The geometry of supersymmetric partition functions

By Cyril Closset, T. Dumitrescu, G. Festuccia, Z. Komargodski. 2013

210 citations recorded by the source corpus.

A bstractWe consider supersymmetric field theories on compact manifolds $ \mathcal{M} $ and obtain constraints on the parameter dependence of their partition functions $ {Z_{\mathcal{M}}} $. Our primary focus is the dependence of $ {Z_{\mathcal{M}}} $ on the geometry of $ \mathcal{M} $, as well as background gauge fields that couple to continuous flavor symmetries. For $ \mathcal{N} $ = 1 theories with a U(1)R symmetry in four dimensions, $ \mathcal{M} $ must be a complex manifold with a Hermitian metric. We find that $ {Z_{\mathcal{M}}} $ is independent of the metric and depends holomorphically on the complex structure moduli. Background gauge fields define holomorphic vector bundles over $ \mathcal{M} $ and $ {Z_{\mathcal{M}}} $ is a holomorphic function of the corresponding bundle moduli. We also carry out a parallel analysis for three-dimensional $ \mathcal{N} $ = 2 theories with a U(1)R symmetry, where the necessary geometric structure on $ \mathcal{M} $ is a transversely holomorphic foliation (THF) with a transversely Hermitian metric. Again, we find that $ {Z_{\mathcal{M}}} $ is independent of the metric and depends holomorphically on the moduli of the THF. We discuss several applications, including manifolds diffeomorphic to S3× S1 or S2× S1, which are related to supersymmetric indices, and manifolds diffeomorphic to S3 (squashed spheres). In examples where $ {Z_{\mathcal{M}}} $ has been calculated explicitly, our results explain many of its observed properties.

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e5308112d7310cc6916989bd94960281e612b9b1

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Citations in the corpus

Cardy formulae for SUSY theories in d = 4 and d = 6 (2014, background): “al mass corresponds to turning on σ= mand D= −mH(see [6]). We then ﬁnd that for such a conﬁguration of background ﬁelds Z d3x √ h D− Aivi−σH = −2m Z d3x √ hH= imr2 3(2π) 2 . (4.4) 9 For a review, see [34]. 17 Combining (4.3) and (4.4) we ﬁnd the asymptotic form of the partition function on the space”
Comments on twisted indices in 3d supersymmetric gauge theories (2016, methodology): “12) (5)See appendix A and especially [43] for a general discussion.”
The Casimir energy in curved space and its supersymmetric counterpart (2015, background): “5 The results of [18, 19] tell us that Z M3×S(1) β is independent of coupling constants and only depends on complex structure parameters of the underlying complex four-manifold M3 × S1.”
Supersymmetry on curved spaces and holography (2012, methodology): “As was discussed in [73], it is equivalent”
Supersymmetric field theories on three-manifolds (2012, background): “Indeed, such an ACMS is equivalent to a structure known as a transversely holomorphic foliation, which has been studied in the mathematical literature (see [48] for additional details and references).”
Localization on Hopf surfaces (2014, background): “J H E P 0 8 ( 2 0 1 4 ) 1 2 3 Perhaps a related question is that of clarifying the dependence of the partition function on the function κ, parametrising the freedom in choosing the background fields Aμ, Vμ [18].”
N$$ \mathcal{N} $$ = 1 supersymmetric indices and the four-dimensional A-model (2017, background): “8) only makes sense for (17)In general, L is also determined by other complex moduli, but no physical observable depend on them due to supersymmetry [10].”
A topologically twisted index for three-dimensional supersymmetric theories (2015, background): “= R2 dθ2 +f(θ)2(dϕ−ςdt)2 +β2dt2 , (4.1) 24More general supersymmetric backgrounds have been constructed in [48, 49], and the dependence of the partition function on the background has been studied in [50]. – 47 – where ds2 S2 = R 2 dθ2 + f(θ)2dϕ2 is a generic metric on S2 with U(1) isometry along ϕ.”
Localization techniques in quantum field theories (2016, background): “2 Euler class of vector bundle and Mathai-Quillen form See Mathai-Quillen [17] and Aityah-Jeffrey [18].”