T-Branes and G_{2} Backgrounds

T-Branes and G2 Backgrounds

Abstract

Compactification of M- / string theory on manifolds with structure yields a wide variety of 4D and 3D physical theories. We analyze the local geometry of such compactifications as captured by a gauge theory obtained from a three-manifold of ADE singularities. Generic gauge theory solutions include a non-trivial gauge field flux as well as normal deformations to the three-manifold captured by non-commuting matrix coordinates, a signal of T-brane phenomena. Solutions of the 3D gauge theory on a three-manifold are given by a deformation of the Hitchin system on a marked Riemann surface which is fibered over an interval. We present explicit examples of such backgrounds as well as the profile of the corresponding zero modes for localized chiral matter. We also provide a purely algebraic prescription for characterizing localized matter for such T-brane configurations. The geometric interpretation of this gauge theory description provides a generalization of twisted connected sums with codimension seven singularities at localized regions of the geometry. It also indicates that geometric codimension six singularities can sometimes support 4D chiral matter due to physical structure “hidden” in T-branes.

UPR-1298-T

T-Branes and Backgrounds

Rodrigo Barbosa***e-mail: barbosa@sas.upenn.edu, Mirjam Cvetiče-mail: cvetic@physics.upenn.edu, Jonathan J. Heckmane-mail: jheckman@sas.upenn.edu,

[4mm] Craig Lawrie§§§e-mail: craig.lawrie1729@gmail.com, Ethan Torrese-mail: emtorres@sas.upenn.edu, and Gianluca Zoccaratoe-mail: gzoc@sas.upenn.edu

 1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA 2Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA 3Center for Applied Mathematics and Theoretical Physics, University of Maribor, Maribor, Slovenia

Abstract

June 2019

1 Introduction

Manifolds of special holonomy are of great importance in connecting the higher-dimensional spacetime predicted by string theory to lower-dimensional physical phenomena. This is because such manifolds admit covariantly constant spinors, thus allowing the macroscopic dimensions to preserve some amount of supersymmetry.

Historically, the most widely studied class of examples has centered on type II and heterotic strings compactified on Calabi-Yau threefolds [1]. This leads to 4D vacua with eight and four real supercharges, respectively. Such threefolds also play a prominent role in the study of F-theory and M-theory backgrounds, leading respectively to 6D and 5D vacua with eight real supercharges. Compactifications on Calabi-Yau spaces of other dimensions lead to a rich class of geometries, and correspondingly many novel physical systems in the macroscopic dimensions. In all these cases, the holomorphic geometry of the Calabi-Yau allows techniques from algebraic geometry to be used.

There are, however, other manifolds of special holonomy, most notably those with and structure. For example, compactification of M-theory on and spaces provides a method for generating a broad class of 4D and 3D vacua, respectively.111In F-theory, there has recently been renewed interest in the use of backgrounds as a way to generate novel models of dark energy [2, 3] (see also [4, 5, 6, 7]). Though less studied, F-theory on backgrounds should also lead to novel 5D vacua [8].

Despite these attractive features, it has also proven notoriously difficult to generate singular compact geometries of direct relevance for physics.222In the non-compact case, there are quite a few known examples of conical -manifolds. A -metric on a seven-dimensional cone is equivalent to a nearly-Kähler structure on the base of the cone, and these are known to exist on , , and . Moreover, recent work [9, 10] as well that in [11, 12] has established the existence of one-parameter families of -metrics deforming the conical -structure on the cone over . Analogous -metrics were constructed in [13] (see also [14] as well as the review [15]). In the case of M-theory on a background, realizing a non-abelian ADE gauge group requires a three-manifold of ADE singularities (i.e., codimension four), and realizing 4D chiral matter requires codimension seven singularities. While there are now some techniques available to realize backgrounds with codimension four singularities, it is not entirely clear whether a smooth compact can be continuously deformed to such singular geometries, as necessary for physics. In the non-compact setup a recent construction of a deformation family of such -structures was developed in the Math PhD thesis [16], using deformations of ADE singularities [17] and Kronheimer’s deformation families of ALE spaces [18]. Once one is given a codimension four ADE singularity, further degenerations at points of the three-manifold should produce the codimension seven singularities required for 4D chiral matter.3334D globally consistent type IIA compactifications with chiral matter [19, 20] and their relation to M-theory on compact holonomy spaces were studied in reference [21]. It seems possible that some of the fibers in the deformation family of [16] could acquire such point-like singularities, however as of now there is no technique available to probe them. Compact ’s with point-like singularities seem even more elusive.

Rather than directly build a global geometry, an alternative approach is to adapt methods from gauge theory to characterize the appearance of such codimension seven singularities. In reference [22], the partial topological twist of a six-brane wrapped on a three-manifold embedded in a manifold was studied in some detail, and we shall refer to it as the “Pantev–Wijnholt” (PW) system. The choice of 4D vacuum is dictated in the six-brane gauge theory by an adjoint-valued one-form and a vector bundle. The eigenvalues of this one-form parameterize normal deformations in the local geometry , and this leads to a natural spectral cover description. Localized matter in this setup is obtained by allowing the one-form to vanish at various locations. In [22] this was used to analyze codimension seven singularities, and in reference [23] this analysis was greatly developed and also extended to the case of codimension six singularities, i.e. non-chiral matter. As argued in [23] a potentially appealing feature of these codimension six singularities is that they provide a way to possibly connect to one of the (few) methods available for building manifolds via twisted connected sums, using Calabi-Yau threefolds as building blocks [24, 25, 26]. The physics of M-theory compactified on such compact TCS manifolds has been studied in [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 23].

In the TCS construction of backgrounds [24, 26], one first begins with a pair of -manifolds of the form , where is a circle we call “external” and is an asymptotically cylindrical Calabi-Yau threefold. This means that outside of a compact region, the Calabi-Yau metric looks like a surface times a cylinder: . One then glues and using a hyperkähler rotation (and similarly for the other two circles) and shows that the resulting compact smooth manifold admits a full metric. Although all known examples result in smooth total spaces, it is reasonable to expect that degenerations in the Calabi-Yau building blocks will provide a way to generate the long sought for codimension six and seven singularities in compact models.

From this perspective, one might ask whether this is the most general starting point one can entertain for realizing localized matter in compactifications. One important clue comes from the structure of local backgrounds in the presence of non-zero fluxes. This leads to manifolds with structure group , and these solutions can often be interpreted in terms of a system of lower-dimensional branes localized on subspaces inside the bulk geometry [37, 38, 39]. It is thus natural to ask about fluxed solutions with lower-dimensional defects.

In this paper we consider the case of non-abelian fluxes of a six-brane wrapped on a three-manifold. To accomplish this, we return to the local gauge theory on a six-brane (see also [40, 41, 23]). To date, most analyses of localized matter have assumed that the adjoint valued one-form is diagonal, and that there are no fluxes present in the three-manifold. Here, we shall relax this assumption and attempt to study a far broader class of situations. This necessarily means that the components of this one-form will not commute. We shall refer to this as a T-brane configuration (even though the matrix components are not upper triangular) since it naturally fits in the broader scheme of T-brane like phenomena. For earlier work on T-branes, see references [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61].

Locally modelling the three-manifold as a Riemann surface fibered over an interval, we show that for each smooth fiber, the gauge theory on the Riemann surface is described by a mild deformation of Hitchin’s system on a complex curve (see figure 1). Since the local Hitchin system directly describes a local Calabi-Yau geometry (see e.g. [62, 63, 64, 47, 54]), we obtain a local deformation of a TCS-like construction which can be interpreted as building up a local background (see figure 2).

An important feature of these structures is the appearance of holomorphic geometry as a guide in constructing these local backgrounds. This points the way to a method for constructing backgrounds using more general holomorphic building blocks than those appearing in the classical TCS construction.

Another feature we study in great detail is the resulting localized matter obtained from such T-brane configurations. We provide examples where we explicitly determine the profile of localized matter fields in a given background. This involves solving a second order differential equation. We also develop algebraic methods for reading off the appearance of localized zero modes by determining the local ring structure of trapped matter. This is similar in spirit to the analysis of localized zero modes in T-brane configurations carried out in references [44, 45].

One of the outcomes of this analysis is that it also provides evidence for the existence of localized matter field configurations which would be “invisible” to the bulk geometry since they originate from degenerate spectral equations. Instead, they would be fully characterized by allow limiting behavior in the four-form -flux of the M-theory background. We present explicit examples exhibiting this behavior. A canonical example is the “standard embedding” namely we embed the spin connection in the gauge bundle, taking our holomorphic vector bundle to be the tangent bundle on dimensionally reduced (after a Fourier-Mukai transform / formally three T-dualities) to the three-manifold. Related examples show up in a number of other T-brane constructions (see e.g. [42, 47]).

The rest of this paper is organized as follows. We begin in section 2 by discussing the PW system, and its relation to geometric engineering. Next, in section 3 we show how to build examples of solutions to the PW system in cases with non-zero gauge field flux. In section 4 we present some general methods for analyzing zero modes in such backgrounds, and then proceed to determine the localized matter field wave functions by explicitly solving the corresponding partial differential equations. Section 5 presents a conjectural proposal for how to algebraically determine the localized zero mode content in fluxed solutions. We present our conclusions in section 6.

2 Six-Brane Gauge Theory on a Three-Manifold

In this section we discuss the gauge theory of a six-brane wrapped on a three-manifold, as obtained from M-theory on a background. Geometrically, we engineer this gauge theory from a three-manifold of ADE singularities with corresponding ADE gauge group - i.e., the local is a fibration of ADE singularities over a three-manifold. The terminology follows from the fact that the “brane” in question is actually a seven-dimensional supersymmetric gauge theory wrapped on the three-manifold, namely a six-brane. Supersymmetry is preserved since we assume the three-manifold is an associated three-cycle of the local manifold. It is also in accord with the type IIA string theory description of D6-branes wrapped on special Lagrangian submanifolds of a Calabi-Yau threefold. For a recent pedagogical discussion of geometric engineering, and the relation between localized gauge theories and singular geometry in the context of string compactification, see the online lectures of reference [65].

In terms of the geometry of the local background, we note that we have an associative three-form which pairs with the three-form potential of M-theory to form the complex moduli . Resolving the ADE fibers and performing a corresponding reduction to the three-manifold, we have a decomposition:

 C=Aα∧ωα, \ \ ρ=ϕα∧ωα (2.1)

where the are harmonic representatives of forms on the resolution of the local ADE singularity, and the and are one-forms on the three-manifold. The index should be thought of as labeling the generators of the Cartan subalgebra of the corresponding ADE gauge group. The remaining generators are obtained from M2-branes wrapped on collapsing two-cycles of the fiber.

As explained in reference [22] (see also [40, 23, 66, 67]), the partial twist of a six-brane with gauge group on a three-manifold retains 4D supersymmetry in the uncompactified directions. The vacua in the macroscopic dimensions are controlled by a three-dimensional gauge theory with gauge field coupled to an adjoint-valued one-form . These quantities combine into a complexified connection which we write as:

 A=A+iϕ, (2.2)

which should be thought of as the bosonic component of a collection of 4D chiral superfields which transform as a one-form on the three manifold. In our conventions, we take anti-hermitian generators for the Lie algebra so that and . We shall also find it convenient to absorb the factor of to define a hermitian Higgs field:

 Φ≡iϕ. (2.3)

We construct various curvatures from the complexified connection , its conjugate as obtained from as well the purely real . In a unitary gauge, we have . Locally these connections may be written as:

 DA=d+A, \ \ D¯¯¯¯A=d+¯¯¯¯¯A, \ \ DA=d+A. (2.4)

We introduce gauge field strengths:

 F=[DA,DA], \ \ D=[DA,D¯¯¯¯A]% , \ \ F=[DA,DA]. (2.5)

Once written in components the various field strengths are:

 Fij =∂iAj−∂jAi+[Ai,Aj] (2.6) Dij =∂i¯¯¯¯¯Aj−∂jAi+[Ai,¯¯¯¯¯Aj] (2.7) Fij =∂iAj−∂jAi+[Ai,Aj]. (2.8)

In terms of the original fields and , the complexified field strengths decompose as:

 ImFij =Fij−[ϕi,ϕj] (2.9) ReFij =(∂i+Ai)⋅ϕj−(∂j+Aj)⋅ϕi (2.10) ReDij =−(∂i+Ai)⋅ϕj−(∂j+Aj)⋅ϕi (2.11) ImDij =Fij+[ϕi,ϕj]. (2.12)

Variation of the 7D gaugino produces the corresponding conditions to have a 4D supersymmetric vacuum. These are conveniently packaged as F-terms and D-terms, which are respectively metric independent and dependent:

 Fij=0 \ \ and \ \ gijDij=0, (2.13)

in the obvious notation. The moduli space of vacua is then given by two equivalent presentations:

 {F =0}/GgaugeC (2.14) {F =0, \ \ gijDij=0}/GgaugeU, (2.15)

where here, refers to complexified gauge transformations and refers to unitary gauge transformations. We refer to both presentations of the moduli space as the defining equations of the Pantev–Wijnholt (PW) system.

The F-term equations of motion are obtained from the critical points of the Chern-Simons superpotential for the complexified connection:

 W[A]=s4π∫TrgC(A∧dA+23A∧A∧A), (2.16)

for a complex parameter. Four-dimensional vacua are labelled by critical points of modulo .

More precisely, in the description specified by F-terms modulo gauge transformations, the appropriate notion of “stability” is that we only look at connections with semisimple monodromy. By the Donaldson-Corlette theorem [68, 69], these automatically solve the harmonic metric equation, i.e. the D-term. An interesting feature is that for any hermitian generator of the algebra, the signature of the real matrix (with an index in the adjoint) must have at least one sign and one sign each. This is simply to satisfy the D-term constraint. We note that this is in accord with the “Hessian condition” of references [22, 23] observed in the special case where the adjoint-valued one-form is diagonal.

Another way to study this system is to first consider stable holomorphic vector bundles on the local Calabi-Yau . These are described by Hermitian-Yang-Mills (HYM) instantons [70, 71]. Taking the linearization of the HYM equations in a neighborhood of the zero section then produces the same equations [22]. Briefly summarizing this approach, we introduce a connection:

 ¯¯¯∂A=¯¯¯∂+A. (2.17)

The conditions to have a stable holomorphic vector bundle are a F-term and a D-term:

 F(0,2)=0 \ \ and \ \ J∧J∧F(1,1)=0, (2.18)

with the Kähler form on . To make contact with the PW equations, we consider as a complexified gauge field which splits up as:

 A=Aidxi+ϕjdpj, (2.19)

where are local coordinates on and are local coordinates in the cotangent direction. The topological twist amounts to making a further identification which introduces an additional factor of thus recovering (2.2). Equivalently, we take the Calabi-Yau structure on to be semi-flat. A helpful feature of this construction is that it also describes the heterotic dual to this local model. More precisely, it is the linearization obtained from a (singular) fibration over .

This alternate presentation already points to an important general point: A priori, there is no reason for the components in equation (2.19) to be simultaneously diagonalizable. Returning to the PW equations, this also means there is no reason to exclude gauge fluxes through the three-manifold. Let us also note that even if such fluxes are present, it does not directly mean there will be a bulk four-form flux in the model. This is because these fluxes are inherently localized on the three-manifold and are “hard to see” from the bulk point of view. Indeed, the local geometry of the background is primarily sensitive to just the eigenvalues of and bulk fluxes, and not to any of these non-abelian local features.

A canonical example is the tangent bundle of . It has the important feature that the spectral cover description is degenerate. On the three manifold we have a vector bundle with structure group and the ’s certainly do not commute. Let us note that in heterotic / F-theory duality, the standard embedding also corresponds to a T-brane configuration of an F-theory compactification [42, 47].

3 Fluxed PW Solutions

In this section we consider fluxed solutions of the PW system. Our strategy for obtaining such configurations will be to consider a local description of the three-manifold as a Riemann surface fibered over an interval, and we shall often further specialize to the case of a Cartesian product with a Riemann surface and an interval. This description will only be valid locally, and so we can either assume these solutions extend outside of the patch in question, or alternatively, we can cut off the solution by allowing singular field configurations at prescribed regions of the three manifold.

To aid in our study of fluxed PW solutions, we shall often assume the metric on the three-manifold takes the form:

where in the above, denotes a local coordinate on , and for denote coordinates on the Riemann surface, and since we often focus on metric independent questions, we shall also sometimes take the metric to be flat in some local patch.

In terms of this presentation, the PW equations take the form:

 Fab−[ϕa,ϕb] =0, (3.2) Daϕb−Dbϕa =0, (3.3) gabDaϕb+gttDtϕt =0, (3.4) Fta−[ϕt,ϕa] =0, (3.5) Dtϕa−Daϕt =0, (3.6)

modulo unitary gauge transformations. In the above, we have written for the covariant derivative. We now observe that the first three equations describe a small deformation of the standard Hitchin system of reference [72]. Indeed, introducing a covariant derivative and a one-form on each fiber, we have:

 FΣ−[ϕΣ,ϕΣ] =0, (3.7) DΣϕΣ =0, (3.8) ∗ΣDΣ∗ΣϕΣ =−gttDtϕt, (3.9)

which would have described the Hitchin system in the special case where .

We remark that these equations tell us that the induced Hitchin system does not describe a Higgs bundle in the mathematician’s sense [73]; indeed, the condition for harmonicity of the bundle metric is [68, 69] , and equation (3.9) tells us that a PW solutions gives a deformation of that condition along the -direction. We point out that the dependence of equation (3.9) on the bundle metric is hidden in the definition of the Hodge star .

The remaining F-term relations can also be interpreted as a flow equation:

 (3.10)

i.e.:

 ∂tAa+[At,Aa]=∂aAt. (3.11)

Geometrically, we interpret the flow equations as a gluing construction for local Calabi-Yau threefolds. To see why, we first recall the correspondence between the Hitchin system on a genus curve and the integrable system associated to a family of non-compact Calabi-Yau manifolds containing as a curve of ADE singularities [64, 63, 54]. Let denote the Calabi-Yau threefold that is the central fiber of the deformation family. Recall that the isomorphism between the integrable systems implies in particular that variations in the complex structure for are described by (spectral curves of) Higgs fields - i.e., adjoint-valued -forms on the curve . With notation as in equation (2.1) for harmonic forms on the ADE singularity, the variations of the holomorphic three-form on the local Calabi-Yau decompose as:

 δΩ(2,1)=φα∧ωα. (3.12)

The Calabi-Yau condition enforces the condition , which translates to one of the Hitchin equations:

 ¯¯¯∂Aφ=0. (3.13)

The deformation of line (3.9) tells us that the righthand side is no longer zero. Translating back to the Calabi-Yau integrable system, we see that we instead get a deformation of a Calabi-Yau manifold that does not respect the Kähler condition, resulting in a “symplectic Calabi-Yau” in the sense of Smith, Thomas and Yau [74] (for a recent discussion see reference [75], and for a review see reference [76] and additional references therein).

From the perspective of the local , we thus see that asymptotically near the boundaries of the interval, we retain an approximate Calabi-Yau geometry, but as we proceed to the interior of the interval, each fiber will instead be described by a symplectic Calabi-Yau.

Let us illustrate the correspondence between the spectral equation for the Higgs field and deformations of its dual local Calabi-Yau in the case of . The spectral equation in the fundamental representation is:

 det(uholIN−φhol)=0, (3.14)

with a local coordinate in the cotangent direction of . Since we also have a Higgs field in the more general case, it is natural to consider two related spectral equations, as generated by the PW system. First, we have the one closely linked to the asymptotic Hitchin system associated with :

 det(uzIN−Φz)=0. (3.15)

This equation only makes sense asymptotically, since in the bulk of the three-manifold, the Higgs field of the PW system is not a holomorphic (or even meromorphic) section of a bundle on the curve . Indeed, more generally we ought to speak of the spectral equations:

 det(uiIN−Φi)=0. (3.16)

where here, for are coordinates in the cotangent direction of . This is a triplet of real equations in which cut out a three-manifold in this ambient space. Geometrically, then, we can interpret the spectral equation of line (3.16) as a special Lagrangian manifold in with boundaries specified by the holomorphic curves dictated by equation (3.15). Indeed, the PW equations ensure that supersymmetry has been preserved.

This also allows us to elaborate on the sense in which having is an example of T-brane phenomena. Even though each takes values in the unitary Lie algebra , it is also natural to consider linear combinations which take values in the complexification . If the complexified combination is a nilpotent element of , then we observe that the holomorphic spectral equation of line (3.15) is degenerate. Indeed, even though and are hermitian (and thus never nilpotent), their complex combination can of course be nilpotent. From the perspective of a global background, this also suggests that such phenomena may be invisible to the geometry, instead being encoded in non-abelian degrees of freedom localized along lower-dimensional subspaces. A related comment is that even though each component of the Higgs field is hermitian, when these matrices do not commute, there is no canonical way in which we can speak of the spectral sheets intersecting along a locus of symmetry enhancement. We return to these issues in section 5.

The plan in the remainder of this section is as follows. First, we discuss some general aspects of background solutions in a local patch. Starting from a solution along a single fiber, we show that this solution extends to a local neighborhood of the three-manifold. We then present some explicit examples of backgrounds, including T-brane configurations.

3.1 Background Solutions in a Local Patch

We first locally characterize solutions of the system in a patch with trivial topology. By abuse of notation, we continue to write for this local patch. The F-term equations of motion tell us that we are dealing with a complexified flat connection, so the most general solution for the gauge connection is of the form:

 A=g−1dg, (3.17)

where takes values in the complexified gauge group. One must remember that here, we are working with a complexified connection, so even though the gauge field appears to be “pure gauge,” we must only permit gauge transformations which do not alter the asymptotic behavior of the gauge field. A related point is that to actually find a solution in unitary gauge, we need to substitute our expression for into the D-term constraint , resulting in a second order partial differential equation for .

As an illustrative example, suppose that is of the form:

 W⋅g=1+h+h22!+..., (3.18)

where is a general element of , and we have introduced an infinitesimal a Lie algebra valued function on the patch. Then, the complexified connection takes the form:

 A=dh+.... (3.19)

Feeding this into the D-term constraint, we get, to leading order in :

 gijDij=gij∂i∂j(h−h†)=0. (3.20)

So in other words, is a Lie algebra valued harmonic map.

There are, of course, more general choices for and we will in fact find it necessary to consider a more general choice of complexified connection to generate novel examples of T-brane configurations with localized matter.

3.2 Power Series Solutions

We now turn to a more systematic method for building up solutions of the PW equations in the presence of flux. The basic idea is that we shall split up our three-manifold as a local product with a Riemann surface (possibly with punctures) and an interval. We introduce a local coordinate on denoted by so that is in the interior of the interval, and coordinates for for coordinates on the Riemann surface. It will sometimes prove convenient to also use coordinates .

We show that if enough initial data is specified at , then we can start to extend this solution in a neighborhood, building up a solution on the entire patch of the three-manifold. The demonstration of this will be to develop a power series expansion in the variable for all fields of the PW system, and solve the expanded equations order by order in this parameter. We shall not concern ourselves with whether the series converges, because we do actually anticipate that there could be singular behavior for the fields at locations on the three-manifold. This is additional physical data, and must be allowed to make sense of the most general configurations of relevance for physics.

We consider a series expansion of the complexified connection around ,

 Ai(t,xa)=∞∑j=0A(j)i(xa)tj. (3.21)

Furthermore, it is convenient to work in “temporal gauge” with:

 A(j)t=0. (3.22)

In this gauge, and with the power series expansion (3.21), the PW equations can be written as non-trivial differential equations on the coefficients

 ∞∑j=0(∂aA(j)b−∂bA(j)a+j∑m=0([A(j−m)a,A(m)b]−[ϕ(j−m)a,ϕ(m)b]))tj=0∞∑j=0(∂aϕ(j)b+j∑m=0[A(j−m)a,ϕ(m)b]−∂bϕ(j)a−j∑m=0[A(j−m)b,ϕ(m)a])tj=0, (3.23)

together with equations which fix the higher order coefficients in terms of the preceding ones,

 ∞∑j=0(gab(∂aϕ(j)b+j∑m=0[A(j−m)a,ϕ(m)b])+(j+1)gttϕ(j+1)t)tj=0∞∑j=0((j+1)A(j+1)a−j∑m=0[ϕ(j−m)t,ϕ(m)a])tj=0∞∑j=0((j+1)ϕ(j+1)a−∂aϕ(j)t−j∑m=0[A(j−m)a,ϕ(m)t])tj=0. (3.24)

At these equations collapse to

 F(0)ab−[ϕ(0)a,ϕ(0)b]=0D(0)aϕ(0)b−D(0)bϕ(0)a=0gabD(0)aϕ(0)b+gttϕ(1)t=0A(1)a−[ϕ(0)t,ϕ(0)a]=0ϕ(1)a−D(0)aϕ(0)t=0. (3.25)

We will assume that and are such that the non-trivial zeroth order differential equations are solved, and the higher order coefficients are fixed by solving the linear equations. The one remaining free parameter is , which sets the “trajectory” of the solution. Once we are given this initial set of data, solving the zeroth order equations, we can show that the PW equations are solved to all orders in .

To show that solving the zeroth order equations leads to a solution at all orders in the power series expansion we will first substitute (3.24) into (3.23). We begin by noticing that the commutators that appear in the differential equations (3.23), for the -term, can always be written as

 j+1∑m=0[λ(j+1−m)a,χ(m)b]=1j+1j∑m=0((j+1−m)[λ(j+1−m)a,χ(m)b]+(m+1)[λ(j−m)a,χ(m+1)b]), (3.26)

where and represent either or . We will then replace, using (3.24), the terms and . Combining this expansion with some double sum identities, together with the Jacobi identity, one finds, after some algebra,

 (3.27)

and

 (3.28)

If we define the shorthand notation where the power series expansion in (3.23) look, respectively, like

 ∞∑j=0G(j)abt(j),∞∑j=0H(j)abtj, (3.29)

then we can immediately see, from (3.27) and (3.28), that, after plugging in the solutions to the linear equations (3.24),

 G(j+1)ab =1j+1j∑k=0[ϕ(k)t,H(j−k)ab] (3.30) H(j+1)ab =−1j+1j∑k=0[ϕ(k)t,G(j−k)ab].

These expressions make obvious the inductive proof that if

 G(0)ab=0,H(0)ab=0, (3.31)

which we assume, then it follows that

 G(j)ab=0,H(j)ab=0, (3.32)

Thus, if we have given and such that the zeroth order equations in (3.23) are solved, then we can construct a full solution of the PW equations by specifying all the higher order coefficients as in (3.24). Note that is unspecified, and we consider this parameter as determining how the solution extends to a solution of the full system. Further note that we did not require the first equation in (3.24) to arrive at the conclusions (3.27) and (3.28).

Resumming this power series can be viewed as a complexified gauge transformation. To see why, consider a (partial) solution of the Hitchin system at and generate a flow along the interval to produce a non-trivial dependence in the direction. This has the advantage of automatically solving the F-term equations of motion and gives a method for iteratively solving the D-term equations of motion. We start with a complexified connection with legs only on which gives a solution to the F-term condition . Note that we do not require it to be a full solution to the PW equations and in particular we shall be interested in the case when does not give vanishing D-terms. This issue can be addressed by considering a complexified gauge transformation by a gauge parameter of the form444To avoid changing the solution at we will take . Moreover we will assume that the Lie-algebra valued functions are suitably chosen to ensure that the gauge choice is enforced.

 expχ=exp[∑iχi(x1,x2)ti]. (3.33)

After performing this complexified gauge transformation the background will still solve the F-term equations. Moreover it also admits a power series expansion around , and borrowing from the results of section 3.2, it is possible to find a solution of the D-term equations by suitably choosing the terms in the complexified gauge transformation.

3.3 Examples of Backgrounds

In this section we present some examples of background solutions. We first discuss some abelian examples in which no gauge field flux is switched on, and then present a novel non-abelian solution with flux.

3.3.1 First Abelian Background

Our first example is a particular case of the solutions already constructed in [22]. We take an gauge theory and write the complexified connection as

 A=[α−α], (3.34)

where:

 α=zd¯¯¯z+¯¯¯zdz−4t=2xdx+2ydy−4tdt, (3.35)

with . This background solves the F-term equations by virtue of the fact that with

 g=exp(z¯z−2t2002t2−z¯z).

The D-term equations are satisfied as well because the function is harmonic in with a flat metric.

In terms of the fields and , the gauge field , and the Higgs field is given by:

 Φ=[df−df] (3.36)

with:

 f=z¯¯¯z−2t2. (3.37)

We note that the Hessian of has signature in the coordinate system, so it translates to a geometry with a codimension seven singularity in the local background.

3.3.2 Second Abelian Background

As another example, we can also take the abelian background:

 A=[β−β], (3.38)

with:

 β=zdz+¯¯¯zd¯¯¯z=2xdx−2ydy. (3.39)

The corresponding Higgs field in this case is:

 Φ=[df−df] (3.40)

with:

 f=12z2+12¯¯¯z2, (3.41)

We note that the Hessian of has signature in the coordinate system, so it translates to a geometry with a codimension six singularity in the local background. Starting from this example, we obtain a codimension seven singularity by adding a perturbation which also has vanishing Hessian:

 fnew=f+fpert. (3.42)

We can again solve the F- and D-terms for this system, and thus obtain a genuine background in this more general case as well.

3.3.3 Fibering a Hitchin System

Building on this previous example, we can also consider more general backgrounds on by first solving the Hitchin system on a curve , and then adding perturbations so that it has a non-trivial profile on the interval. In practice, we accomplish this by starting with a complexified connection defined on which solves the Hitchin equations, and trivially extending this to . This solves our fluxed PW equations in the degenerate limit where , corresponding to the physical limit where we take the curve much smaller than the interval. Adding a perturbation to the complexified connection:

 Anew=AH+Apert, (3.43)

we observe that the power series analysis of section 3.2 ensures that we can consistently add in such perturbations and produce a solution to the full PW system of equations. Indeed, the main freedom we have in specifying this contribution is the -dependence which is wholly absent from .

3.3.4 Non-Abelian Background

We can also consider the limit where the deformation away from a Hitchin system is large. This will be our main example of a T-brane configuration. In this solution we take an gauge theory though the technique employed here can be easily generalized to other non-abelian gauge groups with rank greater than one. Our analysis follows a similar treatment to that presented in reference [45]. When written in the fundamental representation the background fields are:

 ϕ =⎡⎢ ⎢ ⎢ ⎢⎣i3dh−υ¯ze−f(z,¯z)d¯z+εef(z,¯z)dz0υze−f(z,¯z)dz−εef(z,¯z)d¯zi3dh000−2i3dh⎤⎥ ⎥ ⎥ ⎥⎦, (3.44) A =⎡⎢ ⎢⎣a000−a0000⎤⎥ ⎥⎦,wherea=i2(∂¯zf(z,¯z)d¯z−∂zf(z,¯z)dz). (3.45)

We will shortly show that for suitable choices of the functions and , this background indeed solves the F- and D-term equations of the PW system. Our main interest will be in picking background values so that as much as possible can be “hidden” from the classical geometry. In particular, if we take to be a function with a degenerate Hessian (one zero eigenvalue), a PW solution with just this contribution would appear to support a codimension six singularity along the locus . Adding in additional off-diagonal components to the Higgs field need not change this interpretation. For example, if we consider the purely off-diagonal contributions to the component of the Higgs field, namely , we observe that when , the limiting Hitchin system has a degenerate spectral equation, i.e., there is not even a codimension six contribution to matter localization from these holomorphic off-diagonal contributions. Taken together, this suggests that the proper geometric interpretation will appear to contain at most a codimension six singularity.

Let us now turn to an explicit example. The background will satisfy the supersymmetry conditions provided that the two functions and satisfy the differential equations:

 4∂z∂¯zh+∂2th=0, (3.46) ∂z∂¯zf=ε2e2f−υ2|z|2e−2f. (3.47)

The first equation allows for several solutions, and in the following we shall take:

 h=κ8(z+¯z)2−κ2t2. (3.48)

The second equation can be related to a Painlevé III transcendent via suitable change of variables provided that . In this case, the solution has an asymptotic expansion near of the form:

 f(r)=logc+13logυ−23logε+c2r2+O(r4), (3.49)

where and is a real constant. In order to avoid singularities at finite values of one should fix:

 c=31/3Γ[23]Γ[13]∼0.73. (3.50)

In the following we shall be also interested in the case where because in this case some of the effect of the Higgs field background would become nilpotent. When this happens (3.47) becomes a modified Liouville equation with solution:

 f(z,¯z)=−log[21−υ2|z|4]. (3.51)

3.3.5 More General Embeddings

As a final comment, we can also generalize the example of section 3.3.4 to other choices of gauge groups. Observe that the Higgs field of the previous example takes values in an subalgebra of . More generally, we can specify a homomorphism , and a commuting subalgebra. Then, we clearly also generate a broader class of examples of fluxed PW solutions.

4 Examples of Localized Matter

In the previous section we discussed a general method for generating consistent solutions to the PW equations, and its connection to TCS-like constructions of manifolds. Assuming the existence of a consistent background, we would now like to determine whether localized zero modes are present.

To frame the discussion to follow, we assume that the complexified connection takes values in some maximal subalgebra so that the commutant subalgebra is . We specify a zero mode by considering fluctuations around this background. The presence of chiral multiplets can be understood in terms of fluctuations of the complexified connection . We therefore consider the expansion:

 A=A(0)+ψ. (4.1)

In what follows, we shall omit the superscript to avoid overloading the notation. Note that under an infinitesimal gauge transformation, the zero mode solution will shift as:

 ψ→ψ+DAχ, (4.2)

where is an adjoint valued zero-form in the Lie algebra.

We need to study the various representations appearing in the unbroken gauge algebra. This is by now a standard story which is largely borrowed from the case of compactifications of the heterotic string as well as local F-theory models so we shall be brief. Decomposing the adjoint representation of into irreducible representations of , we have:

 gC ⊃hC×h′C (4.3) ad(gC) =⨁i(Ti,Ri). (4.4)

For a prescribed representation of , we therefore need to consider the action of the complexified connection on the zero mode.

There are various ways to analyze the zero mode content of this theory. Most directly, we can return to the PW equations, and expanding around a given background we can seek out zero modes modulo unitary gauge transformations. This approach makes direct reference to the metric on the three manifold and is certainly necessary if we want the explicit wave function profile for the zero modes. The relevant linearization of the PW system of equations is:

 ∂iψj−∂jψi+[ψi,Aj]+[Ai,ψj] =0 (4.5) gij(∂i¯¯¯¯ψj−∂jψi+[ψi,¯¯¯¯¯Aj]+[Ai,¯¯¯¯ψj]) =0 (4.6)

modulo unitary gauge transformations as in line (4.2). In practice, we shall actually demand that our zero modes satisfy a slightly stronger set of conditions:

 ∂iψj−∂jψi+[ψi,Aj]+[Ai,ψj] =0 (4.7) gij(∂iψj+[¯¯¯¯¯Ai,ψj]) =0. (4.8)

Any solution to this set of equations automatically produces a solution to the linearized PW equations. It is convenient to use these zero mode equations to track their falloff and thus to ensure they are actually normalizable. This, for example, is what allows us to determine whether we have a normalizable mode in a representation or the complex conjugate representation .

Now, one unpleasant feature of this approach is that it often requires dealing with coupled partial differential equations. In the special case where is diagonal, this is not much of an issue, but in more general T-brane backgrounds this can lead to significant technical complications.

At a qualitative level, however, it is straightforward to see how to pick appropriate backgrounds which could generate localized matter. First of all, we can attempt to find a localized zero mode in the Hitchin system. In M-theory language, this would produce a 5D hypermultiplet. These multiplets can all be organized according to 4D hypermultiplets. In the presence of a non-trivial field profile on the transverse direction to the 4D spacetime, this leads to a localized chiral mode, producing a single 4D chiral multiplet. We see how this comes about in our zero mode equations by taking small relative to the other factors of the metric on . Indeed, in this case, the leading order behavior is governed by a mild deformation of the zero mode equations on the curve , and then there is a broadly localized mode in the remaining -direction. That being said, it is clear that there is some level of “backreaction” in the form of these zero mode solutions, so obtaining explicit wave functions in this approach is more challenging.

Our plan in the remainder of this section will be to discuss some general features of zero mode solutions in T-brane backgrounds, and to then discuss explicit examples. In section 5 we give a conjectural algebraic method of analysis for detecting the presence of localized zero modes.

4.1 Cohomological Approach

One approach that may be employed in the search for a solution of the wave function equations involves relying on the cohomology555Here and in what follows, we will always consider -cohomology. of the operator . Let denote a complex vector bundle with connection , and the bundle associated to a representation of the gauge group . By virtue of the vanishing of the F-term conditions the operator squares to zero implying that we can consider its cohomology complex, much as in reference [23].

The linearized F-terms modulo complex gauge transformations are solved by a cohomology class , and the linearized D-term constraint requires us to find a representative for that is annihilated by the dual operator . For a closed -manifold, a standard integration by parts argument says that the solution is given by the harmonic representative for (we remind the reader that given any elliptic complex with metric over a compact manifold, there is a Hodge isomorphism .

The situation is trickier when is allowed to be non-compact: not only does one have to choose appropriate decay conditions on the fields in order for the boundary term to vanish, but also one must work with metrics such that the relevant cohomology classes admit harmonic representative(s). In fact, even when one considers the simplest elliptic operator - the Hodge Laplacian - on a non-compact Riemannian manifold, the space of -harmonic forms depends strongly on the metric. For example, is either or depending on whether or . However, one can give a cohomological interpretation for for special metrics; in particular, Atiyah, Patodi and Singer [77] proved that for manifolds with cylindrical ends666A -dimensional Riemannian manifold has cylindrical ends if there is a -dimensional compact submanifold with smooth non-empty boundary , such that is isometric to ., , i.e. the image of compactly supported cohomology in absolute cohomology.

In our present situation with , we have, under mild assumptions, an example of a cylindrical manifold. Assuming our solution to the linearized F-terms vanishes at infinity , we can package our solution as a relative cohomology class . Suppose that the order of vanishing and the metric are chosen compatibly so as to cancel the boundary term, and moreover that the result of Atiyah, Patodi and Singer generalizes to -harmonic one-forms. Then, a solution to the PW equations on is given by a -harmonic representative for .

This suggests a practical way to try and find a solution of the wave function equations of motion: starting with any solution to the F-term wave function equations we handily build other solutions as the gauge transformed via an element , that is . Thus, even if fails to solve the linearized D-term equation it is possible to find a suitable gauge transformed that is a solution of the full system, which allows us to recast the D-term condition as an equation on

 D†Aψχ=0⟹ΔAχ=D†Aψ0, (4.9)

where we defined the Laplacian . While we are not in general able to show that a suitable solving (4.9) exists we can push this analysis further and argue that when a solution to (4.9) exists, then it is sufficient to check that the mode is normalizable to establish existence of a zero mode. This is due to the fact that the harmonic representative in any cohomology class will minimize the norm. Indeed, calling the harmonic representative we find that the norm of any other element in the cohomology class is

 ∫M||ψ||2=∫M||ψhrm+DAχ||2=∫M||ψhrm||2+||DAχ||2>∫M||ψhrm||2, (4.10)

so in other words, the (already normalizable) trial wave function has bigger norm than the harmonic representative.

4.2 First Abelian Example

We first take a look at the zero modes in the background introduced in Section 3.3.1. In this example we took the gauge group to be and its adjoint representation decomposes as:

 su(2) ⊃u(1) (4.11) ad(su(2)) →10+12+1−2. (4.12)

Recall that in this abelian example, the gauge field , and the Higgs field is given by:

 Φ=[df−df] (4.13)

with:

 f=z¯¯¯z−2t2. (4.14)

Consider a candidate zero mode with charge under the . The analysis of zero modes for this case has already been studied in references [40, 41, 22, 23]. Writing out the zero mode as a one-form on the three-manifold:

 ψ=ψzdz+ψ¯zd¯¯¯z+ψtdt, (4.15)

the zero mode equations are:

 0 =∂zψ¯z−∂¯zψz−qzψz+qψ¯z (4.16) 0 =∂zψt−∂tψz+qzψt−4qtψz (4.17) 0 =∂¯zψt−∂tψ¯z+q¯¯¯zψt−4qtψ¯z (4.18) 0 =gz¯z(∂zψ¯z−q¯¯¯zψ¯z)+g¯zz(∂¯zψz−qzψz)+gtt(∂tψt+4qtψt). (4.19)

We note that in the neighborhood where , we must keep non-zero. To see why, observe that if it were zero, then there is a coupled differential equation for and which does not produce a normalizable solution. Since perturbations involving do not really alter this conclusion, we see that a more sensible starting point is to take non-zero and . In this case, the zero mode equations collapse to:

 ∂zψt+qzψt =0 (4.20) ∂¯zψt+q¯¯¯zψt =0 (4.21) ∂tψt+4qtψt =0, (4.22)

which is solved by:

 ψt∼exp(−q(z¯¯¯z+t2)), (4.23)

so we get a normalizable zero mode for but not for .

4.3 Second Abelian Example

As another example using a similar breaking pattern, we next consider the zero modes in the background introduced in Section 3.3.2. In this example we took the gauge group to be and its adjoint representation decomposes as:

 su(2) ⊃u(1) (4.24) ad(su(2)) →10+12+1−2. (4.25)
 A=[β−β], (4.26)

with:

 β=zdz+¯¯¯zd¯¯¯z=2xdx−2ydy. (4.27)

The corresponding Higgs field in this case is:

 Φ=[df