JHEP05(2020)107 Springer May 4, 2020 April 5, 2020 May 22, 2020 : : : = 1 Geometric

JHEP05(2020)107 Springer May 4, 2020 April 5, 2020 May 22, 2020 : : : = 1 geometric . N Accepted Received Published , a ]. The database can be accessed orientifolds.html 1 Published for SISSA by https://doi.org/10.1007/JHEP05(2020)107 webpage/cicy ) as proposed in [ [email protected] , and Alexander Westphal b thraxions . 3 westphal/orientifold ∼ Jakob Moritz 2003.04902 a The Authors. c Diﬀerential and Algebraic Geometry, Superstring Vacua, Flux compactiﬁca-

We present a vast landscape of O3/O7 orientifolds that descends from the , [email protected] www.desy.de/ Department of Physics, CornellIthaca, University, NY 14853, U.S.A. E-mail: [email protected] Deutsches Elektronen-Synchrotron, DESY, Notkestraße 85, 22607 Hamburg, Germany b a Open Access Article funded by SCOAP ArXiv ePrint: here: Keywords: tions descend from a much larger andthe possibly CICYs new set via of conifold CYtransitions threefolds transitions. involving that can colliding We be O-planes. observe reachedproduce from an Finally, examples as interesting of an orientifolds class that application, of of satisfy we ultra-light the use throat topological our requirements axions dataset for ( the to existence famous set of completeof intersection topological Calabi-Yau data threefolds relevant for (CICY). phenomenologythe such We D3-tadpole, as give and the distributions multiplicities orientifold-odd of Hodgeof O3 numbers, these and orientifolds O7-planes. have Somewhatout conifold surprisingly, by almost singularities the all orientifolding. whose However, deformation they branches can be are resolved, projected so most of the orientifolds actually Abstract: Federico Carta, A landscape of orientifold vacua JHEP05(2020)107 3 18 ] is a fruitful 2 30 21 13 20 2 P × 2 P 28 24 11 – 1 – 9 32 17 23 1 7 7 21 14 20 4 2] | 17 5 P ]. Clearly, by charting out the boundary of the landscape one hopes to 7 , 6 ], giving rise to a plethora of low energy effective field theories (EFT). While their 3.6.4 An example with vanishing D33.7.1 tadpole CY3.7.2 hypersurfaces in products of The del-Pezzo Schoen surfaces manifold 3.6.1 The3.6.2 quintic threefold Anti-canonical3.6.3 hypersurface in [ 5 – 3 3.7 The non-favourable CICYs 3.5 The algorithm 3.6 Examples 3.1 Defining a3.2 CICY involution Singularities at3.3 co-dimension one Singularities at3.4 co-dimension three Conifolds on O7 planes & their resolution branches of the landscape itwithin has a been (much) conjectured larger space thatswampland of the [ EFTs landscape that populates areunderstand only not how small realized string islands in theory stringmore constrains theory, modest known the and as set the related of goal low is energy to observables. understand A how somewhat well distinct field theory sectors can The landscape of flux vacuaarena for in model the building type and for IIBand addressing corner fundamental the issues of such as existence critical moduli stringvast of stabilization, theory [ de [ Sitterspectrum vacua and in Wilson string coefficients theory. are expected to The vary landscape seemingly randomly is along believed the bulk to be C Involutions of del Pezzo surfaces 1 Introduction and conclusions 4 Application: ultra light throatA axions Basic properties of del PezzoB surfaces The anticanonical hypersurface in a product of two del Pezzos 3 O3/O7 orientifolds of CICYs Contents 1 Introduction and conclusions 2 Results JHEP05(2020)107 ] 8 X ]). ], and → 14 9 , X 13 : I ] in the same ] of Candelas et ] respectively. 11 , 15 18 = 1 low energy 10 N d 1 restricts the freedom to descriptions, obtained by determine the number of F 2 1 , 1 − ,H 1 , . This data determines a type IIB ] and Donagi et al. [ 2 + del Pezzo surfaces or rational elliptic X 17 ,H or 1 . For example, the dimensions of the , 1 I + ,H 1 , 2 − – 2 – CICYs), H of co-dimension three and one respectively (see together with a holomorphic involution X -invariant deformation classes of CICY embeddings. CICYs). The database is constructed by finding all 2 X in Z I favourable of co-dimension one sets the number of non-abelian gauge of and its involution as a complete intersection of hypersurfaces in certain products F F X parameters include the magnitude of the flux superpotential in KKLT [ either ]. We make use of their most tractable non-favourable ], 15 16 1) on the holomorphic threeform Ω of decoupling − ]). The Ramond-Ramond (RR) tadpoles induced by the O-planes can be canceled by 12 Most of the orientifolds we produce are singular at co-dimension three, with a number In this article, we construct a database of 2,004,513 such orientifolds, and compute a This motivates us to study on a grand scale the CY orientifolds of O3/O7 type which Examples of The descriptions are useful because the divisor classes of the CY threefold are simply inherited by the 1 2 backreaction radii of branescontext). in comparison to the overall volume ofones the of CY the ambient (see space.al. e.g. have Only [ this about property. half of the descriptions in the original database [ For the non-favourable cases,rational elliptic surfaces we found by incorporate Blumenhagen et the al. [ involutions of del-Pezzoof surfaces conifold and singularitiesthat residing would on be the compatible O7 with planes the which orientifold cannot projection. be deformed However, in all a of way these have a by Candelas et al.Anderson [ et al. in [ of projective spaces (henceforth surfaces (henceforth ambient space involutions, and all number of topological invariants that can bebuilding. used These as input include data the forgroups, D3-tadpole, phenomenological the model the number dimensions of of O3 and thepoint O7 is orientifold-odd planes, the cohomology and classic topological database data of of complete O7-divisors. intersection CY The (CICY) starting threefolds, as constructed complex structure moduli, Kählermoduli, closed stringchiral U(1) multiplets vector respectively. multiplets, For and simplistic axionic choicesof of D7 connected brane components configurations, of the number sectors with gauge algebra so(8),choose and world the volume Euler and characteristic bulk of three-form fluxes. effective supergravity theories featuringbroken perturbative supersymmetry, no-scale non-abelian vacua gaugeAn with theories, important spontaneously chiral subset sectorssimple of etc topological the (see data data e.g. of orientifold [ determining even/odd these cohomology effective groups field theories is given by that acts as ( O3/O7 orientifold with O3geometric and fixed O7 point planes locus e.g. residing [ at theintroducing connected appropriate components configurations of of D7bulk the branes three-form with fluxes, or and/or without world-volume mobile fluxes, D3 branes. This generates 4 their (renormalizable) self-interactions can bethis tuned. is related Due to to how the far Dine-Seiberg the problem landscape [ extendsare away a from basic its starting bulk. point forchoice constructing of phenomenologically Calabi-Yau interesting (CY) vacua: threefold consider a be decoupled from one-another at the level of non-renormalizable operators, and how weak JHEP05(2020)107 . ]. 1 are 21 i 2 , 1 are the 2 ]. B 20 . 6 and 2 10 where we explain C × 3 , we display the dis- 2 2 ∼ ]. Here, Thus, the number of distinct 4 23 [ 3 2 τB ]. A histogram is shown in figure for a histogram. − we discuss the distributions of topo- 2 1 2 25 ] (WGC) and the swampland distance , C 1 ], and how many are new. 22 i 2 O3 planes have previously been described in the 6= 0 in the Kreuzer-Skarke database [ Σ : this computes the number of bulk com- 1 we show how to find examples with throat 20 , R 0 is about 17%. 1 1 − , : this determines the number of perturbatively 4 2 − h eats 1 ≡ > , h – 3 – 1 − i 1 , h G 1 − ]. See figure h ] such as [ 2 24 ] for results on orientifolds with 28 , ] (SDC). 7 27 ]. 19 , we use our database to find many orientifolds that satisfy all topological require- 4 plex structure moduli thatstabilized remain by after three-form the fluxes orientifold [ projection. These have to be two-form potentials of the ten-dimensionalthe type orientifold-odd IIB supergravity two-cycles. theory,models and Σ of These axion-monodromy axions [ The are fraction promising of inflaton-candidates orientifolds with in massless axionic chiral multiplets The orientifold-odd Hodge number The orientifold-odd Hodge number This paper is organized as follows. In section We present two applications of our database. First, in section Since the resolution phases are not contained in the CICY database, we produce, as Similar transitions across which an O7See plane also [ 3 4 (a) (b) literature [ In this section we displaylet interesting us features explain of what the topological distributions properties we we have have obtained. computed for First, each orientifold, and why: how the database isaxions. constructed. In section 2 Results jectures such asconjecture the [ weak gravity conjecture [ logical data that we havethe obtained landscape and and use the this to swampland. speculate The about bulk the boundary of between our paper is section properties, display the boundarywhat of features our we expect CICY-orientifoldsection to landscape generalize and beyond speculate the about ments CICY-orientifold landscape. known to Second, us in We to expect realize these models to of be ultralight an throat ideal axions, framework as for proposed testing in and [ challenging swampland con- a byproduct, many CY threefoldsof that conifold are transitions, but connected are towould themselves the be not CICY interesting contained database to in via understand theof a how original CYs number many CICY such of database. as these It the are Kreuzer-Skarke already database contained [ in othertributions lists of topological data that we have obtained, comment on some of their statistical orientifold there belongs aflop number transitions. of Across different such geometric transition loci,of phases the O7 related number divisors of to jumps, O3 in each planessmooth a as other well way geometric via as that phases the preserves topology is the actually D3 charge. much larger than the quoted number of distinct resolution branches that are orientifold preserving, so to each singular JHEP05(2020)107 1 . i, − χ h 200. �� − (of the = χ χ (right) on a �� is shown in ) of the un- 1 CICYs with 1 , , 2 − 2 h = 0. Red curve: CY 1 , h χ �� 1 p , 1 ), h 1 ). For orientifolds of �� - i, − 1 � singular , h 2 �� i , while the computation (left) and c h 1 e , �� 1 , + 2 − 1 − + h 1 h , i 1 b �� − ( h 1 �� , / i + 1 p h ) 1 ]). = 2( �� i, − � 26 h ( CY ∝ χ 1). It is clear that the distributions of � � . � χ ) �� 3 ] and even the tuple ( �� 1 ��

��

- i, − � �� / ) #( 9 �� 15 -�� . h 3 ]. However, as mentioned in the introduction, , 5 = 0. The latter ﬁt results in parameter estimates 29 – 4 – − 1 p 10 -�� × ), 1 � 9 i, − . h (5 -�� q i c ≈ e ) 2 + ) is left for future work. The distribution of -�� i � � , p 1 , b 2 ( 2 + �� - / , c � i 2 , h p �� b

1 )

, � �� 1 � �� 1 + i, − h h ( � ∝ 8) and ( . ) CICYs, the Euler characteristic [ . 5 1 , ’s of both signs appear it is tempting to speculate that our list of CYs contains pairs i, − 2 χ h (71 . Histogram showing the number of orientifolds with given Euler characteristic . Histogram showing the number of orientifolds with given � ≈ singularities that can be resolvedThe in smooth ways threefold compatible thus obtained with hasof the an the orientifold Euler CICY. projection. characteristic Given diﬀerent (a) from andof the (b) one the this also tuple determines ( ﬁgure smooth derlying CY threefold is well-known [ most of the orientifolds that we determine are involutions of ) � � � The Euler characteristic of the CY threefold 1 ��

��

, c ��

� �� / )

#( (c) 1 �� b Figure 2 CY threefold) with binGiven width that 5. Therelated most by mirror negative symmetry. value Theby occurs going mean for through is the conifold shifted transitions quintic to from positive with the values, deformed because to we the reach resolved smooth side, CYs strictly increasing fit with #( ( have a tail heavier than exponential fall-off (compare e.g. [ Figure 1 logarithmic scale. Black dashed curve: fit with #( JHEP05(2020)107 3 ]). Fur- maximal 30 and ]. In figure 31 ). Divisors with D O minimal D, ( χ the induced D3 charge on minus . These determine the induced D3 2 ) D ( 1 c arithmetic genus D . Left: the minimal number of O3 planes from R 3 O ≡ n – 5 – D d degree and O3 planes 7 5 O , for each singular CICY there are many resolution branches which differ from n , and its D 3.4 χ = 1 pure Yang-Mills (YM) sectors confine and generate scalar potentials ). N 4 ) = 1 may generate non-perturbative superpotential terms, via euclidean D3 D O . Histograms showing on a logarithmic scale the number of orientifolds producing a D, ( branes on top of eachfor O7-plane. a We generic compute D7-brane the configuration generically(see for much figure a larger subset D3 of tadpole the orientifolds of smooth CICYs branes is generically negative andfluxes. can be The canceled larger via theO3-planes the tadpole and introduction (which of 7-branes) we three-form the definefluxes. more as freedom For all there orientifolds, is we in compute choosing this different number three-form for configurations with four D7 charge dissolved in the 7-branes,χ and the brane instantons, or thewe above display the mentioned distributions strong ofnumber gauge the of dynamics number O3 O7 [ planes. planes and the groups, with charged matterworld volume living fluxes, on relevant for intersectionthermore, particle curves, Physics and model building chiral (seefor index e.g. Kählermoduli. set [ by For eachwrapped divisor O7 plane, we compute the Euler characteristic of the canceled by introducing D7-branes. Stacks of seven-branes host non-abelian gauge The set of distinct O-planes: the O7 planes contribute a D7-tadpole that can be The D3-tadpole: the total D3 charge of O3 planes and induced charge on seven- Our results indicate strong correlations between the topological quantities that we have As explained in section 5 (e) (d) computed. More so, our CICY-landscape clearly populates regions ineach other the by total the parameter numberO3 of planes extra corresponds O3 to planes a thatof resolution reside O3 branch on planes that the corresponds produces exceptionalnumber to no curves. of extra a The O3 resolved resolution minimal planes, conifolds. branch number while with of the the maximal maximal number number of extra O3 planes equal to the resolution branches that do notfrom produce the extra resolution O3 planes. branches that Right: produce the one maximal extra number of O3 O3 plane planes per conifold singularity. Figure 3 given number of O7 planes JHEP05(2020)107 = 3 132]. D SO(8) , �� Q − �� ) and for each , χ �� 3 36]. Right: D3-tadpole �� D SO(8) , , where we display 3D � 4 Q 5 − �� ). The steep cliffs carve out clear , χ values in [ 1 , 2 − �� h even � ��

��

�� / ) #( �� – 6 – that cannot be explained by the mere ﬁniteness �� boundaries �� (�) �� ). Right: count for each value of ( �� , χ 3 ). The steep cliﬀs delineate clear islands embedded in otherwise empty D SO(8) , χ Q 1 , � 2 − h . Left: histogram showing the number of orientifolds with given D3-tadpole . Histograms showing on a logarithmic scale the number of orientifolds. Left: count for � � ]. ��

�� 20

� �� ) (

� �� / )

#( 4 for local D7-charge cancellation. The tadpole takes �� / f of the landscape andverify the or beginning refute this of by the findinglist swampland. orientifolds [ in Clearly, other it CY-datasets such would as be the interesting Kreuzer-Skarke to space that have pronounced of our sample size.histograms Two showing more the such structures number are ofvalue visible orientifolds of in for ( figure eachregions. value of It ( is tempting to speculate that some of these boundaries in fact mark the end Figure 5 each value of ( islands embedded in large otherwise empty regions. χ for a small setrestricted of ourselves to examples the withan subset non-local ambient of space tadpole smooth divisor. cancellation CICYs The by where tadpole the a takes O7 generic even divisor values D7-brane. obviously in descends a We from significantly have larger range [12 Figure 4 JHEP05(2020)107 r T ) +1 +1 r p n n − (3.2) (3.3) (3.4) (3.1) , the n i j n I ∼ I as local m , . . ., p n 0 n 1 , I p ] n x =( ). Without loss : , ~ I , . . . , x action as p } 2 x as local transverse ... ,... Z , given the ) : j 1 ) = 0 1 , 1 p n j − j n ) p x n ( x ambient space acts as a : P 1 r y, x = n ( − ↔ p P , . . . , x x , with vectors ... ) 0 1 1 − i , 7−→ x i ~ = S n ( : ) , it we may use     p P ~ I x ] ... × · · · × x, y K 1 K r +1 : . . . ˙ ( p ] arise as the common vanishing locus ∪{ 32 1 m m 0 } − n n x n 15 gives rise to the same P , -th polynomial under the scaling of the i F − n p n [ = 0 I P 1 ··· ··· we may use − factors [ × p n 7−→ p – 7 – x 1 1 1 r n n ] F . . . n P P m m . In particular, a section of some line bundle may = x : +1 while near ) . Clearly, p ... −→ r 1 r n . . . − n ( n (1) n ... p n n n = P P P : I indicating the swaps (( P I . The (non-unique) starting point for each such manifold 0 0 r T x     × n x ) [ P n P ≡ { actions on complete intersection Calabi-Yau (CICY) manifolds , ,... ) 2 n 1 +1 as matrices acting on the projective coordinates, and : p P Z , j − n 1 × · · · × n factors n denotes the weight of the i +1 S 1 p + 1. As the latter requirement determines the n −→ j i n − j ˙ ∪F P n n P n m n n = (( p P F ~ homogeneous polynomials in an ambient space formed by a product of S −I = + 3 and in order for its first Chern class to be zero we also need to impose : i j = K K n p m n I p = , and . In order for the resulting manifold to be a complex threefold we need that I i =1 ) i j K i j j n p n ( symmetric in a local frame around the first fixed point locus and anti-symmetric in configuration matrix n I P P 2 i We encode the ambient space involution into a pair Up to linear equivalence, an involution of a We restrict ourselves to geometric involutions of the CICY manifolds that can be ex- =1 Z Q r j -th coordinates that are inverted by be another local frame around the second fixed locus. ' with fixed point locus the diagonal because is the trivial involution.coordinates that Near are the inverted locus by with fixed locus the disjoint union and swaps of two to the ambient space. combination of involutions of individual that first column of the configuration matrix is often omitted. tended to involutions of the ambientwe space. cannot While exclude our the aim possibility is that to exhaust other all CY such involutions involutions exist that cannot be extended where the entry j P of a set of projective spaces is the 3.1 Defining aWe CICY enumerate involution possible which have a fixedor point three. locus The with CICY connected manifolds components of Candelas of et complex al. co-dimension [ one 3 O3/O7 orientifolds of CICYs JHEP05(2020)107 . 1 n 1). P − via − This . , (3.6) (3.5) ~ f T 7 . 1) ) and the − diag(1 , involutions n ,... disjoint com- ] the original −→ +1 ) P . The complex , 1 I 1 16 Kähler-favourable 4 σ N P , n favourable 1 act on distinct m S and otherwise diago- = (+1 6 ~ P = (( c . As the action of an invo- ~ S contains entries +1 for each and +1 ~ p T P − n n 3)) I of the ambient space. We restrict , , and furthermore giving a useful , 1 and the swaps , 1 − ~ f ,    p ) h with I = ((2 g ( c n p f ~ S I , parity vector has an action on the set of polynomials T action on the divisor classes of the CY from i ] in calling such embeddings 0 00 2 21 0 2 2 2 n – 8 – 7−→ R ). 2)) denotes the number of non-trivial 16 Z , S ~ ~ P f i I , 1 1 4 c N Q P P P ~ S : a , = ((1 ineffective splittings a p    ~ g S ) where the I ~ , S ), i.e. factors is equal to ~ a P ~ , I g , ( n T Q c f ~ P ) is block-diagonal with a number of two-dimensional blocks S 2) = R g , ( 0 g f , R blocks correspond to swaps of columns factors, and inverts two projective coordinates of 1 1 = (0 σ P ] has been manipulated to bring all but 70 of the 7890 configuration ~ I ) it follows that the ambient space fixed locus contains 2 15 generator 3.3 2 . Z that exchange polynomials of different scaling weights 1. The ) is again defined up to exchanging g ( 3.7 f 0 1 1 0 R From eq. ( The number of swapped As an example, consider the ambient space involution of If they are the same,For favourable we embeddings, may the rotate CY them Kählercone into contains each the other one in inherited a from way the that ambient space, diagonalizes but = 6 7 1 ponents of different dimension, where contained in the ambient space involution. We will tabulate a givenit ambient can space have additional fixed generators. If the two cones are equivalent, the embedding is called that of the ambientdatabase space. of For ref. this [ purposematrices it to is a very favourablealternative useful description form that of via in the ref. remainingin [ 70 section cases. The latter will be dealt with separately This is encoded in ourselves to configuration matrices thatfrom have the ambient property space that divisors. allallows the We CY us follow divisors [ to descend simply deduce the induced that swaps the two structure moduli are assumed toare be exchanged adjusted under so the that ambient the space second involution, and and third the two first polynomials is mapped to minus itself. for each polynomial thatthus is encoded mapped in to the minus quadruple itself. ( A candidate orientifold of a CICY is combined swaps of rows andis columns necessary must for leave the thepoint-wise). configuration ambient matrix space Given invariant. an involution This to ambientof space the map polynomials involution the in we CY a encodepolynomial threefold pair the that to ( transformation is itself properties either (but swapped not with another one or mapped to itself, and entries lution does not change theinto scaling each weight other of so a that polynomial,σ we may rotate the polynomials nal entries factors. The some matrix representation where of generality we may assume that the involutions JHEP05(2020)107 ) p I 0 ~ and P ≤ -even , (3.7) f 2 0 is the I ~ Z I − k into a g frame adapted is given by the ↔ − f 0 where k of the ambient space ≥ 0 to find a tuple ( − . If any subset of F k 0 +1 of the normal coordinates, + 1 transverse coordinates. p F − 0 I − l I = 0 that descends to a variety n n are inverted, while the action ≤ a ⊥ 0 0 x I 'I F n ), in a frame adapted to the fixed p g I ( ,...,I. is trivial. Given a frame adapted to a f 0 R that intersects the CY at co-dimension = 1 . F l , i a ⊥ – 9 – x i a plus the number of pairs of swapped polynomials c 0 it does not intersect the CY. ~ 0 P =1 I a X < Thus, their intersection with the CY is determined by . We will denote this choice of gauge as the = − containing 1’s and 2’s that indicate which of the two fixed 0 k depends non-trivially on although these combinations do not make sense globally when i − 8 of co-dimension F I g f i -odd combinations of polynomials will vanish identically on − − 0 2 canonical fixed locus N f − l Z F f factor is to be chosen. For a given ambient space involution and the n F P 1) eigenspace of the matrix 1 polynomials are anti-symmetric around of length − − ~q l ≡ 1 corresponds to ~ , we call 1’s in the parity vector and an odd one p = F − g ~q + . We note, however, that solving this problem can be mapped to the following the solution set has a component given by setting 1 f , I 1 . In such a frame, the coordinates transverse to h . This is because the 0 ≤ With a brute-force na¨ıve scan over all subsets of anti-symmetric polynomials we would A connected component of the ambient space fixed locus of some given co-dimension c fewer constraints than one would naively expect. For a consistent O3/O7 orientifold Locally, around the fixed locus, we can combine any pair of swapped polynomials 0 ~ F S 8 I generically descends to a CY fixed locus of co-dimension are sections of different line bundles. − not be able towith complete the orientifold-scan as the computation timefunction grows too quickly g of dimension three or biggereties either once have intersected components with of thecomponents dimension symmetric of bigger polynomials. dimension than three, three Such orThus, that vari- in contain generically order multiple to intersect reducible avoid each singularitiesanti-symmetric of other polynomials co-dimension at one depends we co-dimension non-trivially should one. an require at that each least set of we can write If whose ambient space fixedfeature point such locus singularities. intersectsambient the They space CY may fixed at arise locus one. co-dimension as one Then, follows. and three Consideranti-symmetric polynomials a frame adapted to an sub-variety of the CY of co-dimension one or3.2 three, or completely misses Singularities the at CY co-dimensionIn locus. one this paper wethey will develop not singularities consider of CY co-dimension three-folds one. at However, a many ambient locus space in involutions their moduli space where number of in the ambient space fixedk locus. we need that each connected component of the ambient space fixed locus descends to a on the projective coordinatesfixed that locus parameterize l dimension of the ( point locus in question. If loci in each involuted action on the polynomialfixed vector, locus for we each may connectedsuch always that gauge component fix the redundancy to locus by a vector JHEP05(2020)107 (3.8) (3.9) (3.11) (3.10) , then = 0 in l ) corre- 0 00 ≥ x vanishing I 00 + 1 normal I + p I + = 3. Thus, the ≡ l I l ≥ 00 I Γ as follows. There are 1, and + I coordinates. In this case, ≥ I I , no row or column swaps, graph T ≤ 00 2) I , it is possible to find a , 0 − , -symmetric CY. For this example, 2), so 3 = , 2 l    , If Z , ≡ . 0 = (1 I where each column corresponds to one of   I αi 0 0 1 1 1 1 A

  we can associate a – 10 – 2 01 0 10 0 1 2 1 1 1 =

A = (i.e. complete subgraph) that contains at least one c A c 3 3 1 Γ Γ P P P Γ =    with at least one column, i.e. ). The dependence of the two anti-symmetric polynomials 3 I clique P × 1). We consider its canonical fixed locus. There are two anti- ⊂ 00 − = 1 if the i-th anti-symmetric polynomial depends non-trivially I 1 , 1 P αi , 1 A − , = (1 polynomials depends non-trivially on P I ’th normal coordinate, and zero otherwise. α anti-symmetric polynomials and each row corresponds to one of the vertices, one for each row and one for each column. We connect with an undirected Finding the largest possible vanishing submatrix (with size defined as We solve this problem as follows: to l p , and two normal to + 1 of the column vertices. In our example this is given by the maximal clique, with clique number equal to 3. where blue vertices correspond to rowsvertex and correspond red to vertices the correspond first to rowvertices. and columns. is The thus connected middle by undirected edges to both column sponds to finding the largest edge each pair of rowsif and and each pair only of if columns,the the and following corresponding connect graph, each matrix pair element of vanishes. row and In column our example, this produces and there are co-dimensionthis one is singularities obvious, in this straightforward. but an efficient search for null-submatrices inp larger matrices is not as Clearly, there is a vanishing sub-matrixtwo of dimension anti-symmetric (1 polynomials depend non-trivially only on two transverse coordinates and an orientifoldand specified parity by thesymmetric involution polynomials, and threeP normal coordinates (one normalon to these the normal point coordinates is encoded in the matrix according to what we havean said example: above, consider there the are CICY singularities with of number co-dimension 7734, one. given Here by is the configuration matrix the coordinates. We set on the sub-matrix of dimension the set of different problem: we may define a matrix JHEP05(2020)107 ∗ 1. )). C de- / − E ( l (3.13) (3.14) (3.12) ), one +1 n F dies out tensored ( (1) F ch F n d F ⊗ O ) ∗ P ∗ E N O . ]. N ( ( ) must be 2 symmetric poly- 2 34 g ch c E ( 2 f Z We consider a com- 1) R ) = 2 − ]), though easily imple- 10 l F ( ). Then, the pull back of 33 l ⊗ E These singularities always , ( E symmetric CYs still contain 9 O ) + ( +1 . . In order for this to descend l 2 F m ch . Z ∗ N ( (1) C )) 1 m d c E P ( · resolved ). As the orientifold projection elim- O E -odd eigenspace of associated with the 2 m 1) + S i Z M 1 anti-symmetric sections degenerates at a F ⊗ O − . D ∗ l ⊕ − ( p N l ( − – 11 – 2 (1) c ) n take values in the co-normal bundle B F P l Z N O F ( = 2 n p c I resolution divisors M B cf that is Poincarédual to the Chern class n = F )) = with the divisors F E ( N F = 1. For now, let us assume for simplicity that the anti-symmetric . This step of the computation dominates the overall computational l , but its divisor class need not descend from an ambient space divisor B F ⊗ O ∗ frozen conifold singularities . N preserving way, but instead can be 1 all take values in the same divisor line bundle ( , 2 1 2 k − c h Z f is the same as its tangent bundle, and the latter is isomorphic to n d Mathematica of the ambient space fixed locus of co-dimension ) because the dependence on the local coordinates of the fixed locus P E in a F ( . The system of differentials of the O F The generic appearance of this phenomenon is seen as follows. The maximal clique problem is non-trivial in general (see e.g. [ We thank Fabian Rühlefor a comment that ledFor the us basics to of drop applied the algebraic requirement geometry, see that e.g. the the orientifolds introductory are chapters of [ 9 projective coordinates and the second comes about because the normal bundle of the 10 smooth as orientifolds of the original CICY manifolds. where the first factorp comes becausediagonal the fixed locusas of is an seen involution from isshows that the given Euler by sequence. fixing Furthermore, using This is straightforward to evaluate because near co-dimension two locus along Thus, the number of conifold singularities along the CY divisor is given by nomials is a surface class, unless of course polynomials their differentials to the fixedwith locus inates the deformation (Coulomb)these branch singularities in complex structure moduli space, weponent denote to a divisor inside theThen, CY the the intersection dimension of of the singularities of co-dimension three,formed generically conifold singularities,reside that on cannot top of be deformed O7 CY, planes but which instead wrapthe correspond divisor to conifolds classes the (we new that will divisor have call classes no that them correspondent arise in upon the resolving mented in cost at large 3.3 Singularities atHaving co-dimension three avoided singularities at co-dimension one, most JHEP05(2020)107 } i 12 ) is A { (3.15) smooth 3.12 This de- 11 ): . ]. Globally, however,    ] = 0, i.e. there exists j i 35 E A [ · i (for details see the discus- i orientifolds.html 4 P E j 1 ≤ − i l X is easily determined (and can be associated with either of the swapped B in the configuration matrix. B ) + p F − n N ( P 1 webpage/cicy c · i −→ E n 1 P – 12 – ). The appropriate generalization of eq. ( i i − l X ]. It admits a flat resolution (Higgs) branch if and E ( − 37 ]. Upon resolving, this four-chain turns into a divisor O ) , transversally. Now consider one of the O7 divisors of 37 F 36 [ 1 N 4 P ( ˜ Σ 2 ) are those associated with the omitted columns. c i ∂ one replaces E ), so upon deforming it turns into a four-chain with boundary    n westphal/orientifold p ( = I B i ∼ O Z A 3.4 i = P cf B swap one sums up the two corresponding rows. n n s.t. P 4 1 distinct line bundles ˜ Σ − l www.desy.de/ ) is a disjoint union. In general there can be further resolution divisors that are ), i.e. the number of vector multiplets and Kählermoduli for each orientifold. 1 , 3.3 1 + , h For each pair of swapped columns, either one of the two is omitted. For each For each involution All columns associated with anti-symmetric polynomials are omitted. Let us explain why the frozen conifold singularities can always be resolved in a way 1 Note that the anti-symmetric part of a pair of swapped polynomials makesIts sense dimension only is locally of near course the bounded from below by the number of O7 planes containing conifold singu- , 2 + 11 12 (c) (a) (d) (b) h sion in the next section fixed divisors. Theycolumns, make which cannot sense be as obtained sections as of pullbacks of linelarities. ambient bundles space on line bundles. has shrunken that satisfya at four-chain least a singlethat homology intersects relation the exceptional our setting that passesthe through singularities at it looks least precisely one like conifold the singularity. resolution divisor Locally, Σ around one of CY threefold. Before applying thegeometry orientifold it projection, is in always the possiblethere local to is non-compact deform generically conifold or an resolve obstructionfor the against Strominger’s singularity the hypermultiplets [ resolution [ inonly the if form of at a the D-term singular potential locus in moduli space a collection of at least two three-cycles that involve these, asKnowing well the as dimension the of( dimension the of resolution the branch would resolution allow branch one forthat to future maintains determine work. the the CY pair condition, producing an O3/O7 orientifold of a different scription also allows one todivisors compute the wrapped triple intersection by numbers O7ambient between the planes space resolution divisor and classes. theresolution divisors bulk The wrapped ones, by triple distinct by intersection O7of intersecting planes numbers eq. vanish with that identically, ( because the involve the fixed twonot appropriate locus or wrapped three by O7 planes. We leave the computation of the triple intersection numbers The divisor line bundles A configuration matrix description forfound the here fixed divisor sections of This needs to be generalized to the case where the antisymmetric polynomials are actually JHEP05(2020)107 0. = 0. (3.16) (3.17) (3.18) (3.19) v −→ , which | = Ω . In other u d 4 4 by replacing C and a group ˜ = Σ 1 R R y ⊂ , action, and the | P 1 2 2 = P = C Z | x U(1) ∈ Ω ] , × . 4 A , we could compute its R C A α, β [ ∈ ∼ | ) SU(2) ) , 7−→ − A × = 0 ! v . x, y, u, v ( β α ,

† · ] , T −→ − Z 35 = LZR iφ ! transformation e : 2 – 13 – and so a resolution is possible, as claimed. As x , v B at the tip of the conifold. If the boundary of this Z v y x u 4 −→ ˜

Σ A Z or ≡ −→ − x , ) transforms homogeneously under the ,Z = 0 3.16 = 0 that passes through the singular point ) either by = 0 ! v β α ]). Z =

3.16 34 · x det Z of [ ) acts as : iφ = 2, i.e. it has a spurious A ), and the deformation modulus is projected out by the orientifolding due to r 2 iπ/ L, R, e is the deformation parameter. Consider now the involution , e 3 iσ Locally, there are two ways to resolve the conifold with an exceptional The symmetry group of the singular conifold is SU(2) Example 7.6.4 − , 1 which we will refer to asorientifolding, the these A-type two and resolution B-type looke.g. resolution the respectively. same Locally, and and before are related by a flop transition (see ( its R-charge the defining equation ( element ( on the conifold coordinates. The geometric involution corresponds to the group element Only then, the defining eq.deformation ( of the conifold is projectedresiding out on by the the locus orientifold.Although We are it left contains with the an O7 conifoldwords, plane singularity, at it the singular wraps point, a theis smooth tangent regular, surface bundle while to the the fixed normal surface bundle inside the of CY its threefold embedding into the conifold degenerates. where In order for this to be a symmetry of the conifold, we have to take the singular limit would describe the local descriptionmay of write the the orientifold in deformed the conifold conifold as geometry. the Locally, locus we [ argued in the next section, the orientifolding does3.4 not project out the Conifolds resolution on branch. O7We have planes stated that & many their O3/O7singularities orientifolds resolution contain in branches O7 the planes singular that CY pass through threefold conifold that can be resolved, but not deformed. Here we four-chain were to contain onlyvolume by this integrating single the holomorphic connected threeform component vanishes Ω, because i.e. Ω Vol( is closed,singularity. in contradiction Therefore, to there our must assumptionform that exist the we a had boundary set deformed of of the at the least four-chain two shrunken three-spheres that containing the deformed three-sphere JHEP05(2020)107 2 2 ]. Z B . 19 is the . The (3.20) (3.21) would factors . Away and 3 cf B 2 1 n ω n C 2 P P C ) + B ( χ parameterized by 1 orientifolds.html ) = P and swaps of columns ˜ B ( ~ S χ . As the orientifold-odd 1 , 1 + h 1]. The former is a point on , action, and the fixed locus is that do not act on integrated over the exceptional , transversally intersecting the . 2 2 ~ I 2 ] = 2 hypermultiplet with bosonic Z C webpage/cicy C β , − N is the radial coordinate and and 0] and [0 α, = 0 , 2 r B βy are proportional to the harmonic two-form ] = [ + 2 C where α, β αu – 14 – 3 − [ ω = and ∧ v 2 7−→ fixed points [1 B = = 0 is replaced by an exceptional dr ] ]. For the A-type resolution, the resolved conifold is westphal/orientifold action leaves them invariant as well. Since x ), blown up at the origin. In a compact setting, ∼ Z 2 two points. Its Euler characteristic is α, β α, β Z [ u, y cf B as described in the previous section, so the O7 wraps a surface notebook used to execute the algorithm that we now describe n . As these forms are invariant under the global symmetry group = 1 chiral multiplet formed by the resolution modulus and the 1 B , 1 N T , while the latter is an isolated fixed point. Thus, upon resolving 2 C does not receive contributions from the resolution moduli, we may infer coordinates are left invariant by the coordinates transform as www.desy.de/ 1 ∼ , 1 is proportional to 1 1 − , there are now P P blown up at 4 h 1 Mathematica , and a disjoint O3 plane. Thus, one may think of the singular orientifold P 1 C B P parameterized by ( integrated over the divisor transversally intersecting the exceptional 2 4 whose combined action leaves the configuration matrix invariant. C C while c ~ S that are swapped. 1 , Next, we exhaust all choices of distinct involutions First, we exhaust all possible choices of pairs of swaps of rows 1 Finally, on both resolution branches, we have an The would-be singular point axion contributing to the orientifold even Hodge number T , and defined as (a) 4 (b) 1 ˜ 3.5 The algorithm Having laid out the prerequisites, weentifolds. now explain The how precisely we enumeratecan the be set found of ori- here: are intrinsically odd underand the we orientifold are action, left the withC associated an axions are projectedHodge out, number it from geometrical data of the deformed side CY. P from the tip of theof resolved conifold harmonic three-form of of the conifold, the geometric A-type and B-type resolutions are relatedbrane by stack a eats flop transition, up underD3 an which tadpole. an O3 SO(8) plane, This seven- while is changing similar its to topology a transition incomponents through a furnished orbifold way by the singularities that resolution described preserves modulus, the in [ given by which is be replaced by the surface B the fixed divisor the conifold, weexceptional are left withof an the O7 conifold plane asresolution, wrapping containing the an O3 plane collapsed onto an O7 plane. For the B-type the two projective coordinates [ invariant if the Thus, on the JHEP05(2020)107 , 2 Z factor (3.23) (3.24) (3.25) (3.22) n P -redundan- 2 Z gauge symmetry 2 , Z } } subject to two gauge ] ] . This fixes the gauge n n and will be relevant in P x +1 p − − n n . . , . . . , x ~ P     ,..., +1 ∼ I +1 n entries in 2 +1 p which are mapped to themselves , n I 2 +1       n 2 +1 , x n 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 1 n x self-dual rows I 2 +1 − ,     n , x     2 +1 gauge symmetry, and each column collects − n parity vectors 2), and no row and column swaps. The first 2 , Z 0 −→ , factor, one simply goes through the involutions – 15 – ,..., 1 1 11 0 01 0 1 02 0 0 01 1 0 1 0 1 1 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 0 , n , . . . , x x row reduction 1     0 P 1). , distinct x − 1 1 1 1 3 [ [     gauge equivalences P P P P P , 2 7→ 7→ = (1       Z ] ] +1 n n , I odd, and involutions 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 +1 , n     , . . . , x , . . . , x 0 0 x x [ [ = (+1 { P ↔ { ). The upper bound on the number of inverted coordinates can be seen as factors with actions. These will be referred to as ] n 2 2 +1 P Z n n [ I for -charges of one of the four polynomials. 2 Finally, we remove the cases with co-dimension one singularities. Then, we exhaust all choices of In general, one finds a gauge fixing constraint that fixes a maximal set of Z ,..., n 1 (c) (d) I Thus we gauge fix the cies by taking theand charge fixing matrix, the reducing polynomials associated modulopositive. with two, In the applying the first above row-reduction non-vanishing example over entry we in get each row to be where each row corresponds to an unfixed and orientifold involution with three rows and the last one are self-dual, so we produce a charge matrix and the polynomials transform with chargeentry equal modulo to the 2. corresponding As configuration matrix an example, consider the CICY 7701, of generality. Second, for everythat self-dual exchanges row the there involutions remains an unfixed a gauge fixing constraint forexcept the under the what follows. (c) is donefixing by constraints: listing first, all the combinations polynomials of that are swapped can be chosen even without loss (a) can bethat done is using not brute-force exchanged( with methods. any other (b) is trivial as well: for each JHEP05(2020)107 1 1 , , of 2 + 2 + 1 h h , 1 − (3.27) (3.28) (3.26) − h 1 , and 1 + h 1 , 1 + h 7], subject to the O , the value of . Before we proceed, 1 , it is given by 1 − is the sum of the Euler h B 3.2 f units of D3 charge. This is χ 1 4 . ]. As a consistency check we ). The results match. − 1 38 B corrected CS action of D7 branes − from this information alone. 0 , 1 α , , f 1 + CY χ cf B h χ 1 4 4 n − . It is given by + = also by counting monomials in examples f , or additional isolated fixed points hosting 0 B χ 1 B , χ 2 − 3 + h D SO(8) = locally 1 , – 16 – Q 1 − B h χ − ). We use the Lefschetz fixed point theorem to -invariant divisor in the class 8[ 1 ). As we are considering favourable CICY descrip- = , 2 1 2 − , is the number of conifold singularities that reside on 1 Z , 1 − 2 − , , h cf D 1 1 h , h , , n 1 1 − 2 + , h 1 + h ( h ( factors that are swapped. As we do not know the dimension n , the Euler characteristics that we use are those of the resolution P −→ ), such that the axio-dilaton does not run. In the generic situation, −→ Z 1 is obtained by adding twice the number of conifold singularities to 1 , , , 3.4 2 1 h h in terms of CY in that the monodromy transformations around seven-brane stacks are in 1 χ , and 2 − is the Euler characteristic computed by ignoring contributions from the h 0 B 3.3 χ simplest and O7 planes, and the factthe that each O3 plane carries the center of Sl(2 the D7 branes are split offD7 the brane O7 plane wrapping and a have recombined generic into a single-component in a configuration whereplane, 4 thus D7 canceling branes the sit D7 on tadpole top of the fixed point locus, i.e. the O7 as follows from a straightforward expansion of the the Euler characteristic ofhave the computed CICY the we resulting startwhere value with this of [ method can be applied (seeFor e.g. each appendix orientifold, we compute the induced D3 brane charge on the seven branes where conifold singularities, and the divisor. Depending onis which attributed resolution either branch to points is blown chosen,O3 up on the planes. latter contribution characteristics of the fixedsections divisors and thebranches number of of the O3 conifold planes. singularities. For As each explained fixed divisor in From this, and knowledgefollows, of but the without CYseparately. extra Euler The input Euler characteristic, characteristic we the of value do the fixed of not point know set how to compute of the resolution branch, we cannot determine The splitting compute The splitting tions, and the conifold resolutionthe cycles CICY do coincides not with contribute the to number one of of pairs the of ambient space. This, in turn, is given by the Finally, (d) is solved by the graph theory exercise described in • • • let us comment on the data that we extract for each orientifold: JHEP05(2020)107 } , = 0 =0 4 (3.32) (3.31) (3.29) (3.30) 0 f x ]. The x denotes χ = = 0. The using the 1 4 3 3 ). For the 40 , 1 , 1 O , x , } 1 − 2 − n 39 h = h 3.28 2 ,... x } 1 = , 3 5], number 7890 of ) denote the Eu- ] | i 1 O 4 4 7 x x cf O P , n : . 1 , n 3 i } ∪ { O 7 ··· ~q O : . , = 0 {{ 1 3 , d , 0 i x 7] descends in an obvious way . The resulting distribution is 7 7] } x : O O { O 0 χ 1 = 63 U(1) x ,... } − Q − + 7[ 1 [ 7 orientifolds.html cf O 4 3 200 7−→ , n D SO(8) ] 1 − = (1)), the ambient space divisor 7 4 Q x O − ~ P − : , d 1 – 17 – = 4 = 14. We compute the value of 7 ··· 4 / O ] because we may use the inverted coordinates as : 4 55 + 1 0 U(1) x , χ ) and number of conifold singularities on the i-th webpage/cicy x 1 2 [ Q − 7 ) 1 : O − c ~q , ( ] = 0 + 4 R {{ ··· 40 , hosting an O7 plane. Its Euler number is easily computed P 1 , : 1 2 − , ≡ is the number of O3 planes in this set, and 2 − 1 h D x descends to an isolated fixed point inside the CY hosting an O3 i −→ , h 3 − 1 } 4 , O . 1 − : P n which acts as 4 0 : = 0 , h x 4 1 4 1 [ 4 13 I I x CY encodes the ambient space fixed locus from which the i-th set of westphal/orientifold = (1). Its fixed locus is the disjoint union = , χ 7−→ i ∼ ~ I ~ 3 P 7 ) corresponds to the surface wrapped by the O7 on the A-type resolution branch. ] , i x 4 7 c /O x = 55. Near the second fixed locus it is more useful to write the involution as O ~ 3 S = : O , d 2 D ~q ~ i S, 7 x χ , from an ambient spaceshown class, in we figure also compute resulting D3 tadpole is [ matches the flux-less F-theorysmall result, number and of can orientifolds be where much the larger O7 than divisor ( [ constraint that it intersects the O7 plane only along double curves [ O ··· ~ I χ = : ( 1 0 13 x x plane. Thus, the D3index tadpole theorem, is (55 + 1) descends to a CYto divisor be [ local transverse coordinates toit the is fixed anti-symmetric point. across{ the As fixed the point, weight so of the the co-dimension polynomial four ambient is space odd, locus or shorthand with co-dimension one respectivelyto four be in symmetric the under ambient the space. involution Choosing (i.e. the polynomial 3.6.1 The quinticWe threefold follow traditional practice inthe starting CICY-list. with the Clearly quintic we threefoldfirst cannot [ swap involution is two rows so all orientifolds will have O7 plane divisor, the total numberwww.desy.de/ of O3 planes.3.6 The Examples full list of orientifolds can be found here: where O3 planes orler the characteristic, degree i-th ( O7 plane divisor descends, ( For each orientifold we produce an entry JHEP05(2020)107 , . 2 D P × (3.40) (3.37) (3.33) (3.38) (3.39) (3.34) (3.36) 2 1). The s. There P 2 − P = ( , . . . . ~ } P ] } } } 4 = 6. Moreover, 4 6 6 = 3. The normal / 12 , x , 14 , : , D }} }} . The anti-symmetric 3 {} . χ 5 5 2 }} , x , , = 16 (3.35) 1 } : } H , }} 2 2 2 } 2 9 2 , x 9 ) = : 1 = 47 {{{ {{{ , , , 1 5 + 5 F {{{ 3 − · x , , }} }} = 5 isolated fixed points hosting N } 2 − 0 ( }} 2 16 4 2 , : 16 0 · − c 7 , , {{ 0 H 9 5 , − 5 , x , , } , 4 4 0), acting on only one of the 3 − 2 = 1 0 # and P [ , , 55 , 19 P R 2 } , 0 , 1 H } , } E 200 + 2 × = 1 1 7−→ = (1 2 − 36 3 + ] 1 1 4 1 {{{ , P ~ I 4 so it has Euler number , H } {{{ − {{{ E ) = 2 x , 1 – 18 – , 4 · 47 : 1 4 F P , ) CY 63 (5). Thus, we encounter P P 47 0 R 2 , N {{{ , F = (2), i.e. , ··· ( ⊂ P 0 , 0 " 1 : , 4 , N ~ I c O 2 ( 47 0 168 1 P , x c 200 0 [ 168 − , 5 + 3 + 16 ) = , , so − − − 2 } , , E = 47. The entry is is a , ) 1 } ( } 144 4 1 1 1 (1) F , O { 2 D P − 2 − {− − , , P N , h = 0 + , ( } 1 O 2 1 {} 1 {} divisor with nine conifold singularities along it. The D3 tadpole is , c { , −→ { 2 − , , 2 2 , h 4 P {} P {} = 36. Then, across the second fixed locus, the polynomial is anti- {} P , {} Z , , } , } : D 1 2 4 {} 1 {} χ , , I {{ {{ } } 4 = 12, and 2 1 / , , 0 0 {{ {{ hosting an O7 plane rather than a locus of co-dimension two. Thus, has to beOne chosen symmetric computes acrosssymmetric the giving first a locus(36 which + 3 then + 9) descends to a divisor 3.6.2 Anti-canonical hypersurfaceAs in a second example wewith CICY would number like 7884. to First, considerare we an consider two anti-canonical ambient hypersurface space in fixed loci, of co-dimensions one respectively two. The polynomial with entry Note that thisresolution is of the the singular conifold singularity. limit It of is a given by smooth an CICY orientifold of orientifold, the obtained CICY 7885, via the B- conifold singularities along it. Therefore, the D3 tadpole is (5 + 3 + 16) The corresponding entry in our list is bundle is isomorphic to polynomial takes values in with two ambient space fixedmust loci be of co-dimensions chosen two anti-symmetric respectively acrossD three. the first The polynomial locus sosecond that fixed it locus descends descends again to5 to a O3 a set divisor planes. of The divisor Next, let us consider the involution In total, we produce the following entry for this orientifold, JHEP05(2020)107 . 9 = dP cf has a n (3.47) (3.45) (3.46) (3.42) (3.43) (3.41) 2 +2 P = 39, and = 35. The 1 . . 0 CY , 1 2 − } } , χ so the divisor 2 − h 6 . The omitted 12 , , h d = 3 , } ) H {} d 6 {} , , CY , H , χ }} {} }} 0 , }} 0 , , }} 13 . This gives four ambient 0 12 4 = 6, and , T , / 8 21 − 4 = 6, and , , 1) , 12 / 4 9 , , = 1. The diagonal ) = (1 + , , 24 } } 3 . , 2 F 1 , 1 , , } 1 − = 21 (3.44) = (1 N {{ 1 6 h ( , , 2 , ~ I c ) {{{} } , {{{} d ,    0 }} , {{{ , , 1 H    which pulls back to 6 36 0 , 36 , , } 2 , 39 1 , 2 1 , 36 H , + (6 , 0 , , 2 d } 1 1 1 2 0 3 120 1 + 3 {{ H 120 2 12 0 01 1 1 1 , − – 19 – 136 6 1 − 1 2 2 , } · factors giving , − {{{ 4). The polynomial must be chosen antisymmetric } 3 H P P P 2 2 2 d , , } , , 1 2 } 1 3 P P P , } H , 1 47    P 1 1 3 = 3 , 3 , ,    {− , 0 1 2 , {− − , E { , 2 , d {{ {} , , {} H 144 }} } , 3 3 3 − }} , , , {} 2 2 } , } , 2 } 1 1 , {{ , 1 , 1 , 1 {{ 1 { , }} , } 2 {{{ {{ 0 , {} , 1 , 0 surface hosting 21 conifold singularities. The B-type resolution of this divisor with 9 conifold singularities. The B-resolution leads to the smooth {{ {} 136. No such CY threefold is contained in the CICY list. 2 {{ 2 , , P − } } P 0 0 , , 0 1 , , has been blown up at 21 points producing a surface which is not nef. , and along it one finds 13 conifold singularities. The other fixed loci give rise to 0 divisor has been blown up at nine points, producing the rational elliptic surface 0 1 2 2 {{ P {{ P P Finally, we may swap the two We may also consider involuting both factors, i.e. × 162 + 26 = 1 containing a orientifold gives the CICY with number 7846, conifold singularities. Therefore,orientifold the entry D3 reads tadpole is (3 + 21) normal bundle isomorphic to its tangentcontains bundle so − with orientifold entry The anti-symmetric polynomial is in The Euler characteristic of a resolution of the conifold singularities is space fixed loci of co-dimensionsacross (2 the first locusP so that it descends to3 + a 3 divisor + 1 inside = thethe 7 CY. orientifold O3 It entry planes. reads has the Thus, topology the of D3 tadpole is (4 + 13 + 7) with orientifold-entry The orientifold of CICY number 7875, containing a JHEP05(2020)107 4], | ) = 2 cf D P (3.53) (3.51) (3.52) (3.48) (3.49) (3.50) [ , n × D 8, and its 1 = (2), and , d P − ~ I D , , . = χ } } } the inclusion of 8 8 0 D , , , χ }} }} }} with ( 4 4 8 . , , , , D } } } T of topology } 2 2 2 without 8 1) , D − {{{ {{{ {{{ , }} , , , contained in the CICY list. 4 , }} }} }} } 0 4 0 = (1 2 , not , , , 4 approximately half of the resolved 0 , ~ 1 P , . 16 − 1 , 49 , {{ − 1 # 24 , , , , , h 0 } 28 8 } , T , 4 1 − , } 2) , } 1 , 168 } 1 {{{ 1 , − , 1 {{{ , , 0 14 1 1 1 } = (0 49 – 20 – {{{ 1 , , {{{ 49 4 2] (CICY number 7889) specified by , ~ I 0 | , , 1 1 5 , 5 41 0 , P P , , P }} 1 , 0 0 { 168 , , " # , 0 168 − , , {} − 168 } , , , at larger values of 24 1 − } 1 , , {} , 0 2 4 1 1 − 1 } , } , 1 h } − 1 1 , , 2 { 1 4 2 − , 1 , , , 1 P P 1 1 {} { {{{ { {{ " , , , {} Euler characteristic and are therefore {} {} , , , } 16. There are no conifold singularities along it, so the induced D3 charge 2 {} {} , , − {{ . One may show that there is a single O7 plane divisor } } 2 4 2] 2 T = | , , positive 5 0 1) d 0 P 4] is a genus two Riemann surface. Its Euler characteristic is [ − | {{ {{ , 2 4), as well as four O3 planes. The orientifold entry is , P Note that while the ability to describe at least one of the resolution branches of a In this case, the two different resolutions of the conifold lead to different orientifolds 0 , = (1 ~ It is alsoeither possible D3 to branes or find three-formtadpole. orientifold fluxes. As vacua These an that are example, precisely are consider those the consistent with CICY vanishing 7888 overall and D3 its smooth orientifold specified by given singular CICY orientifold viafrequently a at smooth low orientifold values of of CYs a have different CICY appears quite 3.6.4 An example with vanishing D3 tadpole and four extra O3 planes have been produced. on the seven branesof is 8 +2. isolated The fixed secondcontribution points ambient from in space the the fixed seven CY,fluxes point branes. hosting and locus The D3 O3 descends D3 branes. planes to tadpole The that a therefore orientifold precisely set entry vanishes cancel is in the the positive absence of The first ambient space fixedand point [ locus descendsdegree to is a divisor The B-type resolution leads to where K3 is blown up at four points. The A-type resolution leads to containing a K3 surface with 4 conifold singularities. of the same CY with CICY number 7886, Finally, we consider the orientifold of [ P (24 3.6.3 JHEP05(2020)107 . . ) 2 6 σ 2 ) 2 B ) T B 8 B ¯ T K, ( do not ( (3.54) 1 0 T dP 8 c ( H B 1 dP while the c R ∈ ≡ i B× f B o ’s. Out of the i,CY d n D P with 2 ] which can be ob- is a rational elliptic g 2 is a union of a set of 9 42 f 1 + dP B 1 g ) = 2. Thus, any global section 1 ]). 8 f almost del-Pezzo surface are 41 = hosting conifold singularities. can be obtained as a complete that is Poincarédual to 1 ¯ , where 8 F } , K, dP B 9 ( 7 ) 0 dP 1 o j,CY dP H B ˜ ( D × 1 with can also be written as a CI manifold. c 9 9 · 9 , . . . , dP is one of the ten del Pezzo surfaces and o i dP 1 are del Pezzo surfaces. It is well known dP ]. dP D B 2 , 42 1 × 1 , dP B 1 1 B Z B P – 21 – = 1 orientifolds we discard these. The other 48 can be expanded as × 8 1 ) where N = 12 P . First, consider an involution that acts non-trivially dP involutions of del Pezzo surfaces have been exhibited where , , σ 2 3 2 B 2 P o i,CY involutions of the ambient space come in two classes. I: Z isolated fixed points. Choosing the anti-canonical section B × { D ], 2 ]. Thus, the anti-canonical hypersurface in each pairing χ fp × B Z 16 n 1 41 = 0 that cannot be deformed (i.e. Bertini’s theorem cannot be applied). descend to generically smooth CY O7 divisors B 2 o i g D = 1 blown up at nine points. g 2 = , containing a number of conifold singularities equal to the degree of P 2 and a set of 2 ). Then, a general CY hypersurface has conifold singularities at the . A boring reason for discarding it is that anti-canonical hypersurfaces of 8 f B 8 o i = D There exist 28 pairs ( dP 1 ¯ . As we are interested in f K, dP 2 14 ( 0 ]. T symmetric the . H 2 17 2 × ∈ Z The anti-canonical hypersurfaces in all equivalent to the Schoen manifold (Theorem 3.1 of [ appears in the list of CICYs. tained as an anti-canonical hypersurfacesurface, in i.e. in an ambient space that the del-Pezzo surfaces intersection manifold [ 3 i 15 matrices describe the same manifold, the Schoen manifold [ 33 matrices describe CY threefolds that can be rewritten as favourable hypersurfaces isolated fixed points descend to CY O7 divisors We start with I. All possible g We exclude 'B K of the anti-canonical line bundle on 14 (a) (b) 1 fp appear in the CICY list.F A better reason forand doing so is thatpoints dim where They can, however, be resolvedThe by result blowing is up the the Schoen single manifold point constructed on in ref. [ and vanishing degree. Eachthe isolated topology of fixed point contributes a del Pezzo CY divisor with only on (say) the firstfixed ambient divisors space factor. Itsto fixed be locus inside n It is straightforward to show that the former have Euler characteristic combinations of involutions ofB the two factors, and II: swaps of thein two ref. factors [ whenever is a non-trivial involution, see table 3.7.1 CY hypersurfacesLet in us products first of focus del-Pezzo on surfaces (a). The matrices fall into two classes [ Now we turn todescend the from remaining the CICY ambientare space manifolds not divisors whose favourable via complete with intersection set respect70 with to configuration of the matrices an divisors CY that ambient does threefold, space areor not left that i.e. non-favourable, is they a a total product of of 22 matrices are either 3.7 The non-favourable CICYs JHEP05(2020)107 1 2 j 0. B and > × C 2 ) (3.58) (3.55) (3.56) 1 i 1 defined C ) has at 2 1 ⊂ B −C ], or it has , ) (3.57) 2 1 ) 2 ¯ 2 − C × B C K B ( B 1 · ¯ ∈ C ¯ ¯ K K K have non-trivial ( ¯ · p · K ] = [ 2 is anti-symmetric 2 ⊂ B j , O 2 ¯ j 1 K C 2 C g . σ )( . Therefore, we may 1 )( 1 1 1 2 B × C B C ) B ¯ 1 ¯ 2 K ¯ K . As K · C , B · 1 ]. · ( in the CY if we choose the 1 i vanishes identically the CY 1 i 1 B 1 i . In the latter case, since C c 2 17 C 1 1 C 2 i = 12 with Euler characteristic B X × C Z 2 − C j ! G/f ) + ( 1 2 fp ) + 2( + ¯ 1 fp C K 1 n . The fixed surface has the topology B 2 × C n B B ¯ 1 ¯ K ). If )). Now, pick a point i + K i χ 1 · , we must require that C ˜ 2 · C ∼ ≡ B C ) + C 1 B i ( 1 1 i 1 C ¯ − C C K B ( O fp ( ( · 2 n 2 ¯ 1 ) K Γ( 2 ) 'B j c 1 4 2 j ( ) acting non-trivially on both del-Pezzo factors, 2 , and two, curves for each pairing of isolated j C · 2 C and vice versa, and three, isolated fixed points 2 ∈ 2 . As just explained, if both C O , – 22 – ( + o i 1 ( i 1 2 j , σ B f D − B X B 1 − of topology 1 ) σ ) o i,CY 2 B 1 fp 2 ←→ B o ij B D Z n B 1 ¯ χ D . Thus, we need that the line bundle ¯ K i K + B 2 · · i X 2 j X 2 fp 2 j

C × C C n ( 1 4 ( 2 1 2 1 fp ) C ) = 3 = 1 i n 1 i 0 for all effective curves C 3 to be anti-symmetric across the surface C ( = ( D > F 3 − Q − which is a section of O 2 ·C 2 − ) 1 n ) we get a non-trivial fixed divisor ) with fixed curves in 2 j ¯ 2 2 j K C C 1 C ( ) which is a global anti-canonical section on , are fixed curves inside ( B 2 G/f p 1 2 ) ( ) 2 C 1 , i 1 i 1 i F C C C under the involution, it vanishes at least linearly along ≡ = ( 1 = ( C G o ij cf D The class II of orientifolds comes from involutions that interchange two isomorphic Second, we consider involutions that act non-trivially on both del Pezzo factors. The n χ a vanishing locus which isis another Fano, and effective thus curve This is not met by all the del-Pezzo involutionsambient classified space in del [ Pezzo surfaces via the vanishing of twodefine global sections across globally define threefold is singular along least one non-trivial global section. This section is either constant, i.e. [ In order to avoid singularities ofrequirements. co-dimension smaller Consider than an three involution we ( mustand impose consider additional one of thefixed two curves factors, say ( anti-canonical section conifold singularities. All otherplanes ambient computed space as fixed loci give rise to a total number of O3 containing for each pairings oforientifold, isolated the fixed anti-canonical points section has in togiving the be rise two chosen to ambient anti-symmetric a space across set the factors. first of For locus CY an divisors O3/O7 where the last equality is shown to hold by going throughambient the space list fixed of involutions point onewhere locus by one. has the various differentfixed components. points One, in surfaces The D3 tadpole takes a universal value, JHEP05(2020)107 is an (3.59) (3.64) (3.61) (3.62) (3.63) (3.60) 9 are the i dP e have been = 10 and a ]. 9 1 , 43 1 dP h coordinates. 1 and the P 1 . , ] P , B ) . d 16 z : ) = 6. A list of all = 0 + 2 B y 3 d coordinates and the latter z B blown up at nine points. As − ] , : χ ), which are also sections of 2 2 1 ij x , t P + 2 δ P = 1 : t − 2 B [ 2 3 . t x, y, z )) χ = g : B (2 j ( . 1 . For an O3/O7 orientifold, the anti- + CY e 1 4 orientifolds.html 1 t | i ] B ( c 2 ! 2 T ¯ xz = 7→ K ] , e 2 + (2 ) ) that satisfy z 3 2 , t = [ 9 ) : 1 D – 23 – = 0 t , and the zero section can be identified with the B [ i y Q i ( 2 CY 1 1 2 30 0 0 1 1 1 e : 1 | g ] − c

1 x 2 , . . . , e =1 + webpage/cicy i : ¯ 1 K 3 [ − , l e 2 ) are the projective P x t ) 2 l, e : B − + , t ( = 1 1 1 l t 2 z t 2 c 2 l y :( is the vanishing locus of B = 3 Z 9 B coordinates that we label ( with generically twelve degenerate fibers. It has is a rational elliptic surface, i.e. ¯ K = ≡ − 1) 1 9 2 dP P P − . The fibration can be given a Weierstrass representation as follows: cf P 3) hypersurface in the toric variety specified by the GLSM ( 9 ). dP , n 1 e P 19). A favourable description is given by the anti-canonical hypersurfaces O , westphal/orientifold , generic degree four respectively six polynomials in the ], and we now give a very brief account of them, following [ ) = 20, there must exists two distinct ambient space divisor classes that 9 ∼ where 3 18 (3) g 9 1 dP P dP ) = (19 × O 1 is the proper transform of the hyperplane class of the base and , , 9 × 2 l 2 9 g (2) , h dP , and the normal bundle is isomorphic to The first basic involution is given by 1 ( 1 , B dP P 1 1 , 1 h O with The first twothree are coordinates projective ( ( the anti-canonical class exceptional divisor consider a degree (6 standard basis of divisor classes ( where exceptional divisors associated with the blown up points. The class of the generic fiber is Up to this subtlety,what the we discussion of discussed orientifoldsanalyzed in of in the the [ previous Schoenelliptic manifold section. fibration is over analogous Holomorphic to involutions of ( in h become equivalent once pulled back to the CY threefold, namely [ the non-favourable orientifoldswww.desy.de/ and some of their3.7.2 relevant data The Schoen canThe manifold Schoen be manifold is found represented here: by the remaining 15 configuration matrices, and has canonical section has to beof anti-symmetric conifold across singularities the is fixed surface. given by Therefore, the number The D3 tadpole is thus given by of JHEP05(2020)107 . (3.65) 1 , , while 1 − ξ h given by 9 ]). We have e = 0. Finally, = 0 with the 18 ] such sections 1 . t y = 4. Moreover, 18 1 0 , 1 − 4 8 4 6 D h which is the disjoint · } ¯ , K ) , its intersection with the fp 4 0 4 0 o 1 z = 0 , and the values of N : D z fp y N : = 1 orientifolds.html c x x · . For one such choice, the action - 4 0 4 : 9 o = 2 t D 2 dP t − : o ) i 1 is an involution for any section i t D e } ∪ { e ( B y ). As shown in section 4 of [ =2 =1 8 i ξ 7→ webpage/cicy 1) 9 i = 0 D ( ) - − P B y ∪ ) by noting the following. According to table 1 z P ( – 24 – i : 9 1) e − ◦ = − e l y − ξ l 2 t : =2 t surfaces, the fixed divisor 3 8 i . As these are exceptional divisors, the involution in = 0 in the base and four isolated fixed points on the 2(3 x . 9 is the disjoint union of the zero section fixed divisor 9 has been fully worked out (see table 1 of [ ξ P e : 1 ) = ( t ξ o t dP ξ 2 } ∪ { t defined by the transverse intersection of − ) t ( ) D ↔ l : B y B B ) ) 1 ) = 0 1 α e D ) without isolated fixed points. We summarize these results in B and B 1) t α i ) 1 e t − − 1) B α ]. At the level of the configuration matrices, these are described , the number of isolated fixed points :( ( { ( 1 − − c =1 ◦ 38 ◦ , α westphal/orientifold ( B , denoted ( i 7 · , ξ , ξ ξ B α ∼ o 9 1 9 P , t , t 1) D 9 9 involution involutions of − dP dP − ( l ( exchanges 2 dP dP = 9, and there are no isolated fixed points. ( Z ( B 2 y 1) D = 0. The former is in the anti-canonical class so has self-intersection zero. ] one finds that the involution one ends up after the blow-down indeed has one has no fixed divisor but four isolated fixed points on the fiber − has a fixed divisor that intersects the generic fiber at four points, and whose ( 2 17 B t ◦ B α . Some ξ with no isolated fixed points. By comparing the action on cohomology with the www.desy.de/ ◦ t = 0 with the divisor 1 and 1) is only an involution if . The resulting orientifolds, are grouped with the other orientifolds of non-favourable ξ ] 7 = 0. Thus, we can blow down both exceptional divisors and obtain an involution − − 1 t z B Two further involutions are obtained by composing the above with a translation along The second basic involution is given by ( 18 α 9 dP = ◦ = e ◦ 1 2 9 ξ ξ 4 Application: ultra lightThe throat CICYs axions are famously known tovia form conifold a transitions web [ of geometric phases connected to each other of results of [ fixed divisor in 2(3 table CICYs: t class can be identifiedof with [ 2(3 fact exchanges the twoe exceptional curves which do not intersect the fixed locus because the fiber by at section can be found at particularon loci cohomology of in ( the modulithat space of and the fixed point setunion is of the fiberfiber over over the point x hypersurface. This intersectse the generic fiber at four points, so Table 1 anti-canonical divisor which acts trivially onorigin. the base and Its can fixed be point shown divisor to reflect points in each fiber over the JHEP05(2020)107 ] ], B 17 44 , (4.1) 2 symmetry, . Following 2 Z over the inde- 2 B effective = 2 level, it must be and N 2 C i i ]: the deformation branch can a 37 gauge theory, and the resolution +1 =1 1 n i , 2 h P increases. At the level of type II CY 1 M , h 2 h must also increase by at least one toward 1 −→ , 1 − # h symmetric del Pezzo surfaces that are connected – 25 – resolution cycles must shrink to a set of conifold +1 2 1 n Z = 1 flux compactifications there are arise new inter- jumps at least by one. One caveat that we need to a 1 , N involution associated with the O3/O7 orientifold must = 2 transition locus counts the number of exponentially 1 − ··· ··· h 2 1 N Z ) whose field excursion weakly twists the various throats decreases while 1 ]. , 0 1 independent M a 1 must increase toward the resolved side. Second, the set of extra " h 45 , 1 ]. These axions can be identified with the massless axions one , 2 1 + h thraxions across the 21 , 7 orientifolded 1 1 , 1 O h ] and Greene, Morrison, and Strominger [ 36 ]. Backreaction of fluxes replaces the conifold regions by warped throats with -compatible blow up/down transitions which has been listed in figure 10 of [ 2 5 , Z 2 It is easy to find explicit conifold transitions between orientifolds of anti-canonical sur- However, there are two necessary conditions that have to be satisfied in order for the Moreover, momentarily disregarding the truncation of the spectrum due to orientifold- At the level of determinantal splittings and find transitions acrosspay which attention to is the following.by If a two blow up/down transitionpoints, have the different number number of of fixed O-planesof divisors will the and/or jump transition isolated across locus, fixed the some transition of locus. the stacks Thus, are in on the the vicinity brink of merging or collapsing. While interchange them. faces in products of dela Pezzo surfaces blow that up/down satisfy all of properties.descends exceptional to As a divisors recalled conifold in in transition appendix in eitherweb the of of CY threefold. the Thus, two we del are instructed Pezzo to factors inspect generically the words, across the locus light axions one obtains onout the by resolved the side O7 ofthe orientifold resolved the side. transition projection, must Thus, i.e. not two singularities, be and the fully geometric projected superpotential of order the deformation parameter. thraxions to actually bepossible part to of the cross light thenamely spectrum. transition the First, locus one at we in the would a like way to that use preserves to a define geometric an O3/O7 orientifold projection. In other ing, the change in light complex axions ( against each other [ obtains on thependent resolved resolution side two-cycles. of On the the transition deformed by side integrating they receive an exponentially small esting phenomena: generic three-formand fluxes more stabilize all so, complex genericallyspace structure stabilize [ moduli them [ exponentiallyexponential close red-shifting [ to conifold points in moduli through such a transition, compactifications the low energyby physics Strominger near [ the conifoldbe transition thought loci of was as understood thebranch Coulomb corresponds branch to of the a bulk partial U(1) Higgsing. The manifolds describedtheir by Euler the numbers matrices are on different. both If sides they are, are the inequivalent splitting if is and called only if by JHEP05(2020)107 ). (4.2) by two. ) 1 1 , 1 , decrease by 1) 48) ), such that 2 h , , 1 , P 1 − ( 4.1 h 1 0 4 3 0 2 1 × ) orientifolds.html ) = (2 3 1 and , ) = (27 1 − 1 1 , σ , , # fixed points 2 − 1 1 + , h P h 1 , , h 1 + 1 , × h 2 + 1 ( h P ( ( webpage/cicy 1 1 0 1 1 1 1 −→ 2 # fixed divisors ) 2) 40) 1 = 2 conifold transitions. Upon stabilizing , , , 2 ) N 3 P ) ) ) ( 4 2 2 ) , σ ) = (3 1 × – 26 – 1 , σ , σ , σ 6= 0. In addition, analogously to what we have , σ , ) = (19 P ) 3 2 1 2 1 − 1 3 1 , , P × 2 − ]. ) − 1 ( , h ) dP dP dP , σ 1 1 westphal/orientifold h 1 2 ( ( ( 3 , , h dJ P 1 + 1 ∼ , ( , σ h , σ + 2 dP −→ 6 ( ( 5 521 orientifolds in our list that fulfill this requirement (see h −→ −→ −→ ) ( , 3 ) ) ) dP −→ dP 3 1 2 ( ( ) , σ 3 2 , σ , σ , σ 5 4 3 −→ −→ , σ −→ dP 3 ( ) dP dP dP ) 2 ( ( ( 3 dP ) ( , σ 3) www.desy.de/ 32) , σ 1 −→ 8 : , , 7 , −→ −→ −→ ) 2 2 ) ) ) dP del-Pezzo transitions −→ P dP 1 1 1 ( -invariant del-Pezzo transitions that lead to conifold singularities in the associated ( ( ) , σ 2 2 4 , σ , σ , σ ) = (4 Z × 94% of the cases with 7 6 5 1 , ) = (11 ) , σ dP 1 − 1 ∼ 2 5 , ( dP dP dP 2 − ( ( ( candidates , h , σ dP 1 . The 5 , , h ( 1 + 1 , h 2 + dP For the orientifolds of favourable CICYs, one searches for involutions that swap at Let us look in detail at the CY orientifold transitions obtained from the anti-canonical ( ( h ( collapsed as well. There arethraxion 319 These are said for thether conifold require transitions that the in O3 the and non-favourable O7 cases, planes are one not might transformed want by the to conifold fur- transition. warped throats with onesmall respectively superpotential two independent as complex described axions in with [ exponentially least one pair ofnon-vanishing entry. rows, If each this containing isthe only the second case, one’s row the and that first is zero’s, row obtained can with be after at the collapsed using splitting most ( still one contains common only one’s and can thus be singularities at generic pointsappear in 9 moduli conifold space.O7 singularities planes Across (18 and each in of thatorientifolding. the the can Locally, double subsequently two this looks transitions cover) be justthe there that deformed like deformation the in are modes a not near way one located that or on both is of top compatible the with of conifold the singularities, one obtains strongly which both the numberWe of then expect isolated that fixed the pointsdiscussion. seven-branes and and We O3 connected list planes the fixed do blow divisors not up/down is play maps an invariant. in important table rolehypersurfaces for in the the chain of transitions In each of the three orientifolds the D3 tadpole is equal to 12, and there are no conifold it would be interestingwhen to a analyze CY the orientifold spectrumlike is to of avoid stabilized light this degrees in complication. of such Thus, freedom a we that region will appears of restrict moduli to blow-up/down space, transitions for across now we would Table 2 CY threefold that avoidone. the In O-plane the loci. first five lines, Across two each subsequent transitions transition are both possible leading to jumps of JHEP05(2020)107 2 = (4.5) (4.3) (4.4) / | thrax- ) each ( CY 15 indicating χ ∆ | pair 0) respectively, − , row i orientifolds.html ) = (12 . , 2 ) transform as    , d 1 }} , 2 2 − χ fold , h − , 1 , 1 1 1 2 0 3 o 2 + , j 47) , h , and an index 1 2 2 , 1 , 0) and ( P P P webpage/cicy 1 − , 28 fold 0), and all parities are positive. The CICY , , − j , h    0 o 0 1 { , i , , , 1 + 0 } (3 h , −→ 0 ) = (36 , 1 pair        − −→ , d – 27 – 1 row χ = (1 39) , i , I 0), and again all parities are positive. On both sides , 20 fold 1 , , − 1 o , westphal/orientifold , i 0 00 1 11 1 0 00 1 0 0 1 0 3 1 1 1 0 ∼ (4 = (0 I 1 1 1 2 2 CICY P P P P P i {{        -th pair of rows interchanged by the involution is collapsed. The second www.desy.de/ : pair − row i , the position of the orientifold in our list We leave the phenomenology of this interesting class of axion-models for future work. transitions This list is not complete because we have only considered cases where a suitably obvious set of swaps 15 CICY We are gratefulGendler, for Mariana useful Graña,Arthur discussions Hebecker, with Manki Andreas Kim, Liam Braun, McAllister,of Mehmet rows Nicole Demirtas, and Righi, columnsconfiguration Naomi brings matrices the recorded configuration in the matrix CICY-list. obtained via the splitting to the form of one of the Here, we note only that thekinetic examples matrix. with two Then, axions at we the haveanother considered classical in feature level, a that the diagonal their two axion-sectors scalar are potentials sequestered simply from add one- up. Acknowledgments and on the transition18 locus conifold of singularities with theThese two double-cover can homology of be relations grouped the into amongaction. orientifold a the pair there shrinking of are 9 three-spheres. conifolds each that is interchanged by the orientifold columns are interchanged, there are two smooth O7so divisors the with ( D3-tadpole is 12. The Hodge numbers ( On the resolved side,are the interchanged second by and thedeformed third involution, side row is as reached well by as collapsing the the second two and interchanged third rows. column No further rows and resolution of frozen conifold singularities. Here is an example, i that the entry specifies the deformedpairs side of CICY-orientifold. orientifolds 1,279 of of CICYsto these that orientifolds are are of transitions smooth the between away larger from the set transition of locus, smooth as CYs opposed that are reached from the CICYs via the ion encoded in an entry where the first entry corresponds to the resolved side and contains the CICY number Filtering out these, we obtain a list of 11,533 orientifold conifold transitions JHEP05(2020)107 (A.3) (A.1) (A.2) 8 points, ··· , and Krull , C 0. A del Pezzo . We denote by k = 0 > . In the following k is a complete irre- 2 dP ·C P F , k , so in particular it is ¯ K ' +1 the exceptional divisors, 0 ··· k , i is ample. In the following B e dP X = 1 ' ¯ K k and blow up , F 2 , and i k , 2 e P P , i, j − k =1 is positive, i.e. ] for a more detailed treatment. ij i X δ ¯ 41 K = 9 − − , and let us start with l 2 is B = ) 7 generic points. We will call the first class k j e ··· dP · ) = 3 , ( k i – 28 – 1 . Note that trivially c degree k dP k = 0 ( , e 1 dP dP k c over an algebraically closed field Z . Its = 0 = k = . It is well known that X and i k ¯ e d K 1 F · P × , l 1 P and blow up = 1 any of those 10 del Pezzo surfaces. 1 l P B · l × 1 and the latter P k dP Let us discuss the topological properties of From now on, we will only consider del Pezzo surfaces over the field By definition a Fano variety the proper transform of the hyperplane class of the , the intersection with the anti-canonical divisor and its Euler characteristic is 3 + The anti-canonical class of the del Pezzo surfaces is given by of surfaces enough to consider just we will denote with l associated to the blown-up points. The intersection numbers read surface is a smooth projective Fano variety of Krulldimension dimension coincides 2. with ordinarysurfaces dimension can in be differential classified geometry. asor Complex follows: we del can we Pezzo can take either take ducible algebraic variety suchwe that will its only anticanonical focus sheaf ondefinition Gorenstein it Fano has varieties, to soof be the the anticanonical ample. anticanonical bundle bundle The existsC is Nakai–Moishezon and criterion equivalent by to implies the that requirement the that ampleness for every effective curve A Basic properties ofIn del this Pezzo appendix surfaces properties we of recall del a Pezzo surfaces. quick See collection for of example basic [ facts about algebraic geometrical 647995. FC and AWunder the are HORIZON supported 2020 grant by agreementsupported the no. by ERC 647995. the Consolidator The Deutsche Grant work Forschungsgemeinschaft- under of STRINGFLATION AW EXC Germany’s was 2121 Excellence also partially Strategy “Quantumby Universe” the - National 390833306. Science Foundation Finally, under this Grant work No. was PHY-1748958. supported in part indebted to Andreas Braunmany helpful for discussions. collaboration JMits in and the warm AW early hospitality would likepart stages while to by of this the thank the Simons work the FoundationConsolidator project was KITP Origins Grant Santa finished. and of STRINGFLATION Barbara the for Universe under for The Initiative the and work HORIZON in of part 2020 by JM grant the was agreement ERC no. supported in Fabian Rühle,Raffaele Savelli, Cumrun Vafa, and Timo Weigand. We are particularly JHEP05(2020)107 V da , and (A.4) 6 ,      P'xp 1 is simply the − 2 1 0 1 1 1 1 0 y of ooo = 8, and it also P      d = . i −→ 1 E k dP dP are given in figure . , degree 0 1 P          H this is no longer true. Let us 1

× − k i 1 v V + 2 da aa 1 0 P H ,Q − 1 0 0      and = 2 − 3 pp 1 1 0 1 0 0 1 0 1 1 01 1 11 1 0 10 1 10 0 0 0 0 0 c marked points (the ones that are blown up)               ez of ooo aa ooo k Vz = = 1 3 – 29 – , . . . , dP P 0 dP × 1 dP P DR ) are easily determined from the automorphism group of b e B

, ﬁrst Chern class 0 ,Q = 0 corresponds to the exceptional divisor V v e i H da aa e        1 now. Being the direct product of two projective planes, it has two − 1 and are toric varieties, while for higher ) that leaves the set of ). Their image under the blow-down map ,Q P 1 0 C C k I , H , − ×    dP 1 1 11 0 10 1 0 0 1 1 1 1 0 1 P K y of ooo aa ooo           s.t. the locus = = i 3 the e 2 2 , P . The fans of the toric del Pezzo surfaces. The blue edges are associated with toric 2 dP , Q DR Q 1 , The fans for the toric del Pezzos which is PGL(3 Vs i Y = 0 2

The automorphism groups Aut( P subgroup of PGL(3 the associated GLSM charge matrices are k briefly move to hyperplane classes admit a toric description. Figure 6 coordinates and measures the number of points where two generic anti-canonical divisors intersect. For JHEP05(2020)107 . # (A.9) (A.5) (A.6) (A.7) 2 2 (A.10) , i.e. } which is 0 , 1 2 ¯ ) K P P 2 P " = 2 ]: ) be the degree 2 blown up points = L ) the Todd class d 46 7 k B ( , # T d ) dim Aut( 6= 1 can be written as a ¯ (resp. , dP K − d i , 1 k d B 1 1 1 1 2 3 ( { 2 , d . H } 2 2 3 2 + 1 (A.8) 0 , P P P , d } − . 0 " h 10 ). As each of the ) = max d, ) = = = B − + 1 2 is positive. Choosing B ) + dim B 3 6 ( d d ( ,T ¯ − 2 ) = 10 L K cs { B , B h T d ( 10 ) = ) B 1 ) that in fact . The Grothendieck-Riemann-Roch theorem ( { , dP ¯ ¯ , dP 1 B K K H    ( , i A.8 H – 30 – B ch ( ) = max 0 Z B dim H dim Aut ( 1 0 0 1 1 1 ) = max 2 2 = − ) = dim B − ( ) B ) 1 1 2 dim ( 4 cs ¯ K P P P B P h cs ( , as well. We may also write h . The Kodaira-vanishing theorem implies the vanishing of B cs    h 0 if the line bundle dim Aut( 1 ( B h . We would also like to know the number of complex structure 0 P = = 1 H 2 5 P i > × 1 × P 1 del Pezzo, and in determining their number of Kählerand complex dim P , dP , dP 2 , 1 # # ) for all B 1 1 1 2 ). 2 ⊗ L 1 2 1 2 ) is the Chern class of the anticanonical bundle, and B with P P P P ¯ K 2 ,K ( " " B ( ch i = = (resp × B 1 For the Kählermoduli, we will use the following theorem of Kollár[ Finally, we recall that all the del Pezzo surfaces with degree 4 1 H 1 B B dP dP In the main text,in we are interested instructure CY moduli. threefolds In that thisof arise appendix as we fill anti-canonical in divisors some details. Let B The anticanonical hypersurface in a product of two del Pezzos appear in the CICY list. One of many ways to represent them is as follows, positive for Fano spaces, we deduce from ( complete intersection of hypersurfaces in a product of projective spaces. This is why they where of (the tangent bundledim on) A further important quantitycanonical is line the bundle dimension on of asays the del that space Pezzo surface of global sections of the anti- which again holds for which holds also for moduli of a delis Pezzo labeled surface by two complex coordinates we have invariant. Their dimensions are JHEP05(2020)107 , , 2 ), 2 1 , Z 2 − ¯ K and be a h , (B.5) (B.2) (B.3) (B.4) × B 2 1 1 d X B 1 ( B ), but the d ⊂ 0 + 2 H B.5 Y − intersects the = k 16 . Let dP CY 4 χ ≥ conifold singularities in 2 ) )+ 2 2 B B dim(X) ( )+ ), is employed as a cross-check. ) be the Kleiman-Mori cone of 1 ¯ ) c K ). This way of computing complex structure moduli as co- . -even global sections , 1 2 involution acting only on the first 2 X 2 1 2 B 3.26 , B ( d ) (B.1) 2 2 Z R ( B 1 . , ( h = 1. Thus, upon blowing down an Z X d + cs 2 2 0 ( = i dim Aut( g h d NE + H , the anti-canonical divisor will contain · 2 E . 1 − − d 2 2 · ) that are represented as the vanishing loci NE − ) d p 1 2 ) k 1 1 d ) by dim B k − + B 1 − → ( · (resp. dP k − 1 1 B ) ) cs dP ) ( d g ( ¯ h for a given Y · K dP – 31 – 1 cs Y ( , − c 1 ( 1 h , 1 p 2 − = 20 in ) + B h 1 1 ( 2 NE , = NE 0 ) increases with the degree. This point is given by the dim Aut( B 1 : = 20 D , one obtains ( ¯ f h 1 ∗ H K − 1 cs i , , B 2 1 h B h and . Let ) by the vector space of ( , so there are no conifold singularities. − + be a smooth Fano variety with 0 k 1 ¯ X K ¯ = dim H , D dP K X 1 1 , B ), and replace 2 ( 1 h 0 B ( of the respective divisor line bundles. Because this point is contained H + 2 , 1 p is empty in of the anti-canonical line bundle that can be expanded as 2 ) by the subgroup of the automorphism group that commutes with the f (Kollár)Let D ) implies that we can think of all of the 1 · B 1 ). Then the natural inclusion B.4 D X , we replace 1 B Furthermore, it is important to note that upon blowing down an exceptional divisor It is useful to rewrite this as follows This theorem implies that in our case under study, all the moduli Kähler of the CY The number of complex structure moduli can be determined from the Euler charac- Of course, also before the blow-down the generic section can be expanded as in eq. ( (resp. 16 in the anti-canonical divisor weglobal started section with, the anti-canonical divisor is represented by a intersection anti-canonical divisor at oneexceptional point, divisor to i.e. a point,this point. thus creating This is whyintersection dim of two divisors of global sections action, denoted Aut rather than employing the Lefschetz fixed point formula ( of the first ambientthe space CY. factor This is seen most easily as follows: every exceptional divisor of subtracting one for thephism overall group, scaling and and finally the addingspace. the dimension Clearly, complex if of structure we the want modulifactor ambient to inherited compute space from automor- the ambient replace Aut( Equation ( efficients appearing in a general global section of the anti-canonical line bundle of teristic by subtraction. Usingtherefore the adjunction formula one determines threefold simply descend from the ambient space ones, so smooth divisor in theY class is an isomorphism. Theorem 1. JHEP05(2020)107 . , , . k 3 } } 3 3 3 e e ≤ and dP (C.3) (C.4) (C.1) (C.2) n n , x ,D 2 2 e e ] 2 , x e 1 ,D , i.e. by the e 1 D blown up at . 2 e , x B 3 2 3 , P ,D with coefficients v ] + [ 2 3 1 to each generator , 2 v e is , x . Generically, the decreases by B ρ 2 1 D v k 1 x ,D − , = 1 2 k 1 only the first remains del Pezzo surfaces, i.e. , x v dP h 1 2 ] + [ v dP we delete the second and 3 P ,D x , i v , , , 1 ], and are listed in table 1 { v 0 ) ) ) ' on toric D 1 2 3 17 D dP e e e 0 > 2 { x x x . i L 1 1 1 ] ] ] t , , ] = [ dP 1. Since 3 1 2 − 1 − 2 − 3 ρ 3 ) e e e e 3 − D 0, for , λ , λ , λ v D D D D 3 3 2 2 v v v ]. All divisors have positive volume if , x d and → i x x x 2 . e 1 2 3 v ] + [ ] + [ ] + [ ] 3 [ ] + [ j , e 2 3 3 i 3 1 v v v t e e , x , λ , λ , λ , and for D 1 2 1 1 6= 1 D D D = 0 produces a conifold singularity in the D v v v v D e respectively =0 3 i x x x x 2 ( 1 2 3 involutions of the 1 g 0 P λ λ λ , i L ] = [ ] = [ ] = [ ] + [ 2 ] = [ – 32 – λ ( ( ( −→ 2 3 3 2 3 3 = Z e e e − v e , 3 [ ∼ ∼ ∼ ∼ ] increases by 2 1 D D D e l D ) and set [ ) ) ) ) , g 1 0 3 1 2 3 , t e e e v 2 , . We associate a toric coordinate = h ] + [ ] + [ ] + [ C.1 ] 3 = 1 ] = [ , x , x , x , x = 1 1 2 2 3 2 3 2 3 2 v v v e e producing more severe singularities than conifolds in the v v v v dP p J D D D 1 newly acquired complex structure moduli. Thus, we have D D Σ(1) of the fan. It is useful to first consider the case of [ [ [ , x , x , x , x = 2 1 1 1 − ∈ v v v v , i, j and 2 1 x x x ] + [ ] = [ x ρ 0 d p ( ( ( ( 2 2 2 e 0 and relabel 1). e [ without > D − j , dP → t 1 2 , 2 d , − ] + [ ] ( 1 1 1 i e n , dP v e t are global sections of the anti-canonical line bundle of 1 D . Of course, once the exceptional divisor is blown down, we can deform the D D 2 P − [ 2 g 0 B × t ≡ decreases by one, and 1 ] ] = [ of l 1 1 and P [ , e we delete the last row of ( , 2 1 [ 1 0 d h 2 g dP increases by Let us explain in detail the possible dP 1 , ' 2 We expand the form Kähler as For third rows, set Let us work in a basis of divisor classes that satisfy homology relations To each toric coordinate we associate a toric divisor P of a one-dimensional cone There are four scaling relations among the six toric coordinates h C Involutions of delThe Pezzo involutions surfaces of del Pezzo surfaces have been classified in [ singularity by dialing the described a conifold transitionwhich in the CYgeneric from points, no the pair resolved ofof to them coincides. exceptional the Therefore, divisors deformed we side maythreefold. blow across Using down this, any number one obtains conifold transitions across which that are global sectionscommon of vanishing locus a lineCY bundle because all fourthe sections generic will number vanish of linearly.degree intersections between The two number anti-canonical of divisors such in points is given by where JHEP05(2020)107 ). 0 H,H 2 F F F F F H 1 2 ⊕ ⊕ F F F 1 1 [2pt] ⊕ ⊕ ⊕ F F F 3 1 3 3 5 7 8 ⊕ ⊕ ⊕ F 1 2 2 2 2 3 3 4 ⊕ ⊕ 1 1 ⊕ ⊕ ⊕ F F F (3) (2) (5) (9) (9) F dP dP dP dP dP 1 1 1 1 1 1 1 1 3 3 5 5 5 I I I I I ⊕ 1 2 3 ⊕ ⊕ ⊕ ⊕ ⊕ (3) (2) (5) dP dP dP (5) (5) 1 1 1 dP dP 3 5 5 I I I 1 2 3 I I ) respectively ( (3) dP (5) (5) 1 1 1 n dP dP action on I I I 1 , the number of isolated fixed , , . . . , e 1 1 − 0 1 0 0 0 0 1 0 0 1 2 1 0 2 1 4 2 3 2 4 2 7 5 4 3 8 5 4 1 c h · o 1 l, e D fp involutions of del Pezzo surfaces, the 1 0 4 0 2 0 1 3 1 4 0 2 2 1 3 0 4 0 2 1 3 0 0 2 4 1 1 3 2 N Z 1 c · - - - - 3 4 4 2 4 3 1 3 4 2 2 3 1 4 4 2 3 1 4 4 2 3 3 1 o . D 0 H ) – 33 – 3 o e ]: the possible i i ) ( ←→ i i i i i i e e D 2 2 2 ) e e e e e e 2 2 2 17 0 e e e ) ∪ 0 e e e ( H 1 =1 =1 ) H 1 3 1 7 i 8 i =1 =1 =1 =1 =1 =1 e − − H 2 ∪ ( 5 i 6 i 5 i 7 i 6 i 8 i − − − e e e ( e 1 1 l l l - - - - ) 1 1 1 P P ∪ e e + 1 ∪ P P P P P P − − − e e e 2 3 − ) e l l l ) − − l − − − − − − H 1 − − − ( − − H l l − e l l l l l l l l l ( l l 2 2 l 3 3 3 3 3 3 ( 6 9 , and actions on the divisor classes ( − fixed divisor 1 l , ( 1 − , respectively h j e ) ) ) 3 2 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 2 1 3 2 1 4 3 2 1 2 1 3 2 1 2 1 3 2 1 2 1 ) ←→ , σ , σ , σ G B dJ 1 1 1 i , σ , σ , σ , σ , σ , σ , σ , σ , σ , σ , σ , σ , σ , σ , σ , σ , σ , σ , σ , σ , σ e , σ , σ , σ P P P , σ 1 1 2 2 2 3 3 3 3 4 4 5 5 5 6 6 7 7 7 8 8 , its intersection with the anti-canonical divisor 2 7 8 5 o 1 P × × × ( dP dP dP dP dP dP dP dP dP dP dP dP dP dP dP dP dP dP dP dP dP D flip dP dP dP 1 1 1 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( involution , the values of P P P ( ( ( fp . Table adopted from table 6 of [ N denotes a Table 3 fixed divisor points F JHEP05(2020)107 2 Z .A 3 (C.8) (C.9) (C.5) (C.6) (C.7) e (C.10) (C.11) , so it is , , } 3 3 and e e ) ) 3 2 = 0 2 v is the disjoint t t 3 e invariance fixes } − − x invariance of the 2 1 1 t 2 t Z = = 0 fixed point locus is Z − 1 3 − 2 e x 0 0 Z x t t 2 ( . ( = v invariance of the Kähler , so it is of co-dimension 3 = 2. x . . 1 1 − e , , − 2 2 e e 2 2 x 1 2 3 3 2 , x x + Z e e e e e e 1 − 2 3 2 orientifolded by the above ) 2 ) v h x e 3 − − 3 − − x t t . x 1 1 ←→ 1 2 ... 1 = 2 e e + e e v − e 3 − 3 v 2 x × x 1 2 − − 2 e e action, but the action on the divisor − − t t 2 l l x x l l 2 2 2 1 − is obtained by shrinking − v Z dP x , x } ∪ { 0 . ←→ 0 x 1 t 1 t −→ −→ 2 −→ −→ = 1. ( e 1 P ( = e 3 3 x E 1 1 1 = 0 2 , − . As the requirement of × − 2 e 1 − } ∪ { 3 2 1 1 1 x h t x e e 2 1 P , x , e , e ) , e , e ) ←→ ←→ x – 34 – = 0 3 2 = 3 3 3 i i = t v t 2 1 e e e e 1 , so and acts as v 1 1 x i e E t e − t − 3 − − x 3 3 =3 =3 x 2 2 i i 1 is in the Kählercone. Again, for X X 2 1 2 (2) t dP v t =1 v e e I , x 3 i x J ←→ − − x and − 2 is in the Kählercone. For − − − l l e 1 i P − 0 0 + v l 2 2 l actions with co-dimension one fixed point locus. The first t x t t J 2 admit an analogous x ( 2 ( e = 2 e , so the discussion is analogous. The 1 =1 x , symmetry that exchanges Z 0 3 i 2 x − − 1 0 t −→ −→ −→ −→ 1 v l 2 l ←→ v l l P ) 2 2 and two isolated fixed points. x dP Z dP i x 1 e e { i t v 1 = 0. t = x P 1 i , 0 i 1 − X t } ∪ { X = 1. h action is called 1 − , − 2 is defined by reflecting the fan over the origin, but the fixed point locus 1 − 0 = 0 0 t Z admits a h t 3 1 2 e 2 and an isolated fixed point. The action on our basis of divisors is (3) dP x I 2 1 = (2 dP v P 0 = x 0 J admits two distinct − fixed point locus J 3 2 2 e −→ Z x Finally, dP The second −→ 1 J v J x This maps the Kählerform which is also in theKählerform we Kählercone if have to set two. It acts on the divisor classes as which is in theform cone Kähler we if have totwo set linear combinations of the Kählerparameters, we have of this action is given by two constraints This maps the Kählerform The action on the divisor classes is is the same as{ the one of the disjoint union of a Thus, CY threefolds obtained asaction a will hypersurface have of classes is trivial, so geometric action on theon toric the ambient fan. space coordinates translates to a geometric action The union of a and all exceptional divisors are shrunken. JHEP05(2020)107 , ] 423 , Nucl. 10 , Phys. JHEP , , ]. (1985) 299 JHEP ]. , ]. Phys. Rept. ]. , SPIRE B 162 ]. 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