Counting charged massless states in the (0,2) heterotic CFT/geometry connection
aa r X i v : . [ h e p - t h ] D ec TUW-10-14IPhT-T10/159
Counting charged massless states in the(0,2) heterotic CFT/geometry connection
Matteo Beccaria a ∗ , Maximilian Kreuzer b † and Andrea Puhm bcd ‡ a Physics Department, Salento University and INFN, 73100 Lecce, Italy b Institut f¨ur theoretische Physik, TU Vienna,Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria c Institut de Physique Th´eorique, CEA/Saclay,91191 Gif-sur-Yvette Cedex, France d Kavli Institute for Theoretical Physics, Kohn Hall, UCSB,Santa Barbara, CA 93106, USA
Abstract
We use simple current techniques and their relation to orbifolds with discrete torsionfor studying the (0,2) CFT/geometry duality with non-rational internal N = 2 SCFTs.Explicit formulas for the charged spectra of heterotic SO (10) GUT models are computed interms of their extended Poincar´e polynomials and the complementary Poincar´e polynomialwhich can be computed in terms of the elliptic genera. While non-BPS states contributeto the charged spectrum, their contributions can be determined also for non-rational cases.For model building, with generalizations to SU (5) and SM gauge groups, one can takeadvantage of the large class of Landau-Ginzburg orbifold examples. ∗ matteo.beccaria AT le.infn.it † kreuzer AT hep.itp.tuwien.ac.at ‡ puhma AT hep.itp.tuwien.ac.at or andrea.puhm AT cea.fr ontents N = 2 superconformal field theories . . . . . . . . . . . . . 72.2.1 Minimal models, field identifications and mirror symmetry . . . . . . . . 92.3 Symmetries and projections for (0 ,
2) heterotic models . . . . . . . . . . . . . . 102.3.1 Gepner map and generalized GSO projection in (2,2) models . . . . . . . 112.3.2 The (0 ,
2) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.3 Generalized GSO projection and gauge/SUSY breaking for the (0 ,
2) model 15 he heterotic (0,2) CFT/geometry connection 335 Checks and examples 36 P ( x, , t ) = P ( t − , , x )
47B The quintic ⊗ : 47 A beautiful example of the interplay between world-sheet and space-time techniques is theGreene-Plesser (GP) mirror construction [1], which identifies charge conjugation for (tensorproducts of) minimal models with an orbifold construction and thus establishes the mirrorautomorphism for an exactlty solvable point in the moduli space of a string compactification.Deformation arguments can then be used to extend mirror symmetry to the geometrical realm.The setting of the GP construction is a heterotic string whose compactification geometry isreplaced by a tensor product C int = C k ⊗ . . . ⊗ C k r of N = 2 superconformal minimal models withcentral charge c = 9, for which Gepner [2] was able to construct a modular invariant partitionfunction with space-time supersymmetric massless particle spectrum and gauge group E × E .The relation to geometry proceeds via the Landau–Ginzburg (LG) description [3, 4] of minimalmodels by Fermat-type superpotentials W = Φ K + . . . + Φ K r r with K i = k i + 2, which canthen be identified with the hypersurface equation W = 0 defining a Calabi–Yau variety in aweighted projected space. More precisely, the exactly solvable Gepner point is located at smallvalues of the K¨ahler moduli and can be reached as a certain limit in the parameter space ofthe gauged linear sigma model (GLSM) [5].Mirror symmetry has been a pivotal tool in the study of non-perturbative physics for twodecades and is well understood for heterotic (2,2) compactifications [6–8]. From the phe-nomenological perspective, however, (0,2) world-sheet supersymmetry (with quantized charges)is sufficient for low energy space-time SUSY and much more attractive models with realisticGUT gauge groups arising quite naturally. The GLSM provided an important step for theconstruction of such models as it allowed the study of (0,2) deformations away from the (2,2)locus [5] as well as the construction of large classes of genuine (0,2) models with geometricaland Landau-Ginzburg phases, like the Distler-Kachru (DK) models [9].On the rational CFT side a powerful formalism generalizing Gepner’s construction was3eveloped by Schellekens and Yankielowicz [10] in terms of simple currents [11], which arerelated to certain discrete symmetries and, in a sense, can be regarded as a generalization offree fields. String vacua, from this perspective, are constructed by starting with tensor productsof CFTs and performing a number of projections, like generalized GSO or alignments of Ramondsectors. All of these projections can be realized as simple current modular invariants (SCMIs)of extension type [10, 12, 13] and large classes of (0,2) models can be constructed very naturallywith the same techniques. Moreover, the general classification of SCMIs [14] uncovered theirrelation to orbifolds with discrete torsion, enabling translations of results into geometricallanguage and suggesting generalizations beyond the rational realm [13].Like in the case of (2,2) models, a comparison of particle spectra can be performed to look foridentifications of models that are constructed with geometry and CFT methods, respectively.In [15] Blumenhagen and Wißkirchen (BW) indeed discovered a (0,2) cousin of the quintic with80 generations and gauge group SO(10) that showed up on either side, and the constructioncould be extended to a whole family of identifications [16–18]. On the CFT side it is basedon a Gepner-type tensor product, but with an additional simple current J b that acts as a Z twist breaking the E gauge group of the (2,2) model down to E ∼ = SO (10). On the geometryside this corresponds to a rank 4 vector bundle E on a Calabi-Yau manifold X whose data areconstrained by the anomaly matching condition c ( E ) = c ( X ) and make sense also for certainnon-rational theories C int like Landau-Ginzburg models and orbifolds thereof. More precisely,there is a conjectured identification between certain rational (0,2) heterotic strings constructedwith simple current techniques and (0,2) Landau-Ginzburg models, which can then be deformedto large volume in terms of their GLSM realization. The latter is an interesting topic on itsown but is beyond the scope of this note.In the present note we investigate the non-rational generalization of the CFT/geometryconnection proposed in [12, 16] and develop tools for the computation of their massless spectraon the CFT side. Our starting point is the identification of simple current modular invariantswith orbifolds with discrete torsion [12, 14], which can be used to reformulate the constructionof Blumenhagen et al. [15, 17] in a more geometrical language and to extend it, for example, toarbitrary internal N = 2 SCFTs containing a minimal model factor at odd level. The breakingof E to the gauge group D = SO (10) by a simple current J b is thus attributed to discretetorsions spoiling the algebra extension in the gauge sector and corresponds to a Z orbifolding.The main technical point will be the computation of the spectrum in J b -twisted sectors, forwhich non-BPS states turn out to contribute even to non-gauge-singlet massless states.Our construction also has interesting implications for (0,2) mirror symmetry [18–22] becausecharge conjugation is a simple current modular invariant for (tensor products of) N = 2 minimalmodels. According to the general classification [14], the data defining a SCMI is a simple current(or orbifold) group G together with a choice of discrete torsions (in terms of a fractionallyquantized matrix X G with given symmetrization). Since products of SCMIs are again SCMIsthe mirrors of our (0,2) models can be explicitly constructed within the same framework, whichshould explain the large degree of mirror symmetry for orbifold spectra observed in [18, 19].By our extension of the formalism to non-rational models this (0,2) version of the Greene-Plesser construction extends to the Berglund-H¨ubsch mirror construction for Landau-Ginzburg4rbifolds with minimal transversal superpotentials [23]. The precise mirror map for untwistedminimal LG models has been constructed in [24] and can be extended to arbitrary orbifoldswith discrete torsion using the methods developed in [25] by relating discrete torsion to themodding of quantum symmetries [26]. This generalizes and must be consistent with the SCMImirror construction, but in both versions only an algorithm but no explicit formulas for thetwist groups and torsions of the mirror are available. The universality of these constructionssuggests, however, that a purely group-theoretical description should exist and would be veryinteresting to be unveiled.In section 2 we define our class of models and recollect the basis of our formalism, whichat the same time generalizes and simplifies Gepner’s construction within RCFT, and embedsit beyond rationality to orbifolding techniques via the classification of SCMIs. In section 3 wework out explicit formulas for non-singlet matter spectra. In our class of models the breakingof (2,2) to (0,2) models with GUT gauge is due to a twist that spoils alignment of Ramond andNeveu-Schwarz sectors for the left-movers. As a consequence, it turns out that non-BPS statescontribute even to charged matter. Using the simple current orbit structure and spectral flow wecan determine, however, everything in terms of the finite data given by charge degeneracies ofRamond ground states and excited Ramond states of an arbitrary N = 2 SCFT, as encoded inits extended Poincar´e polynomial (EPP) and the complementary Poincar´e polynomial (CPP).In section 4 we discuss the geometry connection and check the correspondence of spectra fornon-rational examples. Examples and some details of the construction are collected in section5 and the appendices. In this section we recollect the ingredients of our construction, as reviewed in more detailin [12]. The discussion is intended to provide an intuitive picture rather than proofs, whichcan be found in the references. We start with simple currents and their relations to orbifoldsand then discuss their application to projections in arbitrary N = 2 SCFT, with a summary ofwhat we need for the special case of minimal models. Then we define our class of (0,2) modelsand discuss space-time SUSY ( i.e. the generalized GSO projection) and the breaking of thegauge group by a simple current J b , which we will refer to as the “Bonn twist”. The left- chiral algebra (or vertex algebra ) A L of a conformal field theory is the holomorphicsubalgebra of the operator algebra. Similarly, the anti-holomorphic fields define the right-chiralalgebra A R . The Hilbert space of states H can thus be organized into representations of thesymmetry algebra A L ⊗ A R , with chiral and antichiral labels a and ¯ b , respectively, labelingcharacters χ a ( τ ) = tr H a exp (cid:0) πiτ ( L − c ) (cid:1) and their right-moving partners χ ¯ b (¯ τ ). If thedecomposition H = L a, ¯ b H a ⊗ H ¯ b is finite the conformal field theory is called rational and the1-loop partition function Z ( τ ) = M a ¯ b χ a ( τ ) χ ¯ b (¯ τ ) can be written in terms of a finite non-negative5nteger matrix M a ¯ b of multiplicities, called modular invariant, with a unique identity M = 1.It will be important below to distinguish between individual conformal fields , labeled bytheir full set of quantum numbers, and conformal families φ a ¯ b ( z, ¯ z ), which consist of all con-formal fields corresponding to a representation H a ⊗ H ¯ b . For simplicity we can think of thediagonal modular invariant as our starting point and only consider left-moving labels a (or,more rigorously, ignore the “chiral” φ a altogether and only refer to representation labels a ).From the operator product algebra we can then extract the associative and commutative fu-sion algebra φ a × φ b = N abc φ c , whose non-negative integer structure constants N abc denote themultiplicity of the field φ c in the OPE φ a × φ b . Simple currents are conformal families J witha unique fusion product, i.e. for which J × φ a = φ Ja for a unique family φ Ja [10]. Examplesare free fermions or vertex operators of free bosons, so that simple currents can be regarded asa generalization of free fields. They decompose the set of conformal families into orbits whichare of finite length φ a → φ Ja → φ J a → . . . → φ a , (2.1)in a rational CFT. The maximal orbit length N J , called the order of J , is the length of theorbit of the identity because J N J = implies that every other orbit length is a divisor of N J .Since all members of a conformal family have the same conformal weight modulo 1, unique-ness of the fusion product of J implies that all branch cuts originating (with slight abuse ofnotation) from OPE singularities of the form ( z − w ) h Ja − h J − h a have the same monodromy phase e − πiQ J ( φ a ) about the singular point, where Q J ( φ a ) ≡ h J + h a − h Ja mod 1 (2.2)is called the monodromy charge Q J of φ a . The important observation is that Q J is conservedmodulo 1 in operator products and thus implies the existence of a phase symmetry φ a → e − πiQ J ( φ a ) φ a , which is a cyclic group Z N J of order N J because it can be shown that the charges Q J are quantized in units of 1 /N J [11].The set of all simple currents of a rational CFT forms a finite abelian group under fusion,called the center. In order to implement the necessary projections for the construction of ourmodels we will work with a fixed subgroup G of the center, for which we can introduce a setof generators G = h J i i of order N i = N J i . Each current J = Q i ( J i ) α i ∈ G can then be writtenas J = P i α i J i in an additive notation, where we identify J ∼ = [ α ] with an integer vector ~α whose components α i are defined modulo N i . It can then be shown that all conformal weights Modular invariance, in this context, usually refers to the conditions [
M, T ] = [
M, S ] = 0 for the represen-tation matrices T and S of the respective SL (2 , Z ) generators on the characters. The full consistency conditionsof conformal field theories require, in addition, appropriate behavior of all correlation functions under factor-ization and mapping class group transformation of Riemann surfaces of arbitrary genus, which fortunately canbe shown to follow from a finite number of constraints (like 2-loop modular invariance or modularity of 1-pointfunctions on the torus). This multiplicity is usually N abc ∈ { , } , except if the conformal Ward identities do not fix all coefficientsof higher descendents in terms of the coefficient of the most singular contribution of the family φ k to the OPEof two operators ˆ φ a ∈ φ a and ˆ φ b ∈ φ b . N abd C dc is the number of independent 3-point conformal blocks in h φ a φ b φ c i , where the charge conjugation matrix C ab is a symmetric permutation matrix related to the fusioncoefficients by C ab = N ab . G can be parametrized in terms ofa matrix R ij [29], R ij = r ij N i ≡ Q i ( J j ) = Q j ( J i ) , h [ α ] ≡ P i r ii α i − P ij α i R ij α j (2.3)with Q i ≡ Q J i and r ij ∈ Z . The definitions of Q i and R ij , in turn, imply h [ α ] a ≡ h a + h [ α ] − α i Q i ( a ) , Q i ([ α ] a ) ≡ Q i ( a ) + R ij α j . (2.4)If N i is odd we can always choose r ii to be even. With this convention all diagonal elements R ii are defined modulo 2 for both, even and odd N i .A simple current modular invariant (SCMI) is a modular invariant with M ab = 0 only if b is on a simple current orbit of a , i.e. if there is a simple current J with b = J a . Becauseof (2.3) and (2.4) T-invariance [
M, T ] = 0 which requires h a − h [ α ] a ∈ Z , implies that simplecurrents J i of even order can only contribute SCMIs if r ii ∈ Z . Subgroups G of the centerviolating this condition can thus be excluded from further consideration so that r ii ∈ Z and h [ α ] ≡ − α i R ij α j . It can now be shown that the most general SCMI is of the form M a, [ α ] a = µ ( a ) Q i δ Z ( Q i ( a ) + X ij α j ) , (2.5)where X is defined modulo 1 and quantized by gcd( N i , N j ) X ij ∈ Z . The multiplicity µ ( a ) isthe order of the stabilizer of G on the orbit of φ a and δ Z is one on integers and 0 otherwise.The formula (2.5) lends itself to an instructive and useful orbifold interpretation [12], where δ Z ( Q i + . . . ) is identified as the projection to states that are invariant under the Z N i phasesymmetries implied by J i and ~α labels the twisted sectors. A simple calculation shows thatlevel matching h a − h [ α ] a ∈ Z fixes the symmetric part X + X T ≡ R modulo 1 for off-diagonaland modulo 2 for diagonal matrix elements, while the ambiguity due to the choice of a properlyquantized antisymmetric part of X exactly corresponds to the freedom due to the choice ofdiscrete torsions of the orbifolding procedure.In conclusion we note that orbit positions α i in SCMIs (2.5) generalize the shift vectors ofGepner’s construction and, via their identification with the labels of twisted sectors, embed itinto the framework of orbifolds, which we will use to generalize heterotic (0,2) models to thenon-rational realm on the CFT side of the proposed geometry/CFT duality. N = 2 superconformal field theories In non-geometrical supersymmetric compactifications the sigma-model on a Calabi-Yau is re-placed by an “internal” N = 2 SCFT C int with c = 9 and a number of projections like charge The proof in [14] uses factorization and regularity assumptions that exclude unphysical solutions. A state-of-the-art approach is based on modular tensor categories [27]; cf. section 4.2 of [28] and references therein. Discrete torsions can be interpreted as phase ambiguities of the orbifold group action on twisted vacua,which are proportional to α j because of the twist selection rules (also known as quantum symmetries [26]).In fact, the formula (2.5) was motivated by universalities observed in the classification efforts of [29] andthe observation that proper account of quantum symmetries was vital for understanding the relation betweenorbifolds and modular invariants in Gepner models [30]. N = 2 algebra is generated by the Fourier modes of the energy mo-mentum tensor T ( z ), its fermionic superpartners G ± ( z ), and a U (1) current J ( z ). For unitarytheories positivity of expectation values of the anticommutator { G − r , G + s } = 2 L r + s − ( r − s ) J r + s + c ( r − ) δ r + s , (2.6)of the superconformal charges implies the inequalities [4] h R ≥ c/ , h NS ≥ | q/ | with L | h, q i = h | h, q i , J | h, q i = q | h, q i (2.7)for r = s = 0 in the Ramond sector r, s ∈ Z and for r = − s = ± / r, s ∈ + Z , respectively. These inequalities are saturated by the “BPS states” | R i = | h = c , q i , | c i = | h, q = 2 h i , | a i = | h, q = − h i , (2.8)called Ramond ground states and (anti)chiral primary states and are defined by G | R i = 0, G + − / | c i = 0 and G −− / | a i = 0, respectively (in addition to being primary!). For (2,2) heteroticstrings, these states completely determine the charged massless spectrum.The N = 2 algebra admits a continuous family of automorphisms known as spectral flow, L n U θ −→ L n + θJ n + c θ δ n , J n U θ −→ J n + c θδ n , G r U θ −→ G ± r ± θ , (2.9)which interpolates between the Ramond and the NS sector. In particular, U ± / maps Ramondground states to chiral and antichiral primary fields, respectively. Spectral flow is best under-stood by bosonization of the U (1) current J ( z ) = i p c ∂X ( z ) in terms of a free field X withnormalization J ( z ) J ( w ) ∼ c / ( z − w ) . A charged operator O q can thus be written as a normalordered product of a vertex operator with a neutral operator O , O q = e i √ c qX O ( ∂X, . . . , ψ, . . . ) (2.10)with the U (1) charge corresponding to the momentum of the vertex operator, whose contribu-tion to h is q c . The inequalities (2.7) hence imply that the maximal charges of R and c statesin unitary theories are c/ c/
3, respectively.We now have all ingredients to discuss the universal center of N = 2 SCFT’s [13]. Alreadyfor N = 1 the supercurrent G is a simple current, which we denote by J v . Its monodromy chargeis Q v = 0 for NS fields and Q v = 1 / h v = 3 / N = 2, in addition, the Ramond ground state J s = e i √ c/ X with maximal charge c/ Q s ≡ − q modulo 1. If the U (1) charges q are quantized in units of1 /M in the NS sector then c = 3 k/M for some integer k . Since the U (1) charges are shifted by − c/ − k/ M in the Ramond sector, the order N s of J s is 2 M if k ∈ Z and 4 M if k Z and the relation between Q s and Q v modulo 1 implies J Ms = J kv , J v = with c = 3 k/M, h J s , J v i ∼ = ( Z M for k ZZ M × Z for k ∈ Z (2.11)8o that the order of the universal center is 4 M in both cases. The best way to compute themonodromy matrix R J v J v = 1 , R J v J s = 1 / , R J s J s = − c/
12 + ( k ∈ Z k Z (2.12)is by first evaluating Q i ( J j ) modulo 1 using h J v = 3 / h J s = c/ h J s = c/ R ii modulo 2 by imposing h J i ≡ − R ii for r ii ∈ Z . The chiral labels a = ( l, m, s ) of φ a ≡ φ lsm for minimal models C k at level k are best understoodfrom their coset representation C k = ( SU (2) k × U (1) ) /U (1) K with K = k + 2 and c = 3 k/K. (2.13)The labels l = 0 , . . . , k and s mod 4 refer to the factors SU (2) k × U (1) in the numerator,and the U (1) K label m is defined modulo 2 K in accord with the convention that U (1) K has2 K representations. Ramond and NS fields correspond to odd and even s , respectively. Theconformal weights and the U (1) charges obey h ≡ l ( l +2) − m K + s mod 1 , q ≡ s − mK mod 2 (2.14)with exact equality in the standard range | m − s | ≤ l , − ≤ s ≤ U (1) and SU (2) k imply that φ lsm is a simple current if l = 0 or l = k .The branching rule l + m + s ∈ Z of the coset implies the necessity of field identifications φ lsm ∼ φ k − l,s +2 m + K = J id × φ lsm with J id = φ k K ⇒ Q id ≡ ( l + m + s ) / Q id ∈ Z of the iden-tification current J id provides the correct selection rule and, since h id ∈ Z , extends the chiralalgebra [11].After field identification we find that the center of C k is exactly the generic center of an N = 2 SCFT with J s := φ ∼ φ k − K , J v := φ ∼ φ k K and M = k + 2 = K. (2.16)Note that the general parametrization c = 3 k/M of the central charge was chosen above inorder to emphasize the analogy of k with the level of the minimal model, namely that J Ms = J kv determines the group structure (2.11) of the center, while the inverse charge quantum 1 /M isin general unrelated to k .The Landau-Ginzburg description of a minimal model C k requires a simple chiral superfieldΦ with superpotential W = Φ K whose chiral ring [4] is generated by Φ modulo ∂W ∼ Φ K − .9e hence expect k + 1 chiral primary fields Φ l , whose chiral labels are easily checked to be φ l, − l by comparing charges, conformal weights and fusion rules. The remaining BPS states anti-chiral primary: q = − lK R ground states: q = ± ( c − lK ) chiral primary: q = lK φ l l ∼ φ k − l, K + l ∼ Φ l φ l, ± ± ( l +1) ∼ φ k − l, ∓ ∓ ( k − l +1) φ l − l ∼ φ k − l, K − l ∼ Φ l (2.17)can then be identified, for example, by charge conjugation and spectral flow.It is instructive to study the orbit structure of the center for minimal models. Taking intoaccount the selection rule Q id ∈ Z and field identifications we have 2 K ( k + 1) chiral labelsand 4 K simple currents so that we have to expect fixed points for k ∈ Z . Indeed, since J νs J αv × φ l,sm = φ l,s + ν +2 αm + ν the orbits are parametrized by l , which can be restricted to l ≤ k/ l → k − l , which leads to an orbit of length 2 K with multiplicity µ ( l ) = 2 stabilized by φ K = J Ks J k/ v for l = k/ k ∈ Z .Note that in general each orbit contains exactly two BPS states of each type. Considering,for example, chiral primaries φ l − l we use field identification to find its partner with l ′ = k − l at the orbit position φ l ′ − l ′ = J l +1) s J lv φ l − l . For mirror symmetry we, instead, need to implementcharge conjugation φ lsm → φ l, − s − m by fusion with a simple current φ , − s − m = J − ms J m − sv with m mod K and s mod 4. Due to the orbit structure charge conjugation is a SCMI, denotedby C a,Ja , which is determined by the group G and the discrete torsion X of the orbifoldingprocedure. A convenient choice of basis for the generators of the group is G = h J = J s J v = φ , , J = J v = φ , i because the SCMI then splits according to C a,Ja = C m →− m × C s →− s . From(2.27) we can calculate the symmetric part of the torsion matrix X ( ij ) = R ij to be X = K and X = − , while the antisymmetric part corresponding to the discrete torsion in the orbifoldingprocedure vanishes. (0 , heterotic models Let us review the structure of a generic four-dimensional compactification of the (2,2) heteroticstring. The right-moving sector consists of four space-time coordinates and their superpartners( X µ , ψ µ ), a ghost plus superghost system ( b, c, β, γ ), and an ”internal” N = 2, c = 9 SCFT C int which is the abstract version of a supersymmetric sigma model on a Calabi-Yau. Theleft-moving sector is a bosonic string with space-time plus ghost part ( X µ , b, c ) and the sameinternal sector C int so that a left-moving CFT with central charge 13 needs to be added forcriticality. Modular invariance requires this CFT to be either an ˆ E × ˆ D or ˆ D level 1 affine Liealgebra, where we will henceforth ignore the phenomenologically less attractive ˆ D . Insteadof this covariant quantization we can also use light-cone gauge, which amounts to ignoring the(super-) ghosts and restricting the space-time coordinates to transverse directions. We thushave two components ( µ = 2 ,
3) of the space time bosons X µ ( z, z ) and fermions ψ µ ( z ). Theright-moving sector is a conformal field theory with c = 12 composed by10wo copies of the free right-moving SCFT ( X, ψ ) : c = 2 × = 3,an internal N = 2 SCFT with the central charge : c = 9.The left-moving sector is a conformal field theory with c = 24 composed bytwo copies of the free left-moving boson CFT : c = 2,an ( b E ) × ( b D ) Kaˇc-Moody algebra : c = 8+5 = 13,an internal N = 2 SCFT with the central charge : c = 9.In the context of a sigma model on a Calabi-Yau manifold the superstring vacuum is thenobtained by aligning space-time spinors and tensors with internal Ramond and Neveu-Schwarzsectors, respectively, and carrying out the (generalized) GSO projection. This can be under-stood in terms of SCMI’s of extension type which we will discuss below.In order to apply simple current techniques [10], as introduced in the previous sections, toour heterotic (0 ,
2) models, we start with a left-right symmetric theory which can be achievedby applying the so-called Gepner map to the right-movers. This map dates back to [31, 32].The fact that it preserves modular invariance and spin-statistics signs in the partition functionwas proved in the context of the covariant lattice construction [33]. Later, it was applied byGepner in order to relate type-II superstrings to heterotic strings [2].Using the language of simple currents and SCMI’s we will then be able to carry out the(generalized) GSO projection and break the gauge group E of Gepner’s construction [2] to SO (10) by the means of a simple current J b , which we call the Bonn twist . World-sheetsupersymmetry will be accordingly reduced from (2 ,
2) to (0 , The right-moving free space-time fermions form a representation of ( b D ) . The spectrum fallsinto representations of this algebra which must be unbroken being the light-cone gauge remnantof Lorentz invariance. The one loop partition function is a product of the contributions from thespace time fields (bosons and fermions), the internal SCFT and the left-moving ( b E ) × ( b D ) Kaˇc-Moody algebra. For application of SCMI techniques it is convenient to cast the theory ina left-right symmetric form. The asymmetry is focused on the following factorsleft-movers : ( b E ) × ( b D ) , right-movers : ( b D ) (from ψ ( z ))Symmetry can be achieved by exploitng a remarkable map that exchanges space time fermionswith compactified internal bosons while preserving modular invariance [2]. Thus, it can mapa fully bosonic partition function to a superstring or heterotic one. Conversely, starting froma heterotic partition function, we can apply the map to the right-moving sector and obtain aleft-right symmetric theory suitable for simple current techniques [10].The affine algebra ( b D n ) has four integrable highest weight representations, the singlet ,the vector v, the spinor s and conjugate spinor ¯s. The only integrable representation of ( b E )
11s the singlet . The heterotic one loop partition function involves a bilinear combination ofthe ( b D n ) characters of representations with n = 5 from the left-movers and n = 1 from theright-movers. We can arrange the characters in a vector χ = ( , v , s , ¯s). Let us look at themodular transformation properties of χ . Under S : τ → − τ , we have χ ( − τ ) = S n χ ( τ ) , S n = 12 − − − i − n − i − n − − i − n i − n . (2.18)Under T : τ → τ + 1, we have χ ( τ + 1) = T n χ ( τ ) , T n = e − i π n/ diag(1 , − , e i π n/ , e i π n/ ) . (2.19)The singlet of ( b E ) is invariant under S and gets the phase e − π i/ under T .From these relations one sees that it is possible to replace characters of ( b D ) with charactersof ( b D ) while preserving modular invariance. The precise mapping of characters ( b D ) → ( b E ) × ( b D ) is provided by the Gepner map ( , v) → × ( v, ) , (s , ¯s) → − × ( s, s ) . (2.21)Indeed, one can check that defining M = − − , (2.22)one has M S M = S , M T M = e − π i/ T . (2.23)The minus sign of fermionic characters has a double role. On the one hand it is required tofulfill modular T invariance, i.e. level matching, and on the other hand it enforces the spin-statistics condition which requires bosons and fermions to appear in the partition functionwith opposite multiplicities. After the Gepner map states in a (2 ,
2) model have the structureΦ (2 , = φ C int ⊗ χ SO (10) . The construction is completed by two additional steps leading towell-defined spin-structures and space-time supersymmetry. R/NS alignment . Consistent quantization of the gauge fixed N = 1 supergravity theoryrequires that the Ramond and NS sectors of the space-time and internal sectors are aligned.After (2.21) this implies that D spinor representations are aligned with the Ramond sector ofthe internal SCFT. Alignment can be implemented by a SCMI that extends the chiral algebra In general, under T , the affine character of b g k associated to the integrable weight b λ gets the phase e π i m b λ where the modular anomaly m b λ can be expressed in terms of the Weyl vector ρ and dual Coxeter number g of g according to m b λ = | λ + ρ | k + g ) − | ρ | g . (2.20)For the singlet of ( b E ) we have k = 1, λ = 0, g = 30, | ρ | = 620 and one recovers the quoted phase.
12y the current J RNS = J v ⊗ v (which has conformal weight h RNS = + = 2) because Q J v ≡ / Q v ≡ / D spinors. Similarly, in the case of a Gepnermodel, where the internal SCFT C int = N i C k i is a tensor product of N = 2 minimal models,the alignment can be implemented as a SCMI extending the chiral algebra by all bilinears ofthe respective supercurrents J ij = J v i J v j , where h ij = 3. In the following, we shall keep thealignment procedure explicit because we shall be interested in (0 ,
2) models for which the chiralalgebra extension that implements the alignment only takes place in the right-moving sector,where it is needed for consistency.
Space-time supersymmetry . We are interested in four dimensional space-time supersym-metry. Thus, we want to perform a further projection to a theory which admits a conservedsupersymmetry charge exchanging bosonic and fermionic fields. In the string theory, this isnothing but a map between the Neveu-Schwarz and Ramond sectors. In N = 2 SCFT’s, anatural candidate is the total spectral flow operator, i.e. the simple current J GSO = J s ⊗ s .It has integral conformal weight h GSO = c/
24 + 5 / Q GSO = − q , where q refers to the U (1) charge of a state Φ (2 , , this generalized GSO projection implies a projection to even U (1) charges in the bosonic string and,according to (2.21), to odd U (1) charges in the Gepner construction of the superstring [2] whenthe space-time contribution is taken into account.As a final comment, we recall that the mechanism that implements space-time SUSY in thefermionic string is closely related, by the bosonic string map, to the mechanism that extends E × D to the gauge group E × E of a (2 ,
2) compactification. Indeed, the 33 massless vectorbosons that extend the 45 adj of D to the 78 adj of the gauge group E come from the U (1)current of the N = 2 SCFT and 2 ×
16 states associated with ( J GSO ) ± . (0 , model While (2 ,
2) models with E gauge group can be constructed from a 4d bosonic string withinternal CFT given by C int ⊗ SO (10) × E after the Gepner map, the internal CFT needs to besplit into smaller building blocks for (0 ,
2) models in order to be able to break supersymmetryonly in the left-moving sector. We thus decompose C int = C ′ ⊗ F , where C ′ is a general CFTwhile F is a minimal model at odd level k = K −
2. In the Landau-Ginzburg phase F has aFermat-type superpotential W = Φ K and hence will be referred to as Fermat factor . In thegauge sector we start with an SO (8) gauge group which we will then extend to SO (10) in theleft-moving bosonic sector and to E in the right-moving sector which amounts to space-timesupersymmetry after the Gepner map. Our (0 ,
2) models with SO (10) gauge group hence areconstructed from a 4d bosonic string with an internal c = 22 CFT C ′ ⊗ F ⊗ D ⊗ D × E with current algebras D n and E at level 1 and a certain SCMI that will give rise to alignmentof spin structures and the generalized GSO projection. States in a (0 ,
2) model then have thestructure Φ (0 , = φ C ′ ⊗ φ F ⊗ χ D ⊗ χ D .The SCMI that defines the resulting (0 ,
2) models is based on the simple current group generated13y J GSO , J A , J b , J C with J GSO = J s ⊗ J s ⊗ s ⊗ S, J A = 1 ⊗ ⊗ v ⊗ V, J C = J v ⊗ ⊗ ⊗ V (2.24)and the Bonn twist J b = 1 ⊗ ( J Ks J K − v ) ⊗ s ⊗ C ′ ⊗ F ⊗ D ⊗ D since E acts as a spectator.Charges in the Neveu-Schwarz sector are quantized in units of M ′ in C ′ and in units of K in F . The central charge of the minimal model F is c F = kK with k = K − C ′ we can only formally write c ′ = k ′ M ′ . Imposing c ′ + c F = 9 we get c ′ = 6 K +1 K . Since k ′ = M ′ ( K +1) K ∈ Z and K is odd and relatively prime to K + 1 we find that K divides M ′ and that k ′ ∈ Z and hence we get for the order of J GSO N GSO = (cid:26) M ′ for M ′ ∈ Z , M ′ for M ′ ∈ Z + 1 . (2.26)The orders of the alignment currents J A and J C are given by N A = N C = 2 and the order ofthe Bonn twist is N b = 4. Notice, that J b = 1 ⊗ J v ⊗ v ⊗ G is N = 16 M ′ for both, even and odd M ′ , because there is the relation J M ′ GSO = J b amongthe simple currents if M ′ ∈ Z + 1. Thus G can be parametrized by J = J νGSO J αA J βb J γC with α, γ = 0 , β = 0 , , , ν = 0 , ..., M ′ − R , as calculated from the simplecurrent group G , and the torsion matrix X , whose symmetric part X ( ij ) ≡ R ij is determinedby R ij mod 1 for off-diagonal and by R ii mod 2 for diagonal elements while its antisymmetricpart X [ ij ] ≡ X ij − R ij corresponding to the discrete torsion in the orbifolding procedure is apriori subject to choice. Since the right-moving sector of our (0 ,
2) model is equivalent to thatof a (2 ,
2) model we choose X [ ij ] such that we have full Ramond/Neveu-Schwarz alignment inthe right-moving sector. The non-vanishing monodromies between the simple currents J A , J b , J C and J GSO are R Ab ≡ mod 1 and R bb ≡ K − mod 2. This fixes the symmetric part of X and in addition we choose X Ab = and X bA = 0. R J
GSO J A J b J C J GSO J A J b K − J C X J
GSO J A J b J C J GSO J A J b K − J C Table 1: Monodromy matrix R and torsion matrix X While the value of the numerator k ′ , like the level k of a minimal model, determines whether the universalcenter h J s , J v i with J M ′ s = J k ′ v and J v = is cyclic or not, the inverse charge quantum M ′ is completelyunrelated to this generalized “level” [12, 13]. .3.3 Generalized GSO projection and gauge/SUSY breaking for the (0 , model We want to construct heterotic string models with GUT gauge group SO (10) and N = 2supersymmetry only in the right-moving sector where it is needed to obtain space-time super-symmetry after the Gepner map to the heterotic string. This can be implemented by SCMI’sthat extend the left and right chiral algebra in an asymmetric way. Thinking of [ J ] = Q J α j j as the twist in the orbifolding procedure we can write the most general SCMI M [ J ] i,i = µ (Φ) Y j δ Z ( Q j (Φ) + α k X kj ) . (2.27)for a field Φ [ J ] i,i where the left-moving part is obtained by twisting the right-moving partwith the current J = J νGSO J αA J βb J γC . There are two types of invariants. Modular invariantsof automorphism type are permutation matrices that uniquely map representation labels ofthe right-movers to the left-movers, where the permutation is an automorphism of the fusionrules. Let us define the kernel Ker Z X as the set of integral solutions [ α ] of X kj α j ∈ Z where[ α ] = [ ν, α, β, γ ]. If this kernel is trivial then ( Q j (Φ) + α k X kj ) ∈ Z has a unique solution [ α ]for each charge, which defines a unique position [ α ]Φ on the orbit that only depends on thecharge Q i (Φ). This yields an automorphism invariant. If X = 0 we obtain a pure extensioninvariant because all fields with non-integral charges are projected out while all fields on ansimple current orbit are combined to new conformal families. X = 0 is only possible if theconformal weights of all simple currents J ∈ G are integral. Since these currents are in theorbit of the identity they extend the chiral algebras A L and A R so that we obtain a new rationalsymmetric and diagonal CFT.Our (0 ,
2) model is given by X = X T = 0 and is an asymmetric combination of an extensionand automorphism type modular invariant partition function. The extension of the right chiralalgebra A R is defined by Ker Z X which yields the extension ˜ A R = h J A , J b , J C , J GSO i . Weobtain the charge projection rules for the right-moving labels which amounts to the vanishingof all monodromy charges Q A ≡ Q C ≡ Q GSO ≡ Q b ≡ α + K − β modulo1. From the form of ˜ A R and from the charge selection rules we see that there is full alignmentin the right-moving sector which justifies the choice of discrete torsion above. Accordingly, theextension of the left-moving algebra is defined by Ker Z X T , i.e. solutions [ α ] of α k X kj ∈ Z , and yields ˜ A L = h J b , J C , J GSO i for K ≡ A L = h J A J b , J C , J GSO i for K ≡ J A and thepresence of the Bonn twist in the left chiral algebra already indicate that supersymmetry willbe broken in the left-moving sector.Since our asymmetric construction builds on a D = SO (8) gauge group we need an ex-tension mechanism to obtain a D = SO (10) gauge group for the left-movers and a E gaugegroup corresponding to space-time supersymmetry after the Gepner map for the right-movers.Motivated by the free fermion construction of D n = SO (2 n ) in terms of 2 n Majorana fermionswith aligned spin structures where the extension of SO (2 m ) ⊗ SO (2 n ) to SO (2 m + 2 n ) is im- Notice that, as explained in [14] and [12], one can also choose to work with M i, [ J ] i and impose projectionson the left-moving states. Our choice, which projects right-moving states is motivated by the simpler structureof the right-moving sector where we have full R/NS alignment and better BPS properties. J = v D m ⊗ v D n , we will carry out ananalogous ”alignment extension” for our tensor product of (S)CFT’s.In the right-moving sector the extension is 2-fold. First we carry out an extension D ⊗ D → D generated by the alignment current J A which is a prerequisite for a consistent Gepner mapto the heterotic string. The further extension generated by J GSO of D → E on the bosonicversion is then mapped to space-time SUSY on the heterotic side. On the left-moving side ourclass of models avoids the J A extension by an appropriate choice of discrete torsion (Table 1)but uses the J GSO extension to promote the gauge group from D to D = SO (10). Alignment extension.
The right-moving alignment extension D ⊗ D → D is generated by J A = v ⊗ V with the charge projection Q A = Q v + Q V , where Q v = Q V = 0 for fields in theNeveu-Schwarz sector and Q v = Q V = for fields in the Ramond sector. Tensor productsof fields from different sectors are projected out while tensor products of fields from the samesectors get combined to new conformal families with aligned spin structures. These are on theorbit generated by J A and read = ⊗ ⊕ v ⊗ V, v = v ⊗ ⊕ ⊗ V, s = s ⊗ S ⊕ ¯ s ⊗ ¯ S, ¯s = ¯ s ⊗ S ⊕ s ⊗ ¯ S, (2.28)where v , s , ¯s denote characters in D and the choice of s and ¯s is convention. Notice that this D ⊗ D → D extension is just a necessary step before applying the Gepner map and is notrelated to the D = SO (10) gauge group in the left-moving sector which will be obtained byan extension using J GSO . Due to our choice of discrete torsion the alignment current J A is notin the left chiral algebra and hence this alignment extension gets avoided in the left-movingsector. GSO extension.
Contrary to the alignment current, J GSO is in both, the left and right chiralalgebra, yielding an extension of the gauge group to SO (10) in the left-moving sector and afurther extension of D → E in the right-moving sector. The space-time part of the chargeprojection of J GSO , denoted by Q s , follows from the conformal dimensions h = 0 , h v = , h s = h ¯s = n and the monodromy formula (2.2) yielding Q s ( ) = 0 and Q s (v) = 12 (2.29)for fields in the NS sector and Q s (s) = n − (cid:26) n ∈ Z / n ∈ Z + 1 and Q s (¯s) = n − (cid:26) / n ∈ Z n ∈ Z + 1 (2.30)for fields in the Ramond sector. Due to the triality of the Dynkin diagram of SO (8) theextension to SO (10) based on J GSO can be understood in terms of the alignment extensionwith a subsequent exchange of the characters V ↔ S of SO (8). The group structure of character fusion is Z for odd n and Z × Z for even n . The multiplication tableis sv = s , s = s = v n , v = 1. The formulas for the charge projection in the right-moving sector are given by the same expressions exceptfor the exchange Q s ↔ Q s . Computation of the charged massless spectrum
After restricting to the massless part of the spectrum, from the representation of the chiralalgebras only the representation of the zero-mode algebras remain which, in the left-movingsector, contains the linearly realized subgroup SO (10) × U (1) ⊂ E , where the abelian part isthe absolutely defined U (1) int with charges q int of the internal N = 2 algebra C int . Likewisethe quantum numbers of SO (2) = D in the maximal subgroup D × D of D = SO (10)are absolutely defined and we can identify the characters s, , s, v of SO (2) with the labels m = − , , , ± U (1) m charges ( h m , q m ) = ( m , m ) for primaryfields Φ m , as introduced in [17]. The label m contributes to Q GSO with a prefactor − , ascan be seen by evaluating, e.g., the contribution of s = Φ m =1 to the GSO charge projection, Q D s ( s ) ≡ h s − h v ≡ − mod 1. With the U (1) m charges [ − , , , ±
1] of the characters[ s, , s, v ], the D charge q D contributes to Q GSO with a prefactor − and we can write thecharge projection by J GSO as 0 ≡ Q GSO ≡ − ( q int + q D ) + Q D S mod 1.Translating this into the language of Distler–Kachru [9] and of Blumenhagen–Wißkirchen[15,17] can now identify the relevant D = SO (10) decompositions under the maximal subgroup SO (8) × U (1) = = − ⊕ s0 ⊕ = v − ⊕ s1 = v1 ⊕ s − . (3.1)The notation is N χ e q , where N is the dimension of the D representation, χ denotes the SO (8)character and e q = q int + q D is the U (1) charge associated with the U (1) current of the SO (10) ⊃ SO (8) × U (1) decomposition that is a linear combination of the U (1) currents of the N = 2algebra of C int and of SO (2) = D .Given the values for e q from the above decomposition and taking into account the space-timecontribution coming from the D factor we can determine the charge contribution q int from theinternal sector. In the right-moving sector unitarity bounds highly restrict the values of q int andlet us determine the spectrum of massless states completely. In fact, the right-moving statesare all BPS states. The structure of N = 2 minimal models further enables us to derive boundson the internal charge also in the left-moving sector the states in which are on the orbit ofthe BPS states of the right-moving sector. This leaves us with a finite set of possible masslessstates of the heterotic (0 ,
2) string.
After the alignment-extension of D × D to D we can perform the Gepner map on the right-moving side D → D = SO (2) LC to obtain space-time quantum numbers (in light-cone gauge)from the representations of D . The SUSY multiplets yielding space-time matter and space-17ime gauge symmetry generators are then assembled by J GSO . Admissible states are selected byimposing the massless condition ¯ h tot = ¯ h st + ¯ h int = and the GSO projection Q GSO ∈ Z onthe bosonized string. They are summarized in table 2 which in addition shows how the Gepnermap G acts on the characters of D to get the associated space-time representation. From left toright we give the space-time conformal weight, the D part of the GSO charge and the internalquantum numbers which are obtained by the charge selection rule 0 ≡ Q GSO ≡ − ¯ q int + Q D s mod 1 and the unitarity bound | ¯ q int | ≤ c = for states in the Ramond sector and | ¯ q int | ≤ h int for states in the NS sector. χ D G → χ SO (2) LC h st − Q D s h int q int state → v v → ± c , a s → − s
18 12 38 − , R ¯s → − s −
12 38 12 , − R Table 2: Right-moving states with internal and space-time quantum numbersSince on the right-moving side we have full R/NS alignment the SO (2) LC representationsare paired with internal states of the same sector. From the condition for massless states andthe unitarity bound it follows that the only admissible internal states are BPS states. In theNS sector the internal states that fulfill the BPS condition ¯ h int = | ¯ q int | are chiral and antichiralstates, denoted by c and a . In the Ramond sector the internal states that satisfy the analogousunitarity bound are Ramond ground states since ¯ h int = c = and are denoted by R . Note,that the R states with ¯ q int = ± have maximal/minimal U (1) charge, respectively.The SUSY multiplets are now assembled by J GSO as follows. While the D (or SO (2) LC )representations are all on the same orbit, the U (1) charges ¯ q int of the internal contribution tomassless states are shifted under spectral flow J C ′ s ⊗ J F s ⊆ J GSO by ± , which quickly hits theunitarity bound | ¯ q int | ≤ c for Ramond ground states and | ¯ q int | ≤ h int in the NS sector.Chiral multiplets consist of the lowest component of chiral superfields which are scalars withcharge ¯ q int = 1 (see table 2) and their fermionic superpartners whose charge ¯ q int = − isshifted by spectral flow with respect to the scalars by − (a shift by + would yield a U(1)charge which is forbidden by the unitarity bound). Antichiral multiplets consist of the chargeconjugate states of chiral multiplets. Vector multiplets consist of the lowest component ofvector superfields which are gauge bosons of charge ¯ q int = 0 and their superpartners which areleft/right-handed gauginos of charge ¯ q int = ± . See [34] for a more detailed discussion. Thequantum numbers (¯ h int , ¯ q int ) for the massless SUSY multiplets hence are: • Vector multiplets: gauge bosons (0 ,
0) and left/right-handed gauginos (3 / , ± / • Chiral multiplets: fermions (3 / , − / / ,
1) and their charge conjugates. The NS vacuum in the right-moving sector has ¯ h = − . .3 Counting massless states In the right-moving sector the structure of massless states is highly constrained due to R/NSalignment following from supersymmetry while in the left-moving sector, where this alignmentis partially broken, a broader range of possible states is admitted. We can use the restrictedstructure in the right-moving sector and construct admissible left-moving states on orbits ofadmissible right-moving states, the pairings of which give the massless spectrum of the heterotic(0 ,
2) string.In order to break supersymmetry only in the left-moving sector we have to start withsmaller building blocks for the internal CFT as well as for the gauge group as discussed insection 2.3.2. Splitting C int = C ′ ⊗ F and starting with SO (2) × SO (8) ⊆ SO (10) we can writeexplicitly h int = h C ′ + h F and h D = h D + h D . Admissible left-moving states are obtained bytwisting admissible right-moving states by J = J νGSO J αA J βb J γC with α, γ = 0 , β = 0 , , , ν = 0 , ..., M ′ − h tot = h C ′ + h F + h D + h D = 1in the bosonic sector. A generic left-moving state is obtained by a generic right-moving stateby |C ′ F D D > l = J νGSO J αA J βb J γC |C ′ F D D > r (3.2)and the explicit form of the twist current is given by J = J νs J γv ⊗ J ν + βKs J K − βv ⊗ s ν + β v α ⊗ S ν V α + γ . (3.3)Besides organizing the contributions to the spectrum in twisted sectors the exponents ν, α, β, γ determine whether a left-moving (twisted) factor yields the same field as the right-moving factoron the orbit of which it is computed or its superpartner. By choosing a specific SUSY multiplettogether with an SO (10) representation for the gauge multiplet we will study the structure of thecharged massless spectrum of non-singlet matter states. We will use the information obtainedfrom the exact CFT calculations to determine the number of generations, antigenerations andvectors by the means of the extended Poincar´e polynomial and the complementary Poincar´epolynomial. The EPP of a N = 2 SCFT as given by [13] P ( t, t, x ) = X l ≥ X κ =0 , x l ( − κ P l,κ ( t, t ) , (3.4)is the sum of l x -twisted Poincar´e polynomials weighted by an additional change of sign, thatis related to the ambiguity of dealing with a field or its superpartner. The ordinary Poincar´e The NS vacuum in the left-moving sector has h = − P l,κ ( t, t ) = X ( a,a ) ∈R ( c,c ) a = J ls J κv a t q ( a ) t q ( a ) , (3.5)where the sum is over states in the ( c, c ) ring. In the case where the internal sector has alignedspin structures (corresponding to a twist by an even exponent of J b ) the states contributingto the massless spectrum are BPS states. We can determine the number of generations, anti-generations and vectors by looking for particular terms in the EPP that are determined by the U (1) charges of the internal left- and right-moving sector as will be calculated below. In the case where the internal sector has non-aligned spin structures (twist by an odd exponentof J b ) also non-BPS states can contribute to the massless spectrum and we thus need in additionto the information of the (left-moving) internal U (1) charge also the conformal weight. We arethus interested in the complementary Poincar´e polynomial P ( x, q, t ) = X ℓ ≥ X k =0 , X a ∈R a = J ℓs J kv a ( − k x ℓ q H L ( a ) t Q ( a ) , (3.6)where a runs over the Ramond ground states and the sum over a is over all states (includingdescendants) in the conformal family of J ℓs J kv a . This polynomial is complementary to theEPP. It does not involve the right-mover’s charge, but instead keeps track of the conformaldimension of excited left-moving states.We can compute P ( x, q, t ) in terms of the elliptic genus which, for a general N = 2 SCFT,is the trace [35] Z ( q, q, t ) = Tr H ( − F q H L q H R t Q L , (3.7)where H is the full Hilbert space, H L,R are the Hamiltonians of left- and right-movers, Q L is the U (1) charge of left-movers, and F = F L + F R is the total fermion number . Up tospectral flow, we can assume that the left- and right-movers are in the Ramond sector. Bysupersymmetry, the non vanishing contributions to Z come from the states where the right-mover is a ground state H R = 0 and thus Z ( q, q, t ) = Z ( q, , t ) ≡ Z ( q, t ). As discussed in [35],the elliptic genus of a Landau-Ginzburg model can be computed in free field theory. Let thesuperpotential W (Φ , . . . , Φ N ) be a holomorphic function in the chiral superfields { Φ i } i =1 ,...,N such that W ( λ ω Φ , . . . , λ ω N Φ N ) = λ W (Φ , . . . , Φ N ) . (3.8) Note, that the EPP is conventionally defined with right-movers a on the orbit of left-movers a whereas in ouranalysis we choose left-movers on the orbit of right-movers due to the nicer BPS structure in the right-movingsector. The limit q = 0 equals the EPP at t = 1 and with t → /t . This replacement is necessary since we definedthe CPP in (3.6) with left-movers a on the orbit of right-movers a . As usual, we can identify ( − F L,R = e iπ Q L,R . Z LG ( q, t ) = Q i Z ω i ( q, t ) with Z ω ( q, t ) = t − − ω − t − ω − t ω ∞ Y n =1 − q n t − ω − q n t ω − q n t − (1 − ω ) − q n t − ω = ϑ ( q, t − ω ) ϑ ( q, t ω ) , (3.9)where the Jacobi theta function ϑ is given by ϑ ( q, t ) = i X n ∈ Z ( − n q ( n − ) t n − . (3.10)The expression (3.9) is obtained immediately in free field theory. One just keeps track ofthe contributions of the scalar φ and left-moving fermion ψ − in Φ as well as their complexconjugates [35]. The polynomial (3.6) is simply the sum over the twists along the spatialdirection. Notice that the change of sign ( − k in (3.6) due to J v applications is automaticallytaken into account by the fermion sign ( − F in the elliptic genus. The effect of the spatialtwist can be obtained by standard orbifold techniques and gives the contribution [37] q b c ℓ t b cℓ Z LG ( q, q ℓ t ) , (3.11)with b c = c = P Ni =1 (1 − ω i ). The sum over ℓ -twisted factors can include phases as usual inorbifold partition functions [37]. The choice of trivial phases reproduces the EPP at t = 1 inthe q → P ( x, q, t ) = X ℓ ≥ x ℓ q b c ℓ t b cℓ Z LG ( q, q ℓ t ) . (3.12)As a check, a tedious exercise (see appendix) gives indeedlim q → P ( x, q, t ) = P ( t − , , x ) (3.13) We have now assembled all tools that we need in order to compute the charged masslessspectrum. As a representative of the right-moving sector we consider a space-time matterscalar. From table 2 we can read off the right-moving internal conformal weight and charge tobe h int = and q int = ± C int = C ′ ⊗ F we can write the right-moving stateas Ψ right = |C ′ ⊗ F ; D ⊗ D i r = | Φ ⊗ ϕ ℓ, ℓ ; ⊗ V i r (3.14)with ℓ = 0 , . . . , K −
2. With K being the charge quantum in the NS sector we can explicitlycompute the charge of the antichiral state ϕ ℓ, ℓ in the Fermat sector and, hence, we can split q int into contributions form F and C ′ according to q F a = − ℓK and q ′ a = ℓ − KK . (3.15) Our notation is related to [35] by t = e iγ and w = α and to that of [36] by t = y = e πiz . This choice of right-moving representative forbids to further use the field identifications (2.15) for left-movers, since those must be applied simultaneously on both sides in order to yield an admissible state thatcontributes to the spectrum of the heterotic string. Field identifications that are based on the modular propertiesof the labels, however, can still be used. Q A ≡ Q C ≡ Q GSO ≡ Q b (Ψ right ) ≡ h b + h Ψ right − h J b Ψ right involving theBonn twist can be computed to yield Q b Ψ right ≡ (cid:26) ℓ ∈ Z / ℓ Z . (3.16)Comparing this result to the projection rule Q b ≡ α + K − β mod 1 obtained in section 2.3.3,restricts the possible exponents α, β, γ, ν of J = J αA J βb J γC J νGSO by which the admissible right-moving states are twisted to yield admissible left-moving states.Since we want to count generations, as represented by states in , the left-moving states musttransform under SO (8) × U (1) as v − or s according to (3.1). For convenience we stay in theNS sector where states are of the general formΨ left = |C ′ ⊗ F ; D ⊗ V i l . (3.17)With the massless condition h int + h D + = 1 and the charge condition q int + q D = −
1, asfollows from the group theory discussion, we have four possibilities for admissible left-movingstates. Their space-time parts, conformal weights and U (1) charges are | ⊗ V i l with h int = , q int = − | v ⊗ V i l with h int = 0 , q int = − | s ⊗ V i l with h int = , q int = − ; | s ⊗ V i l with h int = , q int = − . (3.18)While in the right-moving sector all factors are aligned, in the left-moving sector this alignmentis partially broken due to the presence of the Bonn twist J b in the extension of the left-chiralalgebra. The remaining alignment between the factors C ′ and D is due to the current J C .Depending on whether the two factors D and D and, hence, also C ′ and F are aligned or not,there is a qualitatively different analysis for counting the number of generations. For an even power of the Bonn twist the internal factors F and C ′ and, hence, also D and D are aligned along the orbit generated by the twist. This case corresponds to the states inthe first line of (3.18). However, taking into account the BPS bound h int ≥ | q int | , states with q int = − cannot appear in the massless spectrum. States with q int = − Ψ left = | Φ ⊗ ϕ ℓ, ℓ ; ⊗ V i l with h int = 12 , q int = − Note, that admissible left-moving states in the F sector could in principle also appear as ϕ K − − ℓ, ℓ + K whichis dual to ϕ ℓ, ℓ under field identification. From the previous footnote, however, it follows that after having fixedthe right-moving representative we cannot use field identifications (2.15) on the left-moving side anymore, andhence we have to discuss both possibilities. Since their U (1) charges are equal we can cover both cases in oneshot by taking into account the two possible labels ℓ and K − − ℓ when counting generations via the EPP. ℓ = 0 , . . . , K −
2. The U (1) charge of the antichiral primary state ϕ ℓ, ℓ in the Fermat factorcan be computed and, hence, the charge contributions from the F and C ′ sectors to q int = − q F a = − ℓK and q ′ a = ℓ − KK . (3.20)The left-moving state (3.19) is on the orbit of the right-moving state (3.14) ifΨ left = J αA J βb J γC J νGSO Ψ right . (3.21)Explicitely, this means Φ = J νs J γv Φ , (3.22) ϕ ℓ, ℓ = J ν + Kβs J β K − v ϕ ℓ, ℓ , (3.23) = s ν + β v α , (3.24) V = S ν V α + γ +1 . (3.25)Using the fusion rules , the last two equations read ν ∈ Z ,α + γ ∈ Z , (3.26) ν + β + 2 α ≡ . These constraints together with the charge projection rule Q b ≡ α K − β ≡ (cid:26) ℓ ∈ Z / ℓ Z . mod 1 (3.27)uniquely determine the possible combinations of twist exponents α, β, γ, ν such that the com-bination of Ψ right and Ψ left = J αA J βb J γC J νGSO Ψ right give contributions to the massless spectrum.In order to count generations we need to find the appropriate terms in the extended Poincar´epolynomial. We use the simple structure in the Fermat sector in order to determine the admis-sible terms for the EPP in the C ′ sector. For a minimal model, like F , the EPP over the chiralring is given by [13] P ( c,c ) ( t K , t K , x ) = K − X l =1 ( t t ) l − − ( − x ) l t K − l − ( − x ) K . (3.28)In general, the order of the spectral flow in C ′ can be larger than that in F . Therefore, we needthe full ’periodic’ expansion of the EPP with arbitrarily high powers of x , which, because of K Z , reads P ( c,c ) ( t K , t K , x ) = K − X l =1 ( t t ) l − (cid:16) − ( − x ) l t K − l (cid:17) ∞ X r =0 ( − r x rK . (3.29) Indeed, the fusion rules imply that any monomial in s , v , and s can be reduced to s p which is iff p ≡ Z structure. In the case of D , any monomial can be reduced to the form S p V q which is iff p and q are even, in agreement with the Z × Z structure. P ( c,c ) ( t, t, x ), which, after identification of the exponentsof t and t with the charges q F = − ℓK and q F = − ℓK of states in the F sector, read F : ( i ) ( − r x rK t − ℓK t − ℓK ℓ = ℓ, ( ii ) ( − r + ℓ x ℓ +1+ rK t − ℓK t − ℓK ℓ = K − − ℓ. (3.30)For each admissible combination of α, β, γ and ν (as follows from (3.26)) the label ℓ and theparameter r can be determined by the charge projection rule (3.27) and by comparison of (3.23)with the generic structure of terms in the EPP of F above. For each of these combinationswe can then determine the generic structure of admissible terms in the EPP over the chiral ringof C ′ yielding C ′ : ( i ) ( − γ x rK t q ′ c t q ′ c ( ii ) ( − γ x ℓ +1+ rK t q ′ c t q ′ c , (3.31)with ℓ and r being determined by the admissible terms in the EPP of F . Note, that the sign( − γ depends on the exponent of J C because it determines whether or not a supercurrent isapplied to states in C ′ , as follows from (3.22). The admissible terms in the EPP P ( c,c ) ( t, t, x )of C ′ are summarized in table 3. Note, that upon spectral flow the U (1) charge of the internalantichiral primary states is shifted to that of chiral primary states by c ′ = K +2 K and weparametrized the exponent of J GSO by ν = 2 n . In table 3 the data necessary for countingaligned generations is collected.
16 - aligned generations σ ′ ℓ l q ′ c q ′ c + ℓ ∈ Z , ≤ ℓ ≤ k l ∈ K Z β = 0 , n ∈ Z l ∈ K Z + 1 β = 2 , n / ∈ Z K +2+ ℓK K +2+ ℓK − ℓ / ∈ Z , ≤ ℓ ≤ k l ∈ K Z + 1 β = 0 , n / ∈ Z l ∈ K Z β = 2 , n ∈ Z K +2+ ℓK K − ℓK Table 3: Left- and right-moving C ′ -sector charges q ′ c and q ′ c , right-moving label ℓ , exponent l of x and sign σ ′ = ( − γ in the EPP of C ′ with terms ∼ σ ′ x l t q ′ t q ′ . For an odd power of the Bonn twist the Fermat sector and the C ′ sector are not aligned anymore.This case corresponds to the states in the second line of (3.18). Due to the remaining alignmentby J C states in C ′ are in the NS sector while states in F are in the Ramond sector. By exploiting Notice, that since the states in the Fermat sector are antichiral states we actually have to sum over theantichiral ring in (3.29) which simply amounts to adding a factor ( tt ) − Kc F = ( tt ) − K . If there is no supercurrent in (3.23), admissible terms have positive coefficient otherwise they have a negativecoefficient. For n = ν ∈ Z the exponent of x must be even and otherwise odd. F and, hence, in C ′ are admissible and which are not. The fields in F are φ ℓsm ∼ φ k − ℓ,s +2 m + K with0 ≤ ℓ ≤ k , m mod 2 K , s mod 4 and ℓ + m + s ≡ i.e. there are 2 K ( K −
1) fields.The ‘generic’ subgroup of the center h J s = φ , J v = φ i has order 4 K . Because of fieldidentification its orbits are labeled by ℓ ∼ k − ℓ and there are fixed points (with ℓ = k/ k . Each orbit of the generic center contains two BPS states of each type( c , a and R ). The left-moving states φ lsm on the orbit of φ ℓ, ℓ are at position J m − ℓs J s − m + ℓ v with ℓ = ℓ and s = ± h ′ + h F = the Ramond sector states in F musthave h F ≤ . A straightforward analysis shows that the BPS bound h ′ ≥ | q ′ | is not satisfied bystates with q int = − . Hence, the only admissible left-moving states with non-aligned F and C ′ factors are of the formΨ left = | Φ ⊗ ϕ ; s ⊗ V i l with h int = 38 , q int = − . (3.32)It is on the orbit of Ψ right if Ψ left = J νGSO J αA J βB J γC Ψ right , (3.33)which explicitely reads Φ = J νs J γv Φ , (3.34) ϕ ℓ,sm = J ν + Kβs J β K − v ϕ ℓ, ℓ = ϕ ℓ,ν +2 βK − βℓ + ν + βK , (3.35) s = s ν + β v α , (3.36) V = S ν V α + γ +1 . (3.37)Using the fusion rules the last two equations read ν ∈ Z ,α + γ ∈ Z , (3.38) ν + β + 2 α ≡ . Together with the charge projection from above Q b ≡ α K − β mod 1 ≡ (cid:26) ℓ ∈ Z / ℓ Z . mod 1 (3.39)the constraints (3.38) strongly restrict the possible combinations of twist exponents and labelsas would follow from (3.35). Using ℓ = ℓ and the BPS condition we can now determine whichleft-moving states in the Fermat sector lead to massless states. In the simplest case they areRamond ground states. Those of the form ϕ ℓ, ℓ +1) need to satisfy the lower bound ℓ ≥ K − , thoseof the form ϕ ℓ, − − ( ℓ +1) must satisfy the upper bound ℓ ≤ K − . Their superpartners ϕ ℓ, ∓ ± ( ℓ +1) are notadmissible since their conformal weights are h = c F +1 > . Apart from ground states there areexcited Ramond states of the form ϕ ℓ, − m with s = − | m | < ℓ in the standard range wherethey have to satisfy the condition ( | m | − ≥ ℓ ( ℓ + 2) + 1 − K , as follows from the BPS bound, Notice, that the label ℓ of ϕ ℓ,sm does not change under fusion with J s or J v . See section 2.2.1. ℓ + m ≡ ℓ ≤ K +36 . Furthermore, thereare excited Ramond states of the form ϕ ℓ, m with | m | > ℓ outside the standard range. We canalways bring these states back to the standard range via field identifications where they need tosatisfy s + 2 = ±
1, the branching rule ℓ + m ≡ | m ± K | ≤ K − − ℓ and ( | m ± K | − ≥ ( K − ℓ − K − ℓ ) + 1 − K . After having determined all admissiblestates in the Fermat sector with conformal weights and U (1) charges as calculated from (2.14)we can compute the conformal weights and charges of admissible states in the C ′ sector. Thenovelty, as compared to the case of aligned generations, is that also non-BPS states in C ′ cancontribute to the spectrum of massless states. Therefore, in addition to the left-moving U (1)charge in C ′ we also need to keep the information of the left-moving conformal weight. Inorder to count generations we are, hence, looking for admissible terms in the complementaryPoincar´e polynomial in the C ′ sector which is, in some sense, complementary to the extendedPoincar´e polynomial. As follows from (3.6), admissible terms in the complementary Poincar´epolynomial have the generic structure( − γ x ν/ q h ′ R − c ′ t q ′ R , (3.40)with the U (1) charges and conformal weights in the Ramond sector. The sign σ ′ = ( − γ is determined by (3.34), i.e. whether or not a supercurrent is applied, and the exponent of x can be read off by comparison of (3.6) with (3.34) to be ν/ The admissible terms inthe complementary Poincar´e polynomial of C ′ are summarized in table 4 which is organized interms of the left-moving Fermat sector states ϕ ℓ,sm as depicted in the first column. The right-moving charge is not necessary. Suppose that our tables find a candidate left-moving state in C ′ along the orbit of the various currents. This means that the right-moving C ′ state is a Ramond ground state(after spectral flow) with charge q ′ obeying q ′ mod 2 = ℓ +1 K with ℓ = 0 , , ..., K −
2. However, q ′ must be suchthat | q ′ | ≤ c ′ / /K . This easily shows that q ′ = ℓ +1 K exactly, i.e. without mod 2. While the information on the exponent of the GSO current can be determined modulo 4 K , all ν withinthe range of 1 , ..., N GSO need to be considered separately in the counting algorithm. In other words, values of ν that are absolutely different but the same modulo 4 K generically appear as exponent of different terms withdifferent coefficients in the complementary Poincar´e polynomial. Fermat K mod 4 σ ′ ℓ m ν mod 4 K h ′ R q ′ R ϕ ℓ, ℓ +1 − ℓ / ∈ Z , K − ≤ ℓ ≤ k m = ℓ + 1 1 − βK ℓ +22 K ℓ +2 K − ℓ ∈ Z , K − ≤ ℓ ≤ k m = ℓ + 1 1 − βK + 2 K ℓ +22 K ℓ +2 K ϕ ℓ, − − ( ℓ +1) ℓ ∈ Z , ≤ ℓ ≤ K − m = − ( ℓ + 1) − − ℓ − βK K − ℓ K K − ℓK ℓ / ∈ Z , ≤ ℓ ≤ K − m = − ( ℓ + 1) − − ℓ − βK K − ℓ K K − ℓK ϕ ℓ, − m ℓ ∈ Z , ≤ ℓ ≤ k m / ∈ Z , | m | < ℓ ( | m | − ≥ ℓ ( ℓ + 2) + 1 − K ( K − ℓ − m − − − βK K +28 K − h ℓ, − m − q ℓ, − m K +22 K − q ℓ, − m ℓ / ∈ Z , ≤ ℓ ≤ k m ∈ Z , | m | < ℓ ( | m | − ≥ ℓ ( ℓ + 2) + 1 − K − ( K + 1)( ℓ − m + 1) + 1 − βK K +28 K − h ℓ, − m − q ℓ, − m K +22 K − q ℓ, − m ϕ ℓ, m − ℓ / ∈ Z , ≤ ℓ ≤ k m ∈ Z , | m ± K | ≤ K − − ℓ ( | m ± K | − ≥ ( K − − ℓ )( K − ℓ ) + 1 − K ( K − ℓ − m + 1) + 1 − βK K +28 K − h K − − ℓ, − m ± K − q K − − ℓ, − m ± K K +22 K − q K − − ℓ, − m ± K − ℓ ∈ Z , ≤ ℓ ≤ kpage m / ∈ Z , | m ± K | ≤ K − − ℓ ( | m ± K | − ≥ ( K − − ℓ )( K − ℓ ) + 1 − K ( K + 1)( − ℓ + m + 1) − − βK K +28 K − h K − − ℓ, − m ± K − q K − − ℓ, − m ± K K +22 K − q K − − ℓ, − m ± K Table 4: Left-moving C ′ -sector charges q ′ R and conformal weights h ′ R (in the Ramond sector), signs σ ′ = ( − γ and constraints for theadmissible terms in the CPP in C ′ ∼ σ ′ x ν/ q h ′ R − c ′ t q ′ R . .5 Counting antigenerations In complete analogy to the counting of generations we have to depict a right-moving represen-tative and compute admissible left-moving states on its orbit. We choose the same space-timematter scalar as in (3.4). In order to count antigenerations, as represented by in (3.1), theleft-moving states must transform under SO (8) × U (1) as v or s − . Again, for convenience,we stay in the NS sector where the admissible states are of the general formΨ left = | Φ ⊗ ϕ ; D ⊗ V i l . (3.41)With the condition for the conformal weights and charges of massless states we get four possi-bilities for admissible left-moving states. Their space-time parts, conformal weights and U (1)charges are | ⊗ V i l with h int = , q int = 1; | v ⊗ V i l with h int = 0 , q int = | s ⊗ V i l with h int = , q int = ; | s ⊗ V i l with h int = , q int = . (3.42) For an even power of the Bonn twist there is alignment within the space-time part. Taking intoaccount the BPS bound the only states that are admissible have are of the formΨ left = | Φ ⊗ ϕ ℓ, − ℓ ; ⊗ V i l with h int = 12 , q int = 1 (3.43)with ℓ = 0 , . . . , K −
2. Internal states are chiral primary states. The U (1) charge of the chiralprimary state ϕ ℓ, − ℓ can easily be computed, and the charge contributions from the F and C ′ sectors to q int = 1, hence, are q F c = ℓK and q ′ c = K − ℓK . (3.44)A similar analysis as in the case of aligned generations can be carried out to yield admissibleterms in the EPP of C ′ which are summarized in table 5. For an odd power of the Bonn twist the alignment in the internal sector is broken. States in C ′ are in the NS sector, while states in F are in the Ramond sector. Repeating the same analysisas for non-aligned generations the only admissible left-moving states turn out to be of the formΨ left = | Φ ⊗ ϕ ; s ⊗ V i l with h int = 38 , q int = 12 . (3.45)Admissible states in the F sector can be derived along the same lines as for non-aligned anti-generations. Using the information about their conformal weights and U (1) charges admissibleterms in the complementary Poincar´e polynomial C ′ can be determine and are listed in table 6.28 - aligned antigenerations σ ′ ℓ l q ′ c q ′ c + ℓ ∈ Z , ≤ ℓ ≤ k l ∈ K Z + 1 β = 0 , n ∈ Z l ∈ K Z β = 2 , n / ∈ Z K +2+ ℓK K − ℓK − ℓ / ∈ Z , ≤ ℓ ≤ k l ∈ K Z β = 0 , n / ∈ Z l ∈ K Z + 1 β = 2 , n ∈ Z K +2+ ℓK ℓ +2 K Table 5: Left- and right-moving C ′ -sector charges q ′ c and q ′ c , right-moving label ℓ , exponent l of x and sign σ ′ = ( − γ in the EPP of C ′ with terms ∼ σ ′ x l t q ′ t q ′ . As before, we choose again the space-time matter scalar as right-moving representative. Inorder to count vectors, as represented by in (3.1), left-moving states must transform under SO (8) × U (1) as , − or s . Let us consider states transforming under the general formof which is given by Ψ left = | Φ ⊗ ϕ ; D ⊗ i l . (3.46)The space-time parts, conformal weights and U (1) charges of admissible states contributing tothe massless spectrum are | ⊗ i l with h int = 1 , q int = 2; | v ⊗ i l with h int = , q int = 1 | s ⊗ i l with h int = , q int = ; | s ⊗ i l with h int = , q int = . (3.47)States with q int = can already be discarded since they do not obey the BPS bound. For an even power of the Bonn twist there are now two possible states, both of which haveinternal chiral primary states. States with q int = 2 are of the formΨ left = | Φ ⊗ ϕ ℓ, − ℓ ; ⊗ i l with q F c = ℓK and q ′ c = 2 K − ℓK , (3.48)while states with q int = 1 are of the formΨ left = | Φ ⊗ ϕ − ℓ, ℓ ; v ⊗ i l with q F c = ℓK and q ′ c = K − ℓK , (3.49)and ℓ = 0 , . . . , k . Admissible terms in the EPP of C ′ are listed in table 7.29 - non-aligned antigenerations Fermat K mod 4 σ ′ ℓ m ν mod 4 K h ′ R q ′ R ϕ ℓ, ℓ +1 ℓ ∈ Z , ≤ ℓ ≤ K − m = ℓ + 1 1 − βK ℓ +2+ K K ℓ +2+ KK ℓ / ∈ Z , ≤ ℓ ≤ K − m = ℓ + 1 1 − βK + 2 K ℓ +2+ K K ℓ +2+ KK ϕ ℓ, − − ( ℓ +1) − ℓ / ∈ Z , K − ≤ ℓ ≤ k m = − ( ℓ + 1) − − ℓ − βK + 2 K K − ℓ K K − ℓK − ℓ ∈ Z , K − ≤ ℓ ≤ k m = − ( ℓ + 1) − − ℓ − βK + 2 K K − ℓ K K − ℓK ϕ ℓ, − m ℓ ∈ Z , ≤ ℓ ≤ k m / ∈ Z , | m | < ℓ ( | m | − ≥ ℓ ( ℓ + 2) + 1 − K ( K − ℓ − m + 1) + 1 − βK K +28 K − h ℓ, m − q ℓ, m K +22 K − q ℓ, m ℓ / ∈ Z , ≤ ℓ ≤ k m ∈ Z , | m | < ℓ ( | m | − ≥ ℓ ( ℓ + 2) + 1 − K − ( K + 1)( ℓ − m + 1) − − βK K +28 K − h ℓ, m − q ℓ, m K +22 K − q ℓ, m ϕ ℓ, m − ℓ / ∈ Z , ≤ ℓ ≤ k m ∈ Z , | m ± K | ≤ K − − ℓ ( | m ± K | − ≥ ( K − − ℓ )( K − ℓ ) + 1 − K ( K − ℓ − m − − − βK K +28 K − h K − − ℓ, m ± K − q K − − ℓ, m ± K K +22 K − q K − − ℓ, m ± K − ℓ ∈ Z , ≤ ℓ ≤ k m / ∈ Z , | m ± K | ≤ K − − ℓ ( | m ± K | − ≥ ( K − − ℓ )( K − ℓ ) + 1 − K ( K + 1)( − ℓ + m −
1) + 1 − βK K +28 K − h K − − ℓ, m ± K − q K − − ℓ, m ± K K +22 K − q K − − ℓ, m ± K Table 6: Left-moving C ′ -sector charges q ′ R and conformal weights h ′ R (in the Ramond sector), signs σ ′ = ( − γ and constraints for theadmissible terms in the CPP in C ′ ∼ σ ′ x ν/ q h ′ R − c ′ t q ′ R . - aligned vectors q int σ ′ ℓ l q ′ c q ′ c − ℓ ∈ Z , ≤ ℓ ≤ k l ∈ K Z + 1 β = 0 , n ∈ Z l ∈ K Z β = 2 , n / ∈ Z K +2+ ℓK K − ℓK ℓ / ∈ Z , ≤ ℓ ≤ k l ∈ K Z β = 0 , n / ∈ Z l ∈ K Z + 1 β = 2 , n ∈ Z K +2+ ℓK K +2+ ℓK ℓ / ∈ Z , ≤ ℓ ≤ k l ∈ K Z + 1 β = 0 , n ∈ Z l ∈ K Z β = 2 , n / ∈ Z K +2+ ℓK K − ℓK − ℓ ∈ Z , ≤ ℓ ≤ k l ∈ K Z β = 0 , n / ∈ Z l ∈ K Z + 1 β = 2 , n ∈ Z K +2+ ℓK ℓ +2 K Table 7: Left- and right-moving C ′ -sector charges q ′ c and q ′ c , right-moving label ℓ , exponent l of x and sign σ ′ = ( − γ in the EPP of C ′ with terms ∼ σ ′ x l t q ′ t q ′ . Due to the BPS bound there are only states of the formΨ left = | Φ ⊗ φ ; s ⊗ i l with h int = 78 ; q int = 32 (3.50)that can contribute to the massless spectrum. Again, states in C ′ are in the NS sector whilestates in F are in the Ramond sector. Those can either be Ramond ground states or excitedstates, as discussed already for non-aligned generations. The complete list of constraints thathave to be satisfied by states in C ′ in order to yield admissible terms in the complementaryPoincar´e polynomial together with their conformal weights and U (1) charges is given in table8. 31 Fermat K mod 4 σ ′ ℓ m ν mod 4 K h ′ R q ′ R ϕ ℓ, ℓ +1 − ℓ ∈ Z , ≤ ℓ ≤ K − m = ℓ + 1 1 − βK ℓ +2+3 K K ℓ +2+2 KK − ℓ / ∈ Z , ≤ ℓ ≤ K − m = ℓ + 1 1 − βK + 2 K ℓ +2+3 K K ℓ +2+2 KK ϕ ℓ, − − ( ℓ +1) ℓ / ∈ Z , K − ≤ ℓ ≤ k m = − ( ℓ + 1) − − ℓ − βK + 2 K K − ℓ K K − ℓK ℓ ∈ Z , K − ≤ ℓ ≤ k m = − ( ℓ + 1) − − ℓ − βK + 2 K K − ℓ K K − ℓK ϕ ℓ, − m − ℓ ∈ Z , ≤ ℓ ≤ k m / ∈ Z , | m | < ℓ ( m − ≥ ℓ ( ℓ + 2) + 1 − K ( K − ℓ − m + 1) + 1 − βK K +28 K − h ℓ, m − q ℓ, m K +22 K − q ℓ, m − ℓ / ∈ Z , ≤ ℓ ≤ k m ∈ Z , | m | < ℓ ( | m | − ≥ ℓ ( ℓ + 2) + 1 − K − ( K + 1)( ℓ − m + 1) − − βK K +28 K − h ℓ, m − q ℓ, m K +22 K − q ℓ, m ϕ ℓ, m ℓ / ∈ Z , ≤ ℓ ≤ k m ∈ Z , | m ± K | ≤ K − − ℓ ( m ± K − ≥ ( K − − ℓ )( K − ℓ ) + 1 − K ( K − ℓ − m − − − βK K +28 K − h K − − ℓ, m ± K − q K − − ℓ, m ± K K +22 K − q K − − ℓ, m ± K ℓ ∈ Z , ≤ ℓ ≤ k m / ∈ Z , | m ± K | ≤ K − − ℓ ( | m ± K | − ≥ ( K − − ℓ )( K − ℓ ) + 1 − K ( K + 1)( − ℓ + m −
1) + 1 − βK K +28 K − h K − − ℓ, m ± K − q K − − ℓ, m ± K K +22 K − q K − − ℓ, m ± K Table 8: Left-moving C ′ -sector charges q ′ R and conformal weights h ′ R (in the Ramond sector), signs σ ′ = ( − γ and constraints for theadmissible terms in the CPP in C ′ ∼ σ ′ x ν/ q h ′ R − c ′ t q ′ R . Distler-Kachru models andthe heterotic (0,2) CFT/geometry connection
In analogy to the case of (2,2) models, a very general framework for the description of (0,2)models can be given in terms of a gauged linear sigma model with (0 ,
2) worldsheet supersym-metry, known as Dister-Kachru models. Since we want to compare the spectra obtained by thecounting algorithm of the previous chapter to that of Dister-Kachru models [9] let us brieflyreview their structure. In (0,2) models there exists an additional structure, as compared to (2,2)models, which is the choice of rank e r stable, holomorphic vector bundle V → M with vanishingfirst Chern class c ( V ) = 0 and c ( V ) = c ( T ), where T is the holomorphic tangent bundle of M . As reviewed in [17], the defining data of a (0,2) sigma model on a Calabi-Yau manifold M is encoded in the superpotentials W j (Φ i ) and F la (Φ i ), where W j (Φ i ) are transversal polynomialsof degree d j which define the base space M of the vector bundle V → M associated to theleft-moving gauge fermions and F la (Φ i ) are polynomials, with degree fixed by requiring chargeneutrality of the action, that define the global structure of the bundle V . The field content isgiven by a set of chiral superfields Φ i with U (1) charges w i with i = 1 , · · · , N i . Neutrality of theaction then requires additional Fermi superfields Σ j with charge − d j with j = 1 , · · · , N j . Theingredients for constructing the bundle V are Fermi superfields Λ a with strictly positive U (1)charges n a with a = 1 , · · · , N a and a chiral superfield P l with charge − m l with l = 1 , · · · , N l such that P l m l = P a n a . The (0,2) superpotential action that summarizes the structure ofthe total bundle is given by S W = Z d zdθ (cid:16) Σ j W j (Φ i ) + P l Λ a F la (Φ i ) (cid:17) . (4.1)The first term ensures that the fields Φ i lie on the hypersurface W j = 0, whereas the secondterm ensures that the gauge fermions λ a (lowest components of the Λ a ) are sections of thebundle V . The (0,2) gauge multiplets are determined by a real superfield V , which contains theright-moving component of the gauge field, and a superfield A , which contains the left-movingcomponent of the gauge field.The structure of the vector bundle V of rank e r = N a − N l is given by the short exactsequence (monad) 0 → V → e r + N l M a =1 O ( n a ) F a −→ N l M l =1 O ( m l ) → c ( V ) = c (cid:0) ⊕ e r + N l a =1 O ( n a ) (cid:1) c (cid:0) ⊕ N l l =1 O ( m l ) (cid:1) . (4.3)Restricting to the case of N l = 1 the exact sequence defines a vector bundle of rank e r = N a − M . The F a are homogeneous polynomials ofdegrees m − n a which do not vanish simultaneously on M . For weighted projective ambientspaces we can write this data as V n ...,n e r +1 [ m ] −→ P w ,...,w Nj +4 [ d , . . . , d N j ] , (4.4) If the n a are not strictly positive, the bundle V is never stable. [9] N j is the codimension of the Calabi-Yau manifold and e r = 4 , SO (10) and SU (5), respectively.The Calabi-Yau condition c ( T ) = 0 and the condition c ( V ) = 0 imply X j d j − X i w i = m − X a n a = 0 . (4.5)Cancellation of gauge anomalies ch ( V ) = ch ( T X ) with the second Chern character ch = c − c implies the quadratic Diophantine constraint X j d j − X i w i = m − X a n a . (4.6)In general there are not many solutions to this equation. In the (2 ,
2) case, which correspondsto F a = ∂ a W and yields the gauge group E , a solution is given by the choice of m = d = P j w j with n a = w a .Note, that the discrete gauge symmetry Z m that survives the breaking of the U (1) in the gaugedlinear sigma model with m defined in (4.2) corresponds to the Z m quantum symmetry [38]resulting from the GSO projection on the CFT side.In [15] R. Blumenhagen and A. Wißkirchen proposed a Gepner-type construction of stringmodels with (0 ,
2) worldsheet supersymmetry based on the simple current construction to ob-tain heterotic compactifications yielding different gauge groups and massless spectra. In [17]they, together with R. Schimmrigk, describe the analog of the (2,2) triality between exactlysolvable conformal field theories, (0 ,
2) Calabi-Yau manifolds and Landau-Ginzburg theories.The suggested CFT/geometry correspondence [15,17] , in particular, associates the vector bun-dle V , , , , [5] over the complete intersection Calabi-Yau P , , , , , [4 ,
4] to a (0,2) cousin of theexactly solvable (2,2) Gepner model 3 , which is described by the Landau-Ginzburg model P , , , , [5] and corresponds, in the sigma model language, to the quintic Calabi-Yau manifold.Note, that the codimension of the Calabi-Yau manifold for the (0 ,
2) cousin has increased ascompared to the (2 ,
2) case. The bundle data of the (0 ,
2) quintic cousin can be expressed bythe exact sequence 0 → M a =1 O (1) → O (5) → . (4.7)The underlying conformal field theory builds on a tensor product of five minimal model factorsand a supersymmetry breaking simple current that acts only on one factor. For this class of(0 ,
2) models the Gepner model data directly determines the vector bundle structure. Since thetwist, that defines the (0 ,
2) model, only acts on one of the minimal model factors, one mightbe tempted to expect that the conjecture can be generalized to a larger picture, where a moregeneral form of an exactly solvable theory directly translates into the bundle data V n ,...,n [ m ].In [12, 16] an ansatz for a solution to 4.5 and 4.6 was made by setting w i = n i for i < w = 2 n V n ,...,n [ m ] → P n ,...,n , n ,w [ d , d ] , (4.8)and imposing (4.5) and (4.6) yielding d + d = m + n + w and d + d = m + 3 n + w . (4.9)34t is quite non-trivial and encouraging that this non-linear system has a general solution w = ( m − n ) / d / d = ( m + 3 n ) / . (4.10)By replacing all minimal model factors of the internal conformal field theory, except the one onwhich the twist acts, by an arbitrary CFT the (0 ,
2) CFT/geometry correspondence needs to beadapted to generic Landau-Ginzburg models. In [12, 16] it was conjectured that there is a non-rational extension of the (0 ,
2) CFT/geometry correspondence between the (0 ,
2) Gepner-typemodels and the Dister-Kachru models defined by the data V n ,...,n [ m ] → P n ,...,n , n , m − n [ m − n , ( m + 3 n ) / , (4.11)where m/n is an odd integer and there exists a transversal polynomial p ( z , . . . , z ) of degree m that is quasi-homogeneous with weights w ( z i ) = n i for i ≤
4. The increase of the codimension ofthe Calabi-Yau manifold may be interpreted as providing an additional field of degree w = d / Z orbifolding due to J b .In order to test the extension of the (0 ,
2) CFT/geometry correspondence to the non-rationalrealm we have to compare the spectra we obtain using the counting algorithm on the CFTside to that of non-linear sigma models at the infrared fixed point which are described byLandau-Ginzburg orbifold models. In particular, we compare the number of generations andantigenerations as arising from both, the CFT and the geometry computations. For a genericchoice of data in a Dister-Kachru model, defined by the stable bundle V n ,...,n e r +1 [ m ] → P w ,...,w Ni [ d , . . . , d N c ] (4.12)of rank e r over a complete intersection space of codimension N c , this can be computed by usingthe elliptic genus Z LG as explained in [18, 39]. Its contribution in the α -th twisted sector isgiven by Z αLG = T r H α ( − F t J q H ∼ χ α + O ( q ) . (4.13)The χ -genus of a bundle of rank e r can be written as χ α = Q a ( − [ α ν a ] ( t ν a q β a / ) { αν a } (1 − t ν a q { αν a } )(1 − t − ν a q − β a ) Q i ( − [ α q i ] ( t q i q β i / ) { αq i } (1 − t q i q { αq i } )(1 − t − q i q − β i ) (cid:12)(cid:12)(cid:12)(cid:12) q t n , (4.14)where ( · · · ) | q t n denotes the evaluation of the q t n terms in the Laurent expansion with integer n and { x } := x − [ x ] , β a := { α ν a } − , β i := { α q i } − . (4.15)The charges of the fields are q i = w i m , ν a = 1 − n a m and ν e r +1+ l = d l m . (4.16)The number of generations is the sum of the positive coefficients of monomials in t (as α varies), while the number of antigenerations is the sum of positive coefficients of monomials in t . 35hese numbers are independent on the defining DK data and are reliable if no extra gauginosor generation/antigeneration pairings occur. In the latter case it turns out that the number ofgenerations n N and antigenerations n N need not be constant over the moduli space. In anycase, the number of net generations n net = n N − n N is constant in moduli space as it is givenby an index theorem n net = (cid:12)(cid:12)(cid:12) R c ( V ) (cid:12)(cid:12)(cid:12) . [9, 18] For vectors it is more subtle. Since mass termsfor states transforming in are not forbidden in the spacetime superpotential, the number ofvectors might jump as we move from the Calabi-Yau phase to the Landau-Ginzburg phase. [9]Further subtleties arise when extra massless gauginos occur in the spectrum which, in [9],is described to be the analog of the destabilization of the vacuum by worldsheet instantons inthe Calabi-Yau phase. In this case the DK model might be sensitive to generic choices of itsdefining data and only certain constraints might lead to “honest” (0 ,
2) SCFTs in the infraredlimit. However, these might not have the desired gauge group. For further reference see, inparticular, [9, 18].
So far the conjectured CFT/geometry correspondence is only based on the existence of a “natu-ral” solution to the anomaly cancellation constraints. We can test it by working out the spectraby two different methods.1. On the CFT side we use the counting algorithm that we have derived in the previoussections and which works for a generic LG model. We can compute the number ofgenerations, antigenerations and vectors.2. On the DK side we use the elliptic genus to compute the Euler characteristic of the bundle.If no extra gauginos or generation/antigeneration pairings occur, it is possible to extractthe number of generations and antigenerations, separately, as explained in [18].In the following we will consider various examples including Fermat-type and non-Fermat-type LG models. As a prominent example of Fermat-type models we show that the number ofgenerations, antigenerations and vectors of the (0,2) cousin of the quintic as computed on theCFT side by our counting algorithm agrees with those first calculated in [15]. A couple of non-Fermat-type examples are shown to give the same numbers of generations and antigenerationson the CFT and the DK side. A couple of non-Fermat-type examples are shown to give thesame numbers of generations and antigenerations on the CFT side as that computed by the χ -genus of DK models. Counting methods for both, Fermat- and non-Fermat LG models havebeen computerized, hence allowing for a large class of LG models to be easily tested. By this, we mean that the χ -genus does not depend on the form of the superpotentials W or F whichare the defining data of a specific DK model. Nevertheless, we will call these (0 ,
2) LG models Distler-Kachrumodels in order to emphasize that they have a geometric and a CFT phase. .1 Fermat-type LG models We consider the following three models of type ( k ′ , ..., k ′ n ; k ) with one minimal model factor F of level k and i = 1 , ..., n minimal model factors of level k ′ i that comprise C ′ . The results of theBlumenhagen-Wißkirchen algorithm carried out in [15, 17, 18] is given in table 9. We computethe number of generations, antigenerations and vectors by determining admissible terms in theextended Poincar´e polynomial and the complementary Poincar´e polynomial of C ′ using the dataand constraints from the tables derived in the previous section. As an illustrative example theEPP and CPP of C ′ for the (0,2) quintic cousin are given in the appendix together with adetailed analysis of the counting of generations, antigenerations and vectors.model N = N A + N NA N = N A + N NA N = ( N A + N A ) + N NA (3 , , ,
3; 3) 80 = 60 + 20 0 74 = (41 + 1) + 32(8 , ,
8; 3) 113 = 85 + 28 5 = 1 + 4 108 = (60 + 0) + 48(2 , , ,
3; 3) 34 = 24 + 10 10 = 8 + 2 40 = (15 + 7) + 18Table 9: Number of generations, antigenerations and vectors for the models 3 ⊗
3, 8 ⊗ ⊗ ⊗ ⊗ A = aligned, AN = non-aligned; A = aligned with q int = 1, A = aligned with q int = 2. Counting in (3 , , ,
3; 3)The number of aligned generations is computed by summing up the coefficients of all admissibleterms in the EPP of C ′ which are characterized by the relevant data listed in the table below.
16 - aligned generations σ ′ ℓ l q ′ c q ′ c N A + 0 0
75 75 + 2 0
95 95 Hence, the number of aligned generations is 40 + 20 = 60. The necessary information in orderto count non-aligned generations in the complementary Poincar´e polynomial is given by thefollowing table.
16 - non-aligned generations
Fermat K mod 4 σ ′ ℓ ν mod 4 K h ′ R q ′ R N NA ϕ ℓ, − − ( ℓ +1)
310 35 ,
2) cousin of the quintic.There are no antigenerations in this model. In order to count aligned vectors we need the dataof the following table.
10 - aligned vectors q int σ ′ ℓ l q ′ c q ′ c N A
85 85
105 105
85 45 Hence, there are 42 aligned vectors. The necessary information in order to count non-alignedvectors in the complementary Poincar´e polynomial is contained in the following table.
10 - non-aligned vectors
Fermat K mod 4 σ ′ ℓ m ν mod 4 K h ′ R q ′ R N NA ϕ ℓ, − − ( ℓ +1) − ϕ ℓ, m There are 32 non-aligned vectors which together with the 42 aligned vectors give a total of 74vectors in the (0 ,
2) cousin of the quintic.
Counting in (8 , ,
8; 3)We can carry out the same analyis as for the quintic cousin with the following results.
16 - aligned generations σ ′ ℓ l q ′ c q ′ c N A + 0 0
75 75 + 2 0
95 95 Hence, the number of aligned generations is 85.
16 - non-aligned generations
Fermat K mod 4 σ ′ ℓ ν mod 4 K h ′ R q ′ R N NA ϕ ℓ, − − ( ℓ +1)
310 35 - aligned antigenerations σ ′ ℓ l q ′ c q ′ c N A + 2 3
35 95 Hence, there is only 1 aligned antigeneration. - non-aligned antigenerations Fermat K mod 4 σ ′ ℓ ν mod 4 K h ′ R q ′ R N NA ϕ ℓ, ℓ +1
910 95 ϕ − ℓ, − ( ℓ +1) −
710 75 There are 4 non-aligned antigenerations and, hence, there are 5 antigenerations in total.
10 - aligned vectors q int σ ′ ℓ l q ′ c q ′ c N A
85 85
105 105 Hence, there are 60 aligned vectors.
10 - non-aligned vectors
Fermat K mod 4 σ ′ ℓ m ν mod 4 K h ′ R q ′ R N NA ϕ ℓ, − − ( ℓ +1) − ϕ ℓ, − − ( ℓ +1) − There are 48 non-aligned vectors and, hence, 108 vectors in total.39 ounting in (2 , , ,
3; 3)
16 - aligned generations σ ′ ℓ l q ′ c q ′ c N A + 0 0
75 75 + 2 0
95 95 + 0 2
75 75 + 2 2
95 95 Hence, the number of aligned generations is 24.
16 - non-aligned generations
Fermat K mod 4 σ ′ ℓ ν mod 4 K h ′ R q ′ R N NA ϕ ℓ, ℓ +1 − ϕ ℓ, − − ( ℓ +1) ϕ ℓ, − − ( ℓ +1) ϕ ℓ, − − ( ℓ +1) The number of non-aligned generations is 10 which, together with the 24 of the aligned gener-ations gives a total of 34 generations. - aligned antigenerations σ ′ ℓ l q ′ c q ′ c N A −
35 85 − + 0 6 1 + 0 11 1 + 0 16 1 + 2 13
35 95 Hence, there are 8 aligned antigenerations. 40 - non-aligned antigenerations Fermat K mod 4 σ ′ ℓ ν mod 4 K h ′ R q ′ R N NA ϕ ℓ, ℓ +1
910 95 ϕ − ℓ, − ( ℓ +1) −
910 95 The number of non-aligned antigenerations is 2. In total there are, hence, 10 antigenerations.
10 - aligned vectors q int σ ′ ℓ l q ′ c q ′ c N A
85 85
85 85 −
75 25 −
95 45
85 45
85 45
85 45 Hence, there are 22 aligned vectors.
10 - non-aligned vectors
Fermat K mod 4 σ ′ ℓ m ν mod 4 K h ′ R q ′ R N NA ϕ ℓ, − − ( ℓ +1) − ϕ ℓ, − − ( ℓ +1) − ϕ ℓ, − m − − ϕ ℓ, m − , +4 38 ϕ ℓ, m − , +4 18 ϕ ℓ, m − , +4 8 There are 48 non-aligned vectors which, together with the 60 aligned vectors, give a total of108 vectors. 41 .2 Non-Fermat-type examples
Counting in P , , , , [10]This model has K = 5 and can therefore be used for checking the case where K ≡ V , , , , [10] −→ P , , , , , [8 , . (5.1)Its χ -genus can be computed by applying (4.14) and we obtain α χ α − t − t + 55 t + 11 t − t t + t − t t − t t − t − t t − t and t , respectively, we get 61 generationsand 1 antigeneration for the DK model. Using our counting method we can compare this resultwith that on the CFT side. The relevant data for admissible terms in the EPP of C ′ for countingaligned generations is listed in the table below.
16 - aligned generations σ ′ ℓ l q ′ c q ′ c N A + 0 0
75 75 + 2 0
95 95 + 0 5
75 75 + 2 5
95 95 Hence, the number of aligned generations is 45. The necessary information in order to countnon-aligned generations in the complementary Poincar´e polynomial is given by42
Fermat K mod 4 σ ′ ℓ ν mod 4 K h ′ R q ′ R N NA ϕ ℓ, − − ( ℓ +1) ϕ ℓ, − − ( ℓ +1)
310 35 ϕ ℓ, − − ( ℓ +1)
310 35 There are 16 non-aligned vectors. In total thera are, hence, 61 vectors which agrees with theprediction from the DK model. In order to count aligned antigenerations we need the followingdata. - aligned antigenerations σ ′ ℓ l q ′ c q ′ c N A + 0 6 1 Since there are no non-aligned antigenerations in this model there is in total only 1 antigener-ation. This agrees with the prediction of the DK model. Moreover, we predict the followingdata for aligned vectors.
10 - aligned vectors q int σ ′ ℓ l q ′ c q ′ c N A
85 85
85 85
85 45
85 45 Hence, there are 32 aligned vectors.
10 - non-aligned vectors
Fermat K mod 4 σ ′ ℓ m ν mod 4 K h ′ R q ′ R N NA ϕ ℓ, ℓ +1) − ϕ ℓ, − − ( ℓ +1) − ϕ ℓ, − − ( ℓ +1) − ϕ ℓ, m − , +4 18 ϕ ℓ, m − , +4 8 Counting in P , , , , [7]This model has K = 7 and is, hence, a check for the case K ≡ V , , , , [7] −→ P , , , , , [6 , . (5.3)Its χ -genus can be computed by applying (4.14) and we obtain α χ α − t − t + 66 t + 11 t − t t − t t − t t − t and t , respectively, we get 66 generationsand 3 antigenerations for the DK model. On the CFT side we use the data in the tables belowto count aligned generations and antigenerations.
16 - aligned generations σ ′ ℓ l q ′ c q ′ c N A + 0 0
97 97 + 2 0
117 117 + 4 0
137 137 Hence, the number of aligned generations is 52.
16 - non-aligned generations
Fermat K mod 4 σ ′ ℓ ν mod 4 K h ′ R q ′ R N NA ϕ ℓ, − − ( ℓ +1)
27 47 There are 14 non-aligned vectors. Hence, we get a total number of 66 vectors which agreeswith the prediction from the DK model. There are no aligned antigenerations in this model.In order to count non-aligned antigenerations we need the following data.44 - non-aligned antigenerations Fermat K mod 4 σ ′ ℓ ν mod 4 K h ′ R q ′ R N NA ϕ ℓ, ℓ +1
57 107 ϕ ℓ, ℓ +1
67 127 There are in total 3 antigenerations which is in agreement with the prediction of the DK model.
In this paper, we have investigated the non-rational generalization [18] of the CFT/geometryconnection proposed for (0 ,
2) heterotic compactifications in [12, 16]. To this aim, we firstreformulated the construction of Blumenhagen et al. [15,17] in terms of simple current modularinvariants identified with orbifolds with discrete torsion [12, 14]. In this language the breakingof E to the GUT gauge group SO (10) is achieved thanks to the discrete torsions associatedwith a simple current J b spoiling the algebra extension in the gauge sector and correspondingto a Z orbifold.We have proposed a simple counting algorithm for charged massless states. Counting inuntwisted sectors goes as in (2 ,
2) compactifications and can be reduced to the sector of BPSstates. Instead, even for non-gauge-singlet states, the spectrum in J b -twisted sectors getscontributions from non-BPS states that we analyzed in detail.The counting algorithm can be used to compare the CFT side with the Distler-Kachrumodels appearing on the geometry side. These are characterized by a rank 4 vector bundle E on a Calabi-Yau manifold X whose data are constrained by the anomaly matching condition c ( E ) = c ( X ) and make sense also for certain non-rational internal superconformal theorieslike Landau-Ginzburg models and orbifolds thereof.While we focus on the SO(10) case, the generalization to E = SU (5) and E = SU (3) × SU (2) gauge groups is straightforward, at least on the CFT side [15]. Since a minimal modelfactor is required in each reduction step, the number of these classes of models becomes slim inRCFT, but the generalization to Landau-Ginzburg orbifolds should partially make up for thisand hopefully create some room for interesting phenomenology.Besides, an important additional topic which could be explored with the methods of thispaper is the singlet spectrum which is interesting for the study of deformations, in particularon the geometry side and in combination with mirror symmetry for (0,2) models that are notdeformations of the tangent bundle [21, 22, 40]. Acknowledgments
M.B. thanks J. Distler and R. Blumenhagen for kind discussions. A.P. wants to thank S. Kachrufor discussions and the Kavli Institute for Theoretical Physics (KITP) Santa Barbara, where45art of this work has been carried out, for their kind hospitality during the workshop “Stringsat the LHC and in the early Universe“. A.P. was partially supported by the Austrian MarshallPlan Foundation. M.K. and A.P. acknowledge support from the Austrian Research Funds FWFunder grants number I192 and P21239.In memoriam of Maximilian Kreuzer, who recently passed away, M.B. and A.P would liketo thank him for his guidance in this work. With great sorrow we heartily remember Max forhis great spirit and the enthusiasm and persistence he taught us in pursuing scientific ideas.46
Proof that P ( x, , t ) = P ( t − , , x ) We want to computelim q → " q b c ℓ t b cℓ N Y i =1 ϑ ( q, t − ω i q ℓ (1 − ω i ) ) ϑ ( q, t ω i q ℓω i ) = lim q → " N Y i =1 q − ωi ℓ t (1 − ω i ) ℓ ϑ ( q, t − ω i q ℓ (1 − ω i ) ) ϑ ( q, t ω i q ℓω i ) . (A.1)Let us consider a specific superfield Φ i and its contribution to the above limit. There are twopossibilities. If ℓω i ∈ N , the exponent of q in ϑ ( q, t − ω i q ℓ (1 − ω i ) ) is minimum for n = − ℓ (1 − ω i )and n = − ℓ (1 − ω i ) + 1 with the same value. Similarly, the exponent of q in ϑ ( q, t ω i q ℓω i ) isminimum for n = − ℓω i and n = − ℓω i + 1 with the same value. The contribution to P ( x, , t − )is a factor ( − ℓ t − − ωi − t − ω i − t ω i , (A.2)If instead ℓω i N , let θ ( ℓ ) i = ℓω i − [ ℓω i ]. Let us assume 0 < θ ( ℓ ) i < / / < θ ( ℓ ) i < q in ϑ ( q, t − ω i q ℓ (1 − ω i ) ) is minimum for n = 1 − ℓ + ℓω i − θ ( ℓ ) i . Similarly, the exponent of q in ϑ ( q, t ω i q ℓω i ) is minimum for n = ℓω i − θ ( ℓ ) i .The contribution to P ( x, , t − ) is a factor( − ℓ − t − − ωi t θ ( ℓ ) i − ω i . (A.3)Taking the product over the superfields and writing Y ℓω i ∈ Z ( − ℓ t − − ωi Y ℓω i Z ( − ℓ − t − − ωi = ( − N − N tw ( ℓ ) t − b c , (A.4)where N tw ( ℓ ) is the number of twisted fields in the ℓ -twisted sector, we recognize the EPPfrom [13] evaluated at t = 1. B The quintic ⊗ : In order to derive the spectrum (80,0,74) of the (0,2) cousin of the quintic [15,17] we decomposethe “quintic Gepner model” 3 into C ′ = 3 and an additional Fermat factor φ , i.e. minimalmodel at level k = 3, on which the Bonn-twist acts [12].We encode the charge degeneracies of the GSO-twisted but unprojected N=2 SCFT C ′ , withalignment between C ′ and the Fermat factor, in its extended Poincar´e polynomial [13]: For theuntwisted sector we obtain the standard Poincar´e polynomial (in the (c,c) ring) P ( t, ¯ t ) = (1 − T ) (1 − T ) = (1 + T + T + T ) = 1 + 4 T + 10 T + 20 T ++31 T + 40 T + 44 T + 40 T + 31 T + 20 T + 10 T + 4 T + T (B.1)with T = ( t ¯ t ) / . In the twisted sectors only the ground states contribute since there are noinvariant fields. Hence the EPP continues with the terms P ( x, t , ¯ t ) = P ( t , ¯ t ) + x ¯ t + x t ¯ t + x t ¯ t + x t + . . . (B.2)47nd then “periodically” with x P ( t , ¯ t ) + x ¯ t + . . . For the (2,2) version of the quintic we would multiply with an additional 1 + T + T + T andobtain the famous 101 = 10 + 20 + 31 + 40 from P ( t, ¯ t ), which is the K¨ahler modulus from the x -term. In order to determine the number of aligned generations, antigenerations and vectorsfor the (0,2) cousin we read off the relevant data from the tables 3,5 and 7 to get N A = 60from the T and T terms, N A = 0 and N A = 74 from the T and T terms in B.1 and fromthe x t t term in B.2.The complementary Poincar´e polynomial P ( x, q, t ) reads (up to O ( q / ) terms) P ( x, q, t ) = 1 t + 4 t + 10 t + 20 t + 31 t + 40 t + 44 + 40 t + 31 t + + 10 t + 4 t + t + x (cid:20) t + q / (cid:0) − t + 4 t (cid:1) + q / (cid:18) t − t + 10 t (cid:19) + q / (cid:18) − t + 24 t −− t + 20 t (cid:1) + q / (cid:18)
60 + 1 t − t − t + 31 t (cid:19) + · · · ++ q / (cid:18) t − t + 168 t − t + 4 t + (cid:19)(cid:21) + x (cid:20) t + q / (cid:18) − t + 4 t (cid:19) + q / (cid:0) t − t (cid:1) + q / (cid:18) t − t + 10 t (cid:19) ++ · · · + q / (cid:18) t − t + 20 t + 28 t − t + 24 t (cid:19)(cid:21) + x (cid:20) t + q / (cid:18) t − t (cid:19) + q / (cid:18) − t + 4 t (cid:19) + q / (cid:18) t − t + 6 t (cid:19) + · · · + q / (cid:18) t − t + 28 t + 20 t − t + t (cid:19)(cid:21) + x (cid:20) t + q / (cid:18) t − t (cid:19) + q / (cid:18) t − t + 6 t (cid:19) + q / (cid:18) t − t ++24 t − t (cid:1) + q / (cid:18)
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