We use a recently proposed formulation of stable holomorphic vector bundles
$V$ on elliptically fibered Calabi--Yau n-fold $Z_n$ in terms of toric geometry
to describe stability conditions on $V$. Using the toric map $f: W_{n+1} \to
(V,Z_n)$ that identifies dual pairs of F-theory/heterotic duality we show how
stability can be related to the existence of holomorphic sections of a certain
line bundle that is part of the toric construction.Comment: 8 pages, harvmac (b

Berglund, P.

Mayr, P.

English

arXiv

We use a recently proposed formulation of stable holomorphic vector bundles $V$ on elliptically fibered Calabi--Yau n-fold $Z_n$ in terms of toric geometry to describe stability conditions on $V$. Using the toric map $f: W_{n+1} \to (V,Z_n)$ that identifies dual pairs of F-theory/heterotic duality we show how stability can be related to the existence of holomorphic sections of a certain line bundle that is part of the toric construction.We use a recently proposed formulation of stable holomorphic vector bundles $V$ on elliptically fibered Calabi--Yau n-fold $Z_n$ in terms of toric geometry to describe stability conditions on $V$. Using the toric map $f: W_{n+1} \to (V,Z_n)$ that identifies dual pairs of F-theory/heterotic duality we show how stability can be related to the existence of holomorphic sections of a certain line bundle that is part of the toric construction

Berglund, Per

CERN Document Server

JHEP12(1999)009Received: October 18, 1999, Accepted: December 10, 1999HYPER VERSIONStability of vector bundles from F-theoryPer BerglundInstitute for Theoretical Physics, University of CaliforniaSanta Barbara, CA 93106, USAE-mail: berglund@itp.ucsb.eduPeter MayrTheory Division, CERN1211 Geneva 23, SwitzerlandE-mail: Peter.Mayr@cern.chAbstract: We use a recently proposed construction of stable holomorphic vectorbundles V on elliptically bered Calabi-Yau n-fold Zn in terms of F-theory compact-ications on local singularities to describe stability conditions on V . Specically, therequirement that the F-theory compactication manifold is Calabi-Yau implies a astability criterion on V which is formulated in terms of the existence of holomorphicsections of certain line bundles.Keywords: Superstrings and Heterotic Strings, F-Theory, String Duality,Superstring Vacua.JHEP12(1999)009The heterotic string compactied on Calabi-Yau n fold Zn gives rise to interest-ing minimal supersymmetric string vacua with a quite realistic spectrum of gaugesymmetries and matter elds [1]. The denition of these vacua, involves the spec-ication of a gauge background V with structure group H 2 G0 embedded in theperturbative heterotic gauge group G0 = E8  E8 or SO(32) such thatc1(V ) = 0 mod 2 , λ(V ) = c2(Z) + [W ] , (1)where λ is the four-dimensional characteristic class of V . The second term [W ]applies to a generalization of heterotic vacua which include non-perturbative ve-branes [2]. Here [W ] is a sum of formal four-forms that integrate to one in thetransverse direction to the ve-branes. Note that the ve-branes necessarily have towrap two-cycles in a 3-fold compactication; such cycles have to be holomorphic inorder to preserve supersymmetry [3].In addition, the connection on V has to satisfyFab = Fa¯b¯ = 0 , gab¯Fab¯ = 0 . (2)The rst equation says that V is holomorphic, while the second equation is inter-preted as some stability condition on V . Although an existence theorem exists undercertain conditions [4], explicit solutions to (2) have been known only in very specialconstructions. The diculty to nd appropriate backgrounds V has hampered thestudy of interesting four-dimensional vacua for a long time.An entirely independent construction of holomorphic, stable vector bundles1 Von Calabi-Yau n-folds Zn has been developed in [6] using type-IIA strings compacti-ed on local Calabi-Yau n+1 fold singularitiesWn+1 and the F-theory limit thereof.2This geometric approach gives a systematic way to obtain families of vector bundlesthat solve (2), with any structure group H 2 G0 and on any toric Calabi-Yau man-ifold Z. Given the fact that it is so hard to nd solutions to (2) in general, theremust be a very good reason why the geometric construction yields a large class ofsolutions without diculties. Now supersymmetry of the F-theory compacticationrequires that the singularity Wn+1 is Calabi-Yau. It is the purpose of this note toshow that it is precisely this Calabi-Yau condition that translates to a stability con-dition on the bundle V on Z, formulated in terms of the existence of holomorphicsections of certain line bundles. The answer to why it was so easy to get solutionsto (2) from the geometric construction is thus that we have replaced the complicatedcondition (2) of the heterotic string, about which we have poor control, with thesimple Calabi-Yau condition on the type-IIA/F-theory side, which is much easier todeal with.1We will loosely call V a bundle in the following, though our construction naturally yieldssheaf generalizations of bundles, which provide the relevant set up for heterotic string compactifi-cations [5].2See [7] for a review.1JHEP12(1999)009The relation of type-IIA strings and holomorphic stable vector bundles can betraced back to a simple extension of mirror symmetry in a K3  T 2 compacti-cation [8], more specically a discrete symmetry in the moduli space of this com-pactication that relates geometric deformations of K3 to Wilson lines on the T 2factor. One can then combine this picture with taking the F-theory limit of thetype-IIA compactication. Specically one can show that the moduli of Wilson lineson Z1 = T2 is the same as the geometric moduli of a local 2-fold singularityW2. Therelevant compactication geometry W2 is the local mirror of a Ka¨hler resolution ofan H singularity of an elliptically bered ALE space. Here H is the structure groupof the bundle V , as before.Fibering this picture, using the adiabatic argument of [9], one obtains an equiva-lence between complex deformations of an elliptic and K3 bered Calabi-Yau singu-larityWn+1 and holomorphic stable vector bundles on an elliptically bered manifoldZn [6]. Holomorphic stable bundles V on smooth elliptic brations Zn have been rstconsidered in [10] and subsequently in [11].The construction in [6] has been formulated in terms of toric geometry whichfor many purposes is the most useful and most general denition for the Calabi-YaumanifoldsWn+1 and Zn. The toric construction of the pair (V, Zn) in terms of an n+1dimensional non-compact Calabi-Yau manifold Wn+1 works for any structure groupH of the bundle. IfH ts into the perturbative heterotic gauge group G0 = E8E8 orSO(32), we can interpret (V, Zn) as a valid perturbative heterotic vacuum. Preciselyin this case one can nd an embedding of the geometryWn+1 into a compact Calabi-Yau manifold Wn+1. The statement that F-theory compactied on the local patchWn+1 of Wn+1 is equivalent to the heterotic string compactication on (V, Zn) thenestablishes F-theory/heterotic duality [12, 13] in the point particle limit.Toric geometry describes a Calabi-Yau n-fold Zn in terms of a convex, toricpolyhedron Wn+1 , which is the convex hull of a set of integral vertices ν?i in astandard integral lattice N .3 It is satisfying to observe [6] that the heterotic physicssuch as the compactication manifold Zn and some dening data of the bundle Vhave a very simple representation in terms of the polyhedron Wn+1 . Moreover alsonon-perturbative dynamics of the heterotic vacuum for singular congurations, inparticular non-perturbative gauge symmetries and non-perturbative ve-branes thatappear in (1), can be read of from the toric data of Wn+1 in a simple way. It isthe purpose of this note to extend the dictionary between physical and toric datadeveloped in [6] to describe stability of the bundle V in terms of convexity of thepolyhedron Wn+1.In particular it was noted in [6], that for a given structure group H of the bundle,convexity of the toric polyhedron ?Wn+1 translates to a bound on the rst Chern classη = c1(N ) of a line bundle N that is an important characteristic of the bundle V .3See [6] for a list of reviews on toric geometry in the physics literature.2JHEP12(1999)009Observing that in six dimensions, the bound on η from convexity of ? translatesto the stability of instantons on K3 with structure group H , these bounds wereinterpreted as a similar stability condition in lower dimensions. A general formulafor the stability of V in terms of bounds on η was subsequently conjectured in ref. [14]which was however not in agreement4 with the original bounds in [6]. In the followingwe derive a general stability bound for η from the toric representation of data (V, Zn)which is consistent and, we believe, gives a correct treatment.We start with a sketch of the description of the pair (V, Zn) in terms of thenon-compact Calabi-Yau n+1-foldWn+1. The latter is elliptically bered with base~Bn and ALE bered with base Bn−1. The manifold Wn+1 is dened as the vanishinglocus of a polynomial p in a toric ambient space with coordinates (y, x, ~z, v, xi), wherexi denote the coordinates of the base Bn−1 and (y, x, ~z, v) are certain coordinates onthe ALE ber. The general form of p isp = p0 + p+ = p0(y, x, ~z; xi) +J∑j=1vjpj+(y, x, ~z; xi) , (3)where p0 and pj+ are quasi-homogeneous polynomials. In particular the v independentpiece p0 describes an elliptically bered Calabi-Yau manifold Zn : p0 = 0, whilethe polynomials pj+ contain the information about the bundle V . More specically,each node in the ane Dynkin diagram Γ(H) with Dynkin index si contributes onemonomial to the polynomial psi+ ; in particular J = max(si). E.g. in the case H =SU(N), we have N nodes of index 1, each of which contributes a monomial to p1+. Theprecise form of the v dependent part of p is p+ = v(aN (xi)~zN+aN−2(xi)~zN−2x+   +a0(xi)xN/2) for N even with the last term being  yx(N−3)/2 for N odd. The complexparameters multiplying the monomials in p+ give a projective parametrization of themoduli space of the SU(N) bundle.Thus the structure group H determines the y, x, ~z, v content of the monomialsappearing in the dening equation p ofWn+1. It does not determine the dependenceon the base variables xi, however. The latter species topological properties of Vthat enter the higher Chern classes ck(V ), k > 1. However note that the dependenceon xi is quite restricted because of the quasi-homogeneousness of p. It was shownin [6] that the dependence of the monomials in p on xi is completely determined byspecifying the action of two C actions acting on (y, x, ~z, v, xi). In other words, wehave to specify two line bundles L and M on the base Bn−1 and the coordinates(y, x, ~z, v, xi) transform as certain sections ofM and L such thaty M3L3 , x M2L2 , ~z M , v M5−N , fc,d MdLc , (4)4Essentially, the discriminant criterion used in [14] is not restrictive enough to fix the structuregroup H .3JHEP12(1999)009where fc,d = fc,d(xi) is the base dependence part in the monomialy(6−2b−c)/3xb~zcvdfc,d(xi)of p. Moreover N is the highest power of ~z that appears in p1+.From the fact that p0 = 0 describes a Calabi-Yau manifold it follows that thebundle L is actually the anti-canonical bundle of Bn−1. On the other handM is animportant intrinsic property of V . In the following we will derive a bound on the rstChern class c1(N ) of the line bundle N =ML6 from convexity of the polyhedronWn+1 , which is the toric way to ensure that the singularity Wn+1 represents a patchof a Calabi-Yau space.The change from M to N is intended to make contact with the notation ofref. [10], where V has been described in terms of sections of a certain weightedprojective bundle W on Bn−1 with coordinates on W being sections of N siL−dj .Here si are the Dynkin indices as above and di are the degrees of the independentCasimir invariants of H . The dependence of higher Chern classes of V on c1(N ) hasbeen determined in [10, 15].To keep the discussion as basic as possible we will work directly with the deningpolynomial p rather than with the toric polyhedron ?Wn+1 that determines p [16]. Inparticular, if we consider a structure group H 2 G0, we have a compact embeddingWn+1 !Wn+1. Moreover the K3 ber (that, in a local patch, contains the ALE berof Wn+1) has a singularity G which is the commutant of H in G0 [13]. The deningpolynomial p^ of Wn+1 can be written in generalized Weierstrass form:p^ = y2 + x3 + yxz^a1 + x2z^2a2 + yz^3a3 + xz^4a4 + z^6a6 . (5)Note that we use (y, x, z^) and (y, x, ~z) to denote the homogeneous coordinates ofthe elliptic ber of the n + 1-dimensional Calabi-Yau Wn+1 and the n dimensionalCalabi-Yau Zn, respectively. The an are functions of the coordinates ~xi of the base~Bn of the elliptic bration of Wn+1. In particular having a singularity of type Gabove a locus, say at z  ~x1 = 0 on the base ~Bn can be phrased in terms of thebehavior of the an near z = 0, an  zδn using Tate’s algorithm [17].To reiterate, the structure group H determines the singularity G of the K3 berof Wn+1. In turn G is determined by the behavior of the coecient functions anin the generalized Weierstrass form p^ near the singularity z = 0. Moreover we areinterested in the singularity of the K3 ber W2 of the K3 bration W2 ! Wn+1 !Bn−1. Thus z is now a coordinate on the base P1 of the elliptically berered K3,pi : W2 ! P1. However note that in the local limit Wn+1 ! Wn+1 where our mapf : Wn+1 ! (V, Zn) applies, the an become nothing but the ~Bn dependent parts ofthe monomials y(6−b−c)/3xb~zcvdfc,d(xi) in the dening polynomial p of Wn+1.To be more explicit let us consider the case G0 = E8E8. The relation betweenthe coordinates in p^ and p, using z  ~x1, w  ~x2 as the coordinates of the base P14JHEP12(1999)009of W2 as above, is then(y, x, ~z; v; xi)p =(y, x, z^zw;wz; ~xi, i > 3)pˆ.In particular the existence of a term an = zδnf(xi) implies that the line bundleN dnLn−6dn (6)has a holomophic section, where we have dened dn = n − δn. Combining theconditions imposed by all the an, n 2 f1, 2, 3, 4, 6g we arrive at the conditionν(G) c1(L)  η ( 12c1(L)) , (7)where ν(G) is the singularity dependent quantityν(G) = max((6dn − n)dn), (8)for all n 2 f1, 2, 3, 4, 6g with dn 6= 0. Note that this argument also tells us that theminimal bundle Nmin is always a power of L, a fact that is non-trivial for h1,1(B) > 1.We have also indicated in (7) an upper bound for η of a very dierent origin. It isnot related to the stability of N but rather to the stability of η0 = c1(N 0)  0, whereN 0 is the equivalent bundle in the second E8 factor.From eqs. (7) and (8) we obtainG = SU(2) : η  143 L G = E8 : η  0  LG = SU(3) : η  92 L G = E7 : η  2  LG = SU(M) : η  5  L G = E6 : η  3  LG = SO(N) : η  4  L G = G2 : η  4  LG = F4 : η  3  L (9)Here N 2 f7, . . . , 12g, M 2 f2, . . . , 6g and the groups with a star denote the sin-gularities associated to III and IV s bers in the list of [17]. Moreover L = c1(L).Note that our arguments for the bound (7) give a necessary but in general not suf-cient criterion for the stability of the bundle. In particular the distinction betweensingularities associated to simply laced (SL) and non-simply laced (NSL) groups isoften the specic form of the sections fc,d in the SL case rather than a dierencein the topological class [17]. Our arguments give a criterion for the existence of aholomorphic section fc,d which is sucient for SL groups, but it does not ensure thatone gets a generic enough section to describe also the NSL case. Thus the boundsgiven in (9) are strict lower bounds only for the simply laced case.5JHEP12(1999)009It would be nice to have a derivation of the bound on η that is directly relatedto the structure group H and in particular also applies to the non-compact case witharbitrary rank. A reasonable proposal based on (9) isH = SU(N) : η  N  L H = E8 : η  5  LH = SO(7) : η  4  L H = E7 : η  143 LH = SO(M) : η  M2 L H = E6 : η  92 LH = Sp(K) : η  2K  L H = G2 : η  72 LH = F4 : η  133 L , (10)where N  2, M  8 and K  2. Here we have used the calculation of the six-dimensional spectrum in [17, table 3] to raise the bounds for the NSL cases, suchthat we get the correct answer in six dimensions. Similarly we could compare thecalculation of the matter spectrum in four dimension for a heterotic compacticationon a Calabi-Yau 3-fold with (9).It is interesting to observe, how the geometric description contains non-trivialinformation about the instanton dynamics on Calabi-Yau manifolds Zn. Moreoverin many cases there is also another point of view, namely in terms of a supersym-metric eld theory in uncompactied space time. In particular, via the heteroticcompactication on K3T 2, the instanton dynamics on K3 is related to the modulispace of N = 2 supersymmetric eld theories. Similarly, instantons on Calabi-Yauthree-folds will be related to the moduli space of N = 1 Super-Yang-Mills theories.It would be very interesting to use the toric construction of V on Zn to study thisrelation in detail.AcknowledgmentsP.M. would like to thank Govindan Rajesh for discussions. P.B. was supported inpart by the Natural Science Foundation under Grant No. PHY94-07194. P.B. wouldalso like to acknowledge LBL, Berkeley for hospitality during the course of this work.References[1] P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum configurations forsuperstrings, Nucl. Phys. B 258 (1985) 46;C.M. Hull and E. Witten, Supersymmetric sigma models and the heterotic string, Phys.Lett. B 160 (1985) 398;E. Witten, New issues in manifolds of SU(3) holonomy, Nucl. Phys. B 268 (1986) 79;6JHEP12(1999)009J. Distler, Resurrecting (2, 0) compactifications, Phys. Lett. B 188 (1987) 431;J. Distler and B. Greene, Aspects of (2, 0) string compactifications, Nucl. Phys. B 304(1988) 1.[2] M.J. Duff, R. Minasian and E. Witten, Evidence for heterotic/heterotic duality, Nucl.Phys. B 465 (1996) 413 [hep-th/9601036].[3] K. Becker, M. Becker and A. Strominger, Five-branes, membranes and nonperturbativestring theory, Nucl. Phys. B 456 (1995) 130 [hep-th/9507158];M. Bershadsky, C. Vafa and V. Sadov, D-branes and topological field theories, Nucl.Phys. B 463 (1996) 420 [hep-th/9511222].[4] K. Uhlenbeck and S.T. Yau, On the existence of hermitian Yang-Mills connections instable vector bundles, preprint 1986.[5] J. Distler, B.R. Greene and D.R. Morrison, Resolving singularities in (0, 2) models,Nucl. Phys. B 481 (1996) 289 [hep-th/9605222].[6] P. Berglund and P. Mayr, Heterotic string/F-theory duality from mirror symmetry,Adv. Theor. Math. Phys. 2 (1999) 1307 [hep-th/9811217].[7] P. Mayr, N = 1 heterotic string vacua from mirror symmetry, hep-th/9904115, to ap-pear in the proceedings of the the Winter School on Vector bundles, mirror symmetry,and lagrangean submanifolds, Harvard University, January 1999.[8] S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solution of 4D N = 2 gaugetheories, I, Adv. Theor. Math. Phys. 1 (1998) 53 [hep-th/9706110].[9] C. Vafa and E. Witten, Dual string pairs with N = 1 and N = 2 supersymmetry infour-dimensions Nucl. Phys. 46 (Proc. Suppl.) (1996) 225 [hep-th/9507050][10] R. Friedman, J. Morgan and E. Witten, Vector bundles and F-theory, Comm. Math.Phys. 187 (1997) 679 [hep-th/9701162].[11] M. Bershadsky, A. Johansen, T. Pantev and V. Sadov, On four-dimensional compact-ifications of F-theory, Nucl. Phys. B 505 (1997) 165 [hep-th/9701165];R.Y. Donagi, Asian J. Math. 1 (1997) 214; MSRI pub. 28 (1992) 65;G. Curio and R.Y. Donagi, Moduli in n=1 heterotic/f theory duality, Nucl. Phys. B518 (1998) 603 [hep-th/9801057].[12] C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022].[13] D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds,I, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114]; Compactifications of F-theory onCalabi-Yau threefolds, II Nucl. Phys. B 476 (1996) 437 [hep-th/9603161].[14] G. Rajesh, Toric geometry and F-theory/heterotic duality in four dimensions, J. HighEnergy Phys. 12 (1998) 018 [hep-th/9811240].7JHEP12(1999)009[15] B. Andreas, On vector bundles and chiral matter in N = 1 heterotic compactifications,J. High Energy Phys. 01 (1999) 011 [hep-th/9802202];G. Curio, Chiral matter and transitions in heterotic string models, Phys. Lett. B 435(1998) 39 [hep-th/9803224].[16] V. Batyrev, Duke Math. Journ. 69 (1993) 349.[17] M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries, Nucl.Phys. B 481 (1996) 215 [hep-th/9605200].8

Stability of vector bundles from F-theory

Per Berglund

Peter Mayr

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Stability of vector bundles from F-theory

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