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...ce, a choice of equivariant lift, and this is the ultimate reason for their quantization in supergravity. We can see Fayet-Iliopoulos parameters as lifts explicitly in the supergravity lagrangians of =-=[32]-=-[chapter 25]. In general, since the fermions χi, ψµ couple to L, L−1, a group action on M must be lifted to an action on L, L−1 in order to uniquely define the theory. (A group action on either of L, ...

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...dim(G) 6 φ∗c1(L) 3 (A.2) Even when (A.2) does not vanish, it is possible to contemplate cancelling the anomaly by adding to the action ”Wess-Zumino”-type terms, whose classical variation is anomalous =-=[51, 52]-=-, but we will not pursue that here. B. Existence of equivariant structures As noted in the text, G-equivariant structures on line bundles do not always exist. In this appendix, we shall work out condi...

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.... This quantized shift is the Fayet-Iliopoulos parameter. The fact that honest lifts, when they exist, are quantized in the fashion above, is a standard result in the mathematics literature (see e.g. =-=[33]-=-[prop. 1.13.1]). Since it also forms the intellectual basis for the central point of this paper, let us give a second explicit argument that differences between lifts are quantized, following4 [34]. A...

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...dim(G) 6 φ∗c1(L) 3 (A.2) Even when (A.2) does not vanish, it is possible to contemplate cancelling the anomaly by adding to the action ”Wess-Zumino”-type terms, whose classical variation is anomalous =-=[51, 52]-=-, but we will not pursue that here. B. Existence of equivariant structures As noted in the text, G-equivariant structures on line bundles do not always exist. In this appendix, we shall work out condi...

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...ifts, possible values of the Fayet-Iliopoulos parameter are quantized. Let us begin by reviewing how one gauges group actions in nonlinear sigma models in general. To preserve supersymmetry (see e.g. =-=[29, 30]-=-), the group action must be generated by holomorphic Killing vectors X(a) ≡ X(a)i ∂ ∂φi where (a) denotes a Lie algebra index, and φ a map in the nonlinear sigma model. To be holomorphic Killing means...

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...t map is uniquely defined7 [44–46]. One can consider also the moduli space of vector multiplets in N = 2 supergravity. Such moduli spaces are described by special geometry, and in this case (see e.g. =-=[47]-=-[equ’n (10)]) the Kähler form on the moduli space arising in Calabi-Yau compactifications is identified with ∂∂ ln〈Ω|Ω〉 and hence unique (as this is invariant under rescalings of the holomorphic top-...

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... in field and string theory, and a discussion of the Fayet-Iliopoulos quantization condition for gerby moduli spaces in supergravity, will appear in [26]. 2. Review of Bagger-Witten Bagger and Witten =-=[27]-=- discussed how the Kähler class of the moduli space1 of scalars of a supergravity theory is quantized, resulting from the fact that ultimately the Kähler class must be the first Chern class of a lin...

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... reviewed above), examples of gerby moduli ‘spaces’ in field and string theory, and a discussion of the Fayet-Iliopoulos quantization condition for gerby moduli spaces in supergravity, will appear in =-=[26]-=-. 2. Review of Bagger-Witten Bagger and Witten [27] discussed how the Kähler class of the moduli space1 of scalars of a supergravity theory is quantized, resulting from the fact that ultimately the K...

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...re but also important point is that L defines a projective embedding of M . The quantization of the Kähler form ω described in [27] means that M is a Hodge manifold. By the Kodaira Embedding Theorem =-=[35]-=-, every Hodge manifold is projective, and L−n, for some n ≫ 0 is the ample line bundle that provides the projective embedding. These features are not characteristic of symplectic quotients, but they a...

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...t characteristic classes be invariant under group actions, but this is not sufficient. Examples of non-equivariant line and vector bundles, with invariant characteristic classes, can be found in e.g. =-=[53, 54]-=- and references therein. Another example is as follows8. Let E be an elliptic curve with a marked point σ ∈ E. Let L = OE(2σ). Let x ∈ E, and let tx : E → E be the translation by x in the group law. T...

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...g. [33][prop. 1.13.1]). Since it also forms the intellectual basis for the central point of this paper, let us give a second explicit argument that differences between lifts are quantized, following4 =-=[34]-=-. Assume the space is connected, and let {Uα} be an open cover, that is ‘compatible’ with the group action5. At the level of Cech cohomology, a G-equivariant line bundle is defined by transition funct...

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...ot widely familiar to physicists, in this section we shall work through a very basic example 15 of a geometric invariant theory (GIT) quotient. (See [36–38] for more information on GIT quotients, and =-=[55]-=-[appendix C] for additional examples.) In principle, given a complex manifold X with very ample line bundle L → X , and the action of some group G on X which has been lifted to a linearization on L, t...

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... hand-in-hand. In the special case of stacks that are gerbes, i.e. the theories discussed in [1], such theories in two dimensions are equivalent to nonlinear sigma models on disjoint unions of spaces =-=[8]-=-, a result named the “decomposition conjecture.” We can understand the decomposition conjecture schematically as follows. Consider a nonlinear sigma model on a space X , for simplicity withH2(X,Z) = Z...

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... image of the moment map, one can recover Pn with any real Kähler class, not necessarily integral. To get an integral Kähler class, the image of the moment map must also be integral. More generally =-=[31]-=-, for abelian G, the moment map takes values in t∗, but only if one reduces on points in T ∗ ⊂ t∗ can one hope to get an integral Kähler form on the quotient. Hence, to get an integral Kähler form o...

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... approach to four-dimensional gauge theories, where one describes the moduli space in terms of its (invariant) chiral rings (see for example [39][section 12.3] and references therein, though also see =-=[40]-=- for a different perspective). Perhaps the best interpretation of the D-terms in supergravity is that the Fayet-Iliopoulos parameter is defined by a choice of linearization, though the rest of the con...

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...clusions 11 8 Acknowledgements 12 A Four-dimensional sigma model anomalies 12 B Existence of equivariant structures 13 C An example of a GIT quotient 15 References 17 1. Introduction The recent paper =-=[1]-=- discussed quantization of Fayet-Iliopoulos parameters in four-dimensional supergravity theories in which the group action on scalars was realized linearly. In this short note, we observe that that qu...

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...ifts, possible values of the Fayet-Iliopoulos parameter are quantized. Let us begin by reviewing how one gauges group actions in nonlinear sigma models in general. To preserve supersymmetry (see e.g. =-=[29, 30]-=-), the group action must be generated by holomorphic Killing vectors X(a) ≡ X(a)i ∂ ∂φi where (a) denotes a Lie algebra index, and φ a map in the nonlinear sigma model. To be holomorphic Killing means...

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... ring of invariant functions, is closely reminiscent of the approach to four-dimensional gauge theories, where one describes the moduli space in terms of its (invariant) chiral rings (see for example =-=[39]-=-[section 12.3] and references therein, though also see [40] for a different perspective). Perhaps the best interpretation of the D-terms in supergravity is that the Fayet-Iliopoulos parameter is defin...

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...der the hypermultiplet moduli space. In rigid N = 2 supersymmetry, that moduli space is a hyperKähler manifold, but in N = 2 supergravity it is a quaternionic Kähler manifold [42]. It was argued in =-=[43]-=-[equ’n (5.16)] that in N = 2 supergravity in four dimensions, the curvature scalar on the quaternionic Kähler moduli manifold is uniquely determined, so that there is not even an integral ambiguity. ...

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...ove is to Gromov-Witten theory, where it has been checked and applied to simplify computations of Gromov-Witten invariants of gerbes, see [17–22]. Another application is to gauged linear sigma models =-=[9]-=-, where it answers old questions about the meaning of the Landau-Ginzburg point in a GLSM for a complete intersection of quadrics, gives a physical realization of Kuznetsov’s homological projective du...

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...ral, and hence (necessarily) undergo chiral rotations across coordinate patches, there is the potential for an anomaly. In this appendix we will briefly outline how the resulting anomalies, following =-=[28]-=- (see also [48–50] for background information on sigma model anomalies). Globally, the fact that the fermions undergo chiral rotations across coordinate patches is encoded in an anomaly, given by the ...