf,Ef)Mg:H2(Xg,Eg)→H2(Xg,Eg), (35)

which act on the string junction spectrum ending on the D3-brane.

\newsec

D3-branes and Duality-required Monodromy Quotients

Let us finally study the D3-brane theory at the codimension two singularity where the seven-brane described by the Weierstrass model

 y2=x3−zax+tb (36)

self-intersects. This is the theory that we called Theory in the introduction, and to study it we will use the geometric action of Section Dualities of Deformed SCFTs from Link Monodromy on D3-brane States.

Recall from the introduction that this D3-brane theory can be naturally thought of in terms of deformations of a simpler seven-brane background, in which the D3-brane realizes an SCFT. These theories, which we call Theory and Theory for brevity, are defined to be the D3-brane theory at in the seven-brane background

 Theory f:y2 =x(x2−za) Theory g:y2 =x3+tb. (37)

Each is an SCFT of Argyres-Douglas [7, 8], Minahan-Nemeschansky [9, 10], or massless Seiberg-Witten [5, 6] type, and in general the flavor symmetries of these theories are different simple Lie groups of different rank. The possible flavor symmetries for Theory and Theory are

 Gf ∈{SU(2),SO(8),E7} Gg ∈{∅,SU(3),SO(8),E6,E8}, (38)

respectively. Deforming each of these seven-brane backgrounds to the same seven-brane background described by (36), keeping the D3-brane fixed at , gives two different descriptions of the deformed D3-brane theory. These dual descriptions must have the same global symmetries, and therefore the deformation must reduce and to some common group . Theory and Theory have massless flavors.

Alternatively, the necessary reduction to a common flavor group can be seen in the fixed background (36). In the -tube the D3-brane is at , with the three-sphere radius. The Weierstrass model over the associated disc centered at is

 y2=x3−Raeiaθzx+tb (39)

where ordinarily would be thought of as a mass deformation with massive flavors in representations of . Then the limit takes the D3-brane to the singularity at and the flavors become massless. Similar statements apply to the -tube theory, which has massive flavors in representations of that become massless as the D3-brane moves to . But the D3-brane theory at does not care about its path to , and thus there must be something wrong with the description of that theory as the massless limit of SCFTs with flavor symmetries and that generally differ.

The resolution is simple: and are not simply mass deformations of an theory with a one-dimensional Coulomb branch, but are both dimensions of space into which the D3-brane may move and the seven-brane may extend. In particular, the deformation breaks the supersymmetry on the D3-brane to since the seven-brane background now preserves supercharges and the D3-brane is half BPS. So there is no paradox, as long as the non-trivial extension of the seven-branes into both directions reduces and to some common group .

It is natural to expect that the reduction arises from the seven-brane monodromy on string junctions. The correct prescription is that the string junctions in the spectrum of the D3-brane at are those junctions from the -tube and the -tube theories that are invariant under the associated monodromies and . Thus, though the flavor symmetries away from are generally different as encoded in the fact that generally444Using the definition from the introduction, the root junction lattice of the flavor algebra associated with is . and , if the spectrum on the D3-brane at is the monodromy-invariant spectrum, one expects an isomorphism

 H2(