ESI The Erwin S hr�odinger International Boltzmanngasse 9Institute for Mathemati al Physi s A-1090 Wien, AustriaOn the Relation between Open and ClosedTopologi al StringsAnton KapustinLev Rozansky

Vienna, Preprint ESI 1599 (2005) February 2, 2005Supported by the Austrian Federal Ministry of Edu ation, S ien e and CultureAvailable via http://www.esi.a .at

ON THE RELATION BETWEEN OPEN AND CLOSEDTOPOLOGICAL STRINGSANTON KAPUSTIN AND LEV ROZANSKYAbstra t. We dis uss the relation between open and losed string orrelators using topologi al string theories as a toy model. Wepropose that one an re onstru t losed string orrelators from theopen ones by onsidering the Ho hs hild ohomology of the ate-gory of D-branes. We ompute the Ho hs hild ohomology of the ategory of D-branes in topologi al Landau-Ginzburg models andpartially verify the onje ture in this ase.1. Introdu tion and SummaryThere exist di�erent viewpoints on the question whether losed oropen strings are more fundamental. The more popular viewpoint is that losed strings are simpler (be ause one does not have to make a hoi eof a boundary ondition) and therefore more fundamental. In this view,the entral problem is to lassify D-branes and onstru t open- losedstring orrelators for a given losed string theory. In pra ti e, om-plete lassi� ation is possible only in very simple (mostly topologi al)examples. The opposite viewpoint is supported by the observation that losed String Field Theory is vastly more ompli ated than the openone. While in the open-string ase the lassi al a tion is ubi [41℄,in the losed-string ase it is non-polynomial [37, 26℄. To write downthe open SFT a tion, one has to spe ify an asso iative produ t on thespa e of states, a di�erential (i.e. a BRST operator), and an invariants alar produ t. Deformations of the losed-string ba kground hangethese data, so there is a map from the spa e of losed-string states tothe spa e of in�nitesimal deformations of the open string theory.One diÆ ulty with the �rst viewpoint is that it is not lear if thespe trum and properties of D-branes are uniquely determined by the losed string theory. In other words, there may exist D-branes whi hare perfe tly onsistent by themselves, but mutually in ompatible, inthe sense that it is not possible to de�ne states whi h orrespond toopen strings stret hed between two di�erent branes. If one regards a D-brane as a boundary ondition for a string worldsheet, su h a situationmay appear absurd, but one must remember that up to now there doesnot exist a pre ise de�nition of the notion of a \boundary ondition" at1

2 ANTON KAPUSTIN AND LEV ROZANSKYthe quantum level. Instead, one hara terizes D-branes as solutions ofa ompli ated set of onditions (see Refs. [27, 18℄), the most nontrivialof whi h is the so- alled Cardy ondition. With su h an axiomati de�nition of a D-brane, the situation des ribed above is not ruled out.If one adopts the se ond viewpoint, then the entral problem is to onstru t losed string orrelators from the open ones. As mentionedabove, losed string states are related to in�nitesimal deformations ofthe open-string theory. Suppose for simpli ity that there is only a sin-gle D-brane in the theory, so that all the information is en oded in anasso iative algebra A equipped with a BRST di�erential Q of ghostnumber one and an invariant s alar produ t. The pair (A;Q) is alleda di�erential graded algebra (DG-algebra) by mathemati ians. Equiv-alen e lasses of deformations of these data are des ribed by a ertain ohomology theory (Ho hs hild ohomology of (A;Q) with oeÆ ientsin itself). The simplest onje ture is that the spa e of physi al losed-string states is isomorphi to the Ho hs hild ohomology of (A;Q).At this point we must be more spe i� about the kind of string the-ory we are talking about. In this paper we will only dis uss topologi alstring theories. Let us re all how they are onstru ted. The startingpoint is a unitary N = 2 d = 2 �eld theory whi h an be twisted to atopologi al �eld theory. To get a string theory, one has to ouple it totopologi al gravity. On the level of the spa e of states, this operationis very simple [42, 15℄: ea h state of the TFT gives rise to an in�-nite sequen e of states of in reasing ghost number. The �rst state inthis sequen e is alled a gravitational primary, and the rest are alledgravitational des endants. Tree level orrelators of primaries an be omputed in the TFT, i.e. the oupling to topologi al gravity plays norole for these orrelators. In this paper we will only dis uss su h orre-lators, and therefore topologi al gravity will play no role. The pre ise onje ture we are making is that for topologi al strings the spe trumof gravitational primaries is given by the Ho hs hild ohomology of the ategory of topologi al D-branes.This onje ture is very appealing, be ause many stru tures of theopen- losed string theory are then automati . For example, the Ho h-s hild ohomology of any DG-algebra is itself a super ommutative al-gebra, whi h may be identi�ed with the algebra of observables in the losed-string TFT. It also has a Lie-type bra ket of degree �1, in agree-ment with the �ndings of Refs. [46, 45, 30, 14, 35, 31℄. Further, onsiderthe ohomology of Q, whi h may be regarded as the spa e of physi alopen-string states. It turns out that this spa e has a natural stru tureof a module over the Ho hs hild ohomology of (A;Q), and this allowsone to de�ne the bulk-boundary map. This will be dis ussed in more

RELATION BETWEEN OPEN AND CLOSED STRINGS 3detail below. Many axioms of open- losed TFT (see Ref. [27℄) are theneasily veri�ed.1The assumption that there is only one D-brane is unrealisti . Forexample, given a D-brane M0, one may onsider dire t sums of several opies of M0, as well as more ompli ated bound states. If there aremany possible D-branes, then one has to take into a ount open stringswith di�erent boundary onditions on the two ends. It is onvenientto think of a D-brane as an obje t of an additive (in fa t C -linear) ategory, and of the spa e of open strings between two D-branes asthe spa e of morphisms. Then the algebra of open string states for aparti ular D-brane is its endomorphism algebra. BRST operators giverise to di�erentials on all spa es of morphisms, so one is a tually deal-ing with a di�erential graded ategory (DG ategory). The grading isgiven by the ghost number. To get the spa e of physi al open-stringstates between any two D-branes, one has to ompute the ohomologyof the BRST operator on the spa e of morphisms. There is a notionof Ho hs hild ohomology whi h lassi�es equivalen e lasses of de-formations of su h ategories. Essentially, one lumps together all theobje ts in the ategory into a single \total obje t" and onsiders theHo hs hild ohomology of its endomorphism algebra. This is equiva-lent to thinking about a C -linear additive ategory as an \algebra withmany obje ts." It is tempting to onje ture that the spa e of physi- al losed strings is isomorphi to the Ho hs hild ohomology of the ategory of D-branes. A heuristi argument for this is explained inthe end of Se tion 2. Again, the spa e of physi al open-string statesbetween any two D-branes is naturally a module over the Ho hs hild ohomology, so the bulk-boundary maps are automati .Even when there are many possible D-branes, one an often �nd a D-brane M0 su h that all other D-branes an be obtained as bound statesof several opies of M0 and its anti-brane. In mathemati al terms,this means that the ategory of D-branes is equivalent to the ategoryof modules of some kind over the endomorphism algebra (whi h isa tually a DG-algebra) of M0. Then the Ho hs hild ohomology ofthe ategory of D-branes oin ides with the Ho hs hild ohomology ofthe endomorphism algebra of M0. In the physi al ase, the role of M0 an be played by a spa e-�lling brane.One example where this pres ription for re onstru ting losed stringsis known to work is the topologi al B-model of a Calabi-Yau manifold1The Cardy ondition is an ex eption in this regard. It seems to be a gener-alization of the Hirzebru h-Riemann-Ro h theorem, and its validity is not at allobvious.

4 ANTON KAPUSTIN AND LEV ROZANSKYX. The algebra of losed string states is given by(1) �p;qHp(�qTX):The ategory of D-branes is believed to be equivalent to the boundedderived ategory of X denoted Db(X). More pre isely, it is a DG- ategory whose derived ategory is believed to be equivalent to Db(X).The appropriate ohomology to ompute is the Ho hs hild ohomol-ogy of the sheaf of algebras OX, and one an show that the latter isisomorphi to Eq. (1) [38℄. This is dis ussed in more detail in the nextse tion.In this paper we study another lass of topologi al string theories:topologi al Landau-Ginzburg models. The losed string se tor has beendes ribed by C. Vafa [39℄. A simple des ription of the ategory of D-branes in LG models has been proposed by M. Kontsevi h and derivedfrom physi al onsiderations in Refs. [21, 22, 5℄ (see also Ref. [28℄).It turns out that the ategory of D-branes an be thought of as the ategory of CDG-modules over a ertain ommutative CDG-algebra,where CDG stands for \ urved di�erential graded." (This notion isexplained in detail in Ref. [22℄ and will be re alled below.) For LGmodels on smooth spa es many orrelators have been omputed [22℄.In this paper we ompute the Ho hs hild ohomology of the ategoryof Landau-Ginzburg branes and show that in this way we an re overthe losed string spa e of states, together with its algebra stru ture ands alar produ t, as des ribed in Ref. [39℄, as well as some open- losed orrelators, as des ribed in Ref. [22℄.One may also onsider LG models on orbifolds. Su h models areparti ularly important be ause they provide an alternative des riptionof ertain Calabi-Yau sigma-models (these are so- alled Gepner mod-els [11℄). The losed-string spe trum for LG orbifolds has been de-s ribed by K. Intriligator and C. Vafa [20℄, but its interpretation inmathemati al terms has been la king. We he k in several examplesthat the Ho hs hild ohomology of the ategory of D-branes on LGorbifolds reprodu es the results of Ref. [20℄ and others. Our examplesin lude the Gepner models for Fermat-type hypersurfa es in proje tivespa es. A ni e feature of the Ho hs hild ohomology approa h is thatthe 3-point losed-string orrelators (i.e. the Yukawa ouplings) andthe bulk-boundary maps ome out automati ally.These results provide eviden e that the onje tural identi� ation ofthe losed string se tor with the Ho hs hild ohomology of the ate-gory of D-branes is orre t, at least for topologi al strings, and that ertain simple open- losed orrelators an be omputed using the al-gebrai stru ture of the Ho hs hild omplex. Hopefully, multi-point

RELATION BETWEEN OPEN AND CLOSED STRINGS 5and higher-genus orrelators an also be extra ted from the Ho hs hild omplex. The te hnique based on the Ho hs hild omplex ould beuseful for the omputation of the spa e-time superpotential for super-string ompa ti� ations with D-branes based on Gepner models [11℄.These open- losed superstring ba kgrounds have been studied in manypapers, see e.g. Refs. [36, 16, 4, 7, 6, 1, 19℄.2. Definitions of the Ho hs hild ohomologyWe begin by re alling various de�nitions of the Ho hs hild ohomol-ogy of algebras, DG-algebras, and aÆne and proje tive varieties. Wewill pi k the most onvenient de�nition and in the next se tion gener-alize it to the ase of CDG algebras relevant for the Landau-Ginzburgmodels.Let A be an asso iative algebra over C . The Ho hs hild o hain omplex (with oeÆ ients in A) is the sequen e of ve tor spa esCn(A) = HomC (An; A); n = 0; 1; : : : ;equipped with a di�erential Æ : Cn(A) ! Cn+1(A) de�ned by theequation(2) (Æf)(a1; : : : ; an+1) = a1f(a2; : : : ; an)+ nXi=1 (�1)if(a1; : : : ; ai�1; aiai+1; ai+2; : : : ; an)+ (�1)n+1f(a1; : : : ; an)an+1:The ohomology of Æ in degree n will be denoted HHn(A) and alledthe Ho hs hild ohomology of A (with oeÆ ients in A). The morestandard notation for it is HHn(A;A). Ea h 2- o y le f(a1; a2) de�nesan in�nitesimal deformation of the asso iative produ t on A. That is,if we de�ne a new produ t bya ? b = ab+ tf(a; b); t 2 C ;it will be asso iative to linear order in t if and only if Æf = 0. Trivial in-�nitesimal deformations (i.e. in�nitesimal deformations whi h lead toan isomorphi algebra) are lassi�ed by 2- oboundaries, i.e. 2- o y lesof the form Æg for some 1- o hain g(a). Thus HH2(A) lassi�es nontriv-ial deformations of the asso iative algebra stru ture on A. One an givea similar interpretation to the total Ho hs hild ohomology HH�(A):it lassi�es in�nitesimal deformations of A in the lass of A1 algebras,asso iative algebras being a very spe ial ase of A1 algebras [34℄.If A is a Z-graded or Z2-graded algebra, the Ho hs hild omplex isde�ned somewhat di�erently. Let Ap be the degree-p omponent of A,

6 ANTON KAPUSTIN AND LEV ROZANSKYso that Ap � Aq � Ap+q. An element f of Cn(A) is said to have aninternal degree p if f(a1; : : : ; an) 2 Ap+k1+���+knwhenever ai 2 Aki. Thus ea h ve tor spa e Cn(A) is graded by theinternal degree, and we de�ne the total degree of an element of C�(A)to be the sum of n and the internal degree. The Ho hs hild omplexis graded by the total degree, and the di�erential is given by(3) (Æf)(a1; : : : ; an+1) = (�1)a�f a1f(a2; : : : ; an)+ nXi=1 (�1)if(a1; : : : ; ai�1; aiai+1; ai+2; : : : ; an)+ (�1)n+1f(a1; : : : ; an)an+1:Here and below whenever a symbol o urs in the exponential of (�1),it is understood as its internal degree.Similarly, let A = (A;Q) be a DG-algebra. The di�erential Q is adegree-1 derivation Q : Ap ! Ap+1 whi h satis�es Q2 = 0. Using Q,we an make Cn(A) into a omplex: one lets(4) (Qf)(a1; : : : ; an) = Q(f(a1; : : : ; an))� nXi=1 (�1)v1+:::+vi�1+f+n�1f(a1; : : : ; ai�1; Qai; ai+1; : : : ; an):Thus on the bigraded ve tor spa e C�(A) we have two di�erentials: Q,whi h has internal degree 1 and n-degree 0, and Æ, whi h has internaldegree 0 and n-degree 1. The total degree for both di�erentials is 1, andit is easy to he k that Q and Æ ommute. The Ho hs hild ohomologyof A is de�ned to be the ohomology of (�1)nQ+ Æ. The ve tor spa eHH2(A) lassi�es in�nitesimal deformations of (A;Q) in the ategoryof DG-algebras, up to quasi-isomorphism. More generally, HH�(A) lassi�es deformations of (A;Q) regarded as an A1 algebra.Sin e Ho hs hild o hains are fun tions taking value in an algebra,the Ho hs hild omplex has an obvious algebra stru ture as well (givenby the multipli ation of fun tions). The orresponding binary produ tis alled the up produ t. The Ho hs hild oboundary operator Æ is aderivation of the up produ t, and therefore the up produ t des endsto Ho hs hild ohomology, making it into a Z-graded algebra. It wasnoted for the �rst time by M. Gerstenhaber [12℄ that the latter algebrais always super ommutative, even if A is non ommutative. Gersten-haber also dis overed that there is a natural graded Lie bra ket onHH�(A) of degree �1. The algebrai stru ture of HH�(A) is formally

RELATION BETWEEN OPEN AND CLOSED STRINGS 7the same as that of the spa e of fun tions on aZ-graded supermanifoldequipped with a Poisson bra ket of degree �1.There is an equivalent de�nition of the Ho hs hild ohomology of analgebra whi h has a ni e geometri interpretation. It is this de�nitionthat we will generalize in the next se tion. Suppose A is ommutative;then one an regard A as the algebra of fun tions on an aÆne s hemeX = Spe (A). Consider further A A, its spe trum, Spe (A A) =X�X, and the diagonal � � X�X. One an think of � as a B-brane(read: obje t of the bounded derived ategory) on X�X, and onsiderits open-string spe trum (read: endomorphism algebra). It turns outthat the resulting algebra of physi al open-string states is pre isely theHo hs hild ohomology of A.An algebrai version of this de�nition is also very on ise. One anthink of A as a left-right bimodule over A, or equivalently as a moduleover A A (here we still assume that A is ommutative). Then theHo hs hild ohomology of A is de�ned asHH�(A) = ExtAA(A;A);that is, it is the endomorphism algebra of A regarded as an obje t ofthe derived ategory of modules over AA. If A is non ommutative,then A is not a module over A A, but it is a module over A Aop,where Aop is the opposite algebra of A. It turns out that in this moregeneral ase we have:HH�(A) = ExtAAop (A;A):To see how this omes about, we need to ompute the endomorphismsof � in Db(X � X): That is, one has to take a proje tive resolutionof A regarded as a module over A Aop, apply to it the operationHomAAop (�; A), and evaluate the ohomology of the resulting omplexof ve tor spa es. The key point is that for any algebra A with a unitthere is a anoni al resolution of A by free AAop modules:� � � ���! A4 ���! A3 ���! A2:Ea h term in this omplex is a bimodule over A, whi h is the same as amodule over AAop. If we use this resolution to ompute Ext�(A;A),we get the Ho hs hild omplex.If we turn our attention to proje tive or quasi-proje tive varieties (ors hemes) and their derived ategories of oherent sheaves, several de�-nitions of the Ho hs hild ohomology are possible. One possibility is toshea�fy the Ho hs hild omplex and take the hyper ohomology of theresulting omplex of sheaves as the de�nition of Ho hs hild ohomol-ogy of Db(X). This is the approa h taken in Ref. [40℄ for Ho hs hild

8 ANTON KAPUSTIN AND LEV ROZANSKYhomology; a similar de�nition of Ho hs hild ohomology is adopted inRef. [13℄. Another possibility is to onsider the diagonal subvariety� � X �X and de�ne the Ho hs hild ohomology of X to be the en-domorphism algebra of � regarded as an obje t of Db(X�X) [38℄. Re-markably, all these de�nitions give the same result for quasi-proje tives hemes [38℄. M. Kontsevi h interpreted the last de�nition of HH�(X)as omputing the spa e of in�nitesimal deformations of the boundedderived ategory of oherent sheaves on X in the lass of A1 ate-gories [25℄.The spe ial thing about the diagonal inX�X is that it is a geometri obje t representing the identity fun tor from the ategory of D-braneson X to itself. That is, if one takes a D-brane on X, pulls it ba kto X � X using the proje tion to the �rst fa tor, tensors with thediagonal obje t, and then pushes down to X using the proje tion to these ond fa tor, then one gets ba k the same D-brane, up to isomorphism.Then endomorphisms of the diagonal D-brane parametrize in�nitesimaldeformations of the identity fun tor, i.e. in�nitesimal deformationsof the ategory of D-branes on X [25℄. Thus it is natural to de�nethe Ho hs hild ohomology of the ategory of D-branes on X as theendomorphism algebra of the diagonal D-brane on X �X.Using the \diagonal brane" de�nition, it is easy to see that forsmooth quasi-proje tive X the Ho hs hild ohomology is(5) HHn(X) = �p+q=nHp(�qTX):Indeed, if we are dealing with a B-brane on a manifold Z whi h or-responds to the stru ture sheaf of a smooth submanifold Y , then itsopen string algebra is given by�p;qHp(�qNY );where NY is the normal bundle of Y in Z. The fermion number (i.e.grade) is p + q. More pre isely, this bigraded ve tor spa e is the �rstterm in a spe tral sequen e whi h onverges to the spa e of physi alopen strings [23℄. In our ase Z = X �X, and Y = �, so NY = TX,and the �rst term of the spe tral sequen e omputing the spa e ofphysi al open string states for � is pre isely Eq. (5). The spe tralsequen e a tually ollapses in this ase, so the Ho hs hild ohomologyof X is given by Eq. (5).More generally, one notes that instead of the anoni al resolutionof the diagonal mentioned above one an take any other proje tiveresolution. This is very useful, be ause the \ anoni al" resolution isin onvenient for omputations: even for A = C [x℄, the algebra of poly-nomials in one variable, it has in�nite length. In the next se tion we

RELATION BETWEEN OPEN AND CLOSED STRINGS 9will be mostly interested in the ase when X is an aÆne spa e V ' C n .In this ase a mu h more onvenient free resolution of � is known,the Koszul resolution. To des ribe it, let x1; : : : ; xn and y1; : : : ; yn beaÆne oordinates on V � V and let us introdu e n fermioni variables�1; : : : ; �n. Consider the ve tor spa e K of polynomial fun tions of allthese variables. It is graded by the negative of the fermion number. Itis easy to see that K is a free graded module over C [V �V ℄ of rank 2n.Consider the following linear map on K:k = nXi=1 (xi � yi) ���iObviously, k is a degree-one endomorphism of K whi h squares tozero. The pair (K; k) is the Koszul omplex for �. One an he kthat its ohomology is on entrated in degree zero and is isomorphi to C [V � V ℄=(x� y) ' C [V ℄.M. Kontsevi h onje tured [25℄ that the \diagonal brane" onstru -tion will work in a similar way for the ategory of A-branes on a Calabi-Yau X. The ategory of A-branes is the derived ategory of the Fukaya ategory, whi h is an A1 ategory whose obje ts are, roughly speak-ing, Lagrangian submanifolds with at ve tor bundles. The diagonalbrane is the diagonal submanifold in X � X, where the se ond opyof X is taken with a symple ti stru ture opposite to that on the �rst opy.2 We will denote the se ond opy of X with a reversed symple ti stru ture by Xop. The ve tor bundle on � � X � Xop is taken tobe the trivial rank one bundle. Kontsevi h onje tured that the en-domorphism algebra of this diagonal brane, regarded as an obje t ofthe derived Fukaya ategory, is isomorphi to the quantum ohomologyring of X (whi h is the algebra of physi al losed string states in theA-model on X). This onje ture remains unproved.One an give a general (although somewhat nonrigorous) argumentthat the Ho hs hild ohomology of the ategory of D-branes de�ned asthe endomorphism algebra of the diagonal brane is the spa e of physi al losed string states. Consider a losed-string spheri al worldsheet � onwhi h there lives some topologi al �eld theory T . We an reinterpret itas an open-string disk worldsheet as follows. Let us draw an equatorial2Apparently, reversing the sign of the symple ti form is analogous to passingfrom A to Aop for B-branes. This analogy would have a natural explanation if theFukaya ategory were somehow related to the deformation quantization of (X;!): hanging the sign of ! has the e�e t of repla ing the quantized algebra of fun tionson X with its opposite. A relation between the Fukaya ategory and deformationquantization has been onje tured by many people, in luding one of the authors ofthe present paper; unfortunately, the nature of su h a relation remains elusive.

10 ANTON KAPUSTIN AND LEV ROZANSKY ir le (seam) on � and identify the upper and lower hemispheres bymeans of an antiholomorphi map. Let T op be the image of T underthis map. (For B-model on a Calabi-Yau, T op is isomorphi to T ; for aLandau-Ginzburg model, T op di�ers from T only by the reversal of thesign of the superpotential; for A-model on a Calabi-Yau, T op di�ersfrom T by the reversal of the sign of the symple ti stru ture.) ThenT living on � is equivalent to T T op living on the upper hemisphere,with some gluing ondition on the boundary. This gluing ondition orresponds to a D-brane on X �Xop, and it is hosen so that afterreinterpreting a T T op �eld on�guration on the upper hemisphere asa T �eld on�guration on the union of upper and lower hemispheres,the �elds join smoothly on the boundary. This is a physi al de�nitionof the diagonal brane. If a single losed-string operator is inserted on�, we may hoose the seam to pass through the insertion point, andthen we should be able to interpret it as an open-string operator inT T op inserted on the boundary of the disk. Continuing this line ofargument, we see that the algebra of physi al losed-string states shouldbe isomorphi to the endomorphism algebra of the diagonal brane,and that this isomorphism should identify the losed-string topologi almetri with the open-string topologi al metri .3 In this paper we he kthese statements in the ase of topologi al Landau-Ginzburg modelson the aÆne spa e and on ertain orbifolds of the aÆne spa e.3. Landau-Ginzburg models on affine spa eConsider a topologi al Landau-Ginzburg model on the aÆne spa eV ' C n with a polynomial superpotential W : V ! C .4 To this dataone an asso iate a CDG-algebra, i.e. a Z2-graded algebra with anodd derivation Q : A ! A and an even element B 2 A su h thatQ2a = [B; a℄ for any a 2 A. The derivation Q is sometimes alled atwisted di�erential (ordinary di�erentials satisfy Q2 = 0). The notionof a CDG-algebra is a slight generalization of the notion of a (Z2-graded) DG-algebra. In the Landau-Ginzburg ase, A = C [V ℄ (thealgebra of polynomials on V ), Q = 0, and B = W . A CDG-moduleover a CDG-algebra (A;Q;B) is a pair (M;D), where M is a gradedmodule over A and D is an odd derivation of M su h that D2m = Bm3Re all that for any algebra or DG-algebra A its Ho hs hild ohomology has thestru ture of a super ommutative algebra [12℄. This is in agreement with the fa tthat the algebra of physi al losed-string states is always super ommutative.4In what follows we will always assume that the riti al set ofW is ompa t; thisis ne essary for the Landau-Ginzburg �eld theory to have a normalizable va uumstate.

RELATION BETWEEN OPEN AND CLOSED STRINGS 11for any element m 2M . In the Landau-Ginzburg ase, Q = 0, so D issimply an odd endomorphism of M satisfying D2 = W .As explained in Refs. [21, 22℄, the ategory of D-branes for the topo-logi al LG model is equivalent to a DG- ategory whose obje ts arefree CDG-modules over the Landau-Ginzburg CDG-algebra, and mor-phisms are morphisms between CDG modules. That is, the spa e ofmorphisms between (M;D) and (M 0;D0) is a di�erential graded ve torspa e, whi h oin ides with the spa e of morphisms between gradedmodules M and M 0, as far as graded ve tor stru ture is on erned,while the di�erential isDMM 0(�) = D0 Æ �� (�1)j�j� ÆD; � 2 HomA(M;M 0):One an make an ordinary ategory out of this DG- ategory by rede�n-ing morphisms between M and M 0 to be the ohomology of DMM 0.This is the homotopy ategory of CDG-modules. In the aÆne ase,this is the same as the derived ategory of CDG-modules. The spa e ofphysi al open-string states is identi�ed with the ohomology of DMM 0.In parti ular, if we take M = M 0, then the algebra of physi al open-string states is identi�ed with the endomorphism algebra of (M;D) inthe derived ategory of CDG-modules.If M is a free Z2-graded module over A = C [V ℄, then the endomor-phism algebra of M (in the ategory of graded A-modules) is simplythe algebra of matri es with entries in C [V ℄. This algebra is graded inan obvious way: endomorphisms whi h preserve the parity of elementsof M are onsidered even, while endomorphisms whi h ip the parityare onsidered odd. If we order the generators of M so that even gener-ators go �rst, then elements of EndA(M) an be written as 2� 2 blo kmatri es with entries in A, where the diagonal blo ks are even, and theo�-diagonal ones are odd. Su h matri es are usually alled supermatri- es (over A = C [V ℄). The problem of �nding solutions to the equationD2 = W an be thought of as the problem of fa torizing W into aprodu t of two identi al odd supermatri es with polynomial entries.We will sometimes refer to it as the matrix fa torization problem.The goal of this se tion is to ompute the Ho hs hild ohomologyof the ategory of D-branes in topologi al LG models and to ompareit with the algebra of losed string states, whi h is known to be theso- alled Ja obi ring of W , i.e.(6) JW = C [V ℄=IW ; IW = (�1W; : : : ; �nW ):But �rst we need to give a de�nition of the Ho hs hild ohomology of aCDG-algebra. By analogy with the previous se tion, we would like tode�ne it as the endomorphism algebra of the \diagonal" brane on V �V ,

12 ANTON KAPUSTIN AND LEV ROZANSKYor more algebrai ally, as the endomorphism algebra of a A = C [V ℄regarded as an obje t of the derived ategory of CDG-bimodules overitself. That is, we would like to regard A as a CDG-module over thetensor produ t of (A; 0;W ) with its opposite. Sin e A is ommutative,one may ask why we are retaining the word \opposite" here. We laimthat in the world of CDG algebras taking an opposite of an algebrain ludes hanging the sign of the spe ial element B. Indeed, supposewe take some arbitrary CDG-algebra (A;Q;B). We annot regard itas a CDG-module over itself by simply setting D = Q, be ause thede�ning relation for a CDG-module is D2(a) = Ba, while Q satis�esQ2a = [B; a℄. Next, if we onsider the tensor produ t of (A;Q;B)with (Aop; Q;B), we still annot regard (A;Q;B) as a CDG-moduleover this tensor produ t algebra, be ause setting D = Q still givesD2(a) = [B; a℄, while the de�ning relation of the CDG-module gives inthis ase D2(a) = Ba+ aB:However, if we take the tensor produ t of (A;Q;B) with (Aop; Q;�B),then setting D = Q does make (A;Q) into a CDG-module over thetensor produ t algebra.In the Landau-Ginzburg ase, this means that we de�ne Ho hs hild ohomology as the endomorphism algebra of (C [V ℄; 0;W ) regarded asa B-brane on V � V with the superpotential W (x) � W (y), wherex; y are aÆne oordinates on the two opies of V . To ompute thisendomorphism algebra, we have to repla e (C [V ℄; 0;W ) by a free CDG-module whi h is isomorphi to (C [V ℄; 0;W ) in the sense of the derived ategory of CDG-modules.We an simply guess the equivalent free CDG-module (a more rigor-ous justi� ation is given in the end of this se tion). IfW were zero, thenwe would have to repla e the stru ture sheaf of the diagonal with its freeresolution. Sin e we are dealing with CDG-modules, we have to �ndinstead a \twisted" resolution, i.e. a sequen e of free C [V �V ℄-modulesand morphisms between them su h that the omposition of su essivemorphisms is W instead of zero. Therefore, we will do the following.We will take the Koszul resolution of the diagonal � � V � V , foldit modulo 2 (sin e our algebra is Z2-graded, omplexes must also beZ2-graded), and then deform in a natural way the di�erential into atwisted di�erential. This will give us a free CDG-module on V � V ,and we will take its endomorphism algebra as the de�nition of theHo hs hild ohomology for the ategory of D-branes.First let us see how this works for the one-variable ase, where V �C , and A = C [x℄. We are onsidering the diagonal submanifold � in

RELATION BETWEEN OPEN AND CLOSED STRINGS 13C 2 given by the equation x = y. The Koszul resolution is very short:C [x; y℄ x�y���! C [x; y℄The folded resolution an be regarded as a Z2-graded module of theform C [x; y℄ � C [x; y℄ and the di�erentialD = � 0 0x� y 0�This is an honest di�erential satisfying D2 = 0. On the other hand,a twisted di�erential would satisfy D2 = W (x) �W (y). The orre tdeformation is obvious:D = � 0 W (x)�W (y)x�yx� y 0 �Note that sin e W (x) and W (y) are polynomials, D has polynomialentries. This matrix fa torization has been previously onsidered in asimilar ontext in Ref. [24℄.Above D was written as an odd supermatrix. Equivalently, we anthink about su h supermatri es as di�erential operators on fun tions ofodd variables. (This is the approa h we took in the last se tion to writedown the Koszul resolution). In the fermioni language the deformedD looks as follows:D = (x� y) ��� + W (x)�W (y)x� y �:A ording to Ref. [21, 22, 5℄, the endomorphism algebra of this braneis the ohomology of the following linear operator on the spa e of su-permatri es with polynomial entries:D : �! D� � (�1)j�j�D;where � is a supermatrix. For the operator D as above, it is easy tosee that the ohomology is purely even and isomorphi to the quotientve tor spa e C [x; y℄=I; I = �x� y; W (x)�W (y)x� y �Obviously, this ve tor spa e is isomorphi to JW = C [x℄=(�xW (x)), as laimed. In fa t, it is easy to see that the isomorphism holds on thelevel of algebras, rather than on the level of ve tor spa es.Next onsider the ase when V ' C n . Let us introdu e onvenient ondensed notations:(7) x = (x1; : : : ; xn); y[i℄x = (y1; : : : ; yi; xi+1; : : : ; xn)

14 ANTON KAPUSTIN AND LEV ROZANSKYand for a polynomial P 2 C [x℄ de�ne(8) �i(P ) = P (y[i�1℄x)� P (y[i℄x)xi � yi 2 C [x;y℄:To write the orre t free CDG-module (the twisted resolution of thediagonal), we �rst write W (x)�W (y) as a sum of polynomials(9) W (x)�W (y) = nXi=1 (xi � yi)�i(W ):We want to writeW (x)�W (y) as D2, whereD is a odd endomorphismof free Z2-graded modules. The key point is that for ea h term �i(W )we an a omplish this by mimi king the one-variable ase. That is, ifwe let(10) Di = (xi � yi) ���i +�i(W ) �i;then Di is a odd endomorphism of free graded C [x; y℄-modules whi hsatis�es D2i = �i(W ). Therefore if we let(11) D = nXi=1 Di;we will get the desired D2 = W (x)�W (y). Moreover, if we take thelimitW ! 0, the di�erentialD will redu e to k, the Koszul di�erentialfrom the previous se tion. The orresponding matrix fa torization hasbeen previously dis ussed in Ref. [24℄.Sin e the di�erential is given by a sum, and we have omputed the ohomology of Di already, the ohomology of D is simplyC [V � V ℄=I; I = �x1 � y1; : : : ; xn � yn;�1(W ); : : : ;�n(W )�;Obviously, this is isomorphi to Eq. (6).We end this se tion by explaining why the D-brane Eq. (11) deservesto be alled the diagonal brane. The general de�nition of the diagonalis that it represents the identity fun tor in the ategory of D-branes on(C n ;W ). In the present ase, this means the following. Let (R;D) bethe matrix fa torization of W (x) �W (y) orresponding to Eq. (11).Thus R is a free graded AA-module, where A ' C [x℄, and D is anodd endomorphism of R satisfying D2 = W (x) � W (y). Let (N;F )be an arbitrary matrix fa torization of W (y) representing some D-brane on (C n ;W ). Thus N is a free graded A-module, and F is an oddendomorphism of N satisfying F 2 =W (y). Consider the graded tensor

RELATION BETWEEN OPEN AND CLOSED STRINGS 15produ t (R A N;D + F ). The module R A N is a free A-module,and D + F satis�es(D + F )2 = D2 + F 2 = (W (x)�W (y)) +W (y) = W (x):Thus the pair (RAN;D+F ) is a D-brane on (C n ;W ) (of in�nite rank).By de�nition, (R;D) is the diagonal i� (RAN;D+F ) is isomorphi to the original �nite-dimensional brane (N;F ), for any (N;F ) (in thehomotopy ategory of CDG-modules). This statement was proved inRef. [24℄ (Prop. 23 of that paper).4. LG orbifolds4.1. Minimal model I. Consider the LG model with target C =Zn:W0 = xn; x � �x; � = e 2�inA ording to Ref. [20℄, the hiral ring is simply the invariant part ofthe Ja obi ring, whi h is C [x℄=xn�1 . In other words, there are no hiral primaries in the twisted se tors. The only invariant is a tuallythe identity element. Let us he k this using the diagonal B-braneapproa h.We onsider a matrix fa torization of the potentialW = xn � yn:W is a fun tion on C 2 invariant under a Zn�Zn a tionx! �kx; y ! �k+ly; k; l 2Z=n:In other words, x has weight (1; 0); while y has weight (1; 1). The orresponding twisted di�erential D isD0(x; y) = � 0 Qn�1i=1 (x� � iy)x� y 0 �D0 is regarded as an odd endomorphism of a free Z2-graded moduleover C [x; y℄ of rank two, whi h we write as M0 = M+0 �M�0 , whereM+0 ' C [x; y℄ is the even omponent, and M�0 ' C [x; y℄ is the odd omponent.We need to de�ne a G =Zn�Zn a tion on M0 so that M0 be omesan equivariant module over C [x; y℄, and the twisted di�erential is anequivariant endomorphism. A Zn a tion on a free module, like M+0or M�0 , is ompletely spe i�ed on e we spe ify the a tion on the unitelement, i.e. its weight. For the �rst Zn, we hoose the weights to be 0forM+0 and 1 forM�0 . This is the only way to makeD0 equivariant, up

16 ANTON KAPUSTIN AND LEV ROZANSKYto an overall shift of weights. For the se ond Zn, no hoi e of weightsworks, so we have to equivariantize D0, i.e. to add \image branes":Dl(x; y) = D0(x; � ly) = � 0 Qn�1i=0;i6=l(x� � iy)x� � ly 0 � ; l = 1; : : : ; n�1:Dl a ts on a moduleMl whi h is isomorphi to M0 as aZ2�Zn-gradedmodule. We take the diagonal brane M to be the dire t sum of allthese branes, with the twisted di�erential D being the sum of all Dl,l = 0; : : : ; n � 1. D is equivariant with respe t to the se ond Zn if welet Zn a t on the summands Ml by y li permutations.Now let us ompute the endomorphism algebra of the diagonal brane(M;D). First we ompute the endomorphisms disregarding theZn�Zna tion, and then proje t onto the invariant part. Sin e M is a dire tsum, we an omputemorphisms between the summands, and then sumthem up. It is also onvenient to onsider separately the morphismsfrom a brane (Ml;Dl) to itself, and to its mirror images (Mk;Dk), k 6= l.An easy omputation shows that the endomorphisms of (Ml;Dl) (i.e.the ohomology of Dl in the adjoint representation) is purely even andis the Ja obi ring of W0. The spa e of morphisms between (Ml;Dl)and (Mk;Dk) is purely odd and one-dimensional. A basis element ofthis odd ve tor spa e will be alled �l;k. Sin e the odd omponentsof Ml have weight 1 with respe t to the �rst Zn, one an easily seethat �k;l has weight 1. Proje ting with respe t to the �rst Zn, we seethat no morphisms between a brane and its image survive. As forendomorphisms of (Ml;Dl), only the invariant part of the Ja obi ring(whi h onsists only of the subspa e spanned by the identity) survives.Thus we get the dire t sum of n opies of the trivial algebra. Finally,we proje t with respe t to the se ond Zn. This Zn y li ally permutesall Ml, and therefore identi�es all the n opies of the trivial algebra.4.2. Minimal model II. This is a generalization of the previous sub-se tion. We onsider a LG model with target C =Zn and a superpoten-tial W0 = xnm; n;m 2 N:The orbifold group a ts byx 7! �kx; k 2Z=n:The analysis is very similar to the previous subse tion. The untwistedse tor is the invariant part of the Ja obi ring. There are still n � 1image branes, and morphisms between di�erent images are odd andhave weight 1 under the �rst Zn a tion. Hen e the twisted se tor

RELATION BETWEEN OPEN AND CLOSED STRINGS 17is killed by the �rst Zn proje tion. This agrees with the results ofRef. [20℄.4.3. Gepner model for a Calabi-Yau 0-fold. This is a LG orbifoldwith a target C 2=Z2 and superpotentialW0 = x21 + x22:TheZ2 a tion is diagonal ( ips the sign of both variables). E. Witten'sargument [44℄ shows that this is a Gepner model for the CY 0-foldgiven as a hypersurfa e x21 + x22 = 0 in P1. The latter onsists of twopoints, so the orresponding hiral ring is two-dimensional (the dire tsum of two opies of the trivial algebra). Let us he k this using thediagonal brane approa h.We onsider the superpotentialW = x21 � y21 + x22 � y22 =W1(x1; y1) +W2(x2; y2):To onstru t the diagonal brane, we fa torize separatelyW1 andW2 andthen take theZ2-graded tensor produ t of the orresponding branes. Ifwe represent D1 and D2 by fermioni di�erential operators, this simplymeans that we let D = D1 +D2, where D21 = W1 and D22 = W2. Sin eD1 and D2 anti ommute, this gives D2 = W .The fa torizations ofW1 and W2 are taken as in the previous subse -tion. Then we have to equivariantize with respe t toZ2�Z2. As above,equivarian e with respe t to the �rst Z2 for es the odd omponent ofM to have weight 1, if the even omponent has weight 0. Hen e oddendomorphisms ofM have weight one, while even endomorphisms haveweight 0. Equivarian e with respe t to the se ond Z2 requires addinga single \image brane" (M 0;D0), where M 0 'M , and D0 isD0(x1; x2; y1; y2) = D(x1; x2;�y1;�y2):The diagonal brane is the dire t sum of (M;D) and its image (M 0;D0).Let us ompute its endomorphism algebra and then proje t onto theZ2 �Z2-invariant part. First onsider the endomorphisms of M andM 0. Sin e both M andM 0 have fa torized form (they are graded tensorprodu ts of M1 and M2), we an ompute the endomorphisms of M1and M2 and then tensor them. From the previous subse tion we knownthat the endomorphism algebras of M1 and M2 are trivial (spanned bythe identity). Hen e the endomorphism algebras of M and M 0 are alsotrivial. They also survive the �rst Z2 proje tion, while the se ond Z2proje tion identi�es them.Now onsider morphisms fromM toM 0. Using the results of the pre-vious subse tion, we see that the spa e of morphisms is one-dimensionaland spanned by �(1)0;1�(2)0;1, where �(i)0;1 spans the spa e of morphisms from

18 ANTON KAPUSTIN AND LEV ROZANSKYMi to its image under the se ond Z2. Thus the spa e of morphismsfromM to M 0 is purely even and one-dimensional. Sin e both �'s haveweight 1 under the �rst Z2, their produ t has weight 0 and survivesthe �rst Z2 proje tion. Finally, the se ond Z2 proje tion identi�esmorphisms fromM to M 0 and from M 0 to M . Thus the twisted se tor ontributes a single even element to the hiral ring. It is easy to seethat this element squares to identity. Thus the endomorphism algebraof the diagonal brane isomorphi to the group ring of Z2. This is thedesired result.4.4. Gepner model for a Calabi-Yau n� 2-fold. We onsider thesuperpotential W = xn1 + : : :+ xnn on C n with a diagonal Zn a tion:xi 7! �xi:This is a Gepner model for the Fermat hypersurfa e xn1 + : : :+ xnn = 0in Pn�1. Let us ompute the endomorphisms of the diagonal brane.The omputation pro eeds along the same lines as in the previous sub-se tion. We now have n � 1 image branes (with respe t to the se ondZn). The untwisted se tor gives us the invariant part of the Ja obiring. Morphisms between a brane and its image have the formn�1i=0 �(i)k;lEa h � is odd and has weight 1 under the �rst Zn, so the produ thas weight 0 and survives the �rst Zn proje tion. It is even or odddepending on whether n is even or odd. The se ond Zn proje tionidenti�es all the twisted se tor states with the same value of k � l.Hen e the twisted se tor ontributes n � 1 states to the hiral ring(labeled by k � l mod n).Let us ompare this with the B-model of the Fermat hypersurfa es.The ase n = 2 has been dis ussed in the previous subse tion. For n = 3the Fermat hypersurfa e is a ubi urve in P2, i.e. an ellipti urve. ItsB-model is isomorphi to the exterior algebra with two odd generators,i.e. it has two-dimensional even subspa e and two-dimensional oddsubspa e. On the LG side, the untwisted se tor is spanned by 1; x1x2x3(the invariant part of the Ja obi ring), while the twisted se tor is oddand two-dimensional. Thus the two agree, as Z2-graded ve tor spa es.One an he k that the ring stru ture also agrees.For n = 4 the Fermat hypersurfa e is a quarti in P3, i.e. a K3surfa e. The B-model is a purely even algebra of dimension 24. Onthe other hand, the Z4-invariant part of the Ja obi ring is easily seento have dimension 21. The twisted se tor gives the other 3 states.

RELATION BETWEEN OPEN AND CLOSED STRINGS 19For n = 5 we are dealing with a quinti in P4. The B-model hasboth an even and odd omponents. The even omponent has dimension204. The odd omponent has dimension 4. It is easy to show that theinvariant part of the Ja obi ring is indeed 204-dimensional. The odd omponent omes entirely from the twisted se tor.5. Open- losed orrelators for some Landau-Ginzburgmodels5.1. Closed topologi al metri from the open one. In this sub-se tion we dis uss how to dedu e the 3-point topologi al losed-string orrelators (the Yukawa ouplings) in the diagonal brane approa h andshow that the result agrees with the onventional approa h [39℄.The losed strings form a super ommutative Frobenius algebra A l:there is a tra e map A l trF�! C , whi h de�nes the s alar produ t on A land determines the 3-point orrelator of losed string operators(12) (a; b) = trF(ab); hab i = trF(ab ):The algebra of open-string states Aop has the A l-linear `boundarytra e' map Aop trb�! A l, whi h expresses the boundary state orre-sponding to a boundary with an open string operator insertion. The omposition of the boundary and Frobenius tra es Tr = trFtrb deter-mines the s alar produ t and the 3-point orrelator for open strings:(13) (A;B) = trb(AB); hABCi = trb(ABC):The onje tured orresponden e between the diagonal D-brane statesand the losed string states implies that the Frobenius tra e on A lshould be equal to the ombined tra e Tr on the diagonal D-branestates. This would also imply the equality between their s alar produ tsand 3-point orrelators.We will verify the equivalen e of Frobenius and ombined tra es fora topologi al LG model with the superpotential W (x) 2 C [x℄. In this ase the algebra of losed strings is the Ja obi algebra JW of Eq. (6)and the Frobenius tra e trx was determined by C. Vafa in Ref. [39℄:(14) trx(p) = 1(2�i)n I p dx1 ^ : : : ^ dxn�1W : : : �nW ; p 2 C [x℄:The integrand in this formula is a meromorphi n-form, and the inte-gration goes over a Lagrangian n- y le de�ned by the equationsj�iW j = �i > 0; i = 1; : : : ; n:

20 ANTON KAPUSTIN AND LEV ROZANSKYThe boundary tra e formula for a B-brane in a LG model, presentedby a matrix fa torization, was derived in Ref. [22℄ (see also Ref. [17℄):(15) trb(O) = (1=n!) STr�(�D)^nO�;where O is an endomorphism regarded as a supermatrix with polyno-mial entries, STr is the supertra e, and(16) (�D)^n = X�2Sn(�1)sign(�) ��(1)D � � � ��(n)D;where Sn is the symmetri group of n elements.Sin e the spa e of endomorphisms of the diagonal B-brane is isomor-phi (up to the BRST equivalen e) to the Ja obi ring JW (x), to provethe equivalen e between losed- and open-string tra es it is suÆ ientto show that(17) trFtrb(f(x)Id) = trx(f):Here f(x) is any polynomial, and Id is the identity endomorphism ofthe diagonal B-brane.The relation (17) has an interesting algebrai orollary. Let A� bethe dual spa e of an algebra A, then the multipli ationmapA m�! AAhas a dual map A� A� ��! A�, whi h is alled omultipli ation. Thes alar produ t (12) establishes a anoni al isomorphism between theFrobenius algebra A l = C [x℄=IW and its dual, so there is a omultipli- ation map(18) A l ��! A l A lwith the property(19) �(a) = (a 1)�(1) = (1 a)�(1):The relation (17) implies that(20) trb(Id) = �(1) 2 (C [x℄=IW ) (C [y℄=IW ):As the �rst step of proving the relation (17), we are going to derivea onvenient formula for trb(Id):(21) trb(Id) = 1(2n)! STr(�D)^2n = det jj�i(�jW )jjni;j=1where �jW (x) = �W (x)=�xj and �i is de�ned by Eq. (8). Indeed, ifwe expand the 2n-th power of the sum (10), then the supertra e of an

RELATION BETWEEN OPEN AND CLOSED STRINGS 21individual monomial is zero, unless it ontains all �'s and all �=��'s.Therefore, if we lump all variables together z = (x;y), then(22) STr(�D)^2n = (2n)!2n X�2S2n(�1)sign(�) nYi=1 Ci[z�(2i); z�(2i+1)℄;where(23) Ci[zj; zk℄ = STri��Di�zj �Di�zk � ;while Di is de�ned by Eq. (10) and STri denotes the tra e over thespa e (1; �i). A dire t omputation shows that if fzj; zkg\fxi; yig = ;,then Ci[zj; zk℄ = 0 and the only non-zero fa tors areCi[xj; yi℄ = �Ci[yi; xj℄ = �i(�jW ); j � i;(24) Ci[xi; yj℄ = �Ci[yj; xi℄ = �i(�jW ); j � i:(25)Thus, a non-zero supertra e STri must ontain at least one derivativeover an i-th variable (x or y). The distributions of derivatives over thesupertra es, whi h do not produ e zero fa tors, an be enumerated bythe elements � 2 Sn: if �(i) = i, then the supertra e (23) in ludesthe derivatives over xi and yi; if �(i) 6= i, then one derivative of (23)should be over the �(i)-th element of the list y[i℄x and the other shouldbe either over xi or over yi depending on whether ��1(i) > i or ��1(i) <i. This ombinatori s together with the expressions (24) leads to theformula (21).It remains to apply the Frobenius tra e to the r.h.s. of Eq. (21) mul-tiplied by f(x). To do this, it is onvenient to assume that all riti alpoints of W are non-degenerate. Sin e the set of su h superpotentialsis an open and dense subset of all admissible superpotentials, and thel.h.s. of Eq. (17) is a ontinuous fun tion of the oeÆ ients of W , itis suÆ ient to prove Eq. (17) in this spe ial ase. For W with onlynon-degenerate riti al points, the tra e fun tion Eq. (14) be omes(26) trF(p) = X rit W p(x)HessW (x) ;where HessW denotes the Hessian ofW , and the sum is over all riti alpoints of W . Thus we simply have to evaluate the r.h.s. of Eq. (21) atthe riti al points of W (x) �W (y), whi h are pairs of riti al pointsof W (x).

22 ANTON KAPUSTIN AND LEV ROZANSKYLet x and y be two riti al points of W . If x 6= y, then trb(Id) = 0in view of Eq. (21)5. Indeed, letri = (�i(�jW ))nj=1be the rows of the matrix of Eq. (21). Obvious an ellations lead tothe following identity:(27) nXi=1 (xi � yi) ri = (�jW (x)� �jW (y))nj=1:Sin e x and y are riti al points ofW , the r.h.s. of this equation is zero,and if x 6= y, then this means that the rows ri are linearly dependent,and the determinant in Eq. (21) is equal to zero.Thus, only the riti al points on the diagonal, i.e. the ones for whi hx = y, ontribute to the sum (26). ExpandingW into a Taylor series toquadrati order, one an easily see that the r.h.s. of Eq. (21) evaluatedon su h a riti al point is equal to HessW . Hen e we get:(28) trFtrb(f(x)Id) = X rit W f(x)HessW (x) = trx(f):This ompletes the proof of Eq. (17).As another example, let us dedu e the topologi al losed-string met-ri for the ellipti urve at the Fermat point, using the LG representa-tion and the diagonal brane approa h. First, let us state the expe tedanswer. The losed-string algebra of an ellipti urve is isomorphi tothe exterior algebra with two generators �1; �2. Thus the spa e of phys-i al losed-string states is four-dimensional, with two-dimensional evensubspa e (the sum of H0(OX) and H1(TX)) and two-dimensional oddsubspa e (the sum of H0(TX) and H1(OX)). The tra e on this algebrais nonvanishing only on the one-dimensional subspa e spanned by �1�2,whi h geometri ally orresponds to H1(TX).In the LG approa h, the states orresponding to �1 and �2 ome fromthe twisted se tor, while 1 and �1�2 span the Z3-invariant part of theJa obi ring of the superpotentialW = x31 + x32 + x33:If we think about the losed-string spa e of states as the endomorphismalgebra of a diagonal brane on (C 3=Z3)� (C 3=Z3), then the endomor-phisms orresponding to �1 and �2 are o�-diagonal (they orrespond tostrings from a brane to one of its images under a Z3), and it followsfrom Eq. (15) that their tra e vanishes. As for the endomorphisms5We are thankful to Christiaan Hofman for pointing out an error in our originalproof and for suggesting the orre t one.

RELATION BETWEEN OPEN AND CLOSED STRINGS 23 orresponding to 1 and �1�2, they are insensitive to the orbifolding andtheir properties are exa tly the same as in the LG model on the aÆnespa e. We already know that the open-string tra e for these endo-morphisms oin ides with the losed-string tra e Eq. (14), provided we orre tly identify endomorphisms with the elements of the Ja obi ring.The endomorphism 1 orresponds to the identity in the Ja obi ring andits tra e is zero, a ording to Eq. (14). The endomorphism �1�2 (whi hgeometri ally represents a basis element for H1(T )) orresponds to theelement x1x2x3 in the Ja obi ring, and its tra e is a nonzero onstant,a ording to Eq. (14). Thus we have re overed the orre t Yukawa ouplings for the ellipti urve by regarding the losed-string algebraas the Ho hs hild ohomology of the ategory of D-branes for the or-responding LG orbifold.5.2. Bulk-boundary maps. Let us onsider the simplest open- losed orrelator: 2-point fun tion on a disk, with one boundary and one bulkinsertion. It an be regarded as a linear fun tionAop A l ! C :Using the topologi al metri on Vo, it an also be regarded as a map� : A l ! Aop:This is known as the bulk-boundary map; it is the dual of the boundarytra e trb. Of ourse, for every hoi e of a D-brane, we get a Aop, andtherefore there are as many maps � as there are D-branes. Axioms ofTopologi al Field Theory with boundaries require � to be an algebrahomomorphism. In this subse tion we will demonstrate that the bulk-boundary map omes for free if we identify A l with the Ho hs hild ohomology of the ategory of topologi al D-branes. We will also he kin some examples that the resulting bulk-boundary map is the orre tone, i.e. it oin ides with the one derived by physi al methods.Let A = (A;Q;B) be a CDG algebra, and suppose the ategory ofD-branes is the derived ategory of CDG modules over A. (The aseof DG algebras and modules is a spe ial ase of this, orresponding toB = 0.) Consider6 the Ho hs hild ohomology of A, de�ned asHH�(A) = Ext�AAop(A;A):That is, it is the endomorphism algebra of A regarded as an obje t ofthe derived ategory of modules overAAop. LetM be some obje t ofthe derived ategory of modules over A, and let � be an endomorphism6The following onstru tion was explained to us by D. Orlov.

24 ANTON KAPUSTIN AND LEV ROZANSKYofM. Then for any element� 2 HH�(A) we may onsider ��, whi his an endomorphism of AAM 2 Db(A):But the latter obje t is isomorphi to M. Hen e we may regard � �as as endomorphism of M. Thus we get a linear map�M : HH�(A) End(M)! End(M):Obviously, this makes End(M) a (left) graded module over HH�(A).Similarly, one an de�ne the right HH�(A)-module stru ture on End(M).More generally, if we onsider any two obje tsM and N , the spa e ofmorphisms between them has a natural stru ture of a graded bimoduleover HH�(A). We de�ne the bulk-boundary map �M to be(29) �M(�) = �M(� 1):It is trivial to see that this is an algebra homomorphism, as required.Let us perform a simple he k of this pres ription for de�ning �. Inthe ase of a LG model on an aÆne spa e V , the disk orrelator hasbeen omputed in Ref. [22℄, and the orresponding bulk-boundary mapis given by(30) �M : f 7! f idMwhere f is an element of the Ja obi ring JW = C [V ℄=�W , and idM isthe identity endomorphism of a free CDG-module M = (M;D) overthe CDG algebra (C [V ℄; 0;W ). It is impli it in this formula that f islifted to a polynomial on V . Although lifting an element of the Ja obiring to an element of C [V ℄ involves a hoi e, the above formula for �Mis well-de�ned. Indeed, if we repla e f byf + nXi=1 gi�iWwhere gi are some polynomials on V , then �M(f) will hange by"D; Xi gi�iD# ;whi h is homotopi to zero (or in other words, is BRST-trivial).On the other hand, we established in Se tion 3 that the Ja obi ringis isomorphi to the endomorphism algebra of the \diagonal brane" onV � V , the isomorphism being given byf 7! f idR;whereR is the twisted Koszul resolution of the diagonal in V �V . Fromthis it follows that the HH�(A)-module stru ture on End(M) is very

RELATION BETWEEN OPEN AND CLOSED STRINGS 25simple: it is indu ed by the A-module stru ture on EndA(M). Hen ethe abstra t map �M de�ned in Eq. (29) oin ides with Eq. (30).6. Dis ussionIn this paper we have shown that for topologi al Landau-Ginzburgmodels on the aÆne spa e (as well as on some quotients of the aÆnespa e) the algebra of physi al losed-string states7 is isomorphi tothe Ho hs hild ohomology of the ategory of D-branes. The algebrastru ture on the Ho hs hild ohomology is given by the standard upprodu t. Moreover, using the open-string topologi al metri (in math-emati al terms, the Serre fun tor on the ategory of D-branes), one ande�ne a s alar produ t on the Ho hs hild ohomology, making it intoa super ommutative Frobenius algebra. We showed that this s alarprodu t agrees with the losed-string topologi al metri .An important question is whether more ompli ated orrelators anbe dedu ed by studying the Ho hs hild ohomology of the ategory ofD-branes. This is not ompletely trivial even at tree level (i.e. in genuszero). While two- and three-point orrelators of gravitational primariesin genus zero are en oded in the Frobenius algebra stru ture, higher-point orrelators an be des ribed by a germ of a Frobenius manifold(whose tangent spa e at the base point is the Frobenius algebra justmentioned.) In the ase of the B-model of a Calabi-Yau manifold, the orresponding formal Frobenius manifold was onstru ted in Ref. [2℄.Frobenius manifolds related to Landau-Ginzburg models are dis ussedin many papers, see e.g. Ref. [32℄. It would be interesting to understandhow to onstru t all these Frobenius manifolds in a uniform manner,starting from natural algebrai stru tures on the Ho hs hild omplex ofa ategory of topologi al D-branes. The key property needed for su ha onstru tion seems to be the existen e of the open-string metri , i.e.a trivial Serre fun tor, on the ategory of D-branes.A further question is how to re onstru t open- losed orrelators. Weonly onsidered the simplest example (disk orrelator with one bound-ary and one bulk insertion) and showed how to re over it from theHH�(A)-module stru ture on the spa e of endomorphisms of any D-brane.Although in this paper we have fo used on topologi al string the-ories, it is tempting to onje ture that something similar holds forbosoni strings. A matter CFT with entral harge = 26 oupledto di�eomorphism ghosts an be thought of as a topologi al �eld the-ory [9, 29, 10, 3, 33, 43℄ and an be oupled to topologi al gravity in7We are talking about gravitational primaries only.

26 ANTON KAPUSTIN AND LEV ROZANSKYthe usual manner. This gives an alternative des ription of the bosoni string, whi h is similar to that of topologi al string theory. However,the physi al spe trum of the bosoni string theory is related not tothe BRST ohomology of the TFT, but to its semi-relative ohomol-ogy [46, 43, 15℄. In parti ular, one annot lassify physi al states asbeing gravitational primaries or des endants, in general, and it is not lear what the relation between physi al losed string states and theHo hs hild ohomology ould be. A good example to look at is non- riti al bosoni string theory with < 1. Some of the bosoni < 1ba kgrounds are equivalent in a nontrivial way to topologi al stringtheories of the kind dis ussed in this paper [42, 9, 8, 29℄, and it wouldbe interesting to rephrase our results in terms of the onventional for-mulation of the bosoni string.A knowledgmentsA.K. would like to thank Volodya Baranovsky, Ezra Getzler, KentaroHori, Dima Orlov, and Sasha Voronov for help at various stages. A.K.is also grateful to the Department of Mathemati s of Northwestern Uni-versity and the Erwin S hr�odinger Institute for hospitality while thiswork was being ompleted. L. R. is very grateful to Mikhail Khovanovfor numerous dis ussions of the ategory of matrix fa torizations. Thiswork was supported in part by the DOE grant DE-FG03-92-ER40701and by the NSF grant DMS-0196131.Referen es[1℄ S. K. Ashok, E. Dell'Aquila and D. E. Dia ones u, Fra tional branes inLandau-Ginzburg orbifolds, arXiv:hep-th/0401135.[2℄ S. Barannikov and M. Kontsevi h, Frobenius manifolds and formality of Liealgebra of polyve tor �elds, Internat. Math. Res. Noti es no. 4 (1998) 201[arXiv:alg-geom/9710032℄.[3℄ M. Bershadsky, W. Ler he, D. Nemes hansky and N. P. Warner, ExtendedN=2 super onformal stru ture of gravity and W gravity oupled to matter,Nu l. Phys. B 401, 304 (1993) [arXiv:hep-th/9211040℄.[4℄ I. Brunner, M. R. Douglas, A. E. Lawren e and C. Romelsberger, D-branes onthe quinti , JHEP 0008, 015 (2000) [arXiv:hep-th/9906200℄.[5℄ I. Brunner, M. Herbst, W. Ler he and B. S heuner, Landau-Ginzburg realiza-tion of open string TFT, arXiv:hep-th/0305133.[6℄ I. Brunner and V. S homerus, On superpotentials for D-branes in Gepner mod-els, JHEP 0010, 016 (2000) [arXiv:hep-th/0008194℄.[7℄ D. E. Dia ones u and M. R. Douglas, D-branes on stringy Calabi-Yau mani-folds, arXiv:hep-th/0006224.[8℄ R. Dijkgraaf and E. Witten, Mean Field Theory, Topologi al Field Theory,And Multimatrix Models, Nu l. Phys. B 342, 486 (1990).

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