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Modularity, quaternion-Kähler spaces, and mirror symmetry
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35.In fact, one can also allow for logarithmic singularities at t = 0 and t = ∞ in and α.17 They give rise to the so called anomalous dimensions, and cα, the numerical coefficients which supplement the twistor data, given by the covering and transition functions, defining uniquely. The anomalous dimension cα plays an important physical role since it encodes the one-loop gs correction to the HM metric.13,18,19 But in the type IIB formulation, for which the formalism developed here is supposed to be applied, it is possible to avoid non-vanishing cα.20
36.In this paper we are interested only in the local metric on . Therefore, we do not need to distinguish between the patches covering the twistor space and their projections to , and in what follows the hats denoting such a projection will be omitted.
37.The shifts of and must be supplemented by a coordinate dependent shift of σ so that these isometries form the Heisenberg group.
38.In arbitrary gauge they will still be given by (3.10) multiplied by the phase of X0.
39.Note that this assumption implies that with k ⩾ 1 are all identical. In particular, . However, we will ignore this subtlety and consider all these patches as different. Otherwise, we would have to pay special attention to coinciding patches and the presentation would become very heavy. We prefer to sacrifice the rigour in favour of clarity. A rigorous analysis is also possible and does not change any results.
40.All variables appearing on the rhs can be expressed through the variables in the patch using the gluing conditions. We prefer to present the constraint in the form (4.8) since it is simpler and more convenient for applications. In fact, the proper meaning of this constraint is that it provides a functional equation on the transition functions which makes them consistent with the modular action.
41.The prefactor in (3.11) agrees with (4.11) since it can be shown that the classical zeros (3.10) satisfy .
42.In fact, the kernel is not the only one possessing the invariance property. Choosing , one obtains two kernels which are also invariant, so is any their linear combination. In particular, . What distinguishes this particular combination, or equivalently k(t) from (4.15), is that it is invariant under the combined action of the antipodal map and complex conjugation ensuring the reality conditions to be satisfied by various quantities. In Ref. 28 it was argued that the kernels K± can nevertheless be used, at least in the linear approximation, due to the property and appear to be useful in the context of the so called large volume limit, where ta → ∞ keeping other coordinates fixed. In this limit all integrals can be evaluated by the saddle point approximation with the saddle points approaching the critical points s±. The problem is that the formulation based on the single kernel K leads to appearance of terms diverging in the large volume limit which, however, can be removed by a coordinate redefinition. The passage to the kernels K± takes care about this redefinition and directly provides final results. Unfortunately, we do not know how to extend these results to the non-perturbative treatment of generic deformations which we are doing here.
43.The prime on the sum indicates that it runs over .
44.Note that the property ς[s+] = s− ensures that .
45.Of course, for generic deformations breaking all continuous isometries the expansion is infinite. We have performed the explicit perturbative check of modular invariance only to the second order. For the deformations preserving two continuous isometries, however, the expansion stops at third order.12
46.One could worry about the situation when some open contours end at the zeros of cξ0 + d, as shown in Fig. 1 and indeed happens in physically interesting cases of D3 and fivebrane instantons,15,28 because the formula (4.35) seems to ignore the contributions of the integrals along the dashed parts of the contours inside the circle. This can be justified as follows. The transition function associated with the closed contour surrounding a zero must ensure the regularity of Darboux coordinates near this point, and therefore it must cancel the discontinuities generated by the open contours. This implies that it should have the same branch cut discontinuities along the dashed parts of the contours as given by the transition functions associated with them. Taking this into account, one can check that the contributions of the dashed parts cancel in all expressions.
47.At this point of the derivation, is still an unknown function. It can be found only after one fixes the action on the coordinate t.
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