arXiv:1003.5217v3 [hep-th] 28 Oct 2010 g lse over classes ogy INTRODUCTION I. h asessetu sgvnb eti cohomology certain by given is dimensions, space-time spectrum four massless to the ten from string heterotic theory. compact- of string classes of various the ifications for in problem from motivation this originates of Our however, appearance problem, own. this its approaching of question mathematical CONTENTS I.SmayadConclusions and Summary III. d) c) b) a) Algorithm Computational A Bundles: Line of Cohomology I loih o iebnl cohomology bundle line for Algorithm II. .Mteaia Background Mathematical A. [email protected] [email protected] [email protected] [email protected] .Introduction I. ti la httecmuaino iebnl cohomol- bundle line of computation the that clear is It is fal nsprymti opcictoso the of compactifications supersymmetric in all, of First .Frhreape 4 examples Further D. .Peiiais2 Preliminaries A. .Apeiiayagrtmcnetr 4 conjecture algorithm preliminary A C. B. .Tefia loih ojcue6 conjecture algorithm final The E. .Sefchmlg 8 3. cohomology Sheaf 2. Sheaves 1. .Idxterm 9 9 theorems divisors Index and 5. bundles line Holomorphic 4. 2) in,weetemnflso neetaecmlt nescin o heterotic intersections complete both are bundles. in interest vector modes of additional manifolds massless the computing c where cohomology for tions, valued point bundle line starting computing basic for algorithm an present We 2010) October 29 (Dated: USA ap Blumenhagen, Ralph 1) ehchmlg 8 cohomology Cech cnwegmn 7 Acknowledgement ehchmlg for cohomology Cech ˇ ˇ de:Pofo h loih 7 algorithm the of Proof Added: e ez- ufc 5 4 surface Pezzo-3 del surface Pezzo-1 del al nttt o hoeia hsc,Kh al CB S UCSB, Hall, Kohn Physics, Theoretical for Institute Kavli a-lnkIsiu ¨rPyi,Forne ig6 80805 6, F¨ohringer Ring f¨ur Physik, Max-Planck-Institut D dmninlvreisi ninteresting an is varieties -dimensional ,2, 1, P 2 a) ejmnJurke, Benjamin ,2, 1, b) 7 7 3 2 1 7 hrtnRahn, Thorsten loih yacul fnntiileape ae on based examples the non-trivial of of computations couple direct a by algorithm it find we non- it. form, after reveal generic eventually now, most to its that appropriate in to so used it economical, algorithm extending and trivially the easier that much us is to appears it However, dles pcsmgtlo ieipiil perdi h papers the toric in general appeared and implicitly to like theorem look Bott’s might spaces of generalization a the such of one instanton. case three-brane in Euclidean a modes is zero matter seven-branes instanton charged and the modes both zero count classes di- manifold. cohomology internal wrapped compact Such seven-brane three-dimensional of a on intersection visors the which to com- curves, due with over arise classes instance cohomology for bundle confronted line is puting one Calabi-Yau Here on B-type fourfolds. compactifications with F-theory in orientifolds or IIB D-branes Type like compactification a manifold in any tricks. classes invoking dependent without cohomology tool i.e. bundle a way, have line straightforward to these important determine be would to it that obvious pretty h rbe a ytmtclyapoce n solved and that approached literature revealed. systematically mathematics was completely the problem in really the was never it recently, was More papers those in heflsi ie ycmlt nescin vrtoric over intersections complete varieties by ambient given is threefolds pcrm ..o etrbnl ooooycass boils cohomology massless classes, the the cohomology determine to of bundle down vector computation of the i.e. How- cases, spectrum, three bundles. vector all holomorphic in stable ever, Calabi- provides of construction fibered class cover large elliptically a spectral for the hand, manifolds, other bun- Yau line the of On sums direct of dles. sequences bundle (exact) vector monad The via the hand, defined which is one for the literature. constructions, on extension the are, and ones in prominent discussed most three been have structions a three-fold Yau loih sfruae sacnetr wihi so is (which conjecture a as formulated is algorithm classes nti etrw ilpoedb oiaigthe motivating by proceed will we letter this In ntesrn hoyltrtr,sm niain how indications some literature, theory string the In string of kinds other in appears also problem same The arXiv:1003.5217 M¨unchen, Germany 2 oee oorkoldeteagrtmused algorithm the knowledge our to however , L naBraa A93106, CA Barbara, anta vrtrcabetvarieties ambient toric over H i 1, ( Y c) ; yesrae ntrcvreissupporting varieties toric in hypersurfaces f V n emtRoschy Helmut and Y ftevco bundle vector the of ) h ags ls fsc Calabi-Yau such of class largest The . X n yeIBFter compactifica- IIB/F-theory Type and o h etrbnlsvroscon- various bundles vector the For . assoe oi aite.Ti sthe is This varieties. toric over lasses ehchmlg.Te the Then cohomology. Cech ˇ 1, d) H X i ( hrfr,i is it Therefore, . X V ; vraCalabi- a over L fln bun- line of ) 1,2 4 1 3 . 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. 1. The region subdivision of the lattice. Note, that it is homogeneous coordinates which are non-vanishing can meant that we have the inclusions Ri ∪ Rj ⊂ Rij for i < j appear with a negative exponent. This leads to and Si Ri ∪ Si m m −p−m −m x 1 x 2 x 1 2 , 2 1 2 3 by itself restricts to Cˇ (UP2 ; F ). Thus we get a sequence m ,m ≥ 0 for U 0 → 1 → 2 → 1 → 0 with trivial cohomology. An 1 2 3 (13) with m ≥ 0, m ≤−p − m for U . element (m1,m2) ∈ Rij \ Ri ∪ Rj first contributes to 1 2 1 2 1 2 Cˇ (UP2 ; F ) and restricts to Cˇ (UP2 ; F ). The induced m ≤−p − m , m ≥ 0 for U 1 2 2 1 sequence 0 → 0 → 1 → 1 → 0 again has trivial cohomol- These monomials can be represented by points ly- ogy. Finally, if ing in different regions Ri in the (m1,m2)-plane, cf. figure 1. (m1,m2) ∈ R123 \ Rij ∪ Ri , (16) i One can study more complicated examples and always D. Further examples finds a similar relationship between the representatives ˇ of Cech cohomology and rationoms of the above form. To get an idea of how this algorithm works in more This observation leads to the following first version of involved cases and where its restrictions are, we now ap- our conjecture, which does not yet cover the most generic ply this conjecture to del Pezzo surfaces arising from a situation. However, it was this form which was sufficient certain number of blowups of P2. to find all the cohomology classes in1,2. del Pezzo-1 surface C. A preliminary algorithm conjecture To get an idea of the simplicity of the counting pro- For the moment we assume that all elements of the cedure once the form of the contributing monomials has Stanley-Reisner ideal are disjoint, i.e. no homogeneous been determined, we first look at dP1 with toric data coordinate xi appears in more than one Sα. given in table I. Note that the condition on the elements In order to determine which rationoms contribute to of the Stanley-Reisner ideal is fulfilled, i.e. they are dis- which cohomology, we have to consider the power set joint. We will see what may happen for more complicated of SR, which we denote as P (SR). Clearly we have a Stanley-Reisner ideals later. splitting of the form |SR| vertices of the coords GLSM charges divisor class 1 2 P (SR) = Pk(SR), (18) polyhedron / fan Q Q k =0 ν1 =( −1, −1) x1 1 0 H where Pk(SR) contains all subsets of SR with k elements. ν2 =( 1, 0) x2 1 0 H Let A = {α ,...,α } ⊂ {1,..., |SR|} and Pk = 1 k A ν3 =( 0, 1) x3 1 1 H + X {S ,..., S } ∈ P (SR) be such an element of P (SR) α1 αk k ν = ( 0, −1) x 0 1 X containing k elements. We now define the set 4 4 k 2 k intersection form: HX − X QA := Sαi (19) i =1 SR(dP1)= x1x2, x3x4 = S1, S2 k of all homogeneous coordinates appearing in PA and the TABLE I. Toric data for the del Pezzo-1 surface c-degree k Forming the power set of the Stanley-Reisner ideal di- k : k NA = QA − k = Sαi − k. (20) rectly yields the following possible contributions to coho- i =1 mology, where the line bundle O(D) = O(m,n) is given The question now is to which cohomology class does a by the divisor D = mH + nX. fixed element Q = Qk with N = N k contribute. 0 A A H (dP1; O(m,n)) : T (x1, x2, x3, x4) , Preliminary Conjecture:: One gets a contribution to 1 T (x3, x4) HN (X, O(D)), which is given just by counting ap- H (dP1; O(m,n)) : , x1x2 W (x1, x2) propriate rationoms, where y ∈Q are the denomi- T (x , x ) nator variables and x ∈ H −Q the complementary 1 2 , numerator variables. The contribution of Q to the x3x4 W (x3, x4) line bundle cohomology is given by: 2 1 H (dP1; O(m,n)) : . x1x2x3x4 W (x1, x2, x3, x4) N T ( x) (22) H (X; O(D)) : . (21) In fact, we would not have to take H2(dP ; O(m,n)) ( yj ) W ( y) 1 into account because of Serre duality, but it will serve as a good test of the procedure to check that it is obeyed T ( x) and W ( y) are monomials of the right degree at the end of the computation. Note that the rationoms to match the weights of D in the line bundle O(D), representing the top cohomology class always involve the i.e. one counts the number of suitable monomials T denominator monomial Q = x of all homogeneous and W . i coordinates xi ∈ H, which is a clear evidence of Serre Now, doing this for all elements in the power set P (SR), duality in “monomial” form, since the canonical class one determines the complete line bundle cohomology. K = − Di is the negative sum of all coordinate di- The whole cohomology computation has been reduced to visor classes. a counting problem, in which the Stanley-Reisner ideal In order to get the dimensions of all cohomology plays an important role. Clearly, this algorithm can eas- groups, one only has to read off the charges of the coordi- ily be computerized. nates from the table to get the degrees of the respective 5 monomials, equate them to the degrees (m,n) of the line It is easy to check that Serre duality holds. Since the bundle and do the remaining combinatorics, i.e. count all canonical divisor is given by the negative sum over all di- possible exponents. Writing hi(m,n) for the dimension visors corresponding to the one-dimensional cones (ver- i of H (dP1; O(m,n)) and xi for the exponent of the co- tices) of the fan ordinate xi in the monomials T and W these steps can be schematically described as follows: K = − Dρ , (28) ρ∈ Σ(1) • Contributions to h0(m,n) from all combinations of exponents with we get K = −3H − 2X from the table of dP1. Serre duality can then be written as deg T (x1, x2, x3, x4) Hi(dP ; O(m,n)) = ( x1 + x2 + x3 , x3 + x4 ) (23) 1 2−i ∨ ! =∼ H (dP ; O(K) ⊗ O(m,n) ) (29) = (m,n) . 1 2−i =∼ H (dP1; O(−m − 3, −n − 2)) . Since the exponents cannot be negative, we only get contributions for m ≥ 0 and n ≥ 0. Defining And indeed, the computed dimensions obviously satisfy r the identity 2 = 0 if r< 2, one calculates hi(m,n)= h2−i(−m − 3, −n − 2) . (30) m +2 m − n +1 h0(m,n)= − . (24) 2 2 In this case the computation could still be performed just with pencil and paper but clearly for more involved • The degrees of the monomials contributing to higher dimensional cases a computer code is necessary.14 h1(m,n) can be evaluated as T (x , x ) del Pezzo-3 surface deg 3 4 x1x2 W (x1, x2) So let’s make two steps further and look at dP3, the del = ( x3 − 2 − x1 − x2 , x3 + x4 ) , (25) Pezzo surface of degree 6 coming from three consecutive T (x1, x2) P2 deg blowups of . This is the first example we considered, x3x4 W (x3, x4) for which nontrivial “region factors” arise from Cechˇ co- = ( x1 + x2 − 1 − x3 , −2 − x3 − x4 ) . homology. Its toric data are given in Table II. Equating the right hand sides to (m,n), this yields vertices of the coords GLSM charges divisor class the two cases polyhedron / fan Q1 Q2 Q3 Q4 1. n ≥ 0 ∧ n ≥ m + 2: ν1 =( −1, −1) x1 1 0 0 1 H + Z 1 n−m −m−1 ν2 =( 1, 0) x2 1 0 1 0 H + Y h (m,n)= 2 − 2 . ν =( 0, 1) x 1 1 0 0 H + X 2. n ≤−2 ∧ n ≤ m − 1: 3 3 1 m−n+1 m+2 ν4 = ( 0, −1) x4 0 1 0 0 X h (m,n)= 2 − 2 . ν5 =( −1, 0) x5 0 0 1 0 Y r ν6 =( 1, 1) x6 0 0 0 1 Z This again makes use of the convention 2 = 0 if r< 2. intersection form: HX + HY + HZ − 2H2 − X2 − Y 2 − Z2 • Contributions to h2(m,n) come from a monomial with degree SR(dP3)= x1x2, x1x3, x1x6, x2x3, x2x5, x3x4, x4x5, x4x6, x5x6 1 | S{z } | S{z } | S{z } | S{z } | S{z } deg 1 2 3 4 9 x1x2x3x4 W (x1, x2, x3, x4) (26) TABLE II. Toric data for the del Pezzo-3 surface. = (−3 − x1 − x2 − x3 , − 2 − x3 − x4 ) . First of all, the elements of the Stanley-Reisner ideal are no longer disjoint. So by our definition of c-degree, Setting this equal to (m,n), we only get contribu- 2 e.g. the element P{1,4} = {x1x2, x2x3} of P2(SR) would tions when m ≤ 0 and n ≤ 0. The result is now contribute to the first cohomology, since the union is −m − 1 n − m h2(m,n)= − . (27) 2 2 2 Q := Q{1,4} = {x1, x2, x3} (31) 6 2 and therefore N{1,4} =3−2 = 1. There arealso two other to determine which rationoms contribute to which coho- combinations in P2(SR) with the same union, namely mology, we have to consider the power set of SR, which 2 2 we denote as P (SR). Clearly we have a splitting, P{2,4} = {x1x3, x2x3} and P{1,2} = {x1x2, x1x3}. This by itself is not a problem, but one further finds out that |SR| the c-degree of Q is not unique any more, since the el- 3 P (SR) = Pk(SR) , (35) ement P{1,2,4} = {x1x2, x1x3, x2x3} of P3(SR) has the k 3 =0 same union but now gives N{1,2,4} =3−3 = 0. Of course one could hope that this kind of combinatorial structure where Pk(SR) contains all subsets of SR with k ele- of the power set is meaningless and a certain element Q ments. Let A = {α1,...,αk} ⊂ {1,..., |SR|} and k only contributes to the cohomology of highest c-degree. PA = {Sα1 ,..., Sαk } be such an element of P (SR) con- As it turns out, this assumption is too naive, since it taining k elements. We now define the set leads to inconsistencies for example when computing the holomorphic Euler characteristic of line bundles, which k k : can also be computed by means of index theorems and is QA = Sαi (36) another useful quantity to cross-check results. Choosing i =1 O(D) with D = −3H − X − Y − Z (equal to the degree k −1 −1 −1 of all homogeneous coordinates appearing in PA and the of x1 x2 x3 ), computing the Todd class of dP3, the c-degree Chern character of the line bundle and doing the integral over X by means of the intersection form, one readily k calculates k : k NA = QA − k = Sαi − k . (37) i =1 χ(dP3; O(D)) (32) = h0(O(D)) − h1(O(D)) + h2(O(D)) = −2 Now, for fixed k we take the disjoint union 0 2 1 and since h (O(D)) = h (O(D)) = 0 we get h (O(D)) = Qk := Qk , (38) 2. But performing our algorithm, the only contribution A −1 −1 −1 A comes from the rationom x1 x2 x3 , so it clearly fails. The rationom must be weighted by a factor of two and the where A runs over all subsets of {1,..., |SR|} with k el- question is how this can be incorporated in the counting ements. The question now is to which cohomology class algorithm. does a fixed denominator monomial Q contribute. To A possible solution is to introduce a sort of “cohomo- decide this one computes a “remnant” or “secondary” logical” sequence that imitates a cochain complex with- cohomology Hi(Q). For its definition, one counts how out specifying the mappings and encodes the additional often the element Q appears in each Qk and with what structure of P (SR) alluded to before. In the case of Q c-degree. Then we can associate the numbers Ci(Q) to this yields such a Q, where Ci(Q) is the number of times the element Q appears in all Qk with c-degree i. ... −→ 0 −→ C0 −→ C1 −→ 0 −→ .... (33) =1 =3 Algorithm Conjecture:: For each non-vanishing cohomol- ogy hi(Q) of the complex and therefore a remnant or secondary cohomology h1 = 3 − 1 = 2 acting as a factor for all corresponding ra- ... −→ C0(Q) −→ ... −→ Cd(Q) −→ ... (39) tionoms, allowing us to schematically write the contribu- tion of Q to line bundle cohomology as one gets a contribution to Hi(X; O(D)), which is given by counting rationoms 1 T (x4, x5, x6) H (dP3; O(D)): 2 × , (34) x1x2x3 W (x1, x2, x3) T ( x) where the monomials T and W have to be chosen such Hi(X; O(D)) : hi(Q) × , (40) that the rationom has the same degree as D. Since it ( yj) W ( y) can be shown that this procedure is suited to reproduce the right factors in many more nontrivial examples in- where y ∈ Q are the denominator variables and cluding higher-dimensional varieties, this motivates the x ∈ H−Q the numerator variables. T ( x) and W ( y) formulation of a more general version of our conjectured have to be taken as monomials of the right degree algorithm. to match the degree of O(D). The total number of these has then to be weighted by a factor of hi(Q). E. The final algorithm conjecture Actually, due to Serre duality it is sufficient to con- To formulate it we have to refine the set-theoretic no- sider only the cohomologies Hi(X; O(D)) with i ∈ d tions from the first version of the conjecture. In order {0, 1,..., [ 2 ]}. We have not really been precise about 7 what the maps in (39) are and therefore have not shown Added: Proof of the algorithm that this is indeed a complex. More mathematical work needs to be done to precisely define this “remnant” co- About three months after the preprint release of this homology. This would mean to see in detail that, after work, a proof of the conjectured algorithm was presented taking the localization of the line bundle cohomologies in15, which also clarifies much of the underlying math- onto the elements of P (SR) into account, one has only ematical structures. At the same time an independant ˇ computed the complete Cech cohomology up to this rem- proof was developed in16 — published in fact a few days nant sequence. earlier — which utilized alternative methods. We have tested the conjecture in many situations, with the most complicated one being line bundles on a Calabi- Yau fourfold with a resolved SU(5) singularity of a cer- Acknowledgement tain divisor. Here the Stanley-Reisner ideal involved about 20 generators with many coordinates appearing We gratefully thank our computers for doing most of more than once. the work for us and R. Blumenhagen acknowledges a brief conversation with J. Distler in 1997. Furthermore, we would like to thank A. Collinucci for a number of use- ful comments on the manuscript. R. Blumenhagen and B. Jurke would like to thank the Kavli Institute for Theo- III. SUMMARY AND CONCLUSIONS retical Physics, Santa Barbara for the hospitality during the final stages of the project. This research was sup- ported in part by the National Science Foundation under Based on earlier observations in1 and2, in this letter Grant No. PHY05-51164. we presented a general working algorithm for the com- putation of line bundle valued cohomology classes over D-dimensional toric varieties. Evaluating the Cechˇ co- homology for certain comparably simple examples, we Note added: During the very final stages of this work, realized that the representatives of the local sections gen- we became aware of17 which partially addresses the same erating Hi(X; L) with i = 0,...,D are given by certain problem. rational functions in the homogeneous coordinates, where the elements of the Stanley-Reisner ideal appeared in the denominator. The algorithm became more involved by Appendix A: Mathematical Background realizing that for the case that homogeneous coordinates appeared more than once in the Stanley-Reisner ideal, This section serves as a reference to the mathematical extra multiplicities had to appear. For the determina- notions and structures used later on. We recall the def- tion of these extra factors, we proposed to evaluate a inition of sheaves, which are abundantly used in math- remnant cohomology, which we suspect can be thought ematical physics in the context of vector bundles, and of as the not yet fully resolved part of the original Cechˇ summarize the basic ideas behind sheaf and Cechˇ coho- cohomology, after the localization on the Stanley-Reisner mology. The language is therefore much more formal ideal sets has been done. compared to the first part of this paper. We checked many examples and found no deviation from known results, Serre duality or the Riemann-Roch- Hirzebruch index theorem. The algorithm also is con- 1. Sheaves sistent with results obtained by the algorithm presented recently in13. Clearly, the advantage of the latter is that In mathematics one often encounters spaces which are it has been mathematically proven, but we want to em- defined by stitching together local patches in a suitable phasize that, in the cases we tested, our algorithm was fashion. Therefore it seems natural to attach additional much more efficient. Indeed, examples that need about data directly to those patches, i.e. to local subsets of 10 minutes of computing time using the “chamber” algo- the described total space. This idea is formalized using rithm could still be done with pencil and paper using the the notion of a sheaf, which can also be treated as a algorithm presented here. generalization of the ordinary bundle concept, where the fiber space attached to every point of the base space may Finally, having available such a simple and easy to vary. implement algorithm paves the way to perform a whole Let us recall the formal definition of a sheaf:18 Given amount of concrete computations in string theory. This a topological space X, a sheaf F of Abelian groups or includes heterotic string compactifications over Calabi- modules on X assigns to each open subset U ⊂ X an Yau manifolds as well as intersecting D-brane models and Abelian group or module F (U), such that F-theory compactifications on Calabi-Yau fourfolds. We will report on the applications to these string theoretic 1. Normalization: F (∅) = 1, the trivial group or cases in a future publication5. module. 8 2. Local uniqueness: If U = {Ui} is an open cover- As one may anticipate from the notion of cohomol- ing of U ⊂ X and s,t ∈ F (U) are two elements ogy, a sheaf homomorphism F −→ G induces a cor- (“sections over U”) that are identical on each local responding homomorphism of sheaf cohomology groups p F p G subset, i.e. s|Ui = t|Ui for all Ui ∈ U, then in fact H (X; ) −→ H (X; ) and a short exact sequence of s = t follows. sheaves gives rise to a long exact sequence of sheaf co- homology groups. Furthermore, from the definition it 3. Gluing: If U, V ⊂ X are two open sets and follows that H0(X; F )= F (X)=Γ(X, F ), i.e. the 0th F F sU ∈ (U) as well as sV ∈ (V ) two elements sheaf cohomology group can be treated as the space of (“sections”) that are identical on the intersection, global sections of the sheaf F over X, which in our cases i.e. sU |U∩V = sV |U∩V , then there exists a glued usually corresponds to global sections of the vector bun- F element s ∈ (U ∪ V ) such that s|U = sU and dle described by the sheaf. s|V = sV . F Note that for some subset U ⊂ X the restriction |U 3. Cechˇ cohomology is itself a sheaf on U, given by F |U (V ) = F (U ∩ V ), whereas F (U) refers to the Abelian group or module attached to the subset. The elegant abstract definition of sheaf cohomology is In order to see that a sheaf generalizes the notion of unfortunately rather unsuitable for actual computations. π However, there is a another cohomology theory based ։ bundles, let E − X be a vector bundle. For U ⊂ X on intersections of open sets covering X that yields an let Γ(U, E) denote the space of sections over U, which equivalent cohomology, called Cechˇ cohomology. naturally has a module structure by addition of sections, m Let UX = {Ui}i=1 be an open covering of the variety i.e. fiberwise addition of the respective vectors, and mul- m X, i.e. X = i=1 Ui and every Ui ⊂ X is an open set. tiplication with holomorphic functions. If we define F F As usual (U i) refers to the Abelian group or module via F (U) := Γ(U, E) it can be shown that all the prop- that the sheaf F associates to the open set Ui. The Cechˇ erties of a sheaf are satisfied. Therefore we observe that cochains are defined by the formal transition from bundles to sheaves is essen- tially nothing else as treating them in terms of the local p Cˇ (U; F )= F (Ui ∩ Ui ∩ ∩ Ui ), (A4) sections. In this sense the notation Γ(U, F ) is frequently 0 1 p 0≤i with the “hatted” index ik being omitted. Basically, d˜0 d˜1 d˜2 0 −→ I 0(X) −→ I 1(X) −→ I 2(X) −→ ... (A2) we are considering the alternating sum with one index omitted, which is familiar from the exterior derivative of satisfying d˜p+1 ◦ d˜p = 0 for all p ≥ 0. Note that e.g. the de Rham complex. Again, one can check the this only implies im(d˜p) ⊆ ker(d˜p+1), whereas for ex- property dˇp+1 ◦ dˇp = 0, such that actness of the sequence the equality must be strictly dˇ0 dˇ1 satisfied. The first term F (X) of the induced com- 0 −→ Cˇ0(U; F ) −→ Cˇ1(U; F ) −→ ... (A6) plex I •(X), d˜• is usually omitted for notational conve- nience and d˜−1 :0 ֒−→ I 0(X) is the zero mapping. The indeed gives the Cechˇ complex Cˇ•(U; F ), dˇ• . The pth sheaf cohomology groups of F are then defined by Cechˇ cohomology group is then defined by ker(d˜p) ker(dˇp) Hp(X; F ) := Hp I •(X) = . (A3) Hˇ p(U; F )= Hp Cˇ•(U; F ) = , (A7) im(d˜p−1) im(dˇp−1) One certainly has to verify a number of mathematical which as before measures the failure of the complex to aspects in order to prove that this is well-defined, see19. be exact. Again, one has to prove several mathematical 9 aspects in order to show that any potential ambiguities fact isomorphic to the set of Cartier divisors under a weak are indeed taken care of in this definition. Like for sheaf smoothness assumption on X, i.e. cohomology one can show that a sheaf homomorphism ∼ 0 ∗ ∗ F −→ G induces a mapping of Cechˇ cohomology groups Div(X) = H (X; MX /OX ). (A10) Hˇ p(U; F ) −→ Hˇ p+1(U; G ) and a short exact sequence of sheaves yields a long exact sequence of Cechˇ cohomology Now consider the set of isomorphism classes of holomor- groups. Furthermore, it follows Hˇ 0(U; F ) = F (X) = phic line bundles over X. The tensor product of vector Γ(X; F ) which establishes the equivalence of both sheaf bundles and the dual vector bundle naturally provide it and Cechˇ cohomology at the 0th level. with an Abelian group structure, which is called the Pi- Naturally the question arises how the sheaf and Cechˇ card group Pic(X). Again, there is an isomorphism cohomology groups for p > 0 are related. Let U = ∼ 1 ∗ m n Pic(X) = H (X; OX ) (A11) {Ui}i=1 and V = {Vj }j=1 be two open coverings of the variety X. Then V is called a refinement of U if there relating to sheaf cohomology. Note that one usually ig- exists an index mapping φ, such that Vj ⊆ Uφ(j) holds nores the fact that Pic(X) consists of the isomorphism for all j =1,...,n — basically every open set of the cov- classes of line bundles instead of the line bundles, which ering U may be broken down into several open subsets leads to a frequent abuse in notation. Now there exists which provide the covering V. This establishes a partial a natural group homomorphism20 ordering “≤” between different coverings and therefore allows to use the notion of direct limit. Then it can be Ξ : Div(X) −→ Pic(X) proved that in general (A12) D → OX (D) Hp(X; F ) =∼ lim Hˇ p(U; F ) (A8) −→U that maps some Weil divisor to a corresponding element sheaf cohomology Cechˇ cohomology of the Picard group, i.e. it maps a divisor to an iso- morphism class of holomorphic vector bundles. A cor- holds for all p ≥ 0, which basically implies that for a responding vector bundle of this class can be defined as sufficiently refined open covering of X both sheaf and follows: ˇ Cech cohomology groups are identical. For the cases rel- Due to (A10) any divisor corresponds to a global sec- evant to physicists the statement can be substantially im- tion ρ of M∗ /O∗ , which can be specified in terms of F m X X proved: Let be a quasicoherent sheaf and U = {Ui}i ∗ =1 functions ρi ∈ MX (Ui) for some covering U = {Ui} an affine open covering of the variety X, then for all p ≥ 0 of X. The line bundle representative corresponding there are natural isomorphisms to OX (D) is then specified by the transition functions −1 ∗ p F ∼ ˇ p F hij := ρi ρj ∈ OX (Uij ). It is necessary to show that H (X; ) = H (U; ). (A9) ˇ 1 ∗ the hij indeed define an element of H (U; OX ), such that Essentially, those statements allow physicists to talk OX (D) is a well-defined Picard generator via (A11) and about sheaf cohomology but carry out all the actual com- (A9). Furthermore, OX (D) satisfies a number of useful putations using Cechˇ cohomology. The huge advantage properties: of Cechˇ cohomology lies in the fact that for a suitable • Abelian group structure: O (D + D′)= O (D) ⊗ choice of the open covering U the necessary explicit com- X X O (D′) putations of the Cechˇ complex are greatly simplified. X ∗ • Inverse elements: OX (−D) =∼ OX (D) 4. Holomorphic line bundles and divisors • Neutral element: OX (0) =∼ OX ∗ ∗ • Naturality: OX (f D) =∼ f OY (D) for a holomor- The primary sheaf of interest in subsequent dealings phic mapping f : X −→ Y . is the sheaf of holomorphic functions on X, denoted by OX . For each open set U ⊂ X the corresponding mod- C ule OX (U) is the set of holomorphic functions U −→ . 5. Index theorems In fact, OX has a natural (commutative) ring structure induced by complex multiplication. In this context O X Due to the numerous sequences and steps one has to is called the structure sheaf of X and turns it into a work through in order to derive the information of inter- ringed space. Furthermore, there are the sheaves O∗ X est, it is useful to have a couple of non-trivial consistency of nowhere zero holomorphic functions, M of mero- X checks at hand. Those are provided by index theorems, morphic functions and M∗ of non-trivial meromorphic X which allow to compute certain topological invariants via functions, which are frequently used in algebraic geome- entirely different means. The Euler characteristic try. Those sheaves are naturally related to divisors, which dimR X are formal sums of irreducible hypersurfaces of X. The χ(X)= (−1)kbk(X)= e(X) (A13) set of such Weil divisors is denoted Div(X) and it is in k=0 X 10 can be defined as either the alternating sum of the Betti have i i numbers b (X) := dimR H (X), or as the integral of a , differential form representing the Euler class e(X), which h0 0 2n 1,0 0,1 is equal to the top Chern class cn(X) ∈ H (X) for some h h complex n-dimensional manifold. h2,0 h1,1 h0,2 For the K¨ahler manifolds that are primarily investi- h3,0 h2,1 h1,2 h0,3 gated via toric geometry, the prior statement can be gen- h2,0 h1,1 h0,2 տ (A19) eralized to a holomorphic version. Basically, the Euler 1,0 0,1 h h տ χ0 class e(X) is replaced by the Todd class Td(X), which is 0,0 defined only for complex vector bundles. The Todd class h տ χ1 can be expanded in terms of Chern classes like տ −χ1 −χ0 2 c1 2c1 +2c2 + c1c2 1 Td=1+ + where χ0 = χ(X; OX ) and χ1 = χ(X;Ω ). 2 24 X −c4 +4c2c + c c +3c2 − c + 1 1 2 1 3 2 4 (A14) 720 −c3c +3c c2 + c2c − c c + 1 2 1 2 1 3 1 4 + ..., 1440 which usually allows for a rather simple computation of this quantity. For holomorphic (0,p)-forms the relation (A13) generalizes to an alternating sum of certain Hodge p,q p,q numbers h := dim H∂¯ (X), i.e. dimC X p p,0 χ(X; OX )= (−1) h (X)= Td(X), (A15) p=0 X whose value minus 1 is called the arithmetic genus or holomorphic Euler characteristic. If bundle-valued forms are taken into account as well, we obtain the Hirzebruch- Riemann-Roch theorem dimC X χ(X; V )= (−1)q hq(X; V )= ch(V ) Td(X). q=0 X (A16) Note that for line bundles L over X the Chern character ch(L) simplifies to ∞ k c1(L) 1 ch(L)=ec1(L) = =1+ c (L)+ c (L)2 + ... k! 1 2 1 k=0 (A17) Some important special cases include the tangent bun- ∗ ∼ 1 dle TX , the cotangent bundle TX = ΩX and its antisym- p ∗ ∼ p metric tensor product Λ TX = ΩX . Note that in general the Hirzebruch genera dimC X q p,q p χp := (−1) h (X)= ch(Ω )Td(X) (A18) q=0 X correspond to the alternating sum of the diagonal ele- ments of the Hodge diamond, e.g. for a K¨ahler 3-fold we 11 1J. Distler, B. R. Greene, and D. R. Morrison, “Resolving sin- University Press, 1993) ISBN 0691000492. gularities in (0,2) models,” Nucl. Phys., B481, 289–312 (1996), 13D. A. Cox, “Lectures on Toric Vari- arXiv:hep-th/9605222. eties,” Unpublished lecture notes, available at 2R. Blumenhagen, “Target space duality for (0,2) com- http://www.cs.amherst.edu/~dac/lectures/coxcimpa.pdf. pactifications,” Nucl. Phys., B513, 573–590 (1998), 14Another possibility to check the validity of these results is to arXiv:hep-th/9707198. perform the “chamber” algorithm described in Chapter 9 of13. 3The cohomology computations in21–23 were also done using this We have checked that the methods of both algorithms coincide 2 algorithm. in some simple examples like P or dP1, but the computing time 4D. A. Cox, J. B. Little, and H. Schenck, “Toric is by a factor of approximately 103 shorter for our algorithm. Varieties,” Unpublished book-in-progress, available at 15H. Roschy and T. Rahn, “Cohomology of Line Bundles: Proof of http://www.cs.amherst.edu/~dac/toric.html. the Algorithm,” (2010), arXiv:1006.2392 [hep-th]. 5R. Blumenhagen, B. Jurke, T. Rahn, and H. Roschy, 16S.-Y. 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