Curvature and the eigenvalues of the Laplacian

McKean, H. P.; Singer, I. M.

doi:10.4310/jdg/1214427880

Cited by 809 publications

(479 citation statements)

References 11 publications

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“…Final comment of this paper concerns the (bulk and boundary) curvature expansion of the Dirichlet Green's function. Classical method of images for this expansion is known [3], and this method for the Dirichlet case is much simpler than for the generalized Neumann boundary conditions. Still further efforts are necessary to convert it into a regular systematic calculational scheme comparable in its universality to the Schwinger-DeWitt technique in spacetime without branes/boundaries [10,11].…”

Section: Discussionmentioning

confidence: 99%

“…This leads to the expansion of the functional trace of the heat kernel in half-integer powers of the proper time parameter [3,19,20,12] …”

Section: Boundary Terms Of the Local Heat Kernel Expansionmentioning

confidence: 99%

“…involve the trace of the extrinsic curvature of the boundary K = g µν K µν [3,12]. For the generalized Neumann (Robin) boundary conditions…”

Section: Boundary Terms Of the Local Heat Kernel Expansionmentioning

confidence: 99%

“…On the contrary, Green's functions with Dirichlet boundary conditions are much easier to obtain -the method of images gives them as relatively simple linear combinations of known propagators in spacetime without boundaries (defined by mirror image continuation across the original boundary [3]). It turns out that Feynman diagrammatic technique based on Neumann type Green's function can be systematically reduced to that of the Dirichlet type, and the goal of the present paper is to develop the needed technique.…”

Section: Introductionmentioning

confidence: 99%

“…In the presence of boundaries its local expansion is modified by additional surface terms which amend easily calculable bulk terms well known in physics context as Schwinger-DeWitt coefficients. Calculation of these surface terms [3] presents a strong challenge of both technical and sometimes conceptual (nonperturbative) nature, especially for the so called oblique boundary conditions [13] which contain derivatives tangential to the brane and arise, in particular, in Born-Infeld context [9]. Interestingly, Neumann-Dirichlet duality relations suggest an alternative method of their calculation, which in view of simplicity and universality has essential advantages as compared to the conventional approach of [13,14,15,12].…”

Section: Introductionmentioning

confidence: 99%

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Quantum effective action in spacetimes with branes and boundaries

Barvinsky

Nesterov

2006

Phys. Rev. D

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We construct quantum effective action in spacetime with branes/boundaries. This construction is based on the reduction of the underlying Neumann type boundary value problem for the propagator of the theory to that of the much more manageable Dirichlet problem. In its turn, this reduction follows from the recently suggested Neumann-Dirichlet duality which we extend beyond the tree level approximation. In the one-loop approximation this duality suggests that the functional determinant of the differential operator subject to Neumann boundary conditions factorizes into the product of its Dirichlet counterpart and the functional determinant of a special operator on the brane -the inverse of the brane-to-brane propagator. As a byproduct of this relation we suggest a new method for surface terms of the heat kernel expansion. This method allows one to circumvent well-known difficulties in the heat kernel theory on manifolds with boundaries for a wide class of generalized Neumann boundary conditions. In particular, we easily recover several lowest order surface terms in the case of Robin and oblique boundary onditions. We briefly discuss multi-loop applications of the suggested Dirichlet reduction and the prospects of constructing the universal background field method for systems with branes/boundaries, analogous to the Schwinger-DeWitt technique.

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Section: Discussionmentioning

confidence: 99%

“…This leads to the expansion of the functional trace of the heat kernel in half-integer powers of the proper time parameter [3,19,20,12] …”

Section: Boundary Terms Of the Local Heat Kernel Expansionmentioning

confidence: 99%

“…involve the trace of the extrinsic curvature of the boundary K = g µν K µν [3,12]. For the generalized Neumann (Robin) boundary conditions…”

Section: Boundary Terms Of the Local Heat Kernel Expansionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%