Supersymmetric derivation of the Atiyah-Singer index and the chiral anomaly

Friedan, Daniel; Windey, Paul

doi:10.1016/0550-3213(84)90506-6

Cited by 248 publications

(161 citation statements)

References 12 publications

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“…In this case we have an interacting one dimensional supersymmetric sigma model that describes a superparticle moving in a non-trivial manifold; the kinetic terms are non-tivial functions of the scalar fields (the position of the superparticle) and are given in terms of the components of the background manifold. A flat measure for the partition function leads to the correct results for the expected index theorems [37,38]. Similarly there are the examples of topological sigma models [39,40] just to mention a few.…”

Section: Jhep07(2017)022mentioning

confidence: 99%

Exact Holography and Black Hole Entropy in N = 8 $$ \mathcal{N}=8 $$ and N = 4 $$ \mathcal{N}=4 $$ String Theory

Gomes

2017

J. High Energ. Phys.

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Abstract:We compute the exact entropy of one-eighth and one-quarter BPS black holes in N = 8 and N = 4 string theory respectively. This includes all the N = 4 CHL models in both K3 and T 4 compactifications. The main result is a measure for the finite dimensional integral that one obtains after localization of supergravity on AdS 2 × S 2 . This measure is determined entirely by an anomaly in supersymmetric Chern-Simons theory on local AdS 3 and takes into account the contribution from all the supergravity multiplets. In Chern-Simons theory on compact manifolds, this is the anomaly that computes a certain one-loop dependence on the volume of the manifold. For one-eighth BPS black holes, our results are a first principles derivation of a measure proposed in arXiv:1111.1161, while in the case of one-quarter BPS black holes our result computes exactly all the perturbative or area corrections. Moreover, we argue that instantonic contributions can be incorporated and give evidence by computing the measure, which matches precisely the microscopics. Along with this, we find a unitary condition that truncates the answer to a finite sum of instantons in perfect agreement with a microscopic formula. Our results therefore solve a number of puzzles related to localization in supergravity and constitute a larger number of examples where holography can be shown to hold exactly.

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Section: Jhep07(2017)022mentioning

confidence: 99%

Exact Holography and Black Hole Entropy in N = 8 $$ \mathcal{N}=8 $$ and N = 4 $$ \mathcal{N}=4 $$ String Theory

Gomes

2017

J. High Energ. Phys.

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“…In particular we discuss mode regularization for the N = 1, 2 nonlinear sigma models. These quantum mechanical models were originally used to calculate chiral [6,7,8] and trace anomalies [9,10] in a simpler way than using standard QFT Feynman rules 1 . They are also very useful to evaluate one-loop effective actions and scattering amplitudes for a Dirac (N = 1) or Maxwell/Proca field and differential forms (N = 2) coupled to scalar, antisymmetric tensor, gauge fields backgrounds [12,13,14,15,16], or to curved space-time (external gravity), as in [17,18,19,20] where i, k = 1, 2 are O(2) indices labeling fermion species; the term proportional to Rψψψψ is dictated by classical supersymmetry, while V contains the quantum counterterms.…”

Section: Introductionmentioning

confidence: 99%

Mode regularization forN= 1, 2 SUSY Sigma model

Bonezzi

Falconi

2008

J. High Energy Phys.

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Worldline N=1 and N=2 supersymmetric sigma models in curved background are useful to describe spin one-half and spin one particles coupled to external gravity, respectively. It is well known that worldline path integrals in curved space require regularization: we present here the mode-regularization for these models, finding in particular the corresponding counterterms, both in the case of flat and curved indices for worldline fermions. For N=1, using curved indices we find a contribution to the counterterm from the fermions that cancels the contribution of the bosons, leading to a vanishing total counterterm and thus preserving the covariance and supersymmetry of the classical action. Conversely in the case of N=2 supersymmetries we obtain a non-covariant counterterm with both curved and flat indices. This work completes the analysis of the known regularization schemes for N=1,2 nonlinear sigma models in one dimension.

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“…The ideas of Witten and other physicists (cf. [15,77]) stimulated a number of new proofs of the local index theorem for Dirac operators, notably the probabilitistic proof due to Bismut [33], the two proofs of Getzler [78,79] inspired by supersymmetry, and the group-theoretic proof given by Berline-Vergne [32]. It turns out that the index density is closely related to the heat kernel of the harmonic oscillator.…”

Section: Local Index Theoremmentioning

confidence: 99%