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reviewed above is of high conceptual importance, it is in practice often easier to construct vector bundles with the desired structure group directly. In the following section, we review

this construction for SU(N) bundles and specializations thereof which has been studied first in [177] and was further developed in [140, 139, 178].

In this sectionE always denotes an elliptic curve with a marked point p. The curve is defined over a general fieldK, which does not necessarily have to be algebraically closed. It is well-known that an elliptic curve with a point phas a representation in the Weierstrass normal form (3.5), where preads[x∶y∶z] = [0∶1∶0]. In general, a degree zero line bundle L Ð→E, i.e. a U(1)-bundle, takes the form

L = O(q) ⊗ O(p)−1= O(q−p), (3.11)

whereqdenotes another arbitrary rational point onE (note that over K=_Cevery point is rational). Furthermore, we note that there is a bijective map φ from the elliptic curve E

onto its Picard group of degree zero which is defined by

φ ∶EÐ→Pic0(E), q↦q−p. (3.12)

In particular, this extends to an isomorphism from the space of line bundles onto Pic0(E), defined by div(L) =q−p. To be more precise, the divisor map ‘div’ is to be applied to a meromorphic section2 of L. For later purposes, we also recall that the addition law in Pic0_{(E) can be identified with the group law on E, which we denote by ⊞, via this}

isomorphism.

A semi-stable SU(N) vector bundle of degree zeroV is then given as the sum3 _of

N holomorphic line bundles L_i, i.e. we have V = ⊕N_i=1L_i = ⊕N_i=1O(q_i−p), such that the 2_{This map is independent of the section chosen.}

3_{If two or more points coincide, the situation is a bit more subtle. In this case the bundle is given by} ⊕N_i=1O(qi−p)Iri, whereridenotes the multiplicity of the pointqiandIris inductively defined by the extension

sequence 0Ð→ O Ð→Ir−1Ð→ O Ð→0. However, one usually only considers bundles up to S-equivalence which identifiesIrwithO⊕r.

determinant ofV is trivial. The latter implies that

⊗N_i=1O(qi−p) = O ⇔ ⊞N_i=1qi=0. (3.13)

An SU(N) vector bundle is therefore determined by the choice of N points on E that sum up to zero. Any such N-tupel is determined by a projectively unique element of

H0(E,O(N p)), i.e. a function withN zeros and a pole of orderN at p. Thus, the moduli

space of SU(N)vector bundles is given by

M_SU(N)=_PH0(E,O(N p)). (3.14)

In the affine Weierstrass form ofE, given by (3.5), the coordinatesx,yhave a pole of order two and three at p, respectively. Accordingly, any element of _PH0(E,O(N p))enjoys an expansion w=c0+c1x+c2y+c3x2+. . .+ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ c_NxN2 ifNis even , c_NxN2−3y ifN is odd , (3.15)

with ci∈K. The section w is called the spectral cover polynomial and has N common points withE, called thespectral cover, which define the desired SU(N)bundle. Counting parameters of (3.15), one is lead to the conclusion that

M_SU(N)=_PN−1. (3.16)

Finally, a comment on rational versus non-rational points is in order. Generically, pis the only point onE over a general field K. However, in such a situation, it is possible to markNpoints in a rational way by the polynomialw=0 which give rise to an SU(N)bundle in the way just described. Nevertheless, under the circumstances that there are additional

structure group reduces in a certain way, as discussed next.

Vector bundles with reduced structure groups

As described in the previous section, the choice of N points on E describes an SU(N) bundle. If we consider just an elliptic curve E over _C, which is the geometry relevant for the construction of heterotic compactifications to eight dimensions, the spectral cover (3.15) can be factorized completely. This corresponds to the 16 possible Wilson lines on T2_.

In contrast, if we consider an elliptic curve over a function field, as it arises in elliptic fibrationsZ_nofEover a baseB_n−1used for lower-dimensional heterotic compactifcations, the N points are the zeros of (3.15), which defines an N-section of the fibration. In non- generic situations, where subsets of theNsheets of thisN-section are well-defined globally, i.e. are monodromy invariant, the structure group of the vector bundle is reduced. For example, a separation into two sets ofkandlsheets (withk+l=N), respectively, results in the structure group S(U(k) ×U(l)). The spectral cover defined by (3.15) is called “split” and defines a reducible variety insideZ_n, see e.g. [165, 166, 167, 168]. In the most extreme case, one could have k=1 and l=N−1. In this case, the elliptic fibration of Z_n has to necessarily have another well-defined section in addition to the section induced by the rational pointp: it is the one marked by the component of the spectral coverw=0 with just one sheet [170]. Thus, the fiberE has a rational point, which we denote byqand one can, as discussed above, define a U(1) line bundleLvia (3.11). As this fiberwise well-defined line bundle is also well-defined globally, it will induce a line bundle on Zn, whose first Chern class is given, up to vertical components, by the difference of the sections induced byqand p, cf. [179]. The structure groupH of the vector bundle is in this case given by

We will see later that this situation will be relevant situation for the construction of U(1) gauge groups in the heterotic string.

We emphasize that for a U(1)-bundle alone there is no spectral cover polynomial (3.15) that would be able to detect this additional rational point. This is due to the fact that there is no function that has only one zero on an elliptic curveE . However, if the rational point is accompanied by further points, rational or non-rational points over the fieldK, it can very well be seen by the spectral cover. For instance, one could construct a spectral cover from qand−q, which would describe a bundle of structure group S(U(1) ×U(1)).

Finally, it needs to be discussed what interpretation should be given to the case that the rational pointqon the curveE happens to be torsion of orderk. In this case the structure groupH reduces further to S(U(N) ×_Zk). To argue for this, we invoke again a fiberwise argument. The fiber at a generic point in B_n−1 admits a line bundle L = O(q−p) with the property that Lk= O. This is clear as the transition functions g_{i j} will be subject to gk_{i j}=1 in ˇCech cohomology ask times the Poincar´e dual of its first Chern class is trivial. However, this is just the statement that the fiberwise structure group ofLis contained in_Zk. Employing thatpandqare globally well-defined sections then suggests that this argument also holds onZn.