A new class of topological amplitudes
Introduction
A special role in extended supersymmetric theories is played by 1/2-BPS couplings that depend only on half of the superspace, generalizing chiral supersymmetric F-terms. Usually, such interactions are easier to study because they are subject to non-renormalization theorems, while they have a variety of interesting physical applications varying from the vacuum structure all the way up to properties of supersymmetric black-holes. Moreover, in string effective field theory, these couplings are expected to be computed by topological amplitudes, depending only on the zero-mode structure of the compactification space [1], [2], [3]. An interesting property is that the half-BPS structure of these terms is broken at the quantum level. On the topological side, this breaking is due to a violation in the conservation of the BRST current described by an anomaly equation [3], [4], [5], while on the string side it is understood from the difference between the Wilsonian and ‘physical’ effective action that includes also the contribution of massless degrees of freedom [2].
The first instance of well-studied 1/2-BPS couplings in supersymmetry is the series , where W is the chiral (self-dual) gravitational Weyl superfield and the coefficients depend on the vector multiplet moduli in the Coulomb phase of the theory [2], [3]. 's are computed by the genus g topological partition function of an twisted σ-model on the six-dimensional Calabi–Yau compactification manifold of type II string theory in four dimensions, subject to a holomorphic anomaly equation that takes the form of a recursion relation. Moreover, the independence of 's from hypermultiplets, which include the string dilaton, implies a non-renormalization theorem for their form. These results have been generalized to supersymmetric compactifications of type II string on , where two series of higher order terms were identified, computed by topological amplitudes: and , where K is a superdescendent of the Weyl superfield [6], [7]. The half-BPS property leads to a harmonicity equation for the moduli dependence of the couplings [8], [9], [10], generalizing holomorphicity, up to anomalous contributions from boundary terms [10]. Despite the bigger supersymmetry, the analysis is more involved than in the case of vector multiplets, since the lack of an ordinary superspace description implies the use of on-shell harmonic superspace [11], [12], [13].
A different question is to study the corresponding couplings when one reduces the supersymmetry by half. On the string side, this can be done in two ways that are dual to each other. Either by considering the ‘semi-topological’ theory obtained by twisting the supersymmetric left-movers of the heterotic string [14], [15], or by applying a world-sheet involution on the type II amplitudes that introduces open string boundaries [16]. In the case of 's, this generates an series of higher order F-terms of the form , where is now the gauge superfield with the gauge indices contracted in an appropriate way [14]. The holomorphic anomaly equation however does not close on 's; it brings new objects that give rise to a double series , where Π denotes generically a chiral projection of a real function of chiral superfields. On the topological side, the same results are obtained upon introducing world-sheet boundaries.3
In this work, we apply the above reduction mechanism to the topological amplitudes and obtain a new series of higher order 1/2-BPS terms with supersymmetry. The novel feature of these terms is that they mix vector multiplets with neutral hypermultiplets, despite the common wisdom. Indeed, starting with , one generates the series , where is now a superdescendent of an vector superfield (with the gauge indices contracted appropriately, as before). The coupling coefficients depend in this case on both analytic vector multiplet as well as on (Grassmann analytic) hypermultiplet moduli, as dictated by the half-BPS structure. Moreover, these coupling share similar properties at the same time with the topological couplings and with the series. More precisely, the appropriate formalism for their study is again (on-shell) harmonic superspace, which complicates the analysis compared to the 's. On the other hand, quantum corrections violate both the holomorphicity condition with respect to the vector moduli, and the harmonicity with respect to the hypermultiplets. Furthermore, the anomaly equation does not closes on 's; it brings new objects generating the double series , where P is an appropriate half-BPS projection.
The organization of the paper and the outline of the results obtained are described below. The next two sections contain the string computation of the new topological amplitudes. In Section 2, we compute the special type of topological amplitudes in type I open string theory, from the topological amplitudes , by applying a world-sheet involution.4 In fact, we evaluate a physical amplitude involving two gauge field strengths, two vector multiplet scalars (with one derivative each) and gauginos with the same four-dimensional chirality, , on a world-sheet with boundaries, and we show that it is reduced to a topological expression within the twisted σ-model on . Then, in Section 3, we compute the same amplitudes on the heterotic side (compactified on ), which turns out to be easier for our subsequent analysis because of the absence of the problematic Ramond–Ramond sector, exploiting heterotic–type I duality. Again, the physical amplitude is expressed as a semi-topological expression, i.e. only for the (supersymmetric) left-movers, while the bosonic part provides the gauge indices appropriately contracted (we are essentially taking products of differences of gauge groups with no charged massless matter).
These two sections are complemented by three appendices. In Appendix A, we review the main properties of the and world-sheet superconformal algebras, Appendix B contains the expressions of the three main vertex operators we use, while Appendix C contains the definitions of the theta-functions and prime forms.
The following section contain the effective field theory description of the topological amplitudes and the study of the generalized analyticity relations and anomaly equations. In Section 4, we study the interpretation of the string results, obtained in Sections 2 Type I open topological amplitudes, 3 Topological amplitudes in heterotic orbifold compactifications, in the context of the effective supergravity. As mentioned above, the appropriate formulation is in terms of the harmonic superspace (for a review see [18]). We first make an analysis in global supersymmetry (Section 4.1), introduce the harmonic variables, define the series of the effective interaction terms and derive the conditions on the moduli dependence of the couplings from their half-BPS structure. These are the usual holomorphicity with respect to the vector multiplet moduli, while the hypermultiplet moduli dependence is subject to two differential constraints, in close analogy with the equations found for the terms: the so-called harmonicity condition, expressing the property that only one combination of the four components of the hypermultiplets enter in the coupling, as well as a second-order constraint. We then study the effects of the curvature of the hypermultiplet scalar manifold (Section 4.2), considering as an example the coset for n hypermultiplets (using the harmonic description of [19]). We show in particular that the second-order differential equation is modified by an additional term linear in and proportional to a R-charge . The generalization to (conformal) supergravity is done in Section 4.3, where the full covariantized expressions of the effective operators are obtained, as well as of the differential equations they obey.
In Section 5, we present a different derivation of the equivalence between string and topological amplitudes which is free of an ambiguity that appears in the computation we perform in Sections 2 Type I open topological amplitudes, 3 Topological amplitudes in heterotic orbifold compactifications. This is achieved by evaluation of a different amplitude related by supersymmetry to the previous one, containing only fermions: two chiral and two antichiral hyperfermions, besides the gauginos. We also generalize the computation from orbifolds considered in the text, to the most general superconformal theory.
In Section 6, we verify explicitly the analyticity equations in string theory, on the heterotic side. Moreover, we evaluate the world-sheet boundary contributions for the holomorphicity and harmonicity equations that give rise to anomalous terms. In contrast to the familiar 's and their generalizations computed by closed topological amplitudes, the anomalous terms do not generate recursion relations for the non-holomorphic/harmonic dependence of the same couplings, because they involve new objects. This is similar to the case encountered in topological amplitudes, irrespectively on which string framework they are defined (heterotic or type I). The new objects involve chiral/half-BPS projections of general non-holomorphic/harmonic functions and generate a double series of higher-dimensional operators with moduli-dependent coefficients , described above. In both equations, the new quantities are proportional to the one-loop threshold corrections to the gauge couplings, on the heterotic side. We argue that the non-holomorphicity appears due to the contribution to the string amplitude (which computes the sum of all connected graphs) from via the elimination of the auxiliary fields. This section is supplemented by Appendix D, where we explicitly compute the string amplitudes generating the double series described above in a generic Calabi–Yau compactification.
Section snippets
Type I open topological amplitudes
In this section we will calculate a special type of topological amplitudes in type I open string theory. They are related to similar objects in the type II theory (see [6], [7], [10]) via a world-sheet involution [20], [21] which we will describe in detail first.
Topological amplitudes in heterotic orbifold compactifications
In Section 2 we have been considering topological amplitudes in the type I theory. However, in order not having to deal with the problem of open string moduli (see e.g. [17]), we rather prefer to transfer the problem to a dual setup, in which the topological amplitudes again compute closed string correlators. One possibility is to exploit the duality between type I and heterotic string theory. Since this duality is perturbative in nature we expect to recover of the type I theory at the
Harmonic description and harmonicity relations
In the previous section we have considered particular amplitudes in heterotic string theory which are captured by correlation functions in a twisted two-dimensional theory. In this section we would like to understand which terms in the heterotic effective action these amplitudes correspond to and whether they have any interesting properties with respect to their moduli dependence. It turns out that the effective action is best formulated in harmonic superspace and we will begin by
Topological amplitudes in generic Calabi–Yau compactifications
In Sections 2 Type I open topological amplitudes, 3 Topological amplitudes in heterotic orbifold compactifications we considered a string amplitude stemming from the effective action (4.17) which involves two self dual field strengths, gauginos and two chiral vector multiplet scalars each carrying one momentum. This amplitude computes the reduced function of Eq. (4.31). However, as pointed out in these sections a direct computation of this term is complicated by the presence of many
Harmonicity relation from string theory
In this section we will discuss special properties of the heterotic couplings , which we have computed in Section 3, from a string theoretic point of view. On the one hand we will check the harmonicity relation (4.33) by explicitly working out all derivatives. On the other hand, as we can see from the coupling (4.17), field theoretic considerations predict a dependence of only on the holomorphic vector multiplets W. The half-BPS nature of these couplings, however, suggests that there might
Concluding remarks
In this work, we have analyzed a new series of topological amplitudes associated to a particular class of higher-dimensional 1/2-BPS terms in the low energy effective superstring action, involving both vector multiplets and neutral hypermultiplets. We computed these amplitudes on both type I and heterotic string side and studied the moduli dependence of the corresponding couplings in string theory, as well as in the effective supergravity using harmonic superspace. Their BPS character
Acknowledgements
We would like to thank Atish Dabholkar, Sergio Ferrara, Boris Pioline and Johannes Walcher for helpful discussions. This work was supported in part by the European Commission under the ERC Advanced Grant 226371 and in part by the French Agence Nationale de la Recherche, contract ANR-06-BLAN-0142. The research of S.H. has been supported by the Swiss National Science Foundation.
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