On Matrix Factorizations, Residue Pairings and Homological Mirror Symmetry
CCERN-TH-2018-062March 2018
On Matrix Factorizations,Residue Pairings andHomological Mirror Symmetry
Wolfgang Lerche
Theoretical Physics DepartmentCERN, Geneva
Abstract
We argue how boundary B -type Landau-Ginzburg models based on matrix factorizations canbe used to compute exact superpotentials for intersecting D -brane configurations on compactCalabi-Yau spaces. In this paper, we consider the dependence of open-string, boundarychanging correlators on bulk moduli. This determines, via mirror symmetry, non-trivial diskinstanton corrections in the A -model. As crucial ingredient we propose a differential equationthat involves matrix analogs of Saito’s higher residue pairings. As example, we compute fromthis for the elliptic curve certain quantum products m and m , which reproduce genuineboundary changing, open Gromov-Witten invariants. a r X i v : . [ h e p - t h ] A p r ontents sans p´eriodes . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Example: the elliptic curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 B -Model 18 B -Model computations . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4.2 B -Model correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4.3 Instantons in the A -Model . . . . . . . . . . . . . . . . . . . . . . . . 34 . Introduction Topological open strings in connection with mirror symmetry (for overviews, see [1–3]) areuseful for understanding certain non-perturbative phenomena related to D -branes on Calabi-Yau manifolds [4], and specifically, for computing exact, instanton-corrected effective super-potentials. So far, substantial progress (initiated in [5–7]) has been made for single ormultiple parallel branes, for which the open strings are associated with “boundary preserv-ing” vertex operators. Most of these works deal with non-compact geometries. There hasbeen much less work on compact geometries (initiated in [8, 9]) and in particular very littlework on intersecting branes, where “boundary changing” open string vertex operators comeinto play and the methods developed so far are not applicable.Such brane configurations are particularly interesting for phenomenological applications,not the least because they naturally give rise to chiral fermions; the boundary changingoperators correspond to matter fields in bi-fundamental gauge representations (for a reviewsee eg. [10]). Such models can be represented by quiver diagrams, where the nodes correspondto branes and the maps between them to boundary changing open strings localized at theintersections. Essentially, the various terms of the effective potential on the world-volumeare given by disk correlators of boundary changing vertex operators, summed over orderingscorresponding to closed paths in the quiver diagram. However, most discussions stop hereat the level of cohomology and charge selection rules.But there is much more to the superpotential than just chasing arrows around a quiver:generically there are moduli from the parent Calabi-Yau space and possibly also open string(location and bundle) moduli from the branes, and the maps, and consequently the effec-tive potential, depend on them. This dependence can be highly non-trivial due to infiniteseries of world-sheet instanton contributions. Obviously, for answering questions like whatthe vacuum structure of the full theory is, one needs to determine the dependence of thesuperpotential on these moduli, over the whole of the moduli space.Mathematically the instanton problem corresponds to the counting of holomorphic mapsfrom “polygon shaped” disks with n boundary components into Calabi-Yau spaces, suchthat these boundaries lie across intersecting special Lagrangian cycles; see Fig. 1 for anillustration adapted to the elliptic curve. There we see that if we allow for generic, ( p, q )types of intersecting branes, the set of possible correlation functions becomes infinitely muchricher as compared to non-intersecting brane configurations!The mathematical framework that is designed to address precisely this kind of questionsis homological mirror symmetry [3, 11]. However, despite of that this has been an importantongoing topic in mathematics since more than 20 years, it has seen little use in physics.Indeed, for example, an explicit method for computing instanton corrections for intersecting2igure 1: In the A -model, the boundary changing correlators get contributions from world-sheet instantons that correspond to holomorphic maps from the disk world-sheet into theintersecting D-brane geometry (with appropriately matching boundary conditions). Noticethat the instanton problem becomes infinitely rich when we allow for intersecting branes,even for the simplest example of the elliptic curve.brane configurations on Calabi-Yau threefolds folds has been missing so far, even for thesimplest amplitudes such as three-point functions.Our purpose in the present paper is to make some modest steps into this direction, froman admittedly simple-minded physicist’s perspective; we hope that the physics intuition mayhelp to further the development of the theory.Let us be more specific. In topological strings, there are two somewhat antagonisticapproaches to correlation functions. One takes a more algebraic, the other a more geometricviewpoint, involving period integrals, the variation of Hodge structures, etc. Obviously onewould like to have a more unified understanding of both perspectives. A sensible first stepwould be to aim at the middle ground, namely by asking how the algebraic open string sectorvaries as fiber over the closed string moduli space.This is what we will discuss in this paper, for open topological strings in the frameworkof twisted N = (2 ,
2) superconformal field theory. More specifically, we will consider openstring correlation functions on the disk D of the form B ( A ,..,A k ) α ....α k ( t ) = (cid:68) Ψ ( A ,A ) α Ψ ( A ,A ) α P (cid:0) (cid:90) ∂D Ψ (1)( A ,A ) α (cid:17) . . . (cid:0) (cid:90) ∂D Ψ (1)( A k − ,A k ) α k − (cid:17) Ψ ( A k ,A ) α k e − (cid:80) t i (cid:82) D φ (2) i (cid:69) , (1)which are perturbed by closed string deformations, φ . Here Ψ ( A i ,A j ) ∗ with i (cid:54) = j are vertexoperators that describe open strings that go from one boundary condition, or D brane L A i ,to another one, L A j . Physically they are localized at the intersection L A i ∩ L A j . Theseare the boundary changing operators, in contrast to the boundary preserving ones, Ψ ( A i ,A i ) ∗ ,3hich are localized on one brane only. The superscripts denote the integrated 1 − or 2 − formdescendants of the respective operators.In terms of these correlators, the effective potential that is induced by the D -brane back-ground is given by summing over all correlators pertaining to the given D -brane configura-tion: W ( A ,..,A k ) eff ( s, t ) = (cid:88) k ≥ k s α s α . . . s α k B ( A ,..,A k ) α ....α k ( t ) , (2)where s α are the (not necessarily commuting) fields in the effective action that source theΨ α . This amounts to summing over appropriate closed paths in the quiver diagram. The general structure of open string correlators on the disk is well-known (see eg. [12, 13]):they can be written as B α ....α k ( t ) = (cid:68)(cid:68) Ψ α , m k (Ψ α ⊗ Ψ α · · · ⊗ Ψ α k ) (cid:69)(cid:69) , (3)where the topological metric (cid:104)(cid:104)∗ , ∗(cid:105)(cid:105) denotes a suitable, non-degenerate and cyclically sym-metric inner product. Moreover, m k : Ψ ⊗ k → Ψ are certain higher multilinear, non-commutative products that take collections of operators as input and produce one operatoras output. Correspondingly, the equations of motion arising from W eff take the form (cid:88) k ≥ m k (Ψ ∗⊗ k ) = 0 , (4)which are nothing but the Maurer-Cartan equations [14] which specify the locus of unob-structed deformations of the theory.The products m k , and thus the correlators built from them, satisfy a host of Ward iden-tities, specifically the A ∞ relations, and have certain cyclicity and integrability properties.These data together with the inner product and some other subsidiary conditions comprisewhat is called a Calabi-Yau A ∞ algebra [15]. If there are several boundary componentspresent, the A ∞ algebra is promoted to an A ∞ category. Moreover, since we have deformedthe correlators by bulk moduli, we encounter deformed m k = m k ( t ), which form what iscalled a curved A ∞ structure. When combined with the bulk sector, this structure is pro-moted to an open-closed homotopy algebra [16]. All this has been discussed at length inthe literature, and we do not want to spend more than a few remarks on this in the presentcontext. Suffice it to refer the reader to refs. [12, 13, 17, 18] for more details, from a physicsperspective.In practice, the explicit evaluation of the correlators (1) is difficult, not the least becauseof contact terms that arise when the integrated insertions hit other operators, or (cid:82) φ (2) hitsthe boundary. While the underlying algebraic structure organizes these contact terms, it is4igure 2: Homological mirror symmetry between A ∞ products m : Ψ ⊗ → ¯Ψ. The valueof the Fukaya product in the A -model is given by the exponentiated disk instanton actionassociated with the intersecting brane geometry. In the B -model the corresponding Masseyproduct is computed by nested trees, involving propagators and lower trees.not of direct help to actually compute the correlators. As we will see, one needs to augmentthe algebraic structure by certain differential equations.An analogous problem appears already for closed strings, where special geometry in con-nection with mirror symmetry comes to the rescue [2]. Specifically, mirror symmetry isthe statement that the topological A -model on a Calabi-Yau space Y is equivalent to thetopological B -model on the mirror manifold, X . This means that instanton corrected corre-lation functions in the A -model can be computed in terms of classical correlation functionsin the simpler B -model, in the framework of topological Landau-Ginzburg models. Fornon-intersecting D -brane geometries, a corresponding geometrical framework of open mirrorsymmetry has been well developed [2, 3] after the initial works [5–7, 19].As said before, our intention is to push the subject to more general, intersecting braneconfigurations, which is the arena of homological mirror symmetry. Given that this is math-ematically highly sophisticated, it would be impossible to give any reasonable account here.Instead we will exhibit only some basic ideas to convey the motivation of what we want todo. In short, the basic statement is an isomorphism [11]: F uk ( Y ) (cid:39) D b ( Coh ( X )) , (5)where F uk ( Y ) and D b ( Coh ( X )) denote the appropriately defined Fukaya category of spe-cial lagrangian cycles on the A -model side, and the bounded derived category of coherentsheaves (crudely: vector bundles on submanifolds) on the B -model side. It has been provedrecently [20] for hypersurfaces in projective space. The relevance for the physics of D -braneshas been recognized early on in refs. [21,22]. Equation (5) implies isomorphisms between the5 ∞ products m k on the two sides, which are called Fukaya and Massey products in the A -and B -model, respectively. See Figure 2 for a visual representation. In physics language thisamounts to an “equality” of the respective correlation functions. We put the word equality inquotation marks, since isomorphism means equality up to maps. The physicists are howeverinterested in explicit expressions, not just in structural existence proofs of isomorphisms. Inthe closed string sector, this has been achieved [23] by applying methods of algebraic geom-etry, in particular the theory of Hodge variations, period integrals and associated flatnessdifferential equations. This leads to an exact map, the mirror map t ( α ) : V B ( X ) (cid:39) H ∗ ( X, ∧ ∗ T X ) → V A ( Y ) (cid:39) QH ∗ ( Y ) , (6)which maps the algebraic modulus α in the B -model into the flat coordinate t of the A -model. Above, V B and V A denote the variation of Hodge structures [24, 25] on the bothsides, respectively. They each comprise of data V = ( H ∗ , ∇ GM , (cid:104)(cid:104)∗ , ∗(cid:105)(cid:105) ), where H ∗ is therelevant cohomology or quantum cohomology, ∇ GM the respective Gauss-Manin connectionand (cid:104)(cid:104)∗ , ∗(cid:105)(cid:105) a suitable inner product, or pairing.A corresponding theory has been developed for homological mirror symmetry, which isthe theory of non-commutative Hodge variations (see eg. [26–28]). The details not beingimportant here, we will be schematic and refer the reader to the readable expositions inrefs. [29, 30]. The basic notion is an A ∞ category C with maps between objects L i : C k ( L , L , . . . L k ) = Hom ∗ ( L , L ) ⊗ Hom ∗ ( L , L ) · · · ⊗ Hom ∗ ( L k − , L k ) . (7)With this one forms the Hochschild chain complex and its homology CC • ( C ) = (cid:77) k C k ( L , L , . . . L k , L ) , (8) HH ∗ ( C ) = H ∗ ( CC • ( C ) , b ) , where b is the Hochschild differential. One may refine this to a semi-infinite variant andconsider the so-called negative cyclic complex, CC −• ( C ), and its cohomology, HC −• ( C ). Thisinvolves the introduction of a spectral parameter, u . Together with the Gauss-Manin-Getzlerconnection ∇ GMG [31] and the pairing [32], this forms a structure V ( C ) = ( HC −• ( C ) , ∇ GMG , (cid:104)(cid:104)∗ , ∗(cid:105)(cid:105) ),which is a non-commutative analog of the semi-infinite variation of Hodge structures V inthe closed string theory. In the present context, there are two versions of this, one for the A -model side and one for the B -model side. Homological mirror symmetry then amounts toan isomorphism, in analogy to (6): V A ( F uk ( Y ) (cid:39) V B ( D b ( Coh ( X )) . (9)It has been shown [29, 30] that (under suitable conditions) open-closed maps exist thatprovide isomorphisms OC : V A ( HC −• ( F uk ( Y ))) → V A ( QH ∗ ( Y )) (10) V B ( HC −• ( D b ( Coh ( X )))) → V B ( H ∗ ( X )) . A - and B -models together. For this, we should impose some extra structurein the form of flatness equations. The question is how to do this in practice.Flatness equations based on the Gauss-Manin-Getzler connection [31], ∇ GMGt , are acentral theme in the mathematical literature. However, the latter has been mostly concernedto map via OC all kinds of quantities to the bulk, closed string sector. This allows to evaluatepairings and higher correlators in the simpler, commutative theory. Specifically, for two-pointfunctions, one might be tempted to write (cid:104)(cid:104) Ψ ( A,B ) , Ψ ( B,A ) (cid:105)(cid:105) D = (cid:104)(cid:104) OC [Ψ ( A,B ) ] , OC [Ψ ( B,A ) ] (cid:105)(cid:105) S , (11)and consider flatness equations of the form0 = (cid:104)(cid:104)∇ GMGt Ψ ( A,B ) , Ψ ( B,A ) (cid:105)(cid:105) D = (cid:104)(cid:104) OC [ ∇ GMGt Ψ ( A,B ) ] , OC [Ψ ( B,A ) ] (cid:105)(cid:105) S (12)= (cid:104)(cid:104)∇ GMt OC [Ψ ( A,B ) ] , OC [Ψ ( B,A ) ] (cid:105)(cid:105) S , where the last line exhibits the intertwining property [29] of ∇ GMGt and ∇ GMt over OC.However, such correlators may make sense for the annulus, but not for the sphere or disk.Moreover, the map OC invariably vanishes on boundary changing operators. Indeed, OC as well as ∇ GMGt act on complete cycles Ψ ( A ,A ) ⊗ · · · ⊗ Ψ ( A k ,A ) , but not on the individualmaps Ψ ( A i ,A j ) when i (cid:54) = j . Thus we cannot capture in this way the boundary changing sectorwe are interested in.Rather, the differential operators we seek should act on the individual operators in awell-defined manner, even if these are boundary changing. This is then finally, in essence,the problem that we want to address: find an explicit construction of flatness differentialequations of the form (cid:104)(cid:104)∇ t Ψ ( A,B ) , Ψ ( B,A ) (cid:105)(cid:105) D = 0 , (13)that make sense especially also for boundary changing sectors. This will be a crucial ingre-dient for computing general correlation functions for intersecting branes. There are two kinds, one associated with moduli deformations, t , plus one associated with the spectralparameter u if we are interested in the semi-infinite extension of Hodge variations; we will suppress thelatter, as for our current purposes this extension is not immediately relevant. .3. Content of the paper Our strategy is guided by a specific realization of the open topological B -model, namely interms of a 2 d , N = (2 ,
2) superconformal Landau-Ginzburg model based on matrix factor-izations. The key point is an isomorphism D b ( Coh ( X )) (cid:39) Cat ( MF, W X ) , (14)where Cat ( MF, W X ) is the category of matrix factorizations of a function W X . Here thisfunction is given by the LG superpotential whose vanishing describes the Calabi-Yau mani-fold X under consideration: W X = 0 (in some weighted projected space). This isomorphismhas been proved under certain assumptions, by Orlov [33], based in earlier ideas of Kontse-vich.For physicists this isomorphism allows to describe topological B type D-branes in termsof a simple field theoretical model, in which all the elements of the abstract category, namelyobjects and maps between them, have a concrete realization in terms of matrix valued fieldoperators. First works on this by physicists include [34]-[36], and a selection of works relevantfor our current purposes is given by [38]-[50].In the next section we will first briefly review well-known aspects of the Landau-Ginzburgmodel in the bulk, and study its deformations. Here we will use a less geometrical and morefield theoretical language in terms of renormalization and contact terms. For this we willintroduce the language of Saito’s higher residue pairings, which so far did not receive a lot ofattention in the physics literature. That is why we will use simple terms, and also provide asample computation for the elliptic curve. It demonstrates how one can formulate differentialequations directly in terms of residue pairings.In Section 3 we will first review some basic facts of matrix factorizations and their de-formations. We then analyze the interplay between bulk-boundary and boundary-boundarycontact terms. This leads in Section 3.3 to a proposal for a flatness differential equation,which is based on a higher variant of the Kapustin-Li supertrace residue that plays an openstring analog of Saito’s first higher residue pairing.In Section 4 we apply these ideas to open strings on the elliptic curve and demonstratehow one can explicitly compute, by solving the differential equation, open string correlatorssuch as in (1). We then match the results of the B -model to the instanton geometry of the A -model mirror, thereby reproducing known results. Indeed, homological mirror symmetryis very well understood for the elliptic curve (see eg. [51–54]), and the structure of bundlesover it and their sections (given by theta functions), are more or less explicitly known.Basically this is because the curve is flat and so one can simply read off the areas of disksusing elementary geometry.The main purpose of the present paper is not to compute new correlation functions,rather than to develop a method how to compute boundary changing, open Gromov-Witten8nvariants directly from the B -model from “first principles”, without relying on prior knowl-edge about the A -model side (we put quotation marks because we present a proposal, nota proof). This will be important for later applications, eg. to Calabi-Yau threefolds, wherethe A -model correlators are not known beforehand.
2. Recapitulation of the closed string, bulk theory
The issue is to determine the dependence of the B -model n -point correlation functions onthe flat complex structure moduli t a of a Calabi-Yau manifold, X . As per (6), via mirrorsymmetry these coincide with the K¨ahler moduli of the A -model on the mirror manifold, Y .Consider for example C abc ( t ) = (cid:104) φ a φ b φ c e (cid:80) t d (cid:82) φ (2) d (cid:105) . (15)Here, φ a are zero-form cohomology representatives and the deformations (cid:90) φ (2) ≡ (cid:90) d z { G −− , [ G + − ,φ ] } (16)are given by their two-form descendants. Expanding out we can write (cid:104) φ a φ b φ c (cid:90) φ (2) d ...... (cid:90) φ (2) d n (cid:105) =: (cid:104)(cid:104) φ a , (cid:96) n +2 ( φ b , φ c , φ d , ...., φ d n (cid:105)(cid:105) , (17)where the maps (cid:96) n are the closed string versions of the m k , which obey certain recursive L ∞ relations. This statement by itself is not useful to actually compute the correlation functions.However, it is here where geometry beats algebra: due to the special geometrical propertiesof (topologically twisted) N = (2 ,
2) superconformal theories, these correlators satisfy a hostof identities [55–57]. One of those is that correlators with n > C abcd ( t ) = ∂∂t d (cid:104) φ a φ b φ c e (cid:80) t e (cid:82) φ (2) e (cid:105) . (18)The property (18) expresses an underlying Frobenius structure [58, 59] and implies crossing-type relations of the form C abe ( t ) C ecd ( t ) = C ade ( t ) C ebd ( t ) . (19)These serve as integrability condition for the existence of a prepotential F such that C abc ( t ) = ∂∂t a ∂∂t b ∂∂t c F ( t ) . (20)9ote that (18) holds for a special choice of “flat” coordinates t a of the moduli space. Oth-erwise the derivatives would need to be replaced by covariant derivatives, whose connectionpieces encode the contact terms. In this way, the flatness of coordinates is tied to thecancellation of contact terms.The important insight of refs. [57, 60–63] and others was that the exact n -point corre-lators, including all contact term contributions, can be computed in an effective B -type,topologically twisted Landau-Ginzburg formulation of the theory. It is characterized by asuperpotential, W ( x ), which is a holomorphic, non-degenerate and quasi-homogeneous poly-nomial that depends on a number of LG fields, x i , i = 1 , ..., N . These are to be viewed ascoordinates of the target Calabi-Yau space X defined by W ( x ) = 0 in some projective (orweighted projective) space.We will consider (mini-versal) deformations of the complex structure described by W ( x, t ) = W ( x ) − (cid:88) g a ( t ) y a ( x ) , (21)where y a ( x ) is some basis of the Jacobi ring, y a ( x ) ∈ Jac( W ) ≡ C N [ x ] /dW . In this language,the flat zero-form operators are represented by polynomials in the LG fields defined by ϕ a ( x, t ) = ∂∂t a W ( x, t ) . (22)To distinguish the various fields, we denote by φ ( x, t ) the marginal operators that are sourcedby the moduli t , and by ϕ ( x, t ) generic elements of the chiral ring that can also includerelevant besides marginal ones. General polynomials are denoted by ξ . In terms of theseoperators, the three-point functions then localize in the IR to the Grothendieck multi-residue[60], ie. , to C abc ( t ) = res x =0 ϕ a ( x, t ) ϕ b ( x, t ) ϕ c ( x, t ) d W . . . d N W = (cid:73) dx ( dW ) N ϕ a ( x, t ) ϕ b ( x, t ) ϕ c ( x, t ) , (23)from which all other correlators can be generated.The issue thus is to determine the dependence of g a ( t ) on the flat coordinates. Themost familiar way to determine the flat coordinates is to solve the Gauss-Manin, or Picard-Fuchs system of differential equations that act on period integrals. This system itself can besystematically derived via the variation of Hodge structures. This procedure `a la Griffith-Dwork [64] is standard since many years (for reviews, see eg. [1, 65]), so we don’t embarkon it further. Suffice it to mention that the core structure consists of the vanishing of theGauss-Manin connection, ∇ GM . Its primary component takes the form ∇ GMt a ( ϕ b ) ≡ ∂∂t a ϕ b ( x, t ) − U ( φ a ϕ b ) = 0 , where (24) U ( φ a ϕ b ) : ≡ d i (cid:18) φ a ϕ b d i W (cid:19) + , (25) We have temporarily suppressed a t -dependent prefactor that is needed for the proper normalization.Moreover, the integration contour is a real cycle supported on | d i W | = (cid:15) . ϕ a ( x, t ). The subscript + denotes aprojection: it instructs to expand the polynomial operator product according to the Hodgedecomposition φ a ( x ) ϕ b ( x ) = C abc ϕ c ( x ) + p abi ( x ) d i W ( x ) , (26)and then to drop negative powers of x ; that is, the result is determined by the exact pieceof the OPE, U ( φ a ϕ b ) = d i p abi ( x ).Mathematically, the U term in (24) arises from integration by parts under the periodintegral in order to reduce the pole order [64]. Physically it is a contact term [66] that arisesfrom integrating out the massive, exact states in the OPE (26). Via (24), their effect issubsumed in the t -dependent renormalization of the coupling functions g a ( t ). Schematically: ϕ a ( x, t ) = ϕ a ( x ) + t U ( (cid:96) ( φ, ϕ a )) | t =0 + 1 / t U ( (cid:96) ( φ, φ, ϕ a )) | t =0 + . . . . (27)The iterative integrating out amounts to going from an off-shell topological string field theoryto an on-shell “minimal model” with non-trivial higher L ∞ products (cid:96) n ; it is the specialgeometry of the N = (2 ,
2) superconformal theories that allows to sum all these terms up inone swoop.
We recapitulate the determination of flat coordinates by using a method that we find espe-cially suitable for the generalization to the boundary theory. This is because it avoids periodintegrals, which we wouldn’t know how to generalize to matrix valued operators when we willface the generalization to open strings. It also seems more natural physicswise, as it makesdirect contact with the underlying field theory. This is the theory of Saito’s higher residuepairings [67–70]. So far it has been rarely discussed in physics (see however: [66, 71–74]).We will be brief and present only a rough outline. For more accurate technical details werefer the reader to the recent expositions [75–77]. The basic objects in Saito theory are thehigher residue pairings K ( ϕ a , ϕ b )( u ) ≡ ∞ (cid:88) (cid:96) =0 u (cid:96) K ( (cid:96) ) ( ϕ a , ϕ b ) , (28)where u is the auxiliary degree 2, spectral parameter that plays an important role in thevariation of semi-infinite Hodge structures [24–26]. For us, it has nothing to do with gravita-tional descendants, rather its significance is to provide a grading with respect to the numberof iterated contact terms; thus only a finite number of powers of u will be relevant.The pairings can be derived and represented in various different ways. We like to out-line a physically motivated, field theoretical derivation which reflects the underlying pathintegral. Following the notation of ref. [78], we define the BRST operator related to B -type11upersymmetry, as well as a deformation of it by u : Q W = ¯ ∂ + ι dW , (29) Q W,u = Q W + u ∂ , (30)where: ¯ ∂ ≡ η ¯ i d ¯ i , ∂ ≡ ∂ θ i d i , ι dW ≡ d i W ∂ θ i . (31)Here, ¯ η ¯ i ∼ dz ¯ i ∈ T ∗ X and θ i ∼ ∂/∂z i ∈ T X denote complex fermions that represent differentialforms on X in the usual manner. Moreover, the Hamiltonian is H λ ≡ (cid:52) ¯ ∂ + (cid:96) W + λ ( L + u ∂ ι dW ) , (32)where (cid:52) ¯ ∂ = [ ¯ ∂, ¯ ∂ † ] is the laplacian, (cid:96) W = d i d j W ∂ η i ∂ θ j , and L = d ¯ i d ¯ j ¯ W ( φ ) η ¯ i θ ¯ j + || d i W ( φ ) || (33)is the zero mode lagrangian. Note that H λ = (cid:2) Q W,u , G λ (cid:3) , where (34) G λ = ¯ ∂ † + λ ι dW (35)¯ ∂ † ≡ d i ∂ η ¯ i , ι dW ≡ d i W θ i . (36)After the reduction to the zero modes of (cid:52) ¯ ∂ in the path integral, the expression for theone-point function, (cid:82) e −H λ ξ , of some arbitrary polynomial ξ takes the form: (cid:104) ξ ( x ) (cid:105) = (cid:90) dxdηdθ ξ ( x ) e − λ ( L + u∂ ι dW ) . (37)Since the Hamiltonian is BRST exact, the correlators are topological and (on-shell) inde-pendent of λ . For λ → ∞ , the second term in L localizes the integral on the critical pointsof W and contributes λ − N | H | − , while the first term yields a factor of the Hessian ¯ H of¯ W . This also comes with a factor λ N so that the N zero modes of each of θ and η of thisterm dominate the path integral, and this implies that the potential θ and η dependence ofany other, non-exponential term drops at λ → ∞ . The sum over the critical points overthe remaining H − can then be converted, via a standard argument, to the usual residueformula. Thus we get (cid:104) ξ ( x ) (cid:105) ( u ) = (cid:73) dx ( dW ) N (cid:88) u (cid:96) ( ∂h ) (cid:96) ξ ( x ) (38)as perturbation expansion in terms of the propagator ∂h ≡ ∂ ι dW || dW || = d i d i W || dW || . x i by writing W → W ( x i )+1 / x i ) ,and then taking the limit x i ∗ → `a l’Hˆopital , we can effectively cancel out the non-holomorphic pieces in h . All-in-all we arrive at (cid:104) ξ ( x ) (cid:105) = (cid:73) dx ( dW ) N L ( u, ξ ) , where (39) L ( u, ξ ) = (cid:88) u (cid:96) (cid:18) d i d i W (cid:19) (cid:96) ξ ( x ) . (40)This then gives the following representation of the higher residue pairings K ( ξ a , ξ b )( u ) = (cid:73) dx ( dW ) N L ( u, ξ a ) L ( − u, ξ b ) . (41)The first, constant term in the expansion in u : K (0) ( ϕ a , ϕ b ) = (cid:73) dx ϕ a ( x ) ϕ b ( x )( dW ) N = η ab , (42)is just the topological two-point function, or metric in field space. The next term can bewritten in the form: K (1) ( ξ a , ξ b ) = 12 (cid:73) dx ( dW ) N (cid:88) j ( d j ξ a ( x )) ξ b ( x ) − ξ a ( x ) d j ( ξ b ( x )) d j W ( x ) . (43)We call this the “integrated operator form” of the residue pairings. We also introduce a“contact term form” of the pairings by writing K C ( ξ a , ξ b )( u ) = K (0) ( L C ( u, ξ a ) , L C ( − u, ξ a ) ) , where (44) L C ( u, ξ ) = (cid:88) u (cid:96) (cid:96) (cid:122) (cid:125)(cid:124) (cid:123) U ( U ( ...U ( ξ ( x ) .. )) , (45)which is supposedly equivalent to the “integrated operator form”, (41). The importance ofthis equality for (cid:96) = 1, K (1) C ( ξ a , ξ b ) = K (1) ( ξ a , ξ b ) , (46)has been emphasized by Losev [66].As will become clear later, the reason for this naming is that integrated operator insertionshave structurally the form (cid:82) ξ (2) ∼ dξ/dW , while contact terms involve a projection topolynomials, U ( ξ ) = d ( ξ/dW ) + , and so can be directly translated to the renormalization ofoperators. One needs to assume isolated critical points for this argument, and as usual this can be achieved bytemporarily resolving the singularity by a small perturbation and invoking continuity. A more rigorousderivation proceeds by introducing a compact support and cutoff functions as explained in detail in [75]. We are unaware of literature where this has been addressed explicitly, except for (cid:96) = 1. .3. Flatness equations sans p´eriodes We now turn to the determination of flat coordinates and field bases in terms of higherresidue pairings, reviewing only aspects that are of immediate interest for our purposes.To start, we preliminarily adopt an overall normalization defined by K (0) ( H,
1) = 1 , (47)where H = det d i d j W denotes the Hessian of the potential W (which is in general not aflat cohomology representative). As is well known, the physical correlators involving flatcohomology representatives will need to have an extra rescaling factor, which is closelyrelated to Saito’s primitive form and is essentially given by the fundamental period ω of theCalabi-Yau X .Next we introduce the notion of a “good basis” [68, 69] of field operators { ϕ a } , which isdefined by K (0) ( ϕ a , ϕ b ) ≡ η ab = const . ,K ( (cid:96)> ( ϕ a , ϕ b ) = 0 . (48)These equations are not quite sufficient for defining flat bases and coordinates. Consider theprimary one:0 = ∂ t K (0) ( ϕ a , ϕ b ) (49)= K (0) ( ∂ t ϕ a , ϕ b ) + K (0) ( ϕ a , ∂ t ϕ b ) − (cid:73) dx ϕ a ( x ) ϕ b ( x )( dW ) N (cid:88) d i φd i W = (cid:0) K (0) ( ∂ t ϕ a , ϕ b ) − K (1) ( φϕ a , ϕ b ) (cid:1) + (cid:0) K (0) ( ϕ a , ∂ t ϕ b ) + K (1) ( ϕ a , φϕ b ) (cid:1) . The peculiar term on the RHS represents in LG language the insertion of the integrated2-form operator (16): (cid:90) φ (2) ←→ dφdW , (50)and for this reason we called (43) the integrated operator form of the residue pairing. Bythe identity (46) we can rewrite it in terms of the contact term form0 = K (0) ( ∂ t ϕ a − U ( φϕ a ) , ϕ b ) + K (0) ( ϕ a , ∂ t ϕ b − U ( φϕ b )) , (51)which reproduces the geometrical equation (24) (we assume the gauge U ( ϕ ∗ ) = 0 here).Either way, the equation can be compactly rewritten as0 = K ( ∇ GMt ϕ a , ϕ b ) + K ( ϕ a , ∇ GMt ϕ b ) , (52)where ∇ GMt = ∂ t − φu . (53)14or complete flatness, requiring the constancy of the topological metric, eq. (49) is notsufficient. Rather one needs to impose the stronger ”chiral square root” of this equation,and this for all higher pairings as well: K ( (cid:96) ) ( ∇ GMt ϕ a , ϕ ∗ ) = 0 . (54)For any given ϕ a , this equation must hold for arbitrary cohomology elements ϕ ∗ in a “good”basis (48), and it can give non-trivial constraints if the degrees of the ϕ ∗ are matchedappropriately to the one of ϕ a , and to (cid:96) .Equation (54) is in disguise what the first order form of the familiar Picard-Fuchs systeminstructs us to do. The essence of the story is this: by scanning over (cid:96) and all possible“spectators” ϕ ∗ , we sample all components of the Gauss-Manin connection. In general, thereare higher order contact terms beyond what we wrote in (24). In the more familiar languageof period integrals, these arise from multiple, iterated partial integrations, which reflect thestructure of the Hodge filtration. These are encoded by the nested propagators U (25), wherethe order of the nesting is measured by (cid:96) . Physically, testing the differential equation (54)against all physical operators samples all possible contact terms. In conjunction with (48),it determines the renormalized coupling functions g a ( t ). Let us illustrate the method of residue pairings by an easy example computation. We willreproduce results that are known since long and have been discussed in the physics literaturein eg. [63, 79–81].We consider the simplest possible case, namely the cubic elliptic curve. It is defined asthe hypersurface W = 0 in CP where W ( x, t ) = ζ ( t ) (cid:20) (cid:0) x + x + x (cid:1) − α ( t ) x x x (cid:21) . (55)Here α ( t ) is the complex structure modulus whose dependence on the flat coordinate t is tobe determined. With hindsight we have also performed an overall rescaling by a function ζ ( t ) which needs to be determined as well. The marginal operator thus is φ ( x, t ) = ∂ t W ( x, t ) = − ζ ( t ) a (cid:48) ( t ) x x x + ζ (cid:48) ( t )3 ζ ( t ) (cid:88) x i d i W ( x, t ) , (56)where we exhibited that it has an exact piece. We are also interested in a perturbation by Such exact pieces correspond to linear combinations of periods in the more familiar geometrical language. ϕ ( x, t ) = ζ ( t ) g ( t ) x , (57) ϕ ( x, t ) = ζ ( t ) g ( t ) x x , (58)and the main task will be to compute t -dependent correlation functions of those. Thisrequires first of all to determine the functions α ( t ) , ζ ( t ) , g a ( t ).We have already used as ansatz an initial good basis of operators. Indeed all the operatorsobey eq. (48), and in particular we have K (1) ( φ, φ ) = 0 , (59)despite that φ has a non-vanishing exact piece. This extra freedom is allowed due to theantisymmetry of K (1) .We now in turn impose the various flatness equations. First the most trivial one: K ( ∇ GMt , φ ) ≡ K (0) ( ∂ t , φ ) − K (1) ( φ, φ ) = 0 . (60)This shows that (59) is indeed a relevant property. Next, imposing K (0) ( ∇ GMt φ, ≡ K (0) ( ∂ t φ, − K (1) ( φ · φ,
1) = 0 , (61)yields the following differential equation: ζ (cid:48) ( t ) ζ ( t ) = α (cid:48)(cid:48) ( t )2 α (cid:48) ( t ) − α ( t ) α (cid:48) ( t )2∆ , (62)where ∆ ≡ α − ζ ( t ) = − i (cid:114) α (cid:48) ( t )3∆ . (63)The rescaling of W indeed coincides with the fundamental period of the elliptic curve: ζ ( t ) = (cid:36) ( α ) ≡ α F (1 / , / ,
1; 1 /α ) , (64)as expected on general grounds. It also relates to Saito’s idea of a primitive form, which inthis context boils down to a rescaling by (cid:36) [68, 69, 75].Moreover, we consider the equation at one level up: K (1) ( ∇ GMt φ, φ ) ≡ K (1) ( ∂ t φ, φ ) − K (2) ( φ · φ, φ ) = 0 (65)(all higher ones being empty). This yields the following non-linear differential equation: (cid:8) z ; t (cid:9) = − − z + 8 z z ∆ z (cid:48) , (66)16here z ≡ /α and { z ; t } ≡ z (cid:48)(cid:48)(cid:48) z (cid:48) − ( z (cid:48)(cid:48) z (cid:48) ) is the Schwarzian derivative. Via standardarguments the solution of this equation is given by:3 α ( α + 8)∆ = j ( t ) / , where j ( t ) = q − + 744 + . . . is the familiar modular invariant function in terms of q = e πit .This relation identifies α ( t ) with the Hauptmodul of the modular subgroup Γ(3). Its inversecoincides with the well-known mirror map of the elliptic curve, given by the ratio of itsperiods: t ( α ) = (cid:36) ( α ) /(cid:36) ( α ) = i/ √ F (1 / , / ,
1; 1 − /α ) / F (1 / , / ,
1; 1 /α ).Finally, we turn to the relevant operators. The equations to consider are K (0) ( ∇ GMt ϕ , ϕ ) ≡ K (0) ( ∂ t ϕ , ϕ ) − K (1) ( φ · ϕ , ϕ ) = 0 , (67) K (0) ( ϕ , ∇ GMt ϕ ) ≡ K (0) ( ϕ , ∂ t ϕ ) + K (1) ( ϕ , φ · ϕ ) = 0 , (68)which give rise to g (cid:48) a /g a = a [ ζ (cid:48) / ζ − α α (cid:48) / ∆]. The solutions are: g a ( t ) = (∆( t ) ζ ( t )) a/ , a = 1 , . (69)We can similarly determine contact terms between the relevant operators, by defining themdirectly in terms of residue pairings, ie. , CT [ ϕ a , ϕ b ] := (cid:88) ∗ K (1) (( ϕ a · ϕ b , ϕ ∗ ) ϕ ¯ ∗ , (70)where ϕ ¯ ∗ is the dual of ϕ ∗ with regard to the inner product defined by K (0) . Explicitly: CT [ ϕ , ϕ ] = 0 , (71) CT [ ϕ , ϕ ] = − ζ (cid:48) ζ ∆ g g · , (72) CT [ ϕ , ϕ ] = − αζ ∆ g · ϕ . (73)Usually these terms are included in the LG potential as higher order corrections, whichleads to a redefinition of the flat fields. We don’t need to present these formulas here, theinterested reader may consult refs. [79–81] for details.Now we are in the position to determine correlation functions. First note that becauseof the flattening rescaling of W by ζ , we need to put a normalization factor to the innerproduct such as to restore (47), ie. (cid:104) H (cid:105) = 1: (cid:104)(cid:104) ξ a , ξ b (cid:105)(cid:105) := 1 ζ K (0) ( ξ a , ξ b ) . (74)This is the natural normalization in the current setup, but might seem peculiar since usuallycorrelators are rescaled by 1 /(cid:36) . However all is fine because the operators have beenconsistently rescaled as well. 17e now simply plug the renormalized operators in and find: (cid:104)(cid:104) φ, (cid:105)(cid:105) ( t ) = (cid:104)(cid:104) ϕ , ϕ (cid:105)(cid:105) ( t ) = 1 , (75) (cid:104) ϕ ϕ ϕ (cid:105) ( t ) = (cid:104)(cid:104) ϕ , ϕ ϕ (cid:105)(cid:105) ( t ) = ζ ( t ) , (76) (cid:104) ϕ ϕ , ϕ , (cid:105) ( t ) = (cid:104)(cid:104) ϕ , ϕ , ϕ , (cid:105)(cid:105) ( t ) = α ( t ) ζ ( t ) , (77)(where ϕ ,i = ϕ | x → x i ). As an example for a higher point function, consider (cid:104) ϕ ϕ ϕ (cid:90) ϕ (2)2 (cid:105) ( t ) = (cid:104)(cid:104) ϕ , (cid:96) ( ϕ , ϕ , ϕ ) (cid:105)(cid:105) , where (78) (cid:96) ( ϕ , ϕ , ϕ ) = 2 CT [ ϕ , ϕ ] · ϕ + CT [ ϕ , ϕ ] · ϕ (79)= − ζ ∆ (cid:20) ζ (cid:48) ζ g g ϕ + α ∆ g ϕ · ϕ (cid:21) . (80)The end result is then (cid:104) ϕ ϕ ϕ (cid:90) ϕ (2)2 (cid:105) ( t ) = 2 α α (cid:48) ∆ − α (cid:48)(cid:48) α (cid:48) ( t ) . (81)All of these correlators reproduce results that are known in the literature [63, 79–82]. Ourpurpose was to re-derive them in terms of a more field theoretical and less geometricallanguage that can be easier generalized to open strings.
3. Open String B -Model We like to generalize the considerations of the previous sections to the open string sector, ie. , to the B -model on the disk. As mentioned in Section 1.3, for the relevant boundaryLG model the various possible B -type topological D -branes are one-to-one to the variouspossible matrix factorizations M ( Q ) of the bulk superpotential: M ( Q ) : Q ( x, t ) · Q ( x, t ) = W ( x, t ) n × n . (82)Here Q ( x, t ) is the boundary BRST operator that can be represented by an odd n × n dimensional matrix (so n is even), whose precise structure encodes the brane geometry wewant to describe. This includes also a specific dependence of closed ( t ) and possibly open( u ) string moduli. The dimension n of the Chan-Paton space can take arbitrary even values,and this reflects that there is in general an infinite number of possible (potentially reducible)brane configurations on a given CY space, X . When n = 2 k for some integer k , one canwrite Q compactly in terms of boundary fermions π, ¯ π that form a Clifford algebra. This iswhat we will do, while not being essential. 18n the following we will focus on the most canonical of such matrix factorizations, for whichwe make use of the quasi-homogeneity of the superpotential, W ( x, t ) = 1 / (cid:80) q i x i d i W ( x, t ).Here the BRST operator is given by Q ( x, t ) = 1 / q i π i x i + ¯ π i d i W ( x, t ) , { π i , ¯ π j } = δ ij , i, j = 1 ...N, (83)where q i are the R -charges of the LG fields. The physical operators at the boundary are then given by the non-trivial cohomologyclasses of Q . A new feature as compared to the bulk LG theory is that they are matrixvalued. Moreover in general there are boundary preserving and boundary changing opera-tors. Boundary preserving operators, denoted by Ψ ( A ) ≡ Ψ ( A,A ) , are each tied to a singlematrix factorization M ( Q ( A ) ), and are represented by n A × n A matrices. Boundary changingoperators Ψ ( A,B ) ∈ Hom ∗ ( M ( Q ( A ) ) , M ( Q ( B ) )) are associated with pairs of matrix factoriza-tions, Q ( A ) and Q ( B ) . They are represented by not necessarily quadratic, n A × n B dimensionalmatrices.For a given pair of boundary conditions ( ∗ , ∗ ), the (“on-shell”) space of physical operatorsis defined by H ( ∗ , ∗ ) P = (cid:8) Ψ ( ∗ , ∗ ) : (cid:2) Q, Ψ ( ∗ , ∗ ) (cid:3) = 0 , Ψ ( ∗ , ∗ ) (cid:54) = [ Q, .... ] (cid:9) , (84)where (cid:2) Q , Ψ ( A,B ) (cid:3) : ≡ Q ( A ) n A × n A Ψ ( A,B ) n A × n B − ( − s Ψ ( A,B ) n A × n B Q ( B ) n B × n B , (85)with s = 0 or s = 1 depending on the statistics of Ψ.Another difference as compared to the bulk theory, where variations of cohomology ele-ments map back to cohomology elements, is that in the boundary theory variations of matrixvalued cohomology elements are in general not BRST invariant: [ Q ( t ) , ∂ t Ψ( t )] (cid:54) = 0. In otherwords, variations of such cohomology elements will generically map into the full, off-shellHilbert space at the boundary. Thus we need to pay attention to the structure of the fullHilbert space, which is given by general, Z graded matrices M with polynomial entries in x .By a basic theorem of Hodge and Kodaira, the latter can always be decomposed as follows: H = H U ⊕ H P ⊕ H E . (86)Here H U comprises the set of unphysical operators that are not annihilated by Q and H E comprises the exact operators that are Q -variations. Note that a priori H P is defined only upto addition of operators in H E , and similarly H U is defined only up to addition of operators For simplicity, we put them all equal to each other, q i = 2 /N . They can be reinstated easily for coveringweighted projected spaces as well. We will always denote by [ − , − ] the graded commutator which acts by definition on even and oddelements with the proper sign, and when boundary changing operators are involved, it implicitly acts fromleft and right in the manner defined here.
19n both H P and H E . A given decomposition is referred to as a cohomological splitting, andcan be viewed as a gauge choice for the off-shell physics (see eg. [12, 13]).This is closely related to the choice of cohomology representatives. We have seen abovethat for the bulk theory, the proper choice of operators including definite exact pieces is animportant ingredient in the determination of flat coordinates. Thus the question arises asto what preferred basis of operators to choose initially.The choice of cohomological splitting corresponds to how the inverse of the BRST operator Q is precisely defined. This inverse, or “homotopy” will also be needed for the computationof higher point correlation functions, in the form of the open string propagator U D that entersin the higher A ∞ products. Its choice corresponds to the choice of an off-shell completionof the theory, and mathematically speaking corresponds to adopting a specific choice of theminimal model of the underlying A ∞ algebra.A good strategy [20] to construct the inverse of Q is to regard the boundary sector asfundamental and the bulk sector as a perturbation by the superpotential, W . That is, wesplit Q = Q S + Q W , Q S = 1 /N x i π i , Q W = ¯ π i d i W , (87)and first invert Q S . We must require that the inversion works properly on the full Hilbertspace, ie. , on arbitrary matrices M ( x ) ∈ H with polynomial entries. To this end, let us makean ansatz U S = κ − ( M ) ¯ π i d i , (88)and introduce projection operators Π ∗ : H → H ∗ with:Π E = Q S · U S Π U = U S · Q S (89)Π P = 1 − Π SE − Π SU , where Q S acts as graded commutator as in (85). We need to determine the coefficients κ ( M )such that these operators are indeed good projectors that satisfy (Π ∗ ) · M = Π ∗ · M . Theydepend on the concrete matrices on which U S acts, and for determining them, we need toadopt some normal ordered form for M to map back to. For example, M = const( a, b, c ) ¯ π a π b x c , (90)where a, b, c are multi-indices labeling all of ¯ π i , π j , x k . The condition of good projectionsthen gives κ ( M ) = (cid:80) ( a i + c i ).The full U D that inverts the complete boundary BRST operator Q can then be obtainedby a simple application of homological perturbation theory, ie. , U D := U S · (cid:88) ( Q W · U S ) l . (91)20he sum terminates at a finite number of steps and so yields an exact result. By construction, Q and U D satisfy (Π ∗ ) = Π ∗ when used in the projectors (89), and so U D is indeed a goodpropagator in the full theory with non-vanishing superpotential W .Note that this construction is not unique: we may equally well define an action of U D from the right and consider a different normal ordering: M = const( a, b, c ) π a ¯ π b x c ; this gives κ ( M ) = (cid:80) ( b i + c i ). In general we may consider linear combinations which also invert Q : U D ( (cid:15) ) := 12 (1 + (cid:15) ) U DL + 12 (1 − (cid:15) ) U DR , (92)where (cid:15) parametrizes the normal ordering ambiguity which leads to an ambiguity of thecohomological splitting of H induced by the projectors Π ∗ ( (cid:15) ). Fortunately, the ambiguitycancels when acting on BRST closed operators in H P ⊕ H E , and won’t play a role in thefollowing. However, we note as a side remark that U D ( (cid:15) ) · Q = − N ( R + 12 (cid:15) ) , (93)where R = (cid:80) ¯ π i π i denotes the (diagonal) matrix of U (1) R charges in the Chan-Paton space.Thus we can formally associate the normal ordering ambiguity with the grade of the D -brane,which too amounts to an arbitrary overall shift of the R -charge.Having defined a cohomological splitting on the space of matrices, we can now write apreferred basis of the physical operators in closed form:Ψ i ..i k := Π P · π i ....π i k ∈ H P , k ≤ N. (94)By construction they are BRST closed but not exact: Q · Ψ i ..i k = 0 = U D · Ψ i ..i k ( ie. , satisfya Siegel-type gauge). This basis is analogous to a polynomial basis { y a ( x ) } of Jac( W ) with U ( y ) = 0, in the bulk theory.To conclude this section, we hasten to clarify a potential confusion: for the operators (94),where are the labels for the boundaries? The point is that LG models describe orbifolds ofgeometrical theories, with self-intersecting branes. As we will review later, the boundarylabels of Ψ ( A,B ) i ..i k appear only after un-orbifolding, which leaves the form of the operatorsinvariant. In the following, we shall be interested in bulk deformations of the open string TFT definedby the matrix factorization (82); see [39, 41, 43, 44, 48–50, 83–86] for some previous workson deformations of matrix factorizations. We will denote the closed string moduli by t a as before and possible open string moduli of a brane by u α (not to be confused with thespectral parameter u ). With “moduli” we refer to operators with R -charge q = 2 in the bulkor q = 1 on the boundary, so that t a and u α are dimensionless and can appear in correlation21unctions in a non-polynomial way. On the other hand, we will denote relevant, “tachyonic”deformations coupling to boundary changing operators Ψ α by s α ; being dimensionful, theycan appear in the effective potential only in a polynomial way.Strictly speaking, because in general an effective potential W eff ( t, u, s ) will be generated,some or all of these deformations won’t be true moduli but will be obstructed (possibly athigher order). The true moduli space consists of the sub-locus of the joint ( t, u, s ) deformationspace that preserves the factorization, ie. , Q ( t, u, s ) = W ( t ). It can be shown [84, 86] thatthis supersymmetry preserving locus coincides with the critical locus, ∂ u,s W eff ( t, u, s ) = 0,of the effective superpotential.In the following, we will consider only unobstructed, supersymmetry preserving factoriza-tion loci, and concentrate on bulk-deformed boundary operators, Ψ = Ψ( t ); we will considerpossible boundary moduli u α as frozen. More specifically, we will work to all orders in thebulk perturbation but only infinitesimally in the boundary changing open string sector: Q ( t, s ) = Q ( t ) + s α Ψ α ( t ) . (95)Note that factorization is preserved to lowest order as long as [ Q ( t ) , Ψ α ( t )] = 0. An ob-struction, and thus a superpotential, can arise only if several boundary changing operatorsare switched on simultaneously. Thus, for the purpose of determining “flat” representativesΨ = Ψ( t ), we can restrict our considerations to linear order in s and consider the theory asun-obstructed at this order (and thus on-shell, making computations well-defined in CFT).We have seen in the previous section how to systematically construct canonical represen-tatives of the boundary cohomology, and even obtain explicit expressions (94) for them forthe class of factorizations we consider. However, in order to obtain actual correlation func-tions, there is more left to do, namely the t -dependent renormalization g ( t ) of the operatorsΨ( t ) still needs to be determined. This is in fact the main part of the work, which it isusually neglected in the discussion of matrix factorizations.Before we will discuss this in the next section, note that the operators that couple tounobstructed, supersymmetry-preserving deformations parametrized by t a are not just givenby the flat bulk fields defined by φ a ( x, t ) = ∂ a W ( x, t ). Indeed, the reason [87] for introducingboundary degrees of freedom in the first place had been to cancel the B -type supersymmetryvariation of the bulk superpotential W that arises on world-sheets with boundaries. This im-plies that the factorization-preserving deformations of W that we consider, must necessarilybe accompanied by simultaneous deformations of Q by certain boundary operators, γ a ∈ H U .Differently said, we deal here with a joint open/closed deformation problem where the de-formations are locked to a common un-obstructed deformation locus via the factorizationcondition (82). By definition, these induced boundary counter terms are not cohomology For this to make sense if Ψ is a boundary changing operator (which we assume), one should view Q asa block matrix containing Q A and Q B in the diagonal, perturbed by the off-diagonal element s a Ψ ( A,B ) a . (cid:2) Q ( t ) , γ a ( t ) (cid:3) = φ a ( t ) | ∂D . (96)That the bulk modulus φ must be BRST exact at the boundary for the deformation tobe un-obstructed, also follows from abstract reasoning (in that degree-two deformations inExt ( ∗ ) determine obstructions [14] and so must be trivial by assumption). From (96) andfactorization it follows that γ a ( t ) = ∂ t a Q ( t ) , (97)up to BRST closed pieces. The point is that only the combined perturbation: (cid:88) t a (cid:90) ˜ φ a ≡ (cid:88) t a (cid:18)(cid:90) D φ (2) a − (cid:90) ∂D γ (1) a (cid:19) , (98)is preserved by the total, B -type BRST operator at the boundary, Q tot = Q W (cid:12)(cid:12) ∂D + (cid:2) Q, ∗ (cid:3) , (99)while the individual terms are not. This follows from (96) and the descent equation,[ Q , φ (2) ] = ∂ ⊥ φ (1) , where φ (1) is a one-form along the boundary of the disk. Physically, (98)implements the subtraction of the “Warner” contact term (cid:82) ∂D φ (1) of φ with the boundary,and mathematically represents the natural invariant pairing on the relative (co-)homologyof the disk.The coupled bulk-boundary perturbation (98) is thus a quite peculiar observable: despiteneither of its individual building blocks belongs to the physical cohomology of the boundarytheory, it deforms correlation functions. Whenever discussing perturbed correlation func-tions such as the one written in (1), one should implicitly use such combined, “relative”perturbations to cancel the bulk-boundary contact terms.A key role is again played by contact terms, in particular the contact terms that arisewhen the insertion (cid:82) ∂D γ (1) hits other operators at the boundary:[ Q , (cid:90) ∂D γ (1) ]Ψ = ( (cid:90) ∂D φ (1) )Ψ + [ γ, Ψ] . (100)This follows from the descent equation [ Q , γ (1) ] = φ (1) + ∂ (cid:107) γ . The second term on the RHSmust then be cancelled by the t -variation of Ψ( t ). Indeed, inverting the BRST operator in ∂ t [ Q, Ψ] = 0 we can write ∂ t Ψ = − U D ([ γ, Ψ]), whose Q -variation reproduces this term. Notethat [ γ, Ψ] ∈ H E and thus the inversion is well-defined.More succinctly, defining Ψ g ( t ) ≡ g ( t )Ψ( t ) one might be tempted to write the followingdifferential equation: ∇ Ut Ψ g ( t ) = g (cid:48) ( t ) g ( t ) Ψ g ( t ) , where (101) ∇ Ut ≡ ∂∂t + U D ([ γ , ∗ ]) , ∇ Ut : H P → H P , Q is determinedonly up to BRST closed pieces in H P ⊕H E , and thus g (cid:48) ( t ) is not really determined. Moreover,by recalling how Ψ( t ) is defined in (94) in terms of U D , and by explicitly following the actionof ∂ t through the latter’s definition in (91), it turns out that (101) is in fact an identity.This means that the contact term that arises if (cid:82) γ hits Ψ is already taken care of by theaction of ∂ t , and so no constraint on g ( t ) can be obtained just from (101) alone. Rather, weexpect that g ( t ) is determined by the interplay between the integrated insertions (98) andthis contact term. See Fig. 3 for a pictorial summary of the contact terms. We now turn to a concrete realization of the desired flatness equations in LG language. Forthis we need to consider correlation functions. The most important quantity is a generaliza-tion of the Grothendieck multi-residue pairing to matrix factorizations, which was found byKapustin and Li [34, 36, 88–90]. It defines an inner product at the boundary as follows: (cid:104) Ψ α , Ψ β (cid:105) ∂D ≡ η αβ = K (0) (1 , str[ dQ ∧ n · Ψ α · Ψ β ]) , (102)24hich is known to be non-degenerate. This pairing is well-defined for cohomology elements,and maps H P ⊗ H P → C . This means that it satisfies trace property and cyclicity of n -pointfunctions only on-shell. Off-shell where Ψ ∗ ∈ H U , cyclicity is violated so this pairing doesnot define a Calabi-Yau structure proper [15], without modifications. This problem has beenaddressed in refs. [49, 91], where correction terms were determined that turn (102) into agood off-shell pairing.In the following, we will not be concerned about off-shell properties of the pairing. Fornow, focusing only on on-shell properties, we like to rewrite the Kapustin-Li supertrace-residue pairing in a symmetric form as follows: K (0) KL (Ψ α , Ψ β ) ≡ K (0) KL (1; Ψ α , Ψ β ) , with (103) K (0) KL ( ξ ; Ψ α , Ψ β ) = K (0) ( ξ, (cid:88) k str[ dQ ∧ k · Ψ α · dQ ∧ ( n − k ) · Ψ β ]) (104) ≡ ( − n ( n + 1)! n (cid:88) k =0 ( − k | Ψ α | n (cid:88) i ∗ =1 (cid:15) i ...i n (cid:73) ξ str (cid:20) d i Qd i W . . . d i k Qd i k W Ψ α d i k +1 Qd i k +1 W . . . d i n Qd i n W Ψ β (cid:21) , where | Ψ | denotes the Z -grade of Ψ. This form obtains naturally when one performs aderivation from the path integral [78], analogous to what we outlined in Section (2.2) for thebulk theory.In view of our previous discussion, an obvious question is how to extend this to higherpairings, for example, by introducing the spectral parameter u . Such an extension has beenconstructed by Shklyarov [90].Note however that the spectral parameter u has degree (or charge) 2, which is matchedto the superpotential W and so matches insertions of d/dW in the higher residue pairings.On the other hand, at the boundary the relevant cohomology is determined by Q , which hasdegree 1, so inversions of Q (resp. contact terms) should formally be counted in terms of adegree 1, and not a degree 2 variable. This is closely tied to the fact that bulk deformationsinvolve two fermionic integrations in (cid:82) D φ (2) , while boundary deformations involve only onein (cid:82) ∂D Ψ (1) . That is, the spectral parameter u seems to be a genuine bulk quantity that doesnot capture all aspects of the boundary theory. It appears that for our purposes we need adifferent extension of (103), not in the direction u , but rather in a different, morally speakinganti-commuting direction.Before we will discuss this, let us mention an instance where an extension in the u -direction appears to be useful for our purposes: namely we can consider an intermediate”bulk-boundary” pairing of the form K ( (cid:96) ) bb ( ϕ a , Ψ β ) := K ( (cid:96) ) ( ϕ a , str[ dQ ∧ n · Ψ β ]) : H closed ⊗ H open → C . (105)That is, we treat the image of the open-closed map, OC [ ∗ ] ≡ str[ dQ ∧ n ∗ ] : H open → H closed , H P → Jac( W ) , (106)25ike any other bulk operator. In analogy to (94) we may propose as further condition for a“good” basis of operators K (0) bb ( ϕ a , Ψ β ) = η aβ = const . (107) K ( (cid:96)> bb ( ϕ a , Ψ β ) = 0 . While we do not have a proof for this requirement, we will see later that these conditionsmake good sense at least in the context of an explicit example.However note, importantly, that the bulk-boundary pairing (105) affects only boundarypreserving operators: Ψ = Ψ ( A,A ) . As mentioned in the introduction, this is because theopen-closed map vanishes identically on boundary changing operators, Ψ ( A,B ) with A (cid:54) = B .Thus K ( (cid:96) ) bb is insensitive to the intrinsically new features of boundaries, namely the ones thatcannot be mapped to the bulk theory.Let us return to our task and try to find a pairing that is useful for our purpose, namelyultimately formulating flatness equations. We will be guided by considering deformations ofcorrelators in LG language, in analogy to what we have reviewed for the bulk theory. Thatis, requiring constancy of the inner product (102) yields0 = ∂∂t K (0) KL (Ψ α , Ψ β ) (108)= K (0) KL ( ∂∂t Ψ α , Ψ β ) + K (0) KL (Ψ α , ∂∂t Ψ β ) − K (0) KL ( (cid:88) i d i φd i W ; Ψ α , Ψ β )+ K (0) (cid:16) , (cid:88) k str (cid:2)(cid:0) ∂∂t dQ ∧ k (cid:1) · Ψ α · dQ ∧ ( n − k ) · Ψ β (cid:3)(cid:17) + K (0) (cid:16) , (cid:88) k str (cid:2) dQ ∧ k · Ψ α · (cid:0) ∂∂t dQ ∧ ( n − k ) (cid:1) · Ψ β (cid:3)(cid:17) . The new ingredient is the action of ∂ t on the dQ ’s. This is in line of what we discussed inthe previous section, namely that we deal here with a coupled bulk-boundary deformationproblem. Eq. (108) explicitly manifests in LG language the heuristic correspondence: (cid:0) (cid:90) D φ (2) − (cid:90) ∂D γ (1) (cid:1) ←→ (cid:0) dφdW − dγdQ (cid:1) . (109)Here “ dγ/dQ (cid:48) ” has the meaning to drop a dQ and replace it with its t -derivative, in all theproper locations. We presented our problem in a way that suggests a generalization of the flatness equationsas follows. We see from the structure of (108) that the kind of higher boundary pairing weare after, should involve d/dQ (of degree −
1) instead of d/dW (degree − Recall that the dQ ’s and thus dγ ’s can be different for the two boundary segments, namely if theseare associated with different matrix factorizations, M ( A ) and M ( B ). Relatedly, implicit in the notation (cid:82) ∂D γ (1) is that there can be different boundary segments of the disk D , and the insertions must be doneaccording to the respective boundary conditions. K (1) KL (Ψ α , Ψ β ) := ( − n +1 ( n + 1)! n (cid:88) k =1 ( − k ( | Ψ α | +1) n (cid:88) i ∗ =1 (cid:15) i ...i n × (110)2 (cid:73) str (cid:34)(cid:18) d i Qd i W . . . d i k − Qd i k − W d k Ψ α d k W d i k +1 Qd i k +1 W . . . d i n Qd i n W Ψ β (cid:19) − (cid:18) d i Qd i W . . . d i k Qd i k W Ψ α d i k +1 Qd i k +1 W . . . d i n − Qd i n − W d n Ψ β d n W (cid:19) (cid:35) , which satisfies K (1) KL (Ψ α , Ψ β ) = ( − | Ψ α || Ψ β | +1 K (1) KL (Ψ β , Ψ α ). In terms of this we can rewriteequation (108) as 0 = ∂∂t K (0) KL (Ψ α , Ψ β )= − K (0) KL ( (cid:88) i d i φd i W ; Ψ α , Ψ β ) (111)+ K (0) KL ( ∂∂t Ψ α , Ψ β ) + K (0) KL (Ψ α , ∂∂t Ψ β )+ K (1) KL ( γ · Ψ α , Ψ β ) + K (1) KL (Ψ α , γ · Ψ β ) . This is supposed to be the boundary analog of eq. (49) in the bulk theory.In fact, this equation represents an identity: the cohomology elements as defined in (94)are already normalized such that all inner products are constant, if the overall normalizationis chosen such the one-point function of the top element is constant: K (0) KL (Ψ ..N , ) = K (0) ( H,
1) = 1. Thus, eq. (111) by itself does not have too a great significance.Indeed, requiring that the topological boundary two-point function (102) (cid:104) Ψ ( A,B ) α Ψ ( B,A ) β (cid:105) ∂D = η αβ , (112)be constant, is of little help for fixing the operators, because it is invariant under the relativerescaling Ψ ( A,B ) → g ( t )Ψ ( A,B ) , Ψ ( B,A ) → g ( t ) − Φ ( B,A ) . Thus, we cannot determine the inde-pendent renormalization factors from (111). However we need to know them because, forexample, the three-point function (cid:104) Ψ ( A,B ) Ψ ( B,C ) Ψ ( C,A ) (cid:105) will be proportional to g ( t ) . Thisis why we need to find differential equations that determine the relative moduli-dependentrenormalization factors for all fields individually, and not just of their products. It also relatesback to what we said in the Introduction: as long as we consider only closed cycles of op-erators, Ψ ( A,B ) Ψ ( B,C ) . . . Ψ ( ∗ ,A ) ∈ HH ∗ ( Cat ( MF, W )), we cancel out important information,namely the one that intrinsically goes beyond the bulk theory.Thus all boils down to one basic and crucial problem, namely to as to how to split equation(111) into two separate, stronger ones. Without any deeper insights, this is an ambiguous27roblem, namely what forbids us to add and subtract terms to the individual pieces suchthat they cancel out in the sum? Alarmingly, in the end, practically all correlations functionsthat we want to compute will depend on this split!The only pragmatic way we see, is to be guided by analogy of the bulk (cf., (54)) and bythe structure of (111), and define “relative” connections by K (0) KL ( ∇ t Ψ α , Ψ β ) := K (0) KL ( ∂ t Ψ α , Ψ β ) + K (1) KL (Ψ α , γ · Ψ β ) − K (0) KL ( (cid:88) i d i φd i W ; Ψ α , Ψ β ) (113) K (0) KL (Ψ α , ∇ t Ψ β ) := K (0) KL (Ψ α , ∂ t Ψ β ) + K (1) KL ( γ · Ψ α , Ψ β ) − K (0) KL ( (cid:88) i d i φd i W ; Ψ α , Ψ β ) . It is convenient to rewrite the differential equations into a common mode and a relative modepart as follows: K (0) KL ( ∇ t Ψ α , Ψ β ) + K (0) KL (Ψ α , ∇ t Ψ β ) = 0 (114) K (0) KL ( ∇ t Ψ α , Ψ β ) − K (0) KL (Ψ α , ∇ t Ψ β ) = 0 . (115)This is what we propose, without proof, for an explicit realization of the flatness equations(13) that we advertized in the Introduction. As said, the first equation can be satisfiedby a judicious common mode normalization, which we assume (ratios of correlators will beinvariant under changes of the overall normalization, anyway). The more non-trivial, newinformation is in the second equation (115) which samples the relative normalization of theoperators.A few remarks are in order.First, note that the covariant derivatives (113) are manifestly written in terms of inte-grated insertions only. As mentioned in the previous section, the contact terms between γ and Ψ are already taken care of by the t derivatives acting on the Ψ’s. One may wonderwhether there could be additional, contact terms directly between φ and the Ψ’s as well.In fact, it is known that the possible stable degenerations of the punctured disk do not in-clude such factorizations, rather degenerations involving bubbling off disks appear only whenboundary operators hit each other; see again Figure 3. So φ can enter only as integratedoperator, which means it should enter symmetrically with respect to the Ψ’s, precisely as itdoes in (113).To see the structure of the differential equations more clearly, it is helpful to disentanglethe scalar-valued renormalization of the operators, g ( t ), from the boundary contact term.Using (101) we can rewrite K (0) KL ( ∇ t Ψ α , Ψ β ) (116)= K (0) KL ( g (cid:48) ( t ) g ( t ) Ψ α , Ψ β ) − K (0) KL ( U D ([ γ, Ψ α ] , Ψ β ) + K (1) KL (Ψ α , γ · Ψ β ) − K (0) KL ( (cid:88) i d i φd i W ; Ψ α , Ψ β ) , α by g ( t ) and Ψ β by 1 /g ( t ). This exhibits the interplaybetween the contact term of γ with Ψ α , and the integrated insertions. In a sense, therenormalization factor g ( t ) is determined by a mismatch between these terms.Second, it may appear counter-intuitive that the middle terms on the RHS in (113) lookswitched. Actually being not contact terms but integrated insertions, there is no reason why γ should stick locally to the operators on which we take derivatives. From their origin, theysample the dQ (cid:48) s rather then the Ψ’s. One can check that precisely the combinations given in(113) have good covariance properties under transformations Q → V ( t ) − QV ( t ). This willbecome evident in the example that we will discuss below. Also, note that the ordering of γ with respect to the Ψ’s does not matter, up to signs.Finally, what about higher pairings K ( (cid:96) ) KL for (cid:96) >
1? Higher versions with more derivativesacting on the Ψ’s can be constructed in analogy to (110). Due to the anti-commuting natureof this kind of pairings and the limited number of dQ ’s, it is clear that there can be just afinite number of them. Such higher versions would play a role in a more thorough treatment.However, at this point we are not sufficiently certain about the mathematical logic of suchhigher pairings, so we prefer to leave this issue to later work. For now, we content ourselvesto test the proposed equation (115) for the simplest possible case, namely for the cubicelliptic curve. B -Model computations We now re-visit open string mirror symmetry for the cubic elliptic curve. In the physicsliterature this has been discussed in refs. [40, 41, 43–45]. We consider the canonical matrixfactorization (83) Q ( x, t ) = 1 / π i x i + ¯ π i d i W ( x, t ) , Q ( x, t ) · Q ( x, t ) = W ( x, t ) , (117) W ( x, t ) = ζ ( t ) (cid:20) (cid:88) x i − α ( t ) x x x (cid:21) , (118)with { π i , ¯ π j } = δ ij , i, j = 1 , ,
3. It corresponds to a special, irreducible point of a continuousfamily of otherwise reducible matrix factorizations; for details see ref. [43]. This means thatthe open string moduli u α (locations of D ie. , obey U D · Ψ i ..i k = 0. Let us put29he relative renormalization factors that we need to determine, as follows:Ψ( t ) = g ( t ) − Ψ i ( t ) = g ( t ) Π P · π i (119)Ψ ij ( t ) = g ( t ) − Π P · π i π j , Ψ ijk ( t ) = g ( t ) Π P · π i π j π k . The top element has charge 1 and represents a marginal operator that couples to the D u (which we suppress). We will need only a few of these operators explicitly: g ( t ) − Ψ ( t ) = π − ζx ¯ π + 32 ζα (cid:104) ( x ¯ π + x ¯ π ) (cid:105) , (120) g ( t )Ψ ( t ) = π π + 94 αζ (cid:104) (cid:0) x + x x α (cid:1) ¯ π ¯ π − (cid:0) x + x x α (cid:1) ¯ π ¯ π (cid:105) (121)+9 ζ (cid:0) x x − x α (cid:1) ¯ π ¯ π − x ζ ¯ π π + 3 x ζ ¯ π π + 32 αζ (cid:104) x ¯ π π − x ¯ π π + x ¯ π π − x ¯ π π (cid:105) , and g ( t ) − Ψ ( t ) = π π π − H ¯ π ¯ π ¯ π (122)+ 94 αζ (cid:104) (cid:0) x x α + 2 x (cid:1) ¯ π ¯ π π + (cid:0) x x α + 2 x (cid:1) ¯ π ¯ π π − (cid:0) x x α + 2 x (cid:1) ¯ π ¯ π π − (cid:0) x x α + 2 x (cid:1) ¯ π ¯ π π + (cid:0) x x α + 2 x (cid:1) ¯ π ¯ π π + (cid:0) x x α + 2 x (cid:1) ¯ π ¯ π π (cid:105) + 94 ζ (cid:104) (cid:0) x x − x α (cid:1) ¯ π ¯ π π + (cid:0) x x − x α (cid:1) ¯ π ¯ π π − (cid:0) x x − x α (cid:1) ¯ π ¯ π π (cid:105) + 32 αζ (cid:104) x ¯ π π π − x ¯ π π π + x ¯ π π π + x ¯ π π π − x ¯ π π π + x ¯ π π π (cid:105) − ζ (cid:104) x ¯ π π π + x ¯ π π π − x ¯ π π π (cid:105) . We also do some modifications: analogous to the rescaling by ζ ( t ) of the superpotential W ( x, t ), we have an corresponding degree of freedom at the boundary, namely a rescaling ofthe boundary fermions as follows: π i → ρ ( t ) π i , ¯ π i → ρ ( t ) − ¯ π i . (123)This in particular affects the boundary counter term as follows: γ ( t ) ≡ ∂∂t Q ( t ) = U D ( φ ( t ) ) − ρ (cid:48) ( t ) ρ ( t ) (cid:2) Q, R ] , (124)where R is the matrix of R -charges and φ is as given in (56). Moreover, in analogy to the bulktheory, where φ had to be shifted by an exact piece (consistent with the condition of good30asis (94)), we allow for a shift of Ψ by an exact piece. It turns out that given the variouspossible Q -exact parameters, their image in the various residues is only one dimensional. Sowe write this extra exact piece conveniently in terms of just one parameter, λ ( t ):Ψ ( t ) → Ψ ( t ) − λ ( t ) (cid:2) Q, Λ (cid:3) , (125)Λ = 27 g ( t ) ρ ( t ) − x ¯ π ¯ π ¯ π π . (126)We now assemble correlation functions from the Kapustin-Li and related pairings. Firstwe rescale them like in the bulk, ie. , ˜ K ( ∗ ) ∗ ( .., ... ) : ≡ ζ − K ( ∗ ) ∗ ( .., ... ), up to a constant factor.In particular, (cid:104) Ψ α , Ψ β (cid:105) D ≡ (cid:104)(cid:104) Ψ α , Ψ β (cid:105)(cid:105) = ˜ K (0) KL (Ψ α , Ψ β ) . (127)It is easy to check the constancy of the boundary inner products: (cid:104) Ψ( t )Ψ ijk ( t ) (cid:105) D = (cid:104) Ψ i ( t )Ψ jk ( t ) (cid:105) D = (cid:15) ijk . (128)Moreover we have for all operators ˜ K (1) KL (Ψ ∗ , Ψ ∗ ) = 0.We now compute the various pairings explicitly, and for this it suffices to consider theoperators in eqs. (120-122). Let us start with Ψ and consider the integrated bulk insertionfirst: K (0) KL ( (cid:88) i d i φd i W ; Ψ , Ψ) = 3 ζ (cid:48) ( t ) ζ ( t ) + 9 α α (cid:48) ( t )4∆ − λ ( t )= 6 η (cid:48) ( t ) η ( t ) − λ ( t ) . (129)Something nice happens here, namely α ( t ) and ζ ( t ) conspire such as to produce the Dedekindfunction ( q = e πit ): η ( t ) ≡ q / (cid:89) n> (1 − q n )( t ) = ( −√ α (cid:48) ( t )) / ∆ / , therefore : (130) ζ (cid:48) ( t ) ζ ( t ) = 2 η (cid:48) ( t ) η ( t ) − α ( t ) α (cid:48) ( t )4∆ . The nicety continues for all the other pairings:˜ K (0) KL ( ∂ t Ψ , Ψ) = g (cid:48) ( t ) g ( t ) + ρ (cid:48) ( t ) ρ ( t ) + 2 η (cid:48) ( t ) η ( t ) − λ ( t ) , (131)˜ K (0) KL (Ψ , ∂ t Ψ) = − g (cid:48) ( t ) g ( t ) , ˜ K (1) KL ( γ · Ψ , Ψ) = 2 η (cid:48) ( t ) η ( t ) − λ ( t ) , ˜ K (1) KL (Ψ , γ · Ψ) = 2 η (cid:48) ( t ) η ( t ) − ρ (cid:48) ( t ) ρ ( t ) + λ ( t )31hus in the covariant derivatives (113) the ρ -dependence cancels out:˜ K (0) KL ( ∇ t Ψ , Ψ) = g (cid:48) ( t ) g ( t ) + η (cid:48) ( t ) η ( t ) + 32 λ ( t ) = − ˜ K (0) KL (Ψ , ∇ t Ψ) . (132)Their sum cancels; thus the first differential equation (114) is satisfied identically, as ex-pected. More interesting is the difference which depends on λ ( t ):˜ K (0) KL ( ∇ t Ψ , Ψ) − ˜ K (0) KL (Ψ , ∇ t Ψ) = 2 g (cid:48) ( t ) g ( t ) + 2 η (cid:48) ( t ) η ( t ) + 3 λ ( t ) (133)This is analogous to what happened in the bulk theory, where φ had to be shifted by anexact piece in order to obtain a trivial relative normalization factor between 1 and φ . To fix λ , we invoke the proposed bulk-boundary pairing conditions (107): K (0) bb (1 , Ψ ) = g ( t ) , (134)˜ K (1) bb ( φ, Ψ ) = g ( t ) (cid:16) η (cid:48) ( t ) η ( t ) + 3 λ ( t ) (cid:17) . The second equation determines λ ( t ) and so the differential equation finally turns into: g (cid:48) ( t ) = 0 , (135)precisely as required. This then also fixes the first condition in (134). We thus see somedegree of consistency of the procedure.Now on to the more interesting sector, where we find:˜ K (0) KL ( ∂ t Ψ , Ψ ) = g (cid:48) ( t ) g ( t ) + ρ (cid:48) ( t )3 ρ ( t ) + 2 η (cid:48) ( t )3 η ( t ) (136)˜ K (0) KL (Ψ , ∂ t Ψ ) = − g (cid:48) ( t ) g ( t ) + 2 ρ (cid:48) ( t )3 ρ ( t ) + 4 η (cid:48) ( t )3 η ( t )˜ K (1) KL ( γ · Ψ , Ψ ) = 2 η (cid:48) ( t ) η ( t ) − ρ (cid:48) ( t )3 ρ ( t )˜ K (1) KL (Ψ , γ · Ψ ) = 2 η (cid:48) ( t ) η ( t ) − ρ (cid:48) ( t )3 ρ ( t )and so ˜ K (0) KL ( ∇ t Ψ , Ψ ) = g (cid:48) ( t ) g ( t ) − η (cid:48) ( t )3 η ( t ) = − ˜ K (0) KL (Ψ , ∇ t Ψ ) . (137)Again, ρ ( t ) cancels and the sum of both equations cancels. This then finally determines, upto a multiplicative constant: g ( t ) = η ( t ) / . (138)32 .4.2. B -Model correlators With the flattening renormalization factors at hand, we are ready to compute correlationfunctions. The simplest one is (cid:104) Ψ i Ψ j Ψ k (cid:105) D ≡ (cid:104)(cid:104) Ψ i , m (Ψ j ⊗ Ψ k (cid:105)(cid:105) = η ( t ) (cid:15) ijk , (139)where the lowest A ∞ product is just matrix multiplication in the chiral ring: m (Ψ j ⊗ Ψ k ) ≡ Ψ j · Ψ k = η ( t ) Ψ jk + [ Q, ∗ ] . (140)Before we will discuss its significance below, let us first consider higher point correlators.As pointed out in the introduction (3), higher point correlators involve higher A ∞ prod-ucts. These can be recursively assembled in terms of the boundary propagator U D defined in(91) and lower products, forming nested trees such as in Figure 2. The functional complexity,on the other hand, is largely governed by the proper “flat” renormalization factors, whichare usually neglected in this context.The first non-trivial A ∞ product is defined by m (Ψ α ⊗ Ψ β ⊗ Ψ γ ) = U D (Ψ α · Ψ β ) · Ψ γ − ( − | Ψ α | Ψ α · U D (Ψ β · Ψ γ ) , (141)whose Q -variation measures the non-associativity of projected OPE’s:[ Q, m (Ψ α ⊗ Ψ β ⊗ Ψ γ )] = Π P ( m (Ψ α ⊗ Ψ β )) · Ψ γ − Ψ α · Π P ( m (Ψ β ⊗ Ψ γ )) . (142)With this a particularly nice correlator can be computed explicitly, by inserting a “weakbounding chain” [14] Ψ s ≡ − / (cid:80) s i Ψ i into (141). This yields m (Ψ s ⊗ Ψ s ⊗ Ψ s ) = η ( t ) W ( s, t ) , (143)where W ( x, t ) is the LG superpotential defined in (118). The interpretation of this is thatthese are the two non-vanishing terms of the Maurer-Cartan equation (4), where the zerothproduct: m = − η ( t ) W ( s, t ) represents the curvature term of the deformed A ∞ alge-bra. Such a solution of the Maurer-Cartan equation with non-zero m is called “weaklyobstructed”.Going back to physics, note that product m in (143) leads to the following correlator (cid:104)(cid:104) Ψ , m (Ψ s ⊗ Ψ s ⊗ Ψ s ) (cid:105)(cid:105) = (cid:104) (cid:90) Ψ (1)123 Ψ s Ψ s Ψ s (cid:105) D = ∂∂u W eff ( s, t, u ) | u =0 (144)= η ( t ) W ( s, t ) , where u is the open string modulus that couples to the marginal boundary operator Ψ .This describes an obstruction that appears if all three s i are switched on simultaneously.33igure 4: Quiver diagram depicting the spectrum of boundary preserving and boundarychanging operators. A -Model So far, we have considered the topological B -model, where the specific brane geometry inquestion is encoded in the canonical matrix factorization (117) of W . All quantities dependon t which is the flat coordinate associated with the algebraic complex structure deformationof W . As it is usual for LG models, the underlying geometry is the one of an orbifold, here T / Z .In the present situation we have a self-intersecting brane configuration, so the opera-tors Ψ i and Ψ ij are localized on intersections despite not literally being boundary-changing.After undoing the orbifold, we obtain an equivariant matrix factorization with 3 differentbranes, L , L , L , with RR charges ( r, c ) given by (2 , − ,
1) and ( − ,
2) , resp. (up tomonodromy). The operators localized at the intersections then gain corresponding equiv-ariant labels that govern the selection rules for correlators, Ψ i → Ψ ( A,B ) i , Ψ ij → Ψ ( B,A ) ij andΨ ijk → Ψ ( A,A ) ijk , etc. This can be visualized with help of the quiver diagram in Figure 4. Forbackground on such equivariant matrix factorizations, see eg. , [38, 43].We now consider the mirror geometry, where t has the interpretation as a K¨ahler modulus.The mirror geometry is given by an orbisphere with three punctures, P (cid:39) T / Z , whereagain there is just one brane: namely the Seidel special lagrangian which intersects itselfthree times [92]. The fundamental domain is just one-third as compared to the one of thecurve, which is why t is implicitly rescaled by a factor of 3. Undoing the orbifold, the Seidellagrangian unfolds into three different, pairwise intersecting special lagrangians L , , on T .These are shown, on the covering plane, in Figure 1.34 -model correlation functions for these brane configurations are well-known [41, 93, 94]essentially because for the flat torus the instanton contributions can be just read-off bymeasuring the areas of polygons. These results are particular examples of the general storylaid out in refs. [51–53]. The correlators have the structure of generalized theta functions,and in our case the three-point functions are very simple: (cid:104) Ψ ( A ,A )1 ( u , u )Ψ ( A ,A )1 ( u , u )Ψ ( A ,A )1 ( u , u ) (cid:105) D = α ( (cid:88) u i , t ) (145) (cid:104) Ψ ( A ,A )1 ( u , u )Ψ ( A ,A )2 ( u , u )Ψ ( A ,A )3 ( u , u ) (cid:105) D = α ( (cid:88) u i , t ) (cid:104) Ψ ( A ,A )1 ( u , u )Ψ ( A ,A )3 ( u , u )Ψ ( A ,A )2 ( u , u ) (cid:105) D = α ( (cid:88) u i , t ) , where α (cid:96) ( u, t ) = e πi ( (cid:96)/ − / Θ (cid:104) (1 − (cid:96) ) / − / − / (cid:12)(cid:12)(cid:12) u, t (cid:105) , (146)Θ (cid:104) c c (cid:12)(cid:12)(cid:12) n u, nt (cid:105) = (cid:88) m q n ( m + c ) / e πi ( n u + c )( m + c ) . (147)If the brane moduli are switched off: u α = 0, these indeed reproduce our result (139) due to α (0 , t ) = 0 and α (0 , t ) = − α (0 , t ) = η ( t ).This gives then the following A -model interpretation of the η -function that we find fromthe B -model: its first term, η ( t ) ∼ q / + .... , measures the area of the smallest triangleas shown in Figure 1, which is 1 / t dependent normalization was put by handin order to fit the known areas of polygons, and was not computed. In a sense, the explicitmirror map between the two sides of the isomorphism (9) was missing. Our point was to makepredictions for the A -model starting from the B -Model, without the need to fix functions byhand. This will be important for later applications, eg. to Calabi-Yau threefolds, where the A -model correlators are not known beforehand.For the four-point amplitude (144), there is a reassuring relation to the works [92]. Theauthors used matrix factorizations as well, however on the A -model side, which is surprisingbecause in physics, matrix factorizations appear for B -type supersymmetry. What they havedone, in a beautiful way, is to interpret the individual matrix entries of Q as maps whichcount polygons, and thereby implicitly reproduce the B -model matrices directly in termsof A -model variables. Thus Q serves as an A ∞ functor that maps from F uk ( T , L ∗ ) into Cat (MF , W ). At some point they had to impose a “quantum” product by hand, and thisis precisely the manifestation of the structure constant η ( t ) in the product m (140), whicharises from our B -model flatness equations. This is the most basic manifestation of the openstring mirror symmetry between quantum products in the A -model, and classical productsin the B -model; see Figure 5. 35igure 5: Quantum and classical products m in the topological A - and B -models, resp. Theformer features genuine boundary changing open Gromov-Witten invariants. The latter isentirely determined by renormalization factors computed by the differential equation (115).It was also shown in [92] that by counting triangles one obtains as curvature term of the A ∞ algebra: m ( s, t ) = (cid:104) ϕ ( t ) (cid:88) s i − ψ ( t ) s s s (cid:105) , (148)where ϕ ( t ) and ψ ( t ) are certain modular functions. By comparison with (143), we reproducethem as follows: ϕ (cid:39) ηζ and ψ (cid:39) ηζα , up to constants. In view of (144), we can explain theresults of [92] by simply taking u -derivatives of (145): ∂ u α ( u ) | u =0 (cid:39) ηζ , and ∂ u α ( u ) | u =0 = − ∂ u α ( u ) | u =0 (cid:39) ηζα . In this way, their results can be directly understood from B -modelopen string correlation functions.
4. Summary and outlook
In this paper we made a proposal as to how to compute correlation functions in B-typetopological strings that involve boundary changing operators. This is important because inphysics this corresponds to computing superpotentials for the largest class of string back-grounds with D -branes, namely ones involving intersecting branes. This class is infinitelyricher than backgrounds without intersecting branes, and extra significance lies also in thefact that, to our knowledge, such moduli-dependent, boundary changing correlators havenever been computed (though determined by hand for the elliptic curve).Summarizing, our main result is a differential equation that is formulated in terms ofcertain residue pairings. It generalizes the flatness equations of the bulk theory on thesphere, whose primary component looks K (0) ( ∇ GMt ϕ, ∗ ) ≡ K (0) ( ∂ t ϕ, ∗ ) − K (1) ( φϕ, ∗ ) = 0 . (149)Here K ( (cid:96) ) ( ∗ , ∗ ) are Saito’s higher residue pairings, where K (0) ( ∗ , ∗ ) is nothing but the topo-logical metric, and K (1) ( ∗ , ∗ ) has extra insertions of ddW in it. One can package all the36airings into one quantity, by summing K ( u )( ∗ , ∗ ) = (cid:80) u (cid:96) K ( (cid:96) ) ( ∗ , ∗ ), where u is the degree2 spectral parameter. Then one can concisely write K ( u )( ∇ GMt ϕ, ∗ ) = 0 , with ∇ GMt = ∂ t − ∂ t Wu . (150)Our generalization to the boundary theory looks formally similar, except that the pairings areformulated in terms of matrix factorizations which underlie B -type topological LG modelson the disk. The basic pairing, namely the topological metric, is given by the Kapustin-Lisupertrace residue formula, K (0) KL given in (102). The next higher pairing that we consider, K (1) KL in (110), has extra insertions of ddQ in it, where Q is the BRST operator that is definedby the factorization Q = W . In terms of these, the flatness equation then looks (cid:104)(cid:104)∇ t Ψ ( A,B ) , Ψ ( B,A ) (cid:105)(cid:105) D = 0 , (151)where the “relative” boundary-bulk connection is given in (113).The key role at the boundary is played not by φ = ∂ t W , but by the boundary counterterm γ = ∂ t Q . This has to do with how stable degenerations of the disk work: there is no directcontact term between φ and boundary fields. This also reflects that at the boundary, therelevant cohomology is given by the one of Q , and not of Q W = ¯ ∂ + ι dW . Morally speaking,this suggests to define a boundary connection by ∇ t = ∂ t − ∂ t Qv , where v is a formal anti-commuting parameter of degree 1 which plays the role of the spectral parameter u in thebulk.Whether there is more flesh to this than a naive analogy to the bulk, depends on whetherone can meaningfully define higher pairings K ( (cid:96) ) KL which would involve more insertions of ddQ . These should reflect the filtrated Hodge structure associated with the boundary BRSToperator Q . Related questions are how the general definition of a “good basis” in terms ofhigher residues would look like, in relation to the operator basis that we defined in (94). Thatshould also include a boundary-bulk pairing K ( (cid:96) ) bb ( ∗ , ∗ ) as defined in (105), as for (cid:96) = 0 it isan important ingredient of the axiomatic definition of open topological strings [15, 95, 96].All-in-all, conjecturally we would have for a “good basis”: K (0) ( ϕ a , ϕ b ) = η ab , K ( (cid:96)> ( ϕ a , ϕ b ) = 0 K (0) bb ( ϕ a , Ψ β ) = η aβ , K ( (cid:96)> bb ( ϕ a , Ψ β ) = 0 (152) K (0) KL (Ψ α , Ψ β ) = η αβ , K ( (cid:96)> KL (Ψ α , Ψ β ) = 0 . The resolution of these questions would require a deeper understanding of the underlyingmathematics, which is beyond the scope of the paper. Sorting them out would likely beimportant for the application to higher dimensional Calabi-Yau spaces. This was our mainmotivation for the present work and we intend to report on it in the future.I thank Manfred Herbst, Hans Jockers, Johanna Knapp, Calin Lazaroiu and JohannesWalcher for discussions over the years, and especially Dmytro Shklyarov for correspondence37nd comments on the manuscript. This research was also supported in part by the NationalScience Foundation under Grant No. NSF PHY17-48958.
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