An elliptic triptych

We study the dependence of supersymmetric partition functions on continuous parameters for the flavor symmetry group, G F , for 2d N = (0, 2) and 4d N = 1 supersymmetric quantum field theories. In any diffeomorphism-invariant scheme and in the presence of G F 't Hooft anomalies, the supersymmetric Ward identities imply that the partition function has a non-holomorphic dependence on the flavor parameters. We show this explicitly for the 2d torus partition function, Z T 2 , and for a large class of 4d partition functions on half-BPS four-manifolds, Z M 4in particular, for M 4 = S 3 × S 1 and M 4 = Σ g × T 2 . We propose a new expression for Z M d−1 ×S 1 , which differs from earlier holomorphic results by the introduction of a non-holomorphic "Casimir" pre-factor. The latter is fixed by studying the "high temperature" limit of the partition function. Our proposal agrees with the supersymmetric Ward identities, and with explicit calculations of the absolute value of the partition function using a gauge-invariant zeta-function regularization.

Section: Outlook and Summarymentioning

confidence: 77%

’t Hooft anomalies and the holomorphy of supersymmetric partition functions

Closset

Pietro

Kim

2019

“…Thus, the volume divergence contaminates the ground state counting and even the supersymmetric index must be interpreted with care. See [31] for a much more detailed discussion of closely related intricacies. Despite the subtleties we encounter, we can draw a number of conclusions at this stage.…”

Section: The Topological Non-compact Modelmentioning

confidence: 99%

“…This is good check on the localization argument we used to compute the metric. Indeed, since there can be subtleties in applying localization arguments to non-compact models, as for instance in the case of the elliptic genus [22,31,36,37], the check is useful. It would be interesting to understand the regularization of volume divergences in three-point correlators in non-compact models better still.…”

Section: The Conformal Metricmentioning

confidence: 99%

Twisted massive non-compact models

Troost

2018

Self Cite

“…Supersymmetry ensures that the other chirality satisfies the Robin boundary condition as required by the APS boundary conditions. See, for example, [10,51] for a more detailed discussion.…”

Section: The η-Invariant and Path Integralsmentioning

confidence: 99%

“…The same result was obtained in [33,34] using localization of the generalized linear sigma model that flows to the cigar geometry. The cigar elliptic genus was re-derived in [35,51] using canonical methods. Our aim is to re-re derive the elliptic genus of an infinite cigar using localization of the path integral directly in the nonlinear sigma model connecting the computation to the earlier discussion of the η-invariant for the boundary of the finite cigar.…”

Section: Mock Modularity and The η-Invariantmentioning

confidence: 99%

APS η-invariant, path integrals, and mock modularity

Dabholkar

Jain

Rudra

2019

We show that the Atiyah-Patodi-Singer η-invariant can be related to the temperature dependent Witten index of a noncompact theory and give a new proof of the APS theorem using scattering theory. We relate the η-invariant to a Callias index and compute it using localization of a supersymmetric path integral. We show that the η-invariant for the elliptic genus of a finite cigar is related to quantum modular forms obtained from the completion of a mock Jacobi form which we compute from the noncompact path integral.(Dedicated to the memory of Michael Atiyah)