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...opological twists When the superpotential vanishes (and, for ordinary chiral multiplets, X = 0), there are no curvature couplings, and the theories admit the same topological twists discussed in e.g. =-=[37]-=-. When the superpotential is nonzero, this story is more complicated. Sometimes those curvature couplings are incompatible with the topological twist (which can sometimes be 10 alleviated by twisting ...

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... be cohomologically trivial. Furthermore, supersymmetric localization techniques have been applied to theories of this form to obtain quantum mechanically exact results such as the partition function =-=[15, 16, 17, 18, 19, 20, 21]-=-. Recently these methods were applied to two-dimensional gauged linear sigma models (GLSMs) to derive exact expressions for partition functions [20, 21], which has quickly led to some interesting new ...

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... be cohomologically trivial. Furthermore, supersymmetric localization techniques have been applied to theories of this form to obtain quantum mechanically exact results such as the partition function =-=[15, 16, 17, 18, 19, 20, 21]-=-. Recently these methods were applied to two-dimensional gauged linear sigma models (GLSMs) to derive exact expressions for partition functions [20, 21], which has quickly led to some interesting new ...

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... be cohomologically trivial. Furthermore, supersymmetric localization techniques have been applied to theories of this form to obtain quantum mechanically exact results such as the partition function =-=[15, 16, 17, 18, 19, 20, 21]-=-. Recently these methods were applied to two-dimensional gauged linear sigma models (GLSMs) to derive exact expressions for partition functions [20, 21], which has quickly led to some interesting new ...

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...uperfields without curvature couplings As twisted chiral multiplets are discussed less commonly than ordinary chiral multiplets, it will be useful to review a few of their basic properties. (See also =-=[41]-=-[section 12.2] for another comparison of ordinary and twisted chiral multiplets.) Given twisted chiral multiplets of the form (ρi, χ̄i+, χ i − ), where the ρ are bosons, the Lagrangian for a theory co...

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...en r →∞, as one would expect. 1 As an aside, terms closely analogous to the curvature-dependent terms in the Lagrangian above have been discussed in two-dimensional theories in a different context in =-=[29]-=-. 4 There are two conditions on the data above. First, the Kähler potential is invariant under the isometry defined by the holomorphic Killing vector X , which means that on each coordinate patch, LX...

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..., the f ’s are constrained to obey∑ i X i∂if(φ) = 0. (7) On a Kähler manifold with a holomorphic isometry X , for data on a good open cover, one can always choose K’s, f ’s to obey these constraints =-=[30]-=-. The second condition says that the superpotential W is homogeneous of degree 2 under the vector-like U(1)R symmetry, meaning 2W − iX iWi = 0 (8) (up to an additive constant). In particular, if X = 0...

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...kly led to some interesting new computational methods and results for Gromov-Witten invariants [22, 23, 24] and the Seiberg-Witten Kähler potential [25], as well as other insights into older results =-=[26]-=-. As part of that work, the papers [20, 21] worked out curvature couplings in two-dimensional linear sigma models whose target spaces are vector spaces. In this paper, we return to [20, 21] to re-exam...

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... be cohomologically trivial. Furthermore, supersymmetric localization techniques have been applied to theories of this form to obtain quantum mechanically exact results such as the partition function =-=[15, 16, 17, 18, 19, 20, 21]-=-. Recently these methods were applied to two-dimensional gauged linear sigma models (GLSMs) to derive exact expressions for partition functions [20, 21], which has quickly led to some interesting new ...

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...1 Introduction In recent years, curvature couplings in rigidly supersymmetric nonlinear sigma models on nontrivial spacetime manifolds of various dimensions have been discussed by several groups, see =-=[1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 14]-=- and references contained therein. Often, supersymmetry on the nontrivial spacetime manifold will constrain the target space in some fashion, by e.g. requiring the Kähler form on the target space to ...

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...tiplet, their components mix holomorphic and antiholomorphic target space indices: χ = ( χi − χı̄+ ) , χ̄ = ( χ̄ı̄ − χ̄i+ ) (11) 2 We would like to thank I. Melnikov for making this observation. 3See =-=[32, 33, 34, 35]-=- for a recent discussion of the Fayet-Iliopoulos parameter in supergravity, and how old obstruction issues summarized in e.g. [36] can be circumvented. 4 We do not attempt to consider H-flux backgroun...

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...undle TM to T ∗M on the right-movers, the effect would be to convert the twisted chiral multiplets back into chiral multiplets. This is a special case of a duality in (0,2) theories discussed in e.g. =-=[42]-=-. Now, let us specialize to the caseW = 0, in which case there are no curvature couplings on any worldsheet, and consider topological twists. It is straightforward to topologically twist the theory of...

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... be cohomologically trivial. Furthermore, supersymmetric localization techniques have been applied to theories of this form to obtain quantum mechanically exact results such as the partition function =-=[15, 16, 17, 18, 19, 20, 21]-=-. Recently these methods were applied to two-dimensional gauged linear sigma models (GLSMs) to derive exact expressions for partition functions [20, 21], which has quickly led to some interesting new ...

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...1 Introduction In recent years, curvature couplings in rigidly supersymmetric nonlinear sigma models on nontrivial spacetime manifolds of various dimensions have been discussed by several groups, see =-=[1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 14]-=- and references contained therein. Often, supersymmetry on the nontrivial spacetime manifold will constrain the target space in some fashion, by e.g. requiring the Kähler form on the target space to ...

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...1 Introduction In recent years, curvature couplings in rigidly supersymmetric nonlinear sigma models on nontrivial spacetime manifolds of various dimensions have been discussed by several groups, see =-=[1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 14]-=- and references contained therein. Often, supersymmetry on the nontrivial spacetime manifold will constrain the target space in some fashion, by e.g. requiring the Kähler form on the target space to ...

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...re Qα and Q̄α are the supersymmetry generators, Jm are the generators of the SU(2) isometry of S2, and R is the generator of the U(1)R symmetry. The space of Killing spinors on S2 is four-dimensional =-=[28]-=-; a useful basis of this space consists of two Killing spinors called positive Killing spinors and two Killing spinors called negative Killing spinors [20]. Let’s denote the positive Killing spinors a...

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...tiplet, their components mix holomorphic and antiholomorphic target space indices: χ = ( χi − χı̄+ ) , χ̄ = ( χ̄ı̄ − χ̄i+ ) (11) 2 We would like to thank I. Melnikov for making this observation. 3See =-=[32, 33, 34, 35]-=- for a recent discussion of the Fayet-Iliopoulos parameter in supergravity, and how old obstruction issues summarized in e.g. [36] can be circumvented. 4 We do not attempt to consider H-flux backgroun...

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...1 Introduction In recent years, curvature couplings in rigidly supersymmetric nonlinear sigma models on nontrivial spacetime manifolds of various dimensions have been discussed by several groups, see =-=[1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 14]-=- and references contained therein. Often, supersymmetry on the nontrivial spacetime manifold will constrain the target space in some fashion, by e.g. requiring the Kähler form on the target space to ...

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... linear sigma models (GLSMs) to derive exact expressions for partition functions [20, 21], which has quickly led to some interesting new computational methods and results for Gromov-Witten invariants =-=[22, 23, 24]-=- and the Seiberg-Witten Kähler potential [25], as well as other insights into older results [26]. As part of that work, the papers [20, 21] worked out curvature couplings in two-dimensional linear si...

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...ke to thank I. Melnikov for making this observation. 3See [32, 33, 34, 35] for a recent discussion of the Fayet-Iliopoulos parameter in supergravity, and how old obstruction issues summarized in e.g. =-=[36]-=- can be circumvented. 4 We do not attempt to consider H-flux backgrounds in this paper. 7 In this section, we will work with Killing spinors ζ , ζ̄ obeying ∇mζ = i 2r γmζ, ∇mζ̄ = − i 2r γmζ̄ , (12) a ...

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...tential is nonzero, this story is more complicated. Sometimes those curvature couplings are incompatible with the topological twist (which can sometimes be 10 alleviated by twisting bosons as in e.g. =-=[38, 39]-=-). However, a more fundamental problem is that even if one can consistently twist the theory, the curvature terms induced by the superpotential break the BRST symmetry, in the sense that the action is...

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...tential is nonzero, this story is more complicated. Sometimes those curvature couplings are incompatible with the topological twist (which can sometimes be 10 alleviated by twisting bosons as in e.g. =-=[38, 39]-=-). However, a more fundamental problem is that even if one can consistently twist the theory, the curvature terms induced by the superpotential break the BRST symmetry, in the sense that the action is...

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... linear sigma models (GLSMs) to derive exact expressions for partition functions [20, 21], which has quickly led to some interesting new computational methods and results for Gromov-Witten invariants =-=[22, 23, 24]-=- and the Seiberg-Witten Kähler potential [25], as well as other insights into older results [26]. As part of that work, the papers [20, 21] worked out curvature couplings in two-dimensional linear si...

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...ns for partition functions [20, 21], which has quickly led to some interesting new computational methods and results for Gromov-Witten invariants [22, 23, 24] and the Seiberg-Witten Kähler potential =-=[25]-=-, as well as other insights into older results [26]. As part of that work, the papers [20, 21] worked out curvature couplings in two-dimensional linear sigma models whose target spaces are vector spac...

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... linear sigma models (GLSMs) to derive exact expressions for partition functions [20, 21], which has quickly led to some interesting new computational methods and results for Gromov-Witten invariants =-=[22, 23, 24]-=- and the Seiberg-Witten Kähler potential [25], as well as other insights into older results [26]. As part of that work, the papers [20, 21] worked out curvature couplings in two-dimensional linear si...

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...1 Introduction In recent years, curvature couplings in rigidly supersymmetric nonlinear sigma models on nontrivial spacetime manifolds of various dimensions have been discussed by several groups, see =-=[1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 14]-=- and references contained therein. Often, supersymmetry on the nontrivial spacetime manifold will constrain the target space in some fashion, by e.g. requiring the Kähler form on the target space to ...

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...1 Introduction In recent years, curvature couplings in rigidly supersymmetric nonlinear sigma models on nontrivial spacetime manifolds of various dimensions have been discussed by several groups, see =-=[1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 14]-=- and references contained therein. Often, supersymmetry on the nontrivial spacetime manifold will constrain the target space in some fashion, by e.g. requiring the Kähler form on the target space to ...

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...1 Introduction In recent years, curvature couplings in rigidly supersymmetric nonlinear sigma models on nontrivial spacetime manifolds of various dimensions have been discussed by several groups, see =-=[1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 14]-=- and references contained therein. Often, supersymmetry on the nontrivial spacetime manifold will constrain the target space in some fashion, by e.g. requiring the Kähler form on the target space to ...

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...2, 2) chiral multiplet in two dimensions contains components (φi, φ̄ı̄, ψi, ψ̄ ı̄, F i, F̄ ı̄). The chiral fields φi, φ̄ı̄ parametrize the target space M , which is a Kähler manifold. As observed in =-=[20, 21]-=-, the U(1)R charges of the chiral fields φ i enter the definition of the Lagrangian of the linear sigma models on S2. To construct nonlinear sigma models on a worldsheet S2, one needs to use the holom...

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...e A-twisted Landau-Ginzburg models realized Mathai-Quillen classes which gave a mathematical understanding of the behavior of renormalization group flow, as a type of Thom class phenomenon. (See also =-=[40]-=- for closely analogous behaviors in elliptic genera.) It would be interesting to see if something analogous arises in localization computations. 11 6 Acknowledgements We thank Robert Bryant, Josh Lapa...

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...1 Introduction In recent years, curvature couplings in rigidly supersymmetric nonlinear sigma models on nontrivial spacetime manifolds of various dimensions have been discussed by several groups, see =-=[1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 14]-=- and references contained therein. Often, supersymmetry on the nontrivial spacetime manifold will constrain the target space in some fashion, by e.g. requiring the Kähler form on the target space to ...

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...1 Introduction In recent years, curvature couplings in rigidly supersymmetric nonlinear sigma models on nontrivial spacetime manifolds of various dimensions have been discussed by several groups, see =-=[1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 14]-=- and references contained therein. Often, supersymmetry on the nontrivial spacetime manifold will constrain the target space in some fashion, by e.g. requiring the Kähler form on the target space to ...

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...1 Introduction In recent years, curvature couplings in rigidly supersymmetric nonlinear sigma models on nontrivial spacetime manifolds of various dimensions have been discussed by several groups, see =-=[1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 14]-=- and references contained therein. Often, supersymmetry on the nontrivial spacetime manifold will constrain the target space in some fashion, by e.g. requiring the Kähler form on the target space to ...

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...tiplet, their components mix holomorphic and antiholomorphic target space indices: χ = ( χi − χı̄+ ) , χ̄ = ( χ̄ı̄ − χ̄i+ ) (11) 2 We would like to thank I. Melnikov for making this observation. 3See =-=[32, 33, 34, 35]-=- for a recent discussion of the Fayet-Iliopoulos parameter in supergravity, and how old obstruction issues summarized in e.g. [36] can be circumvented. 4 We do not attempt to consider H-flux backgroun...

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...1 Introduction In recent years, curvature couplings in rigidly supersymmetric nonlinear sigma models on nontrivial spacetime manifolds of various dimensions have been discussed by several groups, see =-=[1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 14]-=- and references contained therein. Often, supersymmetry on the nontrivial spacetime manifold will constrain the target space in some fashion, by e.g. requiring the Kähler form on the target space to ...

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...del in [20, 21] can also be obtained by applying a decoupling gravity procedure analogous to the one in [1, 2] to the coupled theory of (2,2) gauged linear sigma model and (1,1) supergravity model in =-=[31]-=-. From the supersymmetry transformations above, we can get some insight into the constraint on W . Specifically, note that with the Killing spinor condition, the supersymmetry variation of the auxilia...

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...tiplet, their components mix holomorphic and antiholomorphic target space indices: χ = ( χi − χı̄+ ) , χ̄ = ( χ̄ı̄ − χ̄i+ ) (11) 2 We would like to thank I. Melnikov for making this observation. 3See =-=[32, 33, 34, 35]-=- for a recent discussion of the Fayet-Iliopoulos parameter in supergravity, and how old obstruction issues summarized in e.g. [36] can be circumvented. 4 We do not attempt to consider H-flux backgroun...