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Cohomology of Line Bundles: Proof Of arXiv:1003.5217 Cohomology of Line Bundles: A Computational Algorithm Ralph Blumenhagen,1,2, a) Benjamin Jurke,1,2, b) Thorsten Rahn,1, c) and Helmut Roschy1, d) 1)Max-Planck-Institut f¨ur Physik, F¨ohringer Ring 6, 80805 M¨unchen, Germany 2)Kavli Institute for Theoretical Physics, Kohn Hall, UCSB, Santa Barbara, CA 93106, USA (Dated: 29 October 2010) We present an algorithm for computing line bundle valued cohomology classes over toric varieties. This is the basic starting point for computing massless modes in both heterotic and Type IIB/F-theory compactifica- tions, where the manifolds of interest are complete intersections of hypersurfaces in toric varieties supporting additional vector bundles. CONTENTS classes Hi(Y ; V ) of the vector bundle V over a Calabi- Yau three-fold Y . The largest class of such Calabi-Yau I. Introduction 1 threefolds is given by complete intersections over toric ambient varieties X. For the vector bundles various con- II. Algorithm for line bundle cohomology 2 structions have been discussed in the literature. The A. Preliminaries 2 three most prominent ones are, on the one hand, monad B. Cechˇ cohomology for P2 3 and extension constructions, for which the vector bundle C. A preliminary algorithm conjecture 4 is defined via (exact) sequences of direct sums of line bun- D. Further examples 4 dles. On the other hand, for elliptically fibered Calabi- delPezzo-1surface 4 Yau manifolds, the spectral cover construction provides a large class of stable holomorphic vector bundles. How- delPezzo-3surface 5 ever, in all three cases, the computation of the massless E. The final algorithm conjecture 6 spectrum, i.e. of vector bundle cohomology classes, boils i III. Summary and Conclusions 7 down to determine the cohomology H (X; L) of line bun- dles L over toric ambient varieties X. Therefore, it is Added: Proof of the algorithm 7 pretty obvious that it would be important to have a tool Acknowledgement 7 to determine these line bundle cohomology classes in a straightforward way, i.e. without invoking any manifold A. Mathematical Background 7 dependent tricks. 1. Sheaves 7 The same problem also appears in other kinds of string 2.Sheafcohomology 8 compactification like Type IIB orientifolds with B-type 3. Cechcohomologyˇ 8 D-branes or in F-theory compactifications on Calabi-Yau 4. Holomorphic line bundles and divisors 9 fourfolds. Here one is confronted for instance with com- 5.Indextheorems 9 puting line bundle cohomology classes over curves, which arise due to the intersection of seven-brane wrapped di- visors on a three-dimensional compact internal manifold. I. INTRODUCTION Such cohomology classes count both the charged matter arXiv:1003.5217v3 [hep-th] 28 Oct 2010 zero modes and instanton zero modes in case one of the It is clear that the computation of line bundle cohomol- seven-branes is a Euclidean three-brane instanton. ogy classes over D-dimensional varieties is an interesting In the string theory literature, some indications how mathematical question of its own. Our motivation for such a generalization of Bott’s theorem to general toric approaching this problem, however, originates from the spaces might look like implicitly appeared in the papers1 appearance of this problem in various classes of compact- and2, however to our knowledge the algorithm used ifications of string theory. in those papers was never really completely revealed.3 First of all, in supersymmetric compactifications of the More recently, it was in the mathematics literature that heterotic string from ten to four space-time dimensions, the problem was systematically approached and solved4. the massless spectrum is given by certain cohomology However, it appears to us that the algorithm used in1,2 is much easier and economical, so that now, after non- trivially extending it to its most generic form, we find it appropriate to eventually reveal it. a)[email protected] In this letter we will proceed by motivating the b)[email protected] algorithm by a couple of non-trivial examples based on c)[email protected] direct computations of the Cechˇ cohomology. Then the d)[email protected] algorithm is formulated as a conjecture (which is so 2 simple that it fits less than a single page). We postpone with the extracted set the computation of vector bundle cohomologies and N many further string theoretic applications to a more F = {xα = xα = ... = xα =0} . (4) 5 1 2 |Sα| extensive paper . α=1 The information contained in F is nothing else than the The high-performance C/C++ implementation Stanley-Reisner ideal SR of the toric variety X cohomCalg of the algorithm is available under SR = S , S ,..., S . (5) http://wwwth.mppmu.mpg.de/members/... 1 2 N ...blumenha/cohomcalg/. Note that the more physical definition of a toric variety given here is equivalent to the one in terms of cones, fans and triangulations. A choice of a phase corresponds to II. ALGORITHM FOR LINE BUNDLE COHOMOLOGY the choice of a triangulation of the polytope defined via D the I vertices νi ∈ Z . These vertices satisfy the R linear In this section we will present an algorithm for the relations computation of line bundle cohomologies Hi(X; O(D)) I (r) over a toric variety X. We will motivate the algorithm Qi νi =0 for r =1, . , R. (6) following the original development. The final conjecture i=1 is given in section II E. For further introductions to toric geometry consult7–11 or for a more mathematical treatment12,13. A. Preliminaries Given such a toric space, the hypersurface {xi = 0} naturally defines a divisor Di of X. By the relation of divisors and line bundles summarized in appendix A 4, To define the framework we are working in, let us this also defines a line bundle Li = OX (Di), which we briefly summarize some facts about toric varieties re- also denote as spectively, their physical counter part, the gauged linear (1) (R) sigma models (GLSM). Li = OX Qi ,...,Qi . (7) A toric variety is a generalization of projective spaces, The tangent bundle T of the toric variety is defined as which contains a set of homogeneous coordinates H := X the cokernel of the map β in the sequence {xi : i =1,...,I} equipped with a number R of equiva- I lence relations β ⊕R α ։ (r) (r) 0 −→ OX ֒−→ OX (Di) − TX −→ 0 . (8) Q1 QI (x1,...,xI ) ∼ (λr x1,...,λr xI ), (1) i=1 (r) In order to compute, for instance, the vector bundle co- for r =1,...,R with the weights Q ∈ Z and λ ∈ C∗ = i r homology Hq(X; T ), one needs as input only the line C−{0}. We will always assume that the weights are cho- X bundle cohomology classes Hq(X; O (D )) with q = sen such that one really gets a bona fide toric space. A X i 0,...,D. powerful method to study such spaces is the GLSM6, in The determination of these cohomology classes for gen- which the homogeneous coordinates are chiral superfields (r) eral line bundles L = O (k ,...,k ) is the problem we and the Q are charges under R Abelian U(1) gauge X 1 R i will address in this letter. Due to the Riemann-Roch- symmetries in a two-dimensional N = (2, 2) supersym- Hirzebruch theorem, the cohomology classes Hq(X; L) metric gauge theory. The Fayet-Iliopoulos parameters ξ r satisfy an index theorem (see appendix A 5) of these U(1) gauge symmetries, can be interpreted as the K¨ahler parameters of the geometric space. D The vanishing of the D-terms of a GLSM then splits χ(X; L)= (−1)q hq(X; L)= ch(L) Td(X) , (9) the space of Fayet-Iliopoulos parameters ξ ∈ RR into q=0 X R-dimensional cones, in which the D-flatness conditions where the right hand side can be computed easily, once can be solved. These cones are also called phases and the intersection form on X is known. correspond geometrically to the K¨ahler cones. In each The simplest toric varieties are the projective spaces such cone, for the D-terms to be solvable one finds sets n P , for which F = {x1 = x2 = ... = xn+1 = 0}. For of collections of coordinates these spaces, the line bundle cohomology is known due to Bott’s theorem Sα = xα1 , xα2 ,...,xα|Sα| with α =1,...,N, (2) n+k for q =0, k ≥ 0 which are not allowed to simultaneously vanish. The toric n q Pn −k−1 variety of dimension D = I − R in this phase is then h ( ; O(k)) = n for q = n, k< −n . (10) defined as 0 else I C − F The goal is now to generalize this result to toric ambient X = (3) (C∗)R spaces. 3 B. Cechˇ cohomology for P2 To motivate the algorithm, we consider a simple ex- ample for which the Cechˇ cohomology can really be com- puted with pencil and paper. We want to compute the Cechˇ cohomology of P2. First we need a suitable covering by open sets, which for this projective space is provided by 2 Ui := [x1 : x2 : x3] ∈ P : xi =0 . (11) Then the Cechˇ cochains for the covering UP2 = {U1,U2,U3} take the form 0 Cˇ (UP2 ; F )= F (U1) ⊕ F (U2) ⊕ F (U3) 1 Cˇ (UP2 ; F )= F (U12) ⊕ F (U23) ⊕ F (U13) (12) 2 Cˇ (UP2 ; F )= F (U123), where we used the abbreviation Uij...k = Ui∩Uj ∩···∩Uk. Now we consider a line bundle F = O(−p) with p > 0 and compute the local sections on all open sets, where the FIG.
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