Index theory of Dirac operators on manifolds with corners up to codimension two

Loya, Paul

doi:10.1007/978-3-0348-7850-0_2

Cited by 10 publications

(27 citation statements)

References 46 publications

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“…Here s 1 and s 2 are, as before, the coordinates on the 'square' over M 10 . In what follows we apply some of the results (surveyed) in [28]. Let D : C ∞ (Z 12 , E) → C ∞ (Z 12 , F ) be a Dirac operator on Z 12 which is of product type D = Γ i (∂ si +D i ) near each hypersurface on the collar…”

Section: Formulation Using Dirac Operatorsmentioning

confidence: 95%

Corners in M-theory

Sati¹

2011

J. Phys. A: Math. Theor.

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M-theory can be defined on closed manifolds as well as on manifolds with boundary. As an extension, we show that manifolds with corners appear naturally in M-theory. We illustrate this with four situations: The lift to bounding twelve dimensions of M-theory on Anti de Sitter spaces, ten-dimensional heterotic string theory in relation to twelve dimensions, and the two M-branes within M-theory in the presence of a boundary. The M2-brane is taken with (or as) a boundary and the worldvolume of the M5-brane is viewed as a tubular neighborhood. We then concentrate on (variant) of the heterotic theory as a corner and explore analytical and geometric consequences. In particular, we formulate and study the phase of the partition function in this setting and identify the corrections due to the corner(s). The analysis involves considering M-theory on disconnected manifolds, and makes use of the extension of the Atiyah-Patodi-Singer index theorem to manifolds with corners and the b-calculus of Melrose.

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Section: Formulation Using Dirac Operatorsmentioning

confidence: 95%

Corners in M-theory

Sati¹

2011

J. Phys. A: Math. Theor.

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“…When the kernel of the corner Dirac operator is non-trivial, we have to add a mass perturbation [Loy04]. The induced twisted spinor bundle over Y = M d−2…”

Section: Index Theorymentioning

confidence: 99%

Extended Quantum Field Theory, Index Theory, and the Parity Anomaly

Müller

Szabo

2018

Commun. Math. Phys.

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We use techniques from functorial quantum field theory to provide a geometric description of the parity anomaly in fermionic systems coupled to background gauge and gravitational fields on odd-dimensional spacetimes. We give an explicit construction of a geometric cobordism bicategory which incorporates general background fields in a stack, and together with the theory of symmetric monoidal bicategories we use it to provide the concrete forms of invertible extended quantum field theories which capture anomalies in both the path integral and Hamiltonian frameworks. Specialising this situation by using the extension of the Atiyah-Patodi-Singer index theorem to manifolds with corners due to Loya and Melrose, we obtain a new Hamiltonian perspective on the parity anomaly. We compute explicitly the 2-cocycle of the projective representation of the gauge symmetry on the quantum state space, which is defined in a parity-symmetric way by suitably augmenting the standard chiral fermionic Fock spaces with Lagrangian subspaces of zero modes of the Dirac Hamiltonian that naturally appear in the index theorem. We describe the significance of our constructions for the bulkboundary correspondence in a large class of time-reversal invariant gauge-gravity symmetryprotected topological phases of quantum matter with gapless charged boundary fermions, including the standard topological insulator in 3 + 1 dimensions. • Dai-Freed theories describing chiral anomalies in odd dimensions d have been sketched in [Mon15]. They are an extended version of the field theories constructed in [DF95].• Wess-Zumino theories describing the anomaly in self-dual field theories have been constructed in [Mon15].• The theory describing the anomaly of the worldvolume theory of M5-branes has been constructed as an unextended quantum field theory in [Mon17].

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“…The generalisation of definitions and results to the case of non-trivial vector bundles (which is natural in many cases, e.g., Dirac operators) is straightforward. Note that a recent paper of Loya [12] studied the index of Dirac operators on manifolds with corners. Although there is a similarity between the geometry on infinite halfcylinders which appear in such situations and our case with edges on the infinite cylinder, there is no real intersection with our work.…”

Section: Introductionmentioning

confidence: 99%

“…Although there is a similarity between the geometry on infinite halfcylinders which appear in such situations and our case with edges on the infinite cylinder, there is no real intersection with our work. We are expressing the relative index, while in [12] (and in papers of other authors on the index of Dirac operators on manifolds with boundary) the index itself is considered without reference to its behaviour in different weighted spaces.…”

Section: Introductionmentioning

confidence: 99%

The Relative Index for Corner Singularities

Harutjunjan¹,

Schulze

2005

Integr. equ. oper. theory

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We study pseudo-differential operators on a cylinder R × B where B has conical singularities. Configurations of that kind are the local model of corner singularities with cross section B. Operators in our calculus are assumed to have symbols a which are meromorphic in the complex covariable with values in the algebra of all cone operators on B. We show an explicit formula for solutions of the homogeneous equation if a is independent of the axial variable t ∈ R. Each non-bijectivity point of the symbol in the complex plane corresponds to a finite-dimensional space of solutions. Moreover, we give a relative index formula. (2000). 35J40, 47G30, 58J32. Mathematics Subject Classification

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Index theory of Dirac operators on manifolds with corners up to codimension two

Cited by 10 publications

References 46 publications

Corners in M-theory

Corners in M-theory

Extended Quantum Field Theory, Index Theory, and the Parity Anomaly

The Relative Index for Corner Singularities

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