Phase transitions in GLSMs and defects
aa r X i v : . [ h e p - t h ] J a n Prepared for submission to JHEP
LMU-ASC 01/21
Phase transitions in GLSMs and defects
Ilka Brunner, Fabian Klos, Daniel Roggenkamp Arnold Sommerfeld Center, Ludwig-Maximilians-Universit¨at,Theresienstraße 37, 80333 M¨unchen, Germany Institut f¨ur Theoretische Physik, Universit¨at Heidelberg,Philosophenweg 19, 69120 Heidelberg, Germany Institut f¨ur Mathematik, Universit¨at Mannheim,B6, 26, 68131 Mannheim, Germany
Abstract:
In this paper, we construct defects (domain walls) that connect different phasesof two-dimensional gauged linear sigma models (GLSMs), as well as defects that embedthose phases into the GLSMs. Via their action on boundary conditions these defects giverise to functors between the D-brane categories, which respectively describe the transportof D-branes between different phases, and embed the D-brane categories of the phases intothe category of D-branes of the GLSMs. ontents W ( X, P ) = X d P d ′ T UV N R IR R i T iN P iN RG N The topic of this paper are two-dimensional gauged linear sigma models with U (1) gaugegroups . These are 2d N = (2 ,
2) supersymmetric gauge theories coupled to chiral su-perfields carrying possibly different charges under the U (1) gauge group, such that therespective superpotentials W are U (1) invariant. Indeed, our discussion easily generalizes to models with arbitrary abelian gauge groups. – 1 –s is well known, gauged linear sigma models exhibit different phases for differentranges of the Fayet-Iliopoulos parameter r associated to the U (1) gauge group [1]. Fornon-anomalous gauged linear sigma models, where axial and vector R -symmetries are pre-served at the quantum level, the RG flow drives the GLSM to a (K¨ahler) moduli space ofsuperconformal field theories parametrized by the complexified Fayet-Iliopoulos parameter t . The phases correspond to different domains of this moduli space. In contrast, in theanomalous case, the FI parameter is a running coupling constant, and the different phasescorrespond to fixed points under the RG flow.The phases typically exhibit gauge symmetry breaking. For instance, in geometricphases, in which the theory can be effectively described by a non-linear sigma model, thegauge group is typically completely broken. On the other hand, in phases in which thetheory can be described by Landau-Ginzburg models (Landau-Ginzburg phases), a finitesubgroup of the gauge group remains unbroken and survives as an orbifold group.The question we will address in this paper is how the boundary sectors (i.e. theD-branes) behave under transitions between different phases of GLSMs.The transport of D-branes between different phases of abelian gauged linear sigmamodels has initially been studied in [2] for the non-anomalous “Calabi-Yau” case. Resultson the anomalous “non-Calabi-Yau” case appeared more recently in [3, 4]. With a carefulanalysis, the authors of these papers obtain a prescription of the D-brane transport onthe level of individual D-branes: Starting in one phase, a D-brane is first lifted to thegauged linear sigma model. This lift is a priori not unique, but requires certain choices.These choices correspond to the homotopy classes of paths along which the D-brane canbe smoothly transported in parameter space (“grade restriction rule”). Having lifted aD-brane in such a way that it can be smoothly transported along the chosen path, oneonly has to push down the lift to the other phase.In this paper, we want to revisit the transport of D-branes between different phasesfrom an alternative point of view and in particular give a uniform and functorial descriptionof it. Our basic idea is to construct suitable supersymmetry preserving defect lines (domainwalls) that connect different phases of a GLSM (or a GLSM and one of its phases). Suchdefects can be inserted along lines in space-time which separate space-time domains inwhich different phases of a GLSM are realized.phase phase GLSM phase (1.1)Supersymmetry preserving defects can be merged with boundaries (“fusion”) and in thisway give rise to an action on D-branes. B GLSM D phase GLSM B D ⊗ B phaseThis action is functorial, and hence any supersymmetric defect between two theories yieldsa functor between the respective D-brane categories.In order to find the defects describing the D-brane transport in GLSMs, we will followa strategy analogous to the one employed in [5] to describe the behavior of D-branes under– 2 –G flows. The basic idea of that paper is to consider an RG flow of a UV theory triggeredby a perturbation that is restricted to a domain of space-time.perturbed UV UV RG flow IR UV R (1.2)The RG flow drives the theory to the IR inside this domain, and leaves it at the UV fixedpoint outside of it, thereby creating a defect line on the domain boundary, which separatesthe UV and IR theories. This kind of construction has found applications in a varietyof contexts, for supersymmetric as well as non-supersymmetric theories, for marginal aswell as relevant perturbations, see [6–8]. This “RG defect” encodes the behavior of theboundary sector under the flow. In case that the models are N = (2 ,
2) supersymmetricand that the respective perturbations preserve half of the supersymmetry, the resultingdefects are also supersymmetric. Their fusion with supersymmetric boundary conditionscan easily be calculated, and the respecting functors can be determined.Supersymmetric RG defects have some important characteristic properties that werediscussed in [9]. For example, contraction of small unperturbed UV regions inside IRpatches does not change correlation functions. This can be interpreted in terms of fusion,and one obtains the identityUVIR IRR T
IRIR R ⊗ T ∼ = IRIR I IR where R is the RG defect, T is an adjoint of it and I IR is the invisible identity defect ofthe IR theory. Thus, fusion R ⊗ T ∼ = I IR yields the identity defect in the IR. This alsoimplies that the fusion of R and T in the opposite order gives rise to a projection defect P = T ⊗ R in the UV theory. This defect is idempotent with respect to fusion and can beused to realize the IR theory inside the UV theory.In this paper we will explain how similar ideas can be employed in the context ofabelian GLSMs to obtain transition defects between different phases of GLSMs as well asdefects between GLSMs and their phases. In this way, we derive a novel method for branetransport and in particular recover the grade restriction rule of [2–4] from this point ofview. While we do have compatible results, our derivations are rather different from thoseof [2–4]. In our discussion we decouple all gauge degrees of freedom and merely take intoaccount the matter sector, the only remnant of the gauge symmetry being an equivariancecondition. This subsector is under good control and still captures the physics of the (B-type) supersymmetry preserving sector, including perturbations, boundary conditions anddefects. Our arguments mainly rely on the rigidity of defect constructions in this setting.The defects we construct on this level directly mediate between the different phases anddo not exhibit an explicit t dependence.In section 2 we will outline the general ideas. In particular, we will discuss the con-struction of defects connecting different phases of a GLSM and explain how they factorizeinto defects lifting phases to the GLSM and those pushing down the GLSM to phases. Fur-thermore, we will introduce projection defects which realize the phases inside the GLSM.– 3 –heir action on the D-brane category of the GLSM corresponds, in the language of [2, 4],to the projection of GLSM branes to “grade restricted” representatives.The starting point for the construction of the transition defects is the identity defect ofthe GLSM theory which will be constructed explicitly in section 2.3.3. For this, one needsto generalize the known constructions of orbifold identity defects [5, 10] to continuous(abelian) groups. We show that this can be achieved (in the context of equivariant matrixfactorizations) by introducing new bosonic fields constrained to the defect. Our expectationis that this idea can be applied more generally in topological field theories.In section 3, we will illustrate our general construction in an explicit class of examples,namely the U (1)-gauged linear sigma models with two chiral fields, X and P , and superpo-tential W = P d ′ X d , d ′ < d . These anomalous models have two different Landau-Ginzburgorbifold phases. The UV phase is described by the Landau-Ginzburg orbifold with superpo-tential W = X d and orbifold group Z d , and the IR phase by the Landau-Ginzburg orbifoldwith superpotential W = P d ′ and orbifold group Z d ′ . Along the RG flow, d − d ′ vacuadecouple to a Coulomb branch taking with them a set of D-branes. All of this is encodedin the transition defects we construct here. We recover various results from [2–4] on branetransport. Moreover, in this example the phase transition between UV and IR phase of theGLSM corresponds to a well understood RG flow between the Landau-Ginzburg modelsdescribing the UV and IR phases [11]. RG defects for these flows have been constructedin [5] and we indeed find that our transition defects between UV and IR phase agree withthe respective RG defects. We are considering two-dimensional N = (2 ,
2) gauged linear sigma models with abeliangauge groups [1]. By X i , i = 1 , . . . , n we denote the chiral superfields of the theory. Theirrepresentation under the gauge group U (1) k is specified by the charge matrix Q ai , where i = 1 , . . . , n and a = 1 , . . . , k . For each U (1)-factor of the gauge group the theory containsa field strength multiplet, a twisted chiral field Σ a , a = 1 , . . . , k . We also allow for asuperpotential W , which is a gauge invariant polynomial in the superfields X i .The classical bosonic potential for the scalar parts x i of the chiral superfields X i and σ a of the twisted chiral fields Σ a is given by U = n X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X a =1 Q ai σ a x i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + e k X a =1 n X i =1 Q ai | x i | − r a ! + n X i =1 (cid:12)(cid:12)(cid:12)(cid:12) ∂W∂x i ( x , ..., x n ) (cid:12)(cid:12)(cid:12)(cid:12) . (2.1)Here, r a ∈ R is the Fayet-Iliopoulos (FI) parameter of the a th U (1) gauge factor. Togetherwith the corresponding θ -angle θ a it forms a complex parameter t a = r a − iθ a . (The gaugecouplings, e of the U (1)-factors are assumed to be equal.)The classical vacuum manifold is obtained as the space of solutions to the equation U = 0 modulo gauge transformations. Its nature depends crucially on the specific valuesof ( r , ..., r k ). The subspace parametrized by the expectation values of the matter fields is– 4 –ommonly referred to as the Higgs branch, whereas the scalars σ a parametrize a Coulombbranch. Phases in which the gauge group is completely broken and all modes transverseto { U = 0 } are massive are called geometric phases . In these phases, the Higgs branch iseffectively described by a non-linear sigma model with target space { U = 0 } /U (1) k . If onthe other hand the space of vacua { U = 0 } /U (1) k consists of a single point and all modestransverse to the orbit of the complexified gauge group remain massless, the Higgs branchis effectively described by a Landau-Ginzburg (orbifold) theory. Such phases are called Landau-Ginzburg phases . Besides these extreme ones, GLSMs can also exhibit variousmixed phases. Furthermore, classically at r = 0, all fields can be 0. This means that someof the σ a can be non-zero, and parametrize vacua on another branch, the Coulomb branch.An important quantum effect is the renormalization of the Fayet-Iliopoulus parameters: r a ( µ ) = r aUV + Q atot log µM UV . (2.2)Here M UV denotes a U V energy scale, µ the scale under consideration, and Q atot = P Ni =1 Q ai is the total charge of the respective U (1) factor. If Q atot = 0 for all a , the axial R-symmetryof the theory is non-anomalous and the FI parameters do not run. The t a are genuineparameters of the theory. This case is called the “Calabi-Yau case”.If one of the total charges is non-zero, the respective FI parameter does run under theRG flow. The direction of the running and with it the nature of the low energy IR phaseis determined by the sign of the total charge.In general, the low energy IR phase to which the system is driven by the RG flowconsists of several branches. In the specific example considered in section 3, there is aHiggs branch described by a Landau-Ginzburg model as well as several massive vacualocated on a Coulomb branch.Note, that also in the anomalous case the system can explore various different phases[4]. For this, one chooses r UV such that at some intermediate energy scale the system iswell described by the desired phase. Our main example in section 3 features, besides theIR phase an additional phase corresponding to the UV fixed point. This UV phase is aLandau-Ginzburg phase as well, but in contrast to the IR phase, it is a pure Higgs phase,i.e. it does not have additional Coulomb vacua. We now want to obtain defects describing the transition between different phases of thesame GLSM, much in line with the construction of RG defects reviewed in the introduction.The general idea is to start with the identity or invisible defect I GLSM in the GLSM, andto push the GLSM down to different phases on the two sides of it: I GLSM
GLSM GLSM phase phase RG A priori this requires tuning the t a on the two sides of the defect to different regimes. Weavoid doing this explicitly by going to an extreme UV limit of the theory, in which the– 5 –auge coupling e becomes very small and the gauge sector decouples [2]. In this limit, thetheory reduces to the matter sector, describing the Higgs branch of the original theory.The gauge group still acts on the matter fields, and physical observables must be gaugeinvariant. The defects are B-type supersymmetric, and depend on the parameters t a onlyindirectly through stability conditions that can be added consistently to this sector. Inthis setup, the transition to a phase restricts the allowed field configurations and breaksthe (remnant of the) gauge symmetry to a subgroup. The details strongly depend on therespective phase.For instance, the class of examples we will discuss in section 3 features Landau-Ginzburg phases. In such phases, some of the fields obtain a vacuum expectation value,reducing the spectrum of massless excitations and breaking the gauge symmetry to a finitesubgroup. Pushing down to such a phase then involves setting the respective fields to theirvacuum expectation values and relaxing the invariance condition accordingly. The generalstrategy outlined here should be applicable to any phases of abelian GLSMs. Transitionsto geometric phases will be discussed in a forthcoming paper [12].Note that one obtains a possibly different defect for every homotopy class of pathsconnecting two given phases in the parameter space spanned by the t a . Thus, in generalthere will not be one transition defect descending from the gauged linear sigma model, butmany, and the choices of defects should correspond to choices of paths. On the other hand,there can be more RG flows and with it RG defects between different phases of a GLSMthen the ones described within the GLSM.Indeed, the transition defects RG between two phases of a GLSM factorize over theGLSM, i.e. RG can be obtained as the fusion of a defect T from phase to the GLSMand a defect R from the GLSM to phase : RG ∼ = R ⊗ T . phase GLSMphase R T The defects T and R are obtained from the GLSM identity defect, by pushing down onlyon one side. T is obtained by pushing down I GLSM on the right to phase and R bypushing down I GLSM on the left to phase : I GLSM
GLSM GLSM GLSM phase T I GLSM
GLSM GLSM phase GLSM R The R i encode the push down from the GLSM to phase i and the T i , the embedding ofphase i into the GLSM. The functors associated to those defects describe the respectiveoperation on D-brane categories. – 6 –ote that in the same way that there can be several transition defects RG betweendifferent phases we expect more than one possible defect T i lifting the phase to the GLSM. This will be discussed in more detail for the concrete example in section 3.Since the transition between one and the same phase has to be trivial, the defects R i and T i have to satisfy the condition R i ⊗ T i ∼ = I phase i , (2.3)where I phase i is the invisible defect of phase i . This implies that the combination P i = T i ⊗ R i is a projection defect from the GLSM to itself. That means P i is an idempotent withrespect to fusion, P i ⊗ P i ∼ = P i , and realizes phase i inside the GLSM in the sense of [9].In particular, the corresponding functor projects the category of D-branes of the GLSMonto the image of the functor associated to T i . Thus, the phase i branes are realized by P i -invariant branes in the GLSM. Indeed, the latter precisely play the role of the branescalled grade restricted in [2] and the action of the projection defects corresponds to theoperation of associating to a GLSM brane a grade restricted representative. We will seethis explicitly in the example discussed in section 3.We have collected the various defects and their actions on D-branes in the followingdiagram: GLSM branes P j -invariantsubcategory P i -invariantsubcategoryphase j branes phase i branes push down R i push down R j transition P j lift T j lift T i transition R j ⊗ T i ⊂ ⊂ Note that along an RG flow, Higgs vacua can migrate to the Coulomb branch and becomemassive. Since we only include the Higgs branch in our discussion, we cannot see thisdirectly. Instead, we observe that D-branes attached to those vacua decouple from thetheory. This decoupling of D-branes is encoded in the defects introduced above. They canbe constructed out of the identity defect of the respective GLSM, which will be introducedin the next section.
Starting point of our construction are the identity defects of abelian GLSMs, which havenot appeared in the literature so far. As discussed above, we will focus on the Higgs branch Indeed, one could also push the path dependence on the R j . – 7 –f the respective GLSM. In particular, we will decouple the gauge sector and only considerthe U (1) k -orbifold of the matter sector. The relevant defects and D-branes can then bedescribed by means of U (1) k -equivariant matrix factorizations.Before discussing the identity defects in GLSMs, we will briefly introduce some use-ful facts about matrix factorizations and discuss the identity defects in Landau-Ginzburgmodels. Defects in Landau-Ginzburg (orbifold) models are described by (equivariant) matrix fac-torizations of the superpotential, see the appendix for a brief discussion and references tothe literature. Consider a matrix factorization P : P p p P of a polynomial W in a polynomial ring S = C [ X , . . . , X n ]. Here, P = P ⊕ P is a Z -graded free module over S and d P = p p ! is an odd endomorphism of P such that d P = W · id P . In the following, instead of dealingwith matrix factorizations we will often consider certain associated modules. To P asabove, one can associate the module M P = coker( p : P ⊗ S C → P ⊗ S C )over the respective quotient ring C = S/ ( W ). This module has a free two-periodic resolu-tion defined by the matrix factorization [13] . . . p −→ P ⊗ S C p −→ P ⊗ S C p −→ P ⊗ S C p −→ P ⊗ S C −→ M P → . Isomorphisms between modules M P and M Q associated to matrix factorizations P and Q of the same polynomial W lift to these resolutions and give rise to isomorphisms of therespective matrix factorizations. We will make excessive use of this fact in our analysis.Indeed, the above argument works in the same way for any C -module M P , which has a freeresolution turning, after finitely many steps, into a two-periodic resolutions defined by thematrix factorization P . This point of view carries over to the case of equivariant matrixfactorizations, c.f. the appendix. The identity or invisible defect I W in a Landau-Ginzburg model with superpotential W ∈ C [ X , . . . , X n ] is given by the following Koszul matrix factorization ( I, d I ) [14]. Denote by S ( X )( X ′ ) = C [ X , . . . , X n , X ′ , . . . , X ′ n ] the polynomial ring of the chiral fields on both sidesof the defect. For later use, we will also denote by S ( X )( − ) = C [ X , . . . , X n ] the ring of– 8 –hiral fields on the left and by S ( − )( X ′ ) = C [ X ′ , . . . , X ′ n ] the ring of chiral fields on the rightof the defect. Then I = I ⊕ I = S ( X )( X ′ ) ⊗ Λ ( V ) (2.4)is the tensor product of the algebra of chiral fields S ( X )( X ′ ) with the exterior algebra of avector space V = span C { θ , . . . , θ n } spanned by additional variables θ , . . . , θ n . The lattercorrespond to fermions on the defect, and the Z -degree is inherited from the Z -degree ofthe exterior algebra. The differential can be written asd I = n X i =1 h ( X i − X ′ i ) · θ ∗ i + ∂ X,X ′ i W · θ i i with ∂ X,X ′ i W = W ( X ′ , ..., X ′ i − , X i , ..., X n ) − W ( X ′ , ..., X ′ i , X i +1 , ..., X n ) X i − X ′ i (2.5)where { θ ∗ , . . . , θ ∗ n } denotes the dual to the basis { θ , . . . , θ n } , i.e. θ ∗ i ( θ j ) = δ ij . Physically,the θ i correspond to boundary fermions. Note that ∂ X,X ′ i W is a polynomial in the X i and X ′ i . This defect is the unit under fusion with other defects or boundaries, which meansthat for any defect P , the fusion I W ⊗ P and P ⊗ I W are isomorphic to P . This canbe easily seen using the associated modules. To the identity defect, we can associate the C ( X )( X ′ ) = S ( X )( X ′ ) / ( W ( X , . . . , X n ) − W ( X ′ , . . . , X ′ n ))-module M I = C ( X )( X ′ ) / ( X − X ′ , . . . , X n − X ′ n ) . This module has a free resolution, which after finitely many steps turns into the two-periodic resolution defined by the matrix factorization (2.5), see e.g. [15].Let P ′ be a matrix factorization of W ( X ′ , . . . , X ′ n ). Fusion of I with P ′ is given bythe tensor product of the respective matrix factorizations I ⊗ P ′ [16]. Let C ( − )( X ′ ) = S ( − )( X ′ ) / ( W ( X ′ , . . . , X ′ n )) , and C ( X )( − ) = S ( X )( − ) / ( W ( X , . . . , X n )) . Then the C ( − )( X ′ ) -module M P ′ = coker( p ′ : P ′ ⊗ S ( − )( X ′ ) C ( − )( X ′ ) → P ′ ⊗ S ( − )( X ′ ) C ( − )( X ′ ) )is associated to the matrix factorization P ′ . Now, the tensor product M I ⊗ S ( − )( X ′ ) M P ′ isa C ( X )( − ) -module, which has a free resolution turning, after finitely many steps, into thetensor product matrix factorization I ⊗ P ′ of W ( X , . . . , X n ). Of course M I ⊗ S ( − )( X ′ ) M P ′ isisomorphic to the cokernel of the matrix p , obtained by setting all entries X ′ i in p ′ to X i .After all, the relations X i − X ′ i in M I just set X ′ i to X i . This module has a free resolutiongiven by the matrix factorization P , which is obtained by replacing all the X ′ i in P ′ by X i .Hence, fusion with I maps matrix factorizations to equivalent ones.In case the Landau-Ginzburg model has a symmetry group G W which linearly acts onthe chiral fields, X i g ( X i ) for g ∈ G W such that W ( g ( X ) , . . . , g ( X n )) = W ( X , . . . , X n ),– 9 –ne can construct defects g I that implement the symmetry operation. The matrix factor-izations of these symmetry defects are built on the same module as the identity matrixfactorization, but the differential is twisted by the symmetry [16]d g I = n X i =1 h ( X i − g ( X ′ i )) · θ ∗ i + ∂ X,X ′ i W ( X, g ( X ′ )) · θ i i . In the Landau-Ginzburg phases appearing here, the gauge group is not completelybroken, and its remnant survives as a finite abelian orbifold group, acting by multiplyingthe chiral fields X i by phases.Defects in Landau-Ginzburg orbifolds can be described by matrix factorizations of thesuperpotential which are equivariant with respect to the action of the orbifold group, seethe appendix for a brief discussion and references.The identity defect in Landau-Ginzburg orbifolds can be constructed from the identitydefect of the unorbifolded LG model (2.4), (2.5) using the method of images (i.e. summingimages under the orbifold group and specifying a representation of the stabilizer subgroup).Let G be a finite abelian orbifold group. The identity defect of the orbifold model can beobtained by summing over all symmetry defects g I non-orb of the unorbifolded Landau-Ginzburg model associated to orbifold group elements [5] I orb = M g ∈ G g I non-orb . (2.6)Note here that one has to orbifold by G × G , the product of the orbifold groups on the leftand on the right of the defect, but that the diagonal subgroup acts as an isomorphism onthe non-orbifolded identity defect. Hence only a non-diagonal copy of G , which we take tobe the copy G r acting trivially on the left of the defect contributes to the sum above.The module on which the orbifolded identity matrix factorization is built is thereforea direct sum of | G | copies of the module (2.4) associated to the identity defect in theunorbifolded Landau-Ginzburg model. We can regard it as a tensor product of the module I non-orb with the regular representation V reg of the group G r : I orb ∼ = I non-orb ⊗ V reg .The differential acts diagonally in the standard basis g ∈ G r of the regular represen-tation, while the orbifold group acts in this basis by permuting the copies of the modules I non-orb according to the group law.Since G r is finite and abelian, we can diagonalize the group action on I orb . This canbe accomplished by decomposing the regular representation into irreducibles, which in thecase of abelian G r are all one-dimensional. In this way, we obtain a basis of I orb , in which G r acts diagonally.Any finite abelian group is isomorphic to a product Z d × . . . × Z d r . We will spellout the details for the case, in which it is isomorphic to a single factor G r ∼ = Z d . Thegeneralization to more factors is straight-forward.A basis of V reg corresponding to the irreducible representations can be obtained byperforming the following transformation: e j = X g ∈ Z d ξ − gj g , ≤ j < d , (2.7)– 10 –here ξ = exp(2 πi/d ) is an elementary d th root of unity. g ξ jg is the character of theirreducible representation ρ j defined by ρ j ([ n ] d ) = ξ jn . Hence, e j is the basis vector of theirreducible representation ρ j ([ n ] d ) = ξ jn , which is of course nothing but the j -fold tensorproduct of ρ : ρ j = ρ ⊗ j . Thus, we can write e j = e ⊗ j . Note that ρ ⊗ d = ρ . Writing e j = α − j , the regular representation can be expressed as V reg ∼ = C [ α ] / ( α d − . (2.8)Note that this is not only a vector space, but also a ring, and that multiplication in this ringcorresponds to taking the tensor product of representations. This allows us to rewrite theidentity matrix factorization in the Landau-Ginzburg orbifold with orbifold group G = Z d as I orb = S ( X )( X ′ ) ⊗ Λ( V ) ⊗ C [ α ] / ( α d −
1) (2.9)with differentiald I orb = n X i =1 h ( X i − α Q i X ′ i ) · θ ∗ i + ∂ X,αX ′ i W · θ i i with ∂ X,αX ′ i W = W ( α Q X ′ , ..., α Q i − X ′ i − , X i , ..., X n ) − W ( α Q X ′ , ..., α Q i X ′ i , X i +1 , ..., X n ) X i − α Q i X ′ i . Here, Q i denote the charges of the chiral fields X j under the orbifold group Z d , i.e. [ n ] ∈ Z d acts on the chiral fields as X j ξ Q j n X j . α can be regarded as a new bosonic defect fieldcarrying charg (1 , −
1) under the product Z d × Z d of the left and right orbifold groups. The representation on I orb under G × G is now completely fixed by the choice of aone-dimensional representation of the diagonal subgroup, since the latter left the identitydefect of the non-orbifold theory invariant. We choose it to be trivial to obtain the identitydefect in the orbifold theory. (Other choices lead to defects implementing the quantumsymmetry of the orbifold theory.) The representations of G × G on the module (2.9) isdetermined by the representation on the chiral fields X i , the θ i (which transform like the X i ) and the representation on α i .Let us give an explicit example which will be important later. Consider the Landau-Ginzburg model with a single chiral superfield X , superpotential W ( X ) = X d and orbifoldgroup G = Z d . [ n ] d ∈ Z d acts on X by multiplication with a phase X e πind ′ d X whichleaves W ( X ) invariant. ( X has charge d ′ under Z d .) Following the construction above, oneobtains the identity matrix factorization I orb : S α { [1] d , [0] d } i = ( X − α d ′ X ′ ) i = Q d − i =1 ( X − ξ i α d ′ X ′ ) S α { [0] d , [0] d } , where S α := C [ X, X ′ , α ] / ( α d − {· , ·} denote a shift in Z d × Z d -charges. Theassociated C ( X )( X ′ ) -module is given by S α / ( X − α d ′ X ′ ) . (2.10) The charge under the action of the right group is clear, because α represents the basis vector of theirreducible representation ρ − under the right group. That it has charge 1 under the left group followsbecause V reg was chosen to be invariant under the left-right diagonal subgroup. – 11 –ne can unpack this, by replacing α by a cyclic shift matrix. This yields the equivalentrepresentation I orb : S d { [1] d , [0] d }{ [2] d , [ − d } ... { [ d ] d , [ − d ] d } ı = ( X I d − ǫ d ′ X ′ ) ı = Q d − i =1 ( X I d − ξ i ǫ d ′ X ′ ) S d { [0] d , [0] d }{ [1] d , [ − d } ... { [ d − d , [ − d ] d } (2.11)Here S = C [ X, X ′ ], I d is the d × d -identity matrix, and ǫ d = (2.12)denotes the d × d -shift matrix. For gauged linear sigma models, mainly boundaries were considered in the literature, see[2, 3, 17]. Defects can in principle be discussed along the same lines, for example using thefolding trick, see [18, 19]. In [18] identity defects of Landau-Ginzburg phases of GLSMsare lifted to GLSMs. Using such constructions, one cannot obtain an identity defect of theGLSM itself, because the lifts are matrix factorizations of finite rank.We now use the method presented in the previous section to construct an identitydefect for a U (1)-orbifold of a Landau-Ginzburg model with chiral fields X , . . . , X n andsuperpotential W ∈ C [ X , . . . , X n ]. The action of U (1) on the chiral fields is specified bytheir charges ( Q , . . . , Q n ), where ϕ ∈ U (1) acts on X j by X j e πiQ j ϕ X j . (Generaliza-tions to higher rank abelian gauge groups are straight-forward.)Since the orbifold group is infinite (and not even countable), the method of images can-not be applied in this situation. As it turns out, the formulation with the additional defectfield α however can be adapted. The irreducible representations of U (1) are countable, ρ j ( ϕ ) = e πijϕ , j ∈ Z , with ρ i ⊗ ρ j ∼ = ρ i + j . But in contrast to the case of representationsof Z d , not all representations can be obtained as tensor products of a single representation ρ − . One needs an additional representation ρ to generate all representations by meansof the tensor product. So in the U (1)-case, instead of one additional bosonic defect field α , one has to introduce two fields α, α − which are inverse to each other, i.e. αα − = 1.They carry U (1) × U (1)-charges (1 , −
1) and ( − , I = S ( X ) , ( X ′ ) ⊗ Λ( V ) ⊗ C [ α, α − ] / ( αα − −
1) (2.13)with differentiald I = n X i =1 h ( X i − α Q i X ′ i ) · θ ∗ i + ∂ X,αX ′ i W · θ i i ,∂ X,αX ′ i W = W ( α Q X ′ , ..., α Q i − X ′ i − , X i , ..., X n ) − W ( α Q X ′ , ..., α Q i X ′ i , X i +1 , ..., X n ) X i − α Q i X ′ i . – 12 –he U (1) × U (1)-representation on this matrix factorization is completely determined bythe transformation properties of the fields X i , i.e. their U (1)-charges Q i . The θ i transformas the X i .Note that this matrix factorization is of infinite rank!As in the case of finite orbifolds, there is the possibility of shifting the charges of theorbifold group on one side, relative to the one on the other. The defect with such a shiftimplements a quantum symmetry. Since we are interested in the identity defect, we setthis shift to zero.As alluded to above, the generalization to orbifold groups U (1) k k > α, α − ) for each U (1)-factor.In the discussion of the identity defect of finite Landau-Ginzburg orbifolds in the lastsubsection, we just gave a different but equivalent representation of the known identitydefect. In the case of the U (1)-orbifold, the defect (2.13) is new, and we have to show thatit really is the identity defect, i.e. that under fusion it behaves as the unit.To do so, we use associated modules as explained in section 2.3.2. For this we have tointroduce some notation: S ( X ) , ( X ′ ) = C [ X , . . . , X n , X ′ , . . . , X ′ n ] S ( α,α − )( X ) , ( X ′ ) = S [ α, α − ] / ( αα − − C ( X ) , ( X ′ ) = S ( X ) , ( X ′ ) / ( W ( X , . . . , X n ) − W ( X ′ , . . . , X ′ n )) C ( α,α − )( X ) , ( X ′ ) = C ( X ) , ( X ′ ) [ α, α − ] / ( αα − − . Replacing X or X ′ by − in the subscripts means setting the respective variables to zero.To the matrix factorization I constructed above we associate the C ( X ) , ( X ′ ) -module M I = C ( α,α − )( X ) , ( X ′ ) / (cid:0) X − α Q X ′ , . . . , X n − α Q n X ′ n (cid:1) . (2.14)As in the discussion of the identity defects in unorbifolded Landau-Ginzburg models insection 2.3.2, this module has a free Koszul-type resolution, which after finitely many stepsturns into the two-periodic complex induced by the matrix factorization I .Let P ′ be a U (1)-equivariant matrix factorization of W ( X ′ , . . . , X ′ n ). The analysisof the fusion I ⊗ P ′ now runs in complete analogy of the discussion of the fusion of theidentity defect in the unorbifolded Landau-Ginzburg models in section 2.3.2, except for thefact that the fusion in the orbifold corresponds only to the part of the tensor product matrixfactorization which is invariant under the gauge group associated to the model squeezed inbetween the two defects.Let M P ′ = coker( p ′ : P ′ ⊗ S ( − )( X ′ ) C ( − )( X ′ ) → P ′ ⊗ S ( − )( X ′ ) C ( − )( X ′ ) )be the module associated to P ′ . The matrix factorization given by the fusion of I and P ′ can be extracted from the U (1)-invariant part M U (1) of the C ( X ) , ( − ) -module M = M I ⊗ C ( − )( X ′ ) M P ′ . – 13 –he relations in (2.14) can now be used to replace all X ′ i by α − Q i X i . This eliminates allthe variables X ′ i . Let us next choose generators e r of M P ′ , on which U (1) acts diagonally,with respective U (1)-charge q r . Then M is generated by α i ⊗ e r , where i ∈ Z . Most ofthese generators are not U (1)-invariant though. Only the ˜ e r := α − q r ⊗ e r generate M U (1) .Thus, for each generator e r of M P ′ of U (1)-charge q r there is exactly one generator of M U (1) , which also has U (1)-charge q r . (Recall that α has U (1) × U (1)-charge (1 , − e r in M P ′ become relations between the respectivegenerators ˜ e r , where all the X ′ i are replaced by X i . Thus, M U (1) is isomorphic to themodule M P associated to the matrix factorization P obtained from P ′ by replacing all X ′ i by X i . Therefore, fusion with I maps matrix factorizations to equivalent ones, and thematrix factorization I acts as the identity matrix factorization. W ( X, P ) = X d P d ′ In this section we exemplify our method in a concrete example of a gauged linear sigmamodel with two Landau-Ginzburg phases. The model has a single U (1)-gauge group andtwo chiral fields X and P of U (1)-charges Q x = d ′ , respectively Q p = − d . Its superpotentialis given by W = X d P d ′ . We assume d > d ′ , and for simplicity we restrict to the case where d and d ′ are coprime integers.For this model, the scalar potential (2.1) takes the form U = | σ | (cid:0) Q x | x | + Q p | p | (cid:1) + e (cid:0) Q x | x | + Q p | p | − r (cid:1) + | ∂ x W ( p, x ) | + | ∂ p W ( p, x ) | . The total charge Q tot = d ′ − d < r << r >> r <
0, the D-term constraint coming from the second term above, requires p = 0.This breaks the U (1) gauge symmetry to Z | Q p | = Z d and σ must vanish according to the firstterm. Because of the first superpotential term, x also vanishes and hence | p | = rQ p = − rd .We obtain a Landau-Ginzburg orbifold model with one chiral field X , superpotential X d and orbifold group Z d . This model is well known to give a Landau-Ginzburg realization ofthe N = (2 ,
2) minimal superconformal model with central charge c = ( d − /d .For r >
0, the roles of X and P are interchanged. The D-term constraint yields x = 0,which further implies that σ and p vanish. The U (1) gauge group is broken to Z | Q x | = Z d ′ ,and | x | = rQ x = rd ′ . We arrive at a Landau-Ginzburg orbifold model with chiral field P ,superpotential P d ′ and orbifold group Z d ′ . Again, this yields a Landau-Ginzburg realizationfo the N = (2 ,
2) minimal model with the smaller central charge c = ( d ′ − /d ′ , to whichthe systems flows at low energies.Classically, there is a Coulomb branch emerging at r = 0, parametrized by σ . Due toa twisted superpotential, the values of σ will be restricted to a finite set of d − d ′ massivevacua that appear in the IR phase. – 14 –n the following we will use our general strategy to construct defects describing thetransitions between UV and IR phase of this model, defects embedding the two phases inthe GLSM as well as defects projecting the GLSM to the phases.Note that there is an effective description of the mirror of this GLSM in terms of anordinary Landau-Ginzburg model [20], namely the Landau-Ginzburg model with one chiralfield X and superpotential W = X d + e td X d ′ . The deformation parameter λ = e td of the superpotential is related to the complexifiedFayet-Iliopoulus parameter t = r − iθ of the GLSM. λ runs under the RG flow from λ = 0in the UV to λ = ∞ in the IR. In the UV, the model is therefore described by a LG modelwith superpotential W = X d and in the IR by a LG model with superpotential W = X d ′ .Now, flows of Landau-Ginzburg models triggered by deformations of the superpoten-tials are relatively well under control, and at least some aspects of them can be studiedvery explicitly. For instance, it is not difficult to analyze what happens to the vacua ofthe model, which correspond to critical points of the superpotential. In the case at hand,some of these vacua ( d − d ′ many) move off to infinity under the RG flow, and decouplefrom the theory, taking with them some (A-type) D-branes attached to them.This decoupling of D-branes is well described by RG defects associated to the flows.Indeed, all the RG defects corresponding to flows of Landau-Ginzburg models with a singlechiral superfield but general deformations of the superpotential W = X d + d − X i =1 λ i X i , (3.1)have been constructed in [5]. Thus, the transition defects between UV and IR phasesof the GLSM which we will obtain here can be checked against known results. We findcomplete agreement.
The starting point of our analysis is the identity defect of the GLSM as constructed for thegeneral abelian GLSMs in section 2.3.3. In this case, it is a U (1) × U (1)-equivariant matrixfactorization of the difference W ( X, P ) − W ( Y, Q ) = X d P d ′ − Y d Q d ′ of the superpotentialsof the gauged linear sigma models on either side of the defects. The two U (1)-factorscorrespond to the gauge groups of the models on the left and the right of the defect,respectively.For clarity, we will repeat and spell out some details of the construction of the identitydefect in this example. We first introduce new variables (corresponding to degrees offreedom on the identity defect) α and α − which satisfy αα − = 1 and which carry U (1) × U (1)-charges | α | = (1 , −
1) and | α − | = ( − , More precisely, they have been constructed in the mirror theories. – 15 –ifference of the superpotentials as follows W ( X, P ) − W ( Y, Q ) = X d P d ′ − Y d Q d ′ = X d P d ′ − α dd ′ Y d P d ′ + α dd ′ Y d P d ′ − Y d Q d ′ = (cid:16) X d − ( α d ′ Y ) d (cid:17) P d ′ + Y d (cid:16) ( α d P ) d ′ − Q d ′ (cid:17) = P d ′ d − Y i =0 ( X − ξ i α d ′ Y ) + Y d d ′ − Y i =0 ( α d P − ( ξ ′ ) i Q ) . Here ξ = e πid and ξ ′ = e πid ′ are elementary d th, respectively d ′ th roots of unity.The matrix factorization associated to the identity defect is then given by the Koszul-type matrix factorization associated to ( X − α d ′ Y ) and ( α d P − Q ). More precisely, denotingthe C [ X, P, Y, Q ]-modules S = S ( X,P )( Y,Q ) = C [ X, P, Y, Q ]and ˜ S = S ( α,α − )( X,P )( Y,Q ) = S ( X,P )( Y,Q ) [ α, α − ] / ( αα − − . the identity matrix factorization can be written as I : ˜ S { d ′ , }{ , − d } ! i i ˜ S { , }{ d ′ , − d } ! (3.2)Here {· , ·} indicates the U (1) × U (1)-charge of the respective generator and i = ( X − α d ′ Y ) − ( α d P − Q ) Y d Q d ′ − i =1 ( α d P − ( ξ ′ ) i Q ) P d ′ Q d − i =1 ( X − ξ i α d ′ Y ) ! i = P d ′ Q d − i =1 ( X − ξ i α d ′ Y ) ( α d P − Q ) − Y d Q d ′ − i =1 ( α d P − ( ξ ′ ) i Q ) ( X − α d ′ Y ) . ! This is nothing but the GLSM identity matrix factorization (2.13) spelled out for the specialcase at hand. To it we associate the module M I = C ( α,α − )( X,P )( Y,Q ) / (( X − α d ′ Y ) , ( α d P − Q )) (3.3)over the ring C = C ( X,P )( Y,Q ) = S ( X,P )( Y,Q ) / ( W ( X, P ) − W ( Y, Q )) , where C ( α,α − )( X,P )( Y,Q ) = C ( X,P )( Y,Q ) [ α, α − ] / ( αα − − , c.f. the general case (2.14). The Koszul resolution of M I turns into the two-periodic complexinduced by the identity matrix factorization I after two steps.– 16 – .3 Pushing down the identity defect into phases Going into the phases of the GLSM, one of the two chiral fields gets a vacuum expecationvalue (which we can take to be 1), and the gauge group is broken to the subgroup leavingthis chiral field invariant. We can therefore push down any defect of the GLSM into aphase by setting the respective chiral field to 1 in the associated matrix factorization andconsidering it equivariant with respect to the residual gauge group. In fact, this can bedone on either side of the defect yielding defects from the GLSM into the phases or fromthe phases to the GLSM. Moreover, one can push down to phases on both sides of thedefect, possibly into different phases on the two sides, which gives rise to defects in thephases or from one phase to another.In the next section, we will employ this push-down to the GLSM identity defect.Pushing down to the UV phase on the right side and the IR phase on the left, we willobtain a defect describing the transition from the UV phase to the IR phase. Indeed, wewill reproduce RG defects of [5], as expected.Before we will come to this, as a warm-up we first discuss the simpler case, where theGLSM identity defect is pushed down to the same phase on both sides. We choose the UVphase. The push down to the IR can be dealt with in a similar way.To push down the GLSM identity defect to the UV phase on both sides, we have toset P and Q to 1 in the matrix factorization (3.2) and consider it equivariant with respectto the residual gauge group Z d × Z d . We can do this on the level of the associated module(3.3).It will be useful to introduce some notation. Replacing the name of a variable with a‘ · ’ in the subscripts of the rings S ( X,P )( Y,Q ) , C ( X,P )( Y,Q ) or C ( α,α − )( X,P )( Y,Q ) just means settingthe respective variable to one. For instance S ( X, · )( Y,Q ) = C [ X, Y, Q ] C ( X, · )( Y,Q ) = S ( X, · )( Y,Q ) / ( W ( X, − W ( Y, Q )) C ( α,α − )( X, · )( Y,Q ) = C ( X, · )( Y,Q ) [ α, α − ] / ( αα − − . Pushing down the GLSM identity defect to the UV phase on both sides yields the module M UV UV I = C ( α,α − )( X, · )( Y, · ) / (( X − α d ′ Y ) , ( α d − . Note that due to the relation α d − C ( α,α − )( X, · )( Y, · ) / ( α d − ∼ = C d ( X, · )( Y, · ) { [0] d , [0] d }{ [1] d , [ − d } ... { [ d − d , [ − d + 1] d } , Or to put it differently, by diving the rings by the corresponding ideals. the generator α i is sent to the generator e i – 17 –here {· , ·} denotes the shift in Z d × Z d -charge of the respective generators, one can write M UV UV I as cokernel M UV UV I ∼ = coker ı UV1 : C d ( X, · )( Y, · ) { [ d ′ ] d, [0] d } { [ d ′ + 1] d , [ − d } ... { [ d ′ + d − d , [ − d + 1] d } → C d ( X, · )( Y, · ) { [0] d , [0] d }{ [1] d , [ − d } ... { [ d − d , [ − d + 1] d } of the map ı UV1 = ( X I d − Y ǫ d ′ d ). Here, as before I d denotes the d × d -identity matrix and ǫ d the d × d -shift matrix (2.12).Indeed, ı UV1 ı UV0 = ( X d − Y d ) I d for ı UV0 = d − Y i =1 ( X I d − ξ i Y ǫ d ′ d ) . Hence, ı UV1 is a factor of a matrix factorization of W ( X, − W ( Y, I UV : S d ( X, · )( Y, · ) { [1] d , [0] d }{ [2] d , [ − d } ... { [ d ] d , [ − d + 1] d } ı UV1 ı UV0 S d ( X, · )( Y, · ) { [0] d , [0] d }{ [1] d , [ − d } ... { [ d − d , [ − d + 1] d } (3.4)This matrix factorization corresponds to the identity defect (2.11) in the UV phase. Thus,the module M UV UV I is associated to the identity matrix factorization in the UV phase.Pushing down the GLSM identity defect on both sides to the UV, one therefore producesthe identity defect in the UV phase. Similarly, pushing down the GLSM identity defect tothe IR phase on both sides yields the IR identity defect. This is of course what is to beexpected. Next, we push down the GLSM identity defect to the UV on the right and to the IR onthe left, to obtain a transition defect between UV and IR phase. Note that there is morethan one (homotopy class of) paths from UV to IR. So there should be more than one suchtransition defects.Implementing the push-down involves setting those variables to 1 in the GLSM identitymatrix factorization, which correspond to fields acquiring a vacuum expectation value inthe respective phases. These are X (IR phase on the left of defect) and Q (UV phase onthe right). On the level of C ( · ,P )( Y, · ) -modules this yields M UV IR I = C ( α,α − )( · ,P )( Y, · ) / (( Y − α − d ′ ) , ( P − α − d )) . In contrast to the push down to the same phase on both sides, this module is of infiniterank, leading to a matrix factorization of infinite rank, which does not correspond to one ofthe RG defects between the respective Landau-Ginzburg models. We propose that underthe push-down to the phases, the module (and the respective matrix factorization) hasto be truncated to finite rank. The truncation is not unique, but it turns out, that the– 18 –ifferent choices of truncation exactly correspond to the different paths from UV to IR. Concretely, we introduce an upper bound N on the α -exponents M UV IR I ( N ) = α N C ( · ,P )( Y, · ) [ α − ] / (( Y − α − d ′ ) , ( P − α − d )) α N C ( · ,P )( Y, · ) [ α − ] . This module is now of finite rank. It has generators e i := α N − i for 0 ≤ i < d ′ of Z d ′ × Z d -charges ([ N − i ] d ′ , − [ N − i ] d ). To write down the relations in a convenient way we define a, b ∈ N with b < d ′ such that d = ad ′ + b . (3.5)Then the generators satisfy relations P e i = P α N − i = α N − d − i = Y a α N − d − i + ad ′ = ( Y a e i + b , i + b < d ′ Y a +1 e i + b − d ′ , i + b ≥ d ′ and the module M UV IR I ( N ) is isomorphic to the cokernel of a map rg : C d ′ ( · ,P )( Y, · ) { [ N − d ] d ′ , [ − N ] d }{ [ N − d − d ′ , [ − N + 1] d } ... { [ N − d − d ′ + 1] d ′ , [ − N − d ′ ] d } → C d ′ ( · ,P )( Y, · ) { [ N ] d ′ , [ − N ] d }{ [ N − d ′ , [ − N + 1] d } ... { [ N − d ′ + 1] d ′ , [ − N + d ′ − d } , which can be written as rg = ( P I d ′ − ǫ bd ′ I Y ), where I Y denotes the diagonal d ′ × d ′ -matrixwith Y a as its first d ′ − b diagonal entries and Y a +1 as the last b diagonal entries. Explicitly rg = P − Y a +1 . . . . . .. . . − Y a +1 − Y a . . .. . . . . . − Y a P . Now d ′ − Y i =0 ( P I d ′ − ( ξ ′ ) i ǫ bd ′ I Y ) = ( P d ′ − Y d ) I d ′ , and hence, rg together with rg = Q d ′ − i =1 ( P I d ′ − ( ξ ′ ) i ǫ bd ′ I Y ) defines a matrix factorization RG N : S d ′ ( · ,P )( Y, · ) { [ N − d ] d ′ , − [ N ] d }{ [ N − − d ] d ′ , − [ N − d } ... { [ N − d ′ + 1 − d ] d ′ , − [ N − d ′ + 1] d } rg rg S d ′ ( · ,P )( Y, · ) { [ N ] d ′ , − [ N ] d }{ [ N − d ′ , − [ N − d } ... { [ N − d ′ + 1] d ′ , − [ N − d ′ + 1] d } (3.6) We expect the truncation to be related to the gradability of the resulting matrix factorization withrespect to R-symmetry. The latter ensures definite gluing conditions for the spectral flow operators of therespective SCFTs along the defect. This is needed to impose a stability condition in the sense of [21] onthe level of the defect. Later we will show that N determines the charge window for the grade restriction rule which appearsin [2]. – 19 –ushing down the GLSM identity defect to the UV on the right and to the IR on theleft with truncation N yields a defect between the Landau-Ginzburg models in the UVand in the IR, given by the matrix factorization RG N . Note that N only appears inthe grading of the matrix factorization RG N , and that RG N = RG N + dd ′ . Furthermore,the shift in N corresponds to conjugation with the quantum symmetries of the respectiveLandau-Ginzburg phases, RG N +1 = Q − ⊗ R N ⊗ Q UV .Thus, we obtain dd ′ many different transitions defects between the two phases. Theseindeed correspond to particular renormalization group defects between Landau-Ginzburgorbifolds describing the UV and IR phases [5]. In fact RG defects between these Landau-Ginzburg orbifolds corresponding to general perturbations of type (3.1) would allow for amore generic distribution of powers of Y in the map rg , of the form rg gen = P − Y n b . . . . . .. . . − Y n d ′ − Y n . . .. . . . . . − Y n b − P . where P n a = d and the grades appearing in (3.6) have to be modified accordingly, c.f. [5].The transition defects we obtain from the gauged linear sigma model are special cases ofthese defects which exhibit a maximally homogeneous distribution of powers of Y in rg .Summarizing, by pushing down to UV and IR on the right, respectively left side of theGLSM identity defect with an additional truncation we obtain an RG defect between theUV and IR phases, which is known to describe the transport between UV and IR Landau-Ginzburg models. Different choices of the truncation parameter N only shift the chargesof the matrix factorization, in particular RG N ∼ = RG { [ N ] d ′ , − [ N ] d } ∼ = Q N IR ⊗ RG ⊗ Q − N UV .The charge shifts are quantum symmetries of the Landau-Ginzburg orbifolds in IR and UV.They can be obtained as monodromies of the GLSM upon encircling the Landau-Ginzburgpoints in the K¨ahler parameter space. Thus, up to winding around the limit points, weobtain one defect describing the flow between UV and IR phases of the GLSM. Using the fact that the GLSM identity defect is an idempotent, I ∼ = I ⊗ I , (3.7)we can factorize the RG defects RG N over the GLSM. More precisely RG N ∼ = R IR ⊗ T UV N , where R IR is a defect from the GLSM to the IR phase obtained by pushing down the GLSMidentity defect to the IR on the left side, and T UV N is a defect from the UV phase to the– 20 –LSM model obtained by pushing the GLSM identity defect to the UV on the right andtruncating. Let us discuss the factor defects in turn. T UV N The module associated to T UV N is obtained by setting Q = 1 in (3.3) and then truncatingthe α -spectrum. This yields the C ( X,P )( Y, · ) -module M UV GLSM I ( N ) = α N C ( X,P )( Y, · ) [ α − ] / (( Y − α − d ′ X ) , ( P − α − d )) α N C ( X,P )( Y, · ) [ α − ] . It is finitely generated with generators e i = α N − i , of U (1) × Z d -charge ( N − i, − [ N − i ] d ),where 0 ≤ i < d . The generators satisfy relations Y e i = Y α N − i = Xα N − i − d ′ = ( Xe i + d ′ , i + d ′ < dP Xe i + d ′ − d , i + d ′ ≥ d , (3.8)and M UV GLSM I ( N ) is isomorphic to the cokernel of the map t : C d ( X,P )( Y, · ) { N, − [ N − d ′ ] d }{ N − , − [ N − − d ′ ] d } ... { N − d + 1 , − [ N − d + 1 − d ′ ] d } → C d ( X,P )( Y, · ) { N, [ − N ] d }{ N − , − [ N − d } ... { N − d + 1 , − [ N − d + 1] d } , with t = ( ǫ d ′ d I P X − Y I d ). Here I P is the diagonal d × d -matrix with 1 in the first b = d − d ′ diagonal entries and P in the last d ′ diagonal entries. Explicitly, t = − Y P X . . . . . .. . .
P XX . . .. . . . . . X − Y . Now, d − Y i =0 ( ǫ d ′ d I P X − ξ i Y I d ) = P d ′ X d − Y d , and hence, together with the map t = Q d − i =1 ( ǫ d ′ d I P X − ξ i Y I d ), t defines a matrix factor-ization of W ( X, P ) − W ( Y, M UV GLSM I ( N ) has a free two-periodicresolution induced by the matrix factorization T UV N : S d ( X,P )( Y, · ) { N, − [ N − d ′ ] d }{ N − , − [ N − − d ′ ] d } ... { N − d + 1 , − [ N − d + 1 − d ′ ] d } t t S d ( X,P )( Y, · ) { N, − [ N ] d }{ N − , − [ N − d } ... { N − d + 1 , − [ N − d + 1] d } . (3.9) Note that the truncation could also be implemented on R instead, or on both R and T . As it turnsout, pushing the truncation on T and not on R leads to a nice interpretation of the factor defects. – 21 –his matrix factorization represents the factor defect T UV N . Note that it is of rank d andexhibits U (1)-charge shifts only in the set { N − d + 1 , N − d + 2 , . . . , N } of d consecutiveintegers starting at N − d + 1. Fusion with the defect T UV N therefore lifts D-branes from theUV phase to GLSM branes in the charge window { N − d + 1 , . . . , N } in the terminologyof [2].Indeed, there is another way to arrive at the defects T UV N . One can start with theidentity defect in the UV phase and then lift on the left to the GLSM. Lifting in this casemeans inserting variables P into the rank- d Z d × Z d -equivariant matrix factorization of X d − Y d in such a way as to make it into a U (1) × Z d -equivariant matrix factorization of P d ′ X d − Y d . (Lifting D-branes in such a manner is an important ingredient in the discussionof D-brane transfer between LG and geometric phases of abelian GLSMs in [2].) In thisway one obtains a defect from the UV phase to the GLSM. This defect is automaticallyof finite rank, so a truncation of the kind we had to impose when coming from the GLSMis not necessary. On the other hand, the lift involves many choices. One of the choicescorresponds to the choice of N , the maximal U (1)-charge. When that is fixed there arestill choices left, and only one of them leads to the defects T UV N . In fact, T UV N correspondsto the unique lift of the UV identity defect, which has maximal U (1)-charge N , and whose U (1)-charges populate { N − d + 1 , . . . , N } . That means it is the only such lift, which uponfusion sends all UV branes to GLSM branes in the respective charge window of length d inthe terminology of [2].As a side remark, pushing down the defect T UV N on the left to the IR (setting X = 1),one obtains the RG defect RG N . Pushing down on the left to the UV (setting P = 1),yields the identity defect in the UV phase.Of course, defects T N can be constructed for any phase. Pushing the GLSM identitydefect to the IR on the right, in a similar fashion yields defects T IR N from the IR phase tothe GLSM. R IR R IR is obtained by pushing down to the IR on the left of the GLSM identity defect. Pushingdown on the level of modules yields the C ( · ,P )( Y,Q ) -module M GLSM IR I = C ( α,α − )( · ,P )( Y,Q ) / (( Y − α − d ′ ) , ( P − α − d Q )) . This module is of infinite rank. Note that Y is invertible in this module. One way to lookat it is as a limit of truncated modules M GLSM IR I = lim N →∞ M GLSM IR I ( N )with M GLSM IR I ( N ) = α N C ( · ,P )( Y,Q ) [ α − ] / (( Y − α − d ′ ) , ( P − α − d Q )) α N C ( · ,P )( Y,Q ) [ α − ] . – 22 –he truncated module is finitely generated with generators e i = α N − i , 0 ≤ i < d ′ of Z d ′ × U (1)-charges ([ N − i ] d ′ , − N + i ). They satisfy relations P e i = P α N − i = Qα N − i − d = Y a Qα N − i − b = ( Y a Qe i + b , i + b < d ′ Y a +1 Qe i + b − d ′ , i + b ≥ d ′ Therefore, M GLSM IR I ( N ) is isomorphic to the cokernel of the map r IR1 : C d ′ ( · ,P )( Y,Q ) { [ N − d ] d ′ , − N }{ [ N − − d ] d ′ , − N + 1 } ... { [ N − d ′ + 1 − d ] d ′ , − N + d ′ − } → C d ′ ( · ,P )( Y,Q ) { [ N ] d ′ , − N }{ [ N − d ′ , − N + 1 } ... { [ N − d ′ + 1] d ′ , − N + d ′ − } , with r IR1 = ( P I d ′ − Qǫ bd ′ I Y ). Here I Y is the d ′ × d ′ -diagonal matrix with Y a as the first d ′ − b diagonal entries and Y a +1 as the last b diagonal entries. Explicitly, r IR1 = P − QY a +1 . . . . . .. . . − QY a +1 − QY a . . .. . . . . . − QY a P . Now, Q d ′ − i =0 ( P I d ′ − ( ξ ′ ) i Qǫ bd ′ I Y ) = P d ′ − Y d Q d ′ =: r IR1 r IR0 , and therefore the truncatedmodules have two-periodic resolutions induced by the matrix factorizations R IR N : S d ′ ( X,P )( Y, · ) { [ N − d ] d ′ , − N }{ [ N − − d ] d ′ , − N + 1 } ... { [ N − d ′ + 1 − d ] d ′ , − N + d ′ − } r IR1 r IR0 S d ′ ( X,P )( Y, · ) { [ N ] d ′ , − N }{ [ N − d ′ , − N + 1 } ... { [ N − d ′ + 1] d ′ , − N + d ′ − } One can think of the matrix factorization associated to R IR as the limit lim N →∞ R IR N . Notethat N only shifts the charges of this matrix factorization!As we will see later, left-fusion with the defect R IR just sets X to 1 in the matrixfactorization R IR is fused with.In the following we will also need the defects R UV obtained by pushing the GLSMidentity defect to the UV on the left. The associated module is given by M GLSM UV I = C ( α,α − )( X, · )( Y,Q ) / (( X − α d ′ Y ) , ( α d − Q )) , which can be obtained as a limit N → −∞ of the truncated modules M GLSM UV I ( N ) = α N C ( X, · )( Y,Q ) [ α ] / (( X − α d ′ Y ) , ( Q − α d )) α N C ( X, · )( Y,Q ) [ α ] . Note that compared to the IR case, truncation is implemented in the opposite direction. – 23 –n analysis analogous to the IR case yields an associated matrix factorization R UV N : S d ( X,P )( Y, · ) { [ N + d ′ ] d , − N }{ [ N + 1 + d ′ ] d , − N − } ... { [ N + d − d ′ ] d , − N − d + 1 } r UV1 r UV0 S d ( X,P )( Y, · ) { [ N ] d , − N }{ [ N + 1] d , − N − } ... { [ N + d − d , − N − d + 1 } with r UV1 = ( X I d − Y ǫ d ′ d I Q ), r UV0 = Q d − i =1 = ( X I d − ξ i Y ǫ d ′ d I Q ), where I Q is the diagonalmatrix with 1 in its first d − d ′ diagonal entries and Q in its last d ′ . Explicitly, r UV1 = X − QY . . . . . .. . . − QY − Y . . .. . . . . . − Y X . The matrix factorization R UV can then be thought of as the limit lim N →−∞ R UV N . Left-fusion with R UV implements the push-down to the UV phase, i.e. setting P to 1. In the previous section we have shown that the defects RG N describing the transition fromUV to IR phase factorize as RG N ∼ = R IR ⊗ T UV N . Here T UV N is the defect lifting the UVphase into the GLSM, and R IR is the defect from the GLSM to the IR phase implementingthe push-down to the IR.Indeed, we can also consider the fusion R UV ⊗ T UV N . This defect describes the lift ofthe UV to the GLSM and the subsequent push-down to the same phase. Since identitydefects are idempotent, I ⊗ I ∼ = I , this defect can be obtained by pushing down to theUV phase on both sides of the GLSM identity defect, combined with a truncation. Theuntruncated push-down was calculated in section 3.3. The result is the identity defect ofthe UV phase. Indeed, it is not difficult so see that the truncation essentially does notchange the calculation, and that also the truncated push-down yields the identity defect ofthe UV phase R UV ⊗ T UV N ∼ = I UV . (3.10)Another way to obtain this result is to use the fact (discussed below) that left-fusion withdefects R i just implements the push-down to phase i , i.e. it just sets the variable to 1, whichis associated to the field obtaining a non-trivial vacuum expectation value in phase i . Inthe case of R UV , this is the variable P . Setting P = 1 in the matrix factorization T UV N given in (3.9) indeed yields the UV identity matrix factorization I UV , c.f. (3.4).Of course, one can also straight-forwardly calculate the fusion. As we already used insection 2.3.3, on the level of modules, fusion corresponds to the part of the tensor productwhich is invariant under the gauge group of the model in between the fused defects [5].– 24 –ne obtains (cid:0) M GLSM UV I ⊗ M UV GLSM I ( N ) (cid:1) U (1) ∼ = C ( α,α − )( X, · )( Y,Q ) (( X − α d ′ Y ) , ( α d − Q )) ⊗ C [ Y,Q ] β N C ( Y,Q )( Z, · ) [ β − ](( Z − β − d ′ Y ) , ( Q − β − d )) β N C ( Y,Q )( Z, · ) [ β − ] U (1) ∼ = β N C ( X, · )( Z, · ) [ α, α − , β − ](( αα − − , ( Z − ( αβ ) − d ′ X ) , ( α d − β − d )) β N C ( X, · )( Z, · ) [ α, α − , β − ] ! U (1) ∼ = ( αβ ) N C ( X, · )( Z, · ) [( αβ ) − ](( Z − ( αβ ) − d ′ X ) , (1 − ( αβ ) − d ))( αβ ) N C ( X, · )( Z, · ) [( αβ ) − ] . With the same arguments as in section 3.3 this can be seen to be a module associated tothe identity matrix factorization I UV of the Landau-Ginzburg orbifold in the UV.Analogously one finds R IR ⊗ T IR N ∼ = I IR . Relation (3.10) implies that the defect P UV N = T UV N ⊗ R UV is idempotent, i.e. P UV N ⊗ P UV N ∼ = P UV N . This defect realizes the UV phase inside the GLSMin the sense of [9]. In particular, the category of D-branes in the UV phase is equivalentto the subcategory of GLSM branes invariant under fusion with P UV N .A module associated to P UV N can be obtained as M P UV N = M UV GLSM I ( N ) ⊗ M GLSM UV I = α N C ( X,P )( Y, · ) [ α − ](( Y − α − d ′ X ) , ( P − α − d )) α N C ( X,P )( Y, · ) [ α − ] ⊗ C [ Y ] C ( β,β − )( Y, · )( Z,R ) (( Y − β d ′ Z ) , ( β d − R )) Z d ∼ = α N C ( X,P )( Z,R ) [ α − , β, β − ](( ββ − − , ( P − α − d ) , ( β d − R ) , ( α − d ′ X − β d ′ Z )) α N C ( X,P )( Z,R ) [ α − , β, β − ] ! Z d The Z d -invariant generators are given by ( αβ ) N − i ( β d ) m for i ∈ N and m ∈ Z . They carry U (1) × U (1)-charges ( N − i, − N + i − md ). Defining γ := αβ and δ := β − d of U (1) × U (1)charges (1 , −
1) and (0 , d ) respectively, one can write M P UV N ∼ = γ N C ( X,P )( Z,R ) [ γ − , δ ](( P − γ − d R ) , ( Xγ − d ′ − Z ) , ( δR − γ N C ( X,P )( Z,R ) [ γ − , δ ] . Note that R is invertible in this module! It can be considered as a module over the ring C δ ( X,P )( Z,R ) := C ( X,P )( Z,R ) [ δ ]( δR − C ( X,P )( Z,R ) [ δ ]– 25 –n which R is invertible. Over this ring, M P UV N is finitely generated with generators e i := γ N − d +1+ i , 0 ≤ i < d of U (1) × U (1)-charges ( N − d + 1 + i, − N + d − − i ). They satisfyrelations Xe i = Ze i + d ′ , i + d ′ < dP Xe i = RZe i + d ′ − d , i + d ′ ≥ d Thus, as module over the ring C δ ( X,P )( Z,R ) , M P UV N is isomorphic to the cokernel of the map p : (cid:16) C δ ( X,P )( Z,R ) (cid:17) d { N − d + 1 + d ′ , − N + d − }{ N − d + 2 + d ′ , − N + d − } ... { N, − N + d ′ }{ N − d + 1 , − N + d ′ − }{ N − d + 2 , − N + d ′ − } ... { N − d + d ′ , − N } → ( C δ ( X,P )( Z,R ) ) d { N − d + 1 , − N + d − }{ N − d + 2 , − N + d − } ... { N, − N } , with p = ( XI P − Zǫ d ′ d I R ). Here I P is the diagonal d × d -matrix whose first d − d ′ diagonalentries are 1 and whose last d ′ diagonal entries are P , and I R is the diagonal d × d -matrixwhose first d − d ′ entries are 1 and whose last d ′ entries are R . Concretely, p = X − P Z . . . . . . X . . . P X . . .. . . − P Z − Z . . .. . . . . . − Z P X
Note that d − Y i =0 (cid:16) Xǫ − id ′ d I P ǫ id ′ d − ξ i Zǫ d ′ d I R (cid:17) = X d P d ′ − Z d R d ′ . Hence, p together with p = Q d − i =1 (cid:16) Xǫ − id ′ d I P ǫ id ′ d − ξ i Zǫ d ′ d I R (cid:17) forms a matrix factorization P UV N : (cid:16) S δ ( X,P )( Z,R ) (cid:17) d { N − d + 1 + d ′ , − N + d − }{ N − d + 2 + d ′ , − N + d − } ... { N, − N + d ′ }{ N − d + 1 , − N + d ′ − }{ N − d + 2 , − N + d ′ − } ... { N − d + d ′ , − N } p p (cid:16) S δ ( X,P )( Z,R ) (cid:17) d { N − d + 1 , − N + d − }{ N − d + 2 , − N + d − } ... { N, − N } of W ( X, P ) − W ( Y, Q ) over the ring S δ ( X,P )( Z,R ) = C [ X, P, Z, R, δ ] / ( δR − C [ X, P, Z, R, δ ]of chiral fields of the GLSM on the left and right of the defect, in which the field R is madeinvertible. In fact, this is true for any choice of diagonal matrices I P = diag( P n , . . . , P n d ) and I R =diag( R m , . . . , R m d ) with P n i = d ′ = P m i . – 26 – .7 Action on D-branes Here, we will discuss the action of the defects T iN , R i and P iN on D-branes (boundaryconditions). R i Fusion with a defect R i acts on D-branes by pushing down the respective GLSM matrixfactorizations to phase i by setting the variable obtaining a vacuum expectation value inthe phase to 1. More precisely, let P : P = S r ( Y,Q ) b ... b r p p S r ( Y,Q ) a ... a r = P be a U (1)-equivariant matrix factorization of Y d Q d ′ representing a D-brane in the GLSM.Here, we use the following notation: S ( Y,Q ) = C [ Y, Q ] C ( Y,Q ) = S ( Y,Q ) / ( Y d Q d ) . As before, replacing one of the variables in the subscript with a ‘ · ’ means that we set therespective variable to 1. So, in particular S ( Y, · ) = C [ Y ] and C ( Y, · ) = C [ Y ] / ( Y d ). To thismatrix factorization we associate the C ( X,P ) -module M P = coker P ′ := C r ( Y,Q ) b ... b r p −→ C r ( Y,Q ) a ... a r =: P ′ . We can now calculate the fusion b P = R UV ⊗ P . On the level of modules we obtain anassociated C ( X, · ) -module M b P = (cid:16) M GLSM UV I ⊗ S ( Y,Q ) M P (cid:17) U (1) = C ( α,α − )( X, · ) , ( Y,Q ) (( Y − α − d ′ X ) , ( Q − α d )) ⊗ S ( Y,Q ) M P U (1) The U (1)-invariant generators of this module are given ˆ e i := α a i e i , where the e i , 1 ≤ i ≤ r are the generators the module P ′ of U (1)-charge a i . Note that α has Z d × U (1)-charge([1] d , − Z d -charge of ˆ e i are just the induced Z d -charges [ a i ] d . The relationsfrom the first tensor factor set the variable Y to α − d ′ X and Q to α d . The relations fromthe second tensor factor, coming from the matrix p can then be written in terms of thematrix b p = p ( Y = X, Q = 1) obtained from p by setting Y to X and Q to 1. Oneobtains M b P ∼ = coker b P ′ := C r ( X, · ) [ b ] d ...[ b r ] d b p −→ C r ( X, · ) [ a ] d ...[ a r ] d =: b P ′ . – 27 –his module is associated to the Z d -equivariant matrix factorization b P : b P = S r ( X, · ) [ b ] d ...[ b r ] d b p b p S r ( X, · ) [ a ] d ...[ a r ] d = b P of X d . The matrices b p i are obtained from the respective p i by setting Y to X and Q to 1.An analogous result holds for the action of R IR .Thus, R i indeed fuses with GLSM branes by setting the variables acquiring a non-trivial vacuum expectation value in phase i to 1 in the respective matrix factorization, andbreaking the gauge symmetry accordingly. T iN Fusion with T UV N lifts Z d -equivariant matrix factorizations of X d to U (1)-equivariant matrixfactorizations of P d ′ X d . Since R UV ⊗ T UV N ∼ = I UV and R UV acts by setting P = 1,the lifted matrix factorization has to reduce to the original one upon setting P = 1.Thus, such lifts are obtained by inserting P ’s into the matrices of the original matrixfactorizations in such a way that the Z d -representations on the matrix factorizations lift to U (1)-representations. In fact, for a given matrix factorization there are many possible lifts.As it turns out, fusion with T UV N produces lifts whose U (1)-representations have charges in { N − d + 1 , N − d + 2 , . . . , N } .Let us illustrate this in the example of Z d -equivariant linear rank-1 factorizations L UV[ a ] d : S ( Y, · ) { [ a + d ′ ] d } YY d − S ( Y, · ) { [ a ] d } of X d . These matrix factorizations generate the category of Z d -equivariant matrix factor-izations of X d , i.e. the category of UV D-branes.Now, any of the U (1)-equivariant rank-1 matrix factorizations L GLSM c,m : S ( X,P ) { c + d ′ − md } Y Q m Y d − Q d ′ − m S ( Y,Q ) { c } of X d P d ′ is a lift of L UV[ a ] d for c ∈ a + d Z and 0 ≤ m ≤ d ′ . Namely, R UV ⊗ L GLSM c,m ∼ = L UV[ a ] d , or to put it differently, setting Y = X and Q = 1 in L GLSM c,m produces L UV[ a ] d .Next, we will compute to which of the lifts L GLSM c,m a matrix factorization L UV[ a ] d ismapped under fusion with T UV N . As before we will compute the fusion on the level ofmodules. To L UV[ a ] d we associate the C ( Y, · ) -module M L UV[ a ] d = C ( Y, · ) { [ a ] d } / ( Y ) . – 28 –he fusion T UV N ⊗ L UV[ a ] d is then given by the matrix factorization associated to the C ( X,P ) -module given by the Z d -invariant part of the tensor product (cid:16) M UV GLSM I ⊗ C [ Y ] M L UV a (cid:17) Z d = α N C ( X,P )( Y, · ) [ α − ](( Y − α − d ′ X ) , ( P − α − d )) α N C ( X,P )( Y, · ) [ α − ] ⊗ C [ Y ] C ( Y, · ) { [ a ] d } ( Y ) ! Z d ∼ = α N C ( X,P ) [ α − ] { a } (( P − α − d ) , α − d ′ X ) α N C ( X,P ) [ α − ] { a } ! Z d . There is just one Z d -equivariant generator of this module over C ( X,P ) , namely α N −{ N − a } d of U (1)-charge N − { N − a } d . Here {·} d denotes the representative of the rest class [ · ] d modulo d in the range { , . . . , d − } . There is one relation, namely P n Xα N −{ N − a } d = 0 , where n = ( , d ′ − { N − a } d ≤ , d ′ − { N − a } d > . (3.11)Hence, (cid:18) M UV GLSM I ⊗ C [ Y ] M L UV[ a ] d (cid:19) Z d ∼ = C ( X,P ) /P n XC ( X,P ) . which is associated to the matrix factorization S ( X,P ) { N − { N − a } d + d ′ − nd } XP n X d − P d ′ − n S ( X,P ) { N − { N − a } d } , This is nothing but L GLSM N −{ N − d } d ,n , where the value of n depends on a as stated in (3.11).Hence: T UV N ⊗ L UV[ a ] d ∼ = L GLSM N −{ N − d } d ,n . Note that due to the specific dependence of n on a , the U (1)-charges of the generators (ofthe module of) the matrix factorization lie in the set { N − d + 1 , N − d + 2 , . . . , N } of d consecutive integers ≤ N .Indeed, this is the way T UV N acts on any boundary condition . It lifts the Z d -equivariant matrix factorization of X d to a U (1)-equivariant matrix factorization of P d ′ X d by inserting factors of P into the matrix factorization in such a way that the Z d -representationlifts to U (1), and that furthermore the U (1)-charges of the lifted representation all lie in { N − d + 1 , N − d + 2 , . . . , N } . More precisely, let P : S r ( Y, · ) [ b ] d ...[ b r ] d p p S r ( Y, · ) [ a ] d ...[ a r ] d or more generally defects – 29 –e a rank- r Z d -equivariant matrix factorization of Y d . Then one can show that T UV N ⊗ P is given by the U (1)-equivariant matrix factorization b P : S r ( X,P ) N − { N − b } d ... N − { N − b r } d b p b p S r ( X,P ) N − { N − a } d ... N − { N − a r } d of X d P d ′ , where the matrix b p is obtained from p by replacing each monomial Y r in thematrix entry ( p ) ij by P n X r , with n = max { , − (cid:0) { N − a i } d − d ′ r (cid:1) div d } . ‘div’ denotes the division with (non-negative) remainder. Similarly b p is obtained from p by replacing monomials Y r in ( p ) ij by P n X r with n = max { , − (cid:0) { N − b i } d − d ′ r (cid:1) div d } . One arrives at a similar conclusion for the action of T IR N , where however the U (1)-chargesof the lifted matrix factorization have to lie in the smalle set { N − d ′ + 1 , . . . , N } of d ′ consecutive integers ≤ N . P iN Since fusion is associative, the last two sections imply the following action of the projectiondefects P UV N ∼ = T UV N ⊗ R UV . Let P : S r ( Y,Q ) b ... b r p p S r ( Y,Q ) a ... a r be a U (1)-equivariant matrix factorization of Y d Q d ′ . Then P UV N ⊗ P is isomorphic to the U (1)-equivariant matrix factorization b P : S r ( X,P ) N − { N − b } d ... N − { N − b r } d b p b p S r ( X,P ) N − { N − a } d ... N − { N − a r } d of X d P d ′ . Here the matrix b p is obtained from p by replacing each monomial Y r Q s in thematrix entry ( p ) ij by X r P n , with n = max { , − (cid:0) { N − a i } d − d ′ r (cid:1) div d } . Similarly b p is obtained from p by replacing monomials Y r Q s in ( p ) ij by X r P n with n = max { , − (cid:0) { N − b i } d − d ′ r (cid:1) div d } . Thus, the matrix factorization b P is obtained from P by shifting all U (1)-charges into therange { N − d + 1 , . . . , N } by adding integer multiples of d , setting all Q in the matrices to1 and inserting factors of P in a way ensuring U (1)-equivariance of b P .One finds an analogous result for P IR , where the charges are shifted by integer multiplesof d ′ into the smaller set { N − d ′ + 1 , . . . , N } , Y is set to 1 and factors of X are insertedin a way ensuring U (1)-equivariance. – 30 – .7.4 RG N As alluded to above, the defects RG N describing the transitions between UV and IR phaseare special RG defects between the Landau-Ginzburg orbifolds in the UV and the IR. Theaction of general RG defects have been discussed at length in [5]. In particular, there is aninstructive picture of the D-brane transport coming from the corresponding flow betweenunorbifolded Landau-Ginzburg models in the mirror theory. These flows are tiggered bylower order perturbations of the superpotential W ( X ) = X d + P i
8, respectively. The transition defects RG N describe a certain RG flowsbetween the Landau-Ginzburg orbifolds X / Z and P / Z . For simplicity we will discussthe action of RG , i.e. we set N = 0. Let us first consider the action on linear rank-1factorizations P : S ( Y, · ) { [ − b + 5] } YY S ( Y, · ) { [ − b ] } (3.15)for 0 ≤ b <
8. Under the action of T these are mapped to the matrix factorizations P ′ : S ( X,P ) {− b + 5 } XX P S ( X,P ) {− b } , (3.16)for 5 ≤ b <
8, and to P ′ : S ( X,P ) {− b − } XPX P S ( X,P ) {− b } , (3.17)for 0 ≤ b <
5. These are the lifts of the Z -equivariant matrix factorizations (3.15)of X to U (1)-equivariant matrix factorizations of X P whose charges are contained in {− , − , . . . , } . Acting with R IR essentially sets X = 1 and breaks the U (1) to Z . Inthe first case, 5 ≤ b <
8, the matrix factorizations (3.16) are mapped to the trivial matrixfactorizations P ′′ : S ( · ,P ) {− [ b ] } P S ( · ,P ) {− [ b ] } . The D-branes corresponding to (3.15) for 5 ≤ b < ≤ b <
5, on the other hand, the matrix factorizations (3.17) are mapped to the linearfactorizations P ′′ : S ( · ,P ) {− [ b + 3] } PP S ( · ,P ) {− [ b ] } . The corresponding D-branes do not decouple.Next, let us discuss the action on quadratic matrix factorizations P : S ( Y, · ) { [ − b + 2] } Y Y S ( Y, · ) { [ − b ] } . (3.18) N can be shifted by a quantum symmetry. This is a charge shift which can be implemented by a chargeshifted versions of the identity defect in the respective LG orbifold [5]. – 32 –cting on them with T , one obtains P ′ : S ( X,P ) {− b + 2 } X PX P S ( X,P ) {− b } , (3.19)for 2 ≤ b < P ′ : S ( X,P ) {− b − } X P X P S ( X,P ) {− b } , (3.20)for 0 ≤ b <
2. Again, the matrix factorization P ′ is the lift of the matrix factorization P in (3.18) to the GLSM whose charges lie in {− , . . . , } . Acting with R IR then yields thelinear matrix factorizations P ′′ : S ( · ,P ) {− [ b + 3] } PP S ( · ,P ) {− [ b ] } . for the case 2 ≤ b < P ′′ : S ( · ,P ) {− [ b + 1] } P P S ( · ,P ) {− [ b ] } . for 0 ≤ b <
2. In the latter case, a quadratic matrix factorization is mapped to a quadraticmatrix factorization under the action of RG . In the case 2 ≤ b <
8, the degree decreasesfrom 2 to 1. Indeed, this can be completely understood in terms of the linear matrixfactorizations. Namely, the quadratic matrix factorizations P in (3.18) can be written as acone of two linear matrix factorizations as in (3.15), one specified by the same label b andone specified by { b − } . In case both of those linear matrix factorizations survive the flow,i.e. for 0 ≤ b < P is again mapped to a quadraticmatrix factorization under RG . For the other cases, 2 ≤ b <
8, however, one of the twolinear matrix factorizations is mapped to the trivial one under RG . Under the RG flow,the quadratic matrix factorization decays into the two constituent linear factorizations andone of them decouples. Thus, the quadratic matrix factorization flows to a linear matrixfactorization.In this way, one can explain the action of RG on any rank-1 matrix factorization P : S ( Y, · ) { [ − b + 5 r ] } Y r Y − r S ( Y, · ) { [ − b ] } . (3.21)The result can be read off from the general formulas above. We summarize it in the– 33 –ollowing table:degree of P : r charge shift of P specified by b charges of lift T N ⊗ P degree of P ′′ : n ≤ b < − b, − b + 5 01 0 ≤ b < − b, − b − ≤ b < − b, − b + 2 12 0 ≤ b < − b, − b − ≤ b < − b, − b + 7 13 0 ≤ b < − b, − b − ≤ b < − b, − b + 4 24 0 ≤ b < − b, − b − ≤ b < − b, − b + 1 35 0 ≤ b < − b, − b − ≤ b < − b, − b + 6 36 0 ≤ b < − b, − b − ≤ b < − b, − b + 3 47 0 ≤ b < − b, − b − RG N , which has its originin the mirror dual of the branes [22] and flows described by the GLSM. Consider a diskpartitioned into d segments labelled (clockwise) by the rest classes [0] d , [ d ′ ] d , [2 d ′ ] d , . . . , [( d − d ′ ] d , c.f. figure 1. Each of these corresponds to a linear matrix factorization, namely thematrix factorization (3.15) with r = 1 and [ a ] d given by the respective label of the segment.Now, the linear matrix factorizations corresponding to consecutive segements form non-trivial cones. More precisely, the matrix factorization (3.12) with r > r linear matrix factorizations labelled by [ a ] d , [ a + d ′ ] d , . . . , [ a + rd ′ ] d .In the picture we represent it by the union of the respective segments. The pictorialrepresentation of matrix factorizations of the IR theory consists of a disk partitioned into d ′ segments. [0] d [ d ′ ] d [2 d ′ ] d [3 d ′ ] d [4 d ′ ] d [5 d ′ ] d Figure 1 . Pictorial representation of rank-1 matrix factorizations (3.12). Disk segments labelledby [ id ′ ] d correspond to linear matrix factorizations, i.e. r = 1 with the respective charge label[ a ] d = [ id ′ ] d . Unions of r neighboring segments correspond to matrix factorizations with r >
1. Forinstance the union of the two green segments above corresponds to the matrix factorization 3.12with r = 2 and [ a ] d = [ d ′ ] d . – 34 –nder the action of RG N all linear factorizations labelled by [ a ] d with { N − a } d ≥ d ′ are mapped to trivial matrix factorizations, while all the other ones are mapped tolinear matrix factorizations of the IR theory, labelled by [ N − { N − a } d ] d ′ . Note thatlinear factorizations in the UV theory labelled by [ a ] d and [ a + d ′ ] d , which are mapped tolinear matrix factorizations under the action of RG N , and which can form bound states(and are therefore represented by neighboring disk segments) are mapped to linear matrixfactorizations in the IR theory, which can form bound states as well (and are thereforerepresented by consecutive segements in the pictorial representation of matrix factorizationsof the IR theory). [0] [0] [5] [2] [2]
7→ ∅ [7] [4] [4] [1] [1]
7→ ∅ [6] [3] [3]
7→ ∅
Figure 2 . Pictorial representation of action of RG in the example of d = 8, d ′ = 5. Thered segments correspond to linear matrix factorizations which are mapped to the trivial matrixfactorization under RG . They are collapsed under the action. The arrows ‘ ’ indicate to whichlinear matrix factorizations in the IR the respective UV linear matrix factorization is mapped. Forinstance [0] [0] means that the linear factorization with label [0] in the UV is mapped to thelinear factorization [0] in the IR. From this, one can read off, how any rank-1-matrix factorizationis mapped. For instance the matrix factorization with r = 3 and [ a ] = [5] corresponds to theunion of consecutive segments [5] , [2] and [7] , marked with yellow dashed boarder in the figure.Under the flow, segment [2] is collapsed, and the other two obtain labels [2] and [4] . They areneighboring in the IR, and their union corresponds to the matrix factorization in the IR with r = 2and [ a ] = [2] . Thus, one arrives at the following pictorial representation of the action of RG N , c.f. fig-ure 2. RG N acts by shrinking the disk segments corresponding to decoupling linear matrixfactorizations. These are the segments labelled by [ a ] d with { N − a } d ≥ d ′ . In this wayone arrives at a disk partitioned into d ′ segments. This is a pictorial representation of thematrix factorizations describing D-branes in the IR theory. A segment labelled by [ a ] d inthe original UV picture gets label [ N − { N − a } d ] d ′ in the IR. One can now read off, what This follows from the fact that { N − a − d ′ } d − { N − a } d is either − d ′ if { N − a } d ≥ d ′ or it is d − d ′ in case { N − a } d < d ′ . Hence,[ N − { N − a } d ] d ′ − [ N − { N − a − d ′ } d ] d ′ = ( [ d ] d ′ , { N − a } d < d ′ [0] d ′ , { N − a } d ≥ g ′ . In the first case, both linear matrix factorization are mapped to linear matrix factorizations under RG N ,which can form a cone. In the second case the first one is mapped to a trivial matrix factorization. – 35 –appens to a general matrix factorization under the action of RG N from this pictorial rep-resentation. A matrix factorization (3.12) corresponds to a union of r consecutive segmentsin the UV picture. After shrinking the respective segments, it is represented by the unionof n < r consecutive segments in the IR picture, which is the representation of the matrixfactorization (3.14). For more details see [5]. D-Brane transport between phases of abelian gauged linear sigma models has been inves-tigated before with very different methods. The non-anomalous “Calabi-Yau” case wasstudied in [2]. A discussion going beyond abelian gauge groups as well as an extension toanomalous models can be found in the more recent work [3, 4].In [3, 4], hemisphere partition functions are computed in curved backgrounds withB-type boundary conditions on the equator by means of path integral localization. As aresult of the curvature of the background, these precisely capture the dependence of B-typeboundary conditions on the parameters appearing in the gauge sector. A thorough analysisof analytic and convergence properties of hemisphere partition functions, then allows todetermine the brane transport between different phases. This as well as the arguments in[2] rely on a detailed analysis of the boundary conditions imposed in the gauge sector.The approach taken in the present paper is very different. We decouple the gaugesector, and boundary conditions in this sector are not taken into account. Essentially ,we only consider information accessible to the B-twisted model. That means that we cannotcontrol any analycity or explicit dependence on t . Remarkably, our approach still yieldsmany similar results that we highlight in the following.A crucial ingredient in the discussion of D-brane transport in [2] as well as [3, 4] areso called “charge windows”. A D-brane whose U (1)-charges all lie in this window canbe transported smoothly from one phase to another. Partition functions of these graderestricted branes are well behaved in both phases involved. Any D-brane in the GLSM hasa grade restricted representative, which can be obtained by binding D-branes which aretrivial in the phase in which the transport starts. The charge window is determined by thechoice of the homotopy class of paths in parameter space, along which the D-branes aretransported.In our approach, the defect RG N automatically takes care that branes are transportedthrough such windows. Indeed the defect T iN lifting a phase i to the GLSM automaticallymaps D-branes from phase i to grade restricted GLSM branes, where the exact window isdetermined by the truncation parameter N . The projection defect P iN realizing phase i inthe GLSM projects the category of GLSM branes on the grade restricted subcategory, i.e.it maps every D-brane to the respective grade restricted representative.Note that a in the treatment of [3, 4] a D-brane transport between two phases actu-ally involves two charge windows, a “large window” which ensures smooth transport asalluded to above, and a “small window” lying in the large one . D-branes, whose charges with the exception of the truncation, which we introduced to obtain the RG defects from the GLSMidentity defect, and which presumably is related to stability The two windows coincide in the Calabi-Yau case. – 36 –ompletely lie in the small window flow to the new conformal fixed points, while D-branes,whose charges lie in the large window, but not completely in the small one undergo somekind of decay. (In [3, 4] this is determined by analyzing the asympotics of the hemispherepartition functions.)In our approach both these windows appear naturally and on the same footing. Thelarge window is determined by the projection P UV N associated to the phase, in which thetransport starts, and the small window comes from the projection P IR N associated to thephase, in which the transport ends. Indeed, on the level of the GLSM the transport fromphase i to phase j can be described by the fusion P IR N ⊗ P UV N of the respective projectiondefects.Transporting branes from one phase to another can involve monodromies. In [3, 4]these are naturally associated with shifts in the two windows, either the large windowas a whole, or the small window inside the large window. In our case, the windows aredetermined by the truncation parameters N , which can be shifted by a quantum symmetry,which exactly realizes the monodromy around the fixpoint of the respective phase.Transporting branes from the GLSM to a phase can be done using two different func-tors. The authors of [4] consider geometric phases and define two functors F flow and F geom .The first one corresponds to the actual flow from the GLSM to the phase, the second one toa restriction to field configuration allowed by the deleted sets of the toric geometry/GLSMdescription. In our case, we have two defects from the GLSM to a given phase i , R iN and R i , the truncated and the untruncated descent defects. R iN depends on the the truncationparameter, and hence a path in parameter space, whereas R i merely sets certain fields to1. So these are precisely the analogues of F flow and F geom . In the same way as in [4],where the two functors agree on grade restricted branes, we have R jN ⊗ P iN ∼ = R j ⊗ P i .This is also the reason, why R IR N does not feature more prominently in our discussion: Thelifts T UV N directly lift the UV phase to grade restricted branes, and we chose to factor-ize RG N = R IR ⊗ T UV N . We could have used the cutoff version of R IR as well, writingequivalently RG N = R IR N ⊗ T UV N .One reason, why our approach, which is essentially based on the B-twisted model, stillcaptures all this information might be the fact that functoriality is a strong constraint.Functoriality is inherent in the defect approach, and B-type defects seem to be ratherrigid. With the exception of the truncation, which we introduced in an ad-hoc fashion toobtain RG defects from the GLSM identity defect, and which probably has its origins instability considerations, there were no choices involved in our construction. Furthermore,this choice exactly aligns with the choice of paths between the respective phases.It would be very interesting, to understand the relation of our approach to the onesin [2–4] even better. For one thing, in [3, 23] the D-brane central charge and concretedependence on the twisted chiral moduli is investigated quite explicitly. In particular, in[23] the mathematics of central charges in Landau-Ginzburg orbifolds is studied in detail.By general arguments, we expect that RG (or deformation) defects act on these objects viafusion, and it should be possible to formulate this operation in a natural way. On the otherhand, one could try to incorporate the functoriality constraint directly into the approach of– 37 –3, 4] by applying their analysis to the GLSM identity defect constructed in section 2.3.3. In this paper, we have constructed defects that concretely describe the behavior of D-branes under transitions between phases of abelian gauged linear sigma models. They acton objects and morphisms of the respective D-brane categories via fusion, and this actionis automatically functorial. A key ingredient is the new construction of the identity defectin gauged linear sigma models presented in section 2.3.3. Our approach gives a novelperspective on earlier work [2–4] on D-brane transport in GLSMs. We conclude this paperwith a list of interesting points for future investigation. • The starting point for the construction of our defects RG N that implement the tran-sition between a UV and IR Landau-Ginzburg phase of a U (1)-gauged linear sigmamodel is the identity defect of the GLSM. The bosonic defect fields that we use toconstruct it create an infinite dimensional Chan-Paton-like space. In other words, themodules on which the associated equivariant matrix factorization is built are of infi-nite rank. Introducing a finite cutoff N for these modules, we obtain defects RG N inagreement with expectations and earlier results [5]. The choice of cutoff correspondsto a choice of homotopy class of paths in K¨ahler parameter space. While we formu-late all defects and boundaries on the level of the B-twisted model, which decouplesfrom K¨ahler parameters, a (mild) K¨ahler dependence sneaks back in via the cutoff.We expect the choice of cutoff to be related to stability, one of the indicators beingthat the cutoff is necessary to ensure consistent gluing conditions on a spectral flowoperator of an IR conformal field theory. It would be very interesting to investigatefurther, whether stability conditions in phases can be discussed on the level of theGLSM, and how this relates to defects. • It would be very interesting to combine our approach with the one of [2–4]. Apply-ing their methods to the GLSM identity defect would at the same time explicitlyincorporate the constraint of functoriality in their approach as well as elucidate theprecise origin of the cutoff appearing in our construction. • In section 3 we applied the general approach outlined in section 2 to a specific classof U (1)-gauged linear sigma models which only exhibit Landau-Ginzburg phases.It would be very interesting to apply it to more interesting models, in particularthose featuring geometric or mixed phases. Indeed, a paper, in which we employ ourmethods to models with geometric phases is already in preparation [12]. • The construction of the identity defect should also generalize to non-abelian gaugedlinear sigma models. It would be very interesting to spell this out and obtain transi-tion and monodromy defects also for phases of non-abelian GLSMs. • While in two dimensions our methods are particularly powerful, as the fusion ofdefects is well-controllable, our basic ideas are not limited to this and it would be– 38 –uite interesting to discuss phase transitions and possibly dualities from this pointof view also in higher dimensions.
Acknowledgements
IB thanks Lukas Krumpeck for discussions and Christoph G¨artlein for comments on themanuscript. FK is thankful to Friedrich-Naumann-Stiftung for supporting this project.DR is supported by the Heidelberg Institute for Theoretical Studies. DR also thanks theMSRI in Berkeley for its hospitality, where part of this work was done. (Research at MSRIis partly supported by the NSF under Grant No. DMS-1440140.) IB is supported by theExcellence Cluster Origins and the DFG.
A Defects in LG models and their orbifolds
In this appendix, we summarize some aspects of the description of defects in Landau-Ginzburg models and their orbifolds in terms of (equivariant) matrix factorizations, payingparticular attention to defect fusion. The exposition is very brief. For a more detailedexposition we refer to the literature. In particular, matrix factorizations where related tocategories of D-branes in [24] from a mathematical point of view, and in [25–27] from aphysical point of view. We refer to [28] for a brief summary of the basic physical aspects.The description of defects in Landau-Ginzburg models by means of matrix factorizationshas been established in [5, 16, 29], see [30] for an exposition emphasizing categorical aspects.
Defects in LG models.
A defect D : W → V from a LG model with chiralfields X , . . . X n and superpotential W ∈ C [ X , ..., X n ] to a LG model specified by V ∈ C [ Z , ..., Z m ] is described by a matrix factorization of the difference of the potentials V − W over S := C [ Z , ..., Z m , X , ..., X n ]. That is, there is a Z -graded free module D = D ⊕ D over S with an odd endomorphismd D = D d D ! such that d = W · id D . Matrix factorizations can also be regarded as two-periodic complexes twisted by WD : D d D d D D with d D · d D = W · id D and d D · d D = W · id D . (Right) boundary conditions are the special class of defects, for which the right LGmodel is trivial, i.e. does not feature any chiral fields.To a matrix factorization D as above one can associate an R = S/ ( W )-module M = coker(d D : D ⊗ S R → D ⊗ S R ) , As is customary in the literature, D refers to the free module as well as the defect and matrix factor-ization. – 39 –hich has a two-periodic free resolution induced by the matrix factorization . . . d D −−→ D ⊗ S R . . . d D −−→ D ⊗ S R d D −−→ D ⊗ S R −→ M −→ . More generally, it is useful to consider R -modules which have free resolutions which, afterfinitely many steps turn into the resolutions induced by the matrix factorizations. Such R -modules can be used for instance to find isomorphisms between different matrix factor-izations. This is explained in section 2.3.1 and we also refer to [15] for an exposition thatis useful for the line of arguments in the current paper.The fusion of defects is given in terms of the tensor product of matrix factorizations[16]. More precisely, consider superpotentials U ∈ C [ X , . . . , X m ], V ∈ C [ Y , . . . , Y n ], W ∈ C [ Z , . . . , Z o ] together with matrix factorizations D of V − W , and D ′ of U − V . Thetensor product D ′ ⊗ D is then a matrix factorization of U − W , with modules( D ′ ⊗ D ) = D ′ ⊗ C [ Y i ] D ⊕ D ′ ⊗ C [ Y i ] D , ( D ′ ⊗ D ) = D ′ ⊗ C [ Y i ] D ⊕ D ′ ⊗ C [ Y i ] D (A.1)and differential d D ′ ⊗ D = d D ′ ⊗ id D + id D ′ ⊗ d D . (A.2)This differential is to be understood with Koszul signs, meaning that(id D ′ ⊗ d D )( ν ⊗ ω ) = ( − deg ( ν ) ⊗ d D ( ω ) . While this tensor product a priori has infinite rank over C [ X i , Z j ] it can be shown that itis always isomorphic to a finite rank matrix factorization. Defects in LG orbifolds.
A symmetry of a Landau-Ginzburg model is a homo-morphism of the ring of chiral fields, which leaves the superpotential invariant. Given agroup of symmetries of a Landau-Ginzburg model, one can take the orbifold by that group.Defects and boundaries in Landau-Ginzburg orbifolds are well understood in the case offinite orbifold groups.Let V ∈ C [ X , . . . , X n ] and W ∈ C [ Y , . . . , Y m ] be two superpotentials and G V and G W orbifold groups of the respective LG models. Then B-type defects between these modelscan be described by G = G V × G W -equivariant matrix factorizations of V − W [5, 31, 32].The defects are equipped with a representation ρ D of G that is compatible with the actionof the combined polynomial ring S := C [ X , . . . , X n , Y , . . . , Y m ] on the modules D , D .Denoting by ρ the representation of G = G V × G W on S this means that for all g ∈ Gρ D ( g )( s · p ) = ρ ( g )( s ) · ρ D ( g )( p ) , ∀ s ∈ S, p ∈ D = D ⊕ D ,ρ D ( g ) ◦ d D = d D ◦ ρ D ( g ) . We are interested in abelian orbifold groups here. The action of the latter gives the polyno-mial ring S the structure of a graded ring, and the representations ρ D turn D and D intograded S -modules. The map d D respects the grading and one can associate grade 0 to it.In all our discussions in this paper, the group action can be diagonalized on the modules,and we will often pick generators of D and D on which ρ D acts diagonal. The action of– 40 – = G V × G W can then be specified by assigning G V × G W grades to the generators. Wedenote these grades (or rather their shifts) using curly brackets. To illustrate our notation,we would specify a rank k Z d × Z d ′ equivariant matrix factorization as follows D : S k { [ l k ] d , [ r k ] d ′ }{ [ l k +1 ] d , [ r k +1 ] d ′ }{ [ l k +2 ] d , [ r k +2 ] d ′ } ... d D d D S k { [ l ] d , [ r ] d ′ }{ [ l ] d , [ r ] d ′ }{ [ l ] d , [ r ] d ′ } ... . Here l i and r i are integers, [ . ] d denotes the rest class modulo d , and { [ l i ] d , [ r i ] d ′ } signifythat the respective generator in S k carries Z d × Z d ′ -charge ([ l i ] d , [ r i ] d ′ ).Defect fusion can be extended to orbifold LG theories in a straight-forward way [5].Let U ∈ C [ X , . . . , X m ], V ∈ C [ Y , . . . , Y n ] and W ∈ C [ Z , . . . , Z o ] be polynomials invari-ant under actions of groups G U , G V , G W on the respective polynomial rings. Consider a G W × G V equivariant matrix factorization D of U − V and a G V × G U -equivariant matrixfactorizations E of V − W . The tensor product D ⊗ E a priori yields a G U × G V × G W -equivariant matrix factorization of U − W . The fusion in the orbifold theory is now givenby the G V invariant part of ( D ⊗ E ) G V of the tensor product factorization D ⊗ E . Thisyields a G U × G W equivariant factorization of U − W over C [ X , . . . , X m , Z , . . . Z o ]. References [1] E. Witten,
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