APS η-invariant, path integrals, and mock modularity

Dabholkar, Atish; Jain, Diksha; Rudra, Arnab

doi:10.1007/jhep11(2019)080

Cited by 18 publications

(13 citation statements)

References 74 publications

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“…For a sigma-model with a compact target space (or for any supersymmetric field theory with a discrete spectrum), the elliptic genus is a holomorphic function of the modular parameter τ. However, for a sigma-model with a noncompact target space, the elliptic genus can have a holomorphic anomaly [6][7][8][9][10][11][12][13][14][15]. The elliptic genus defined by a path integral on the torus is then still modular invariant, but it is no longer holomorphic.…”

Section: Introductionmentioning

confidence: 99%

Duality and mock modularity

2020

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We derive a holomorphic anomaly equation for the Vafa-Witten partition function for twisted four-dimensional \mathcal{N} =4𝒩=4 super Yang-Mills theory on \mathbb{CP}^{2}ℂℙ2 for the gauge group SO(3)SO(3) from the path integral of the effective theory on the Coulomb branch. The holomorphic kernel of this equation, which receives contributions only from the instantons, is not modular but ‘mock modular’. The partition function has correct modular properties expected from SS-duality only after including the anomalous nonholomorphic boundary contributions from anti-instantons. Using M-theory duality, we relate this phenomenon to the holomorphic anomaly of the elliptic genus of a two-dimensional noncompact sigma model and compute it independently in two dimensions. The anomaly both in four and in two dimensions can be traced to a topological term in the effective action of six-dimensional (2,0)(2,0) theory on the tensor branch. We consider generalizations to other manifolds and other gauge groups to show that mock modularity is generic and essential for exhibiting duality when the relevant field space is noncompact.

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Section: Introductionmentioning

confidence: 99%

Duality and mock modularity

2020

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“…The relation between physical states and boundary conditions are mentioned in (2. 19). The index depends on the boundary condition, so let us denote it as index(D X , β).…”

Section: Generalization To Other Boundary Conditionsmentioning

confidence: 99%

“…The original proof is mathematically rigorous, but it is technically complicated. See [18][19][20][21][22] for work which studies this theorem further from different points of view.…”

Section: Introductionmentioning

confidence: 99%

Atiyah-Patodi-Singer index theorem from axial anomaly

Kobayashi,

Yonekura

2021

Preprint

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We give a very simple derivation of the Atiyah-Patodi-Singer (APS) index theorem and its small generalization by using the path integral of massless Dirac fermions. It is based on the Fujikawa's argument for the relation between the axial anomaly and the Atiyah-Singer index theorem, and only a minor modification of that argument is sufficient to show the APS index theorem. The key ingredient is the identification of the APS boundary condition and its generalization as physical state vectors in the Hilbert space of the massless fermion theory. The APS η-invariant appears as the axial charge of the physical states.

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“…Recently, non-perturbative generalization of anomaly inflow has been discussed in terms of the eta-invariant [6][7][8]. It has been applied to high energy physics and condensed matter physics [9][10][11][12][13][14][15][16][17][18][19]. The APS index theorem above describes a special case of anomaly inflow [20].…”

Section: Jhep12(2021)096mentioning

confidence: 99%

Comments on the Atiyah-Patodi-Singer index theorem, domain wall, and Berry phase

Onogi

Yoda

2021

J. High Energ. Phys.

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It is known that the Atiyah-Patodi-Singer index can be reformulated as the eta invariant of the Dirac operators with a domain wall mass which plays a key role in the anomaly inflow of the topological insulator with boundary. In this paper, we give a conjecture that the reformulated version of the Atiyah-Patodi-Singer index can be given simply from the Berry phase associated with domain wall Dirac operators when adiabatic approximation is valid. We explicitly confirm this conjecture for a special case in two dimensions where an analytic calculation is possible. The Berry phase is divided into the bulk and the boundary contributions, each of which gives the bulk integration of the Chern character and the eta-invariant.

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APS η-invariant, path integrals, and mock modularity

Cited by 18 publications

References 74 publications

Duality and mock modularity

Duality and mock modularity

Atiyah-Patodi-Singer index theorem from axial anomaly

Comments on the Atiyah-Patodi-Singer index theorem, domain wall, and Berry phase

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