We present a refinement of a recently found gauge-gravity relation between
one-loop effective actions: on the gauge side, for a massive charged scalar in
2d dimensions in a constant maximally symmetric electromagnetic field; on the
gravity side, for a massive spinor in d-dimensional (Euclidean) anti-de Sitter
space. The inclusion of the dimensionally regularized volume of AdS leads to
complete mapping within dimensional regularization. In even-dimensional AdS, we
get a small correction to the original proposal; whereas in odd-dimensional
AdS, the mapping is totally new and subtle, with the `holographic trace
anomaly' playing a crucial role.Comment: 6 pages, io

A Montecinos

Albin P

Basar G

Branson T

D E Díaz

Diaz D E

Dunne G V

Graham C R

Henningson M

R Aros

English

arXiv

Abstract We present a reﬁnement of a recently found gauge–gravity relation between one-loop effective actions: on the gauge side, for a massive charged scalar in
2d dimensions in a constant maximally symmetric electromagnetic ﬁeld; on the gravity side, for a massive spinor in d-dimensional (Euclidean) anti-de Sitter space. The inclusion of the dimensionally regularized volume of AdS leads to a complete mapping within dimensional regularization. In even-dimensional AdS, we acquire a small correction to the original proposal, whereas in dimensional AdS, the mapping is totally new and subtle, with the ‘holographic trace anomaly’ playing a crucial role.http://iopscience.iop.org/1751-8121/43/29/295401/pdf/1751-8121_43_29_295401.pd

Aros, R.

Díaz, D.E.

Montecinos, A.

Repositorio Institucional Académico (RIA) de la Universidad Andrés Bello

A note on a gauge–gravity relation and functional determinantsThis article has been downloaded from IOPscience. Please scroll down to see the full text article.2010 J. Phys. A: Math. Theor. 43 295401(http://iopscience.iop.org/1751-8121/43/29/295401)Download details:IP Address: 200.27.72.254The article was downloaded on 30/08/2012 at 23:27Please note that terms and conditions apply.View the table of contents for this issue, or go to the journal homepage for moreHome Search Collections Journals About Contact us My IOPscienceIOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICALJ. Phys. A: Math. Theor. 43 (2010) 295401 (6pp) doi:10.1088/1751-8113/43/29/295401A note on a gauge–gravity relation andfunctional determinantsR Aros, D E Dı´az and A MontecinosDepartamento de Ciencias Fı´sicas, Universidad Andre´s Bello, Repu´blica 220, Santiago, ChileE-mail: raros@unab.cl, danilodiaz@unab.cl and alejandramontecinos@unab.clReceived 15 April 2010, in final form 19 May 2010Published 17 June 2010Online at stacks.iop.org/JPhysA/43/295401AbstractWe present a refinement of a recently found gauge–gravity relation betweenone-loop effective actions: on the gauge side, for a massive charged scalar in2d dimensions in a constant maximally symmetric electromagnetic field; on thegravity side, for a massive spinor in d-dimensional (Euclidean) anti-de Sitterspace. The inclusion of the dimensionally regularized volume of AdS leads toa complete mapping within dimensional regularization. In even-dimensionalAdS, we acquire a small correction to the original proposal, whereas in odd-dimensional AdS, the mapping is totally new and subtle, with the ‘holographictrace anomaly’ playing a crucial role.PACS numbers: 11.10.Kk, 04.62.+v, 11.15.KcFunctional determinants of geometric differential operators, such as the Laplacian andDirac’s operator, are ubiquitous in mathematical and theoretical physics (cf [1, 2]). Inparticular, in quantum field theory, the one-loop effective action is given by the determinantof the ‘propagating’ differential operator. Exact results for these determinants in gauge theorydate back at least to the works of Heisenberg and Euler and of Weisskopf, for constantelectromagnetic backgrounds. In special gravitational backgrounds, as in anti-de Sitter, thehigh degree of symmetry also allows for exact expressions.Even though such one-loop effective actions have been known for quite some time, itwas only very recently that a remarkable gauge–gravity relation was identified [3]. On thegauge side, one considers the one-loop effective action for a massive charged scalar in 2ddimensions in a constant maximally symmetric electromagnetic field, and on the gravity side,the one-loop effective action for a massive spinor in d-dimensional (Euclidean) anti-de Sitterspace. The observation was inspired by factorization of graviton amplitudes in terms of gaugeamplitudes and the prospects for the inclusion of one-loop effects, while the similarity wasrevealed by the presence of Barnes’ multiple gamma function [6, 7]. Possible directions,in which the observation might be extended, include relaxing the maximal symmetry of theelectromagnetic field or its homogeneity, and considering higher loops. In addition, one mightalso contemplate including finite-temperature effects.1751-8113/10/295401+06$30.00 © 2010 IOP Publishing Ltd Printed in the UK & the USA 1J. Phys. A: Math. Theor. 43 (2010) 295401 R Aros et alOur concern here, however, is with a seemingly overlooked detail, namely that themapping is only established for even-dimensional AdS. On the gauge side, as long as thenumber of dimensions is twice larger than that of AdS, the effective action will be given interms of Barnes’ multiple gamma function. In odd-dimensional AdS, in turn, the renormalizedeffective Lagrangian is polynomial in the mass parameter and little hope is left to meet Barnes’multiple gamma in this case. In this paper we show how to amend this apparent mismatch bytaking into account the dimensionally regularized volume of AdS space.Gauge sideOn the gauge side, one considers a massive (M) charged scalar field φ in flat Euclidean2d-dimensional space in the constant background of a maximally symmetric electromagneticfield1. The field strength can be written in block form:Fμν = f diagd{(0 1−1 0)}. (1)The dramatis persona on the gauge side is the functional determinant arising in the one-loopeffective actionS2dgauge = ln det{−D2 + M2}. (2)Standard manipulations lead to an expression in terms of the Green’s function G, schematically(−D2 + M2)G = −1,S2dgauge = −∫ M2tr G2d , (3)where the trace tr means volume integral (large box of length L) of the diagonal entry G2d (x, x).The heat kernel gives a useful representation of this diagonal entry, and a starting point to dealwith the inherent short-distance divergence:−∫ M2tr G2d = L2d∫ ∞0dss1(4πs)d(f ssh(f s))de−sM2. (4)The divergence at the lower integration value of the proper-time integral could be cured if thedimension d were negative enough. This observation is the key to dimensional regularizationwhere the dimension is D ≡ d −  and one can work out the value of the effective action inclosed form:S2Dgauge = −(fL22π)D(1 − D)∫ M22fdμ(D2 + μ)(1 − D2 + μ) . (5)The divergence as D → d comes entirely from the (1 − D) factor for any integer value ofd, even or odd.Now, the crucial observation in [3] is related to the presence of Barnes’ multiple gammafunction in the renormalized value of this expression. A quick way to understand the Barnes’multiple gamma function is to go back to the heat kernel representation and perform a‘Weierstrass regularization’, that is, subtract successive terms of the Taylor expansion of theintegrand to achieve convergence in the lower integration limit. We stick to the conventions in[6, 7] for Barnes’ multiple gamma function (A.1), modulo renormalization scheme-dependent1 We refer to [3] for conventions and further references.2J. Phys. A: Math. Theor. 43 (2010) 295401 R Aros et alterms which are polynomial and logarithmic in M (cf [3]), and obtain the finite renormalizedvalueS2dgauge =(fL22π)d· ln d(d2+M22f). (6)Alternatively, expanding in  and cancelling the pole from (1 − D) with the linear-in-term in the expansion of the integrand in equation (5), one obtainsS2dgauge =(fL22π)d(−1)d−12(d − 1)!∫ M22fdμ(d2 + μ)(1 − d2 + μ)(ψ(d2+ μ)+ ψ(1 − d2+ μ))+ · · ·=(fL22π)d(−1)d−1(d − 1)!∫ M22fdμ⎧⎨⎩d−2∏j=0(d2+ μ − 1 − j)⎫⎬⎭ψ(d2+ μ)+ · · ·=(fL22π)d· ln d(d2+M22f)+ · · · . (7)Any subtraction or renormalization prescription will eventually lead to Barnes’ multiplegamma, modulo the aforementioned logarithmic and polynomial terms related to arenormalization scheme ambiguity.Gravity sideLet us now turn to the gravity side and consider a massive (m) Dirac spinor ψ minimallycoupled to gravity in the Euclidean AdSd background, i.e. hyperbolic space Hd. The dramatispersona on the gravity side is the functional determinant in the one-loop effective actionSdgravity = − ln det{/∇ + m}. (8)Again, standard manipulations bring in the Green’s function, (/∇ + m)D = −I,Sdgravity =∫ mtrDd , (9)where now the spinor indices are also traced out.The spinor propagator in AdSd is a well-known quantity [4], involving Gauss’hypergeometric function, and its diagonal entry can be worked out in dimensionalregularizationDD(x, x) = − I√(4π)D/2(1 − D2) (D2 + m√)(1 − D2 + m√) , (10)with I denoting the identity matrix in spinor indices, whose dimensionality is 2[D/2]. Theanalogy with the gauge result is already apparent, but there is a seemingly small mismatchin the gamma factor: (1 − D2)instead of (1 − D). This makes a tiny difference whend = even; both expressions have a pole and the finite piece involves Barnes’ multiple gamma.However, when d = odd, there is a huge difference and there is no mapping of gauge andgravity quantities. On the gauge side there is still a pole, but the gravity side is finite in thelimit D → d. Our contribution in this paper is to amend this discrepancy by taking intoaccount the dimensionally regularized volume of Euclidean AdS which is finite for d = even,whereas for d = odd one finds the missing pole in the above discussion.To compute the dimensionally regularized volume of HD, start with the metricds2 = R2r2[dr2 +(1 − r2)24d	2D−1], (11)3J. Phys. A: Math. Theor. 43 (2010) 295401 R Aros et alwhere R = 1/√ is the radius of AdS. The volume is then [5]vol(HD) = RD 21−D vol(SD−1)∫ 10dr r−D (1 − r2)D−1= π D2 − 12 RD (12− D2). (12)Note that the volume has a pole for D → odd and is finite for D → even, so that whencombined with the effective Lagrangian there is always a divergence as D goes to the physicaldimension, even or odd, as in the gauge side. Using Legendre’s duplication formula for thegamma functions we end up withSDgravity = −2[D/2] (1 − D)∫ m√dμ(D2 + μ)(1 − D2 + μ) . (13)Now one can appreciate the mapping to the gauge one-loop effective action (5).Even dimensions: d = 2nLet us now go to the physical dimension d = 2n, as first considered in [3]. The volumeasymptotics, D = 2n −  as  → 0, is finite:vol(HD) = V2n + o(1), (14)with V2n = πn−1/2 R2n ( 12 − n) = (−1)n πn+1/2(n+ 12 )R2n.Note that our renormalized volume differs by a factor (−1)n from the volume element usedin [3]. We welcome this extra factor for it makes more precise the gauge–gravity mapping.The important cancelation occurs in the expansion of the effective Lagrangian, very muchas in the gauge side,S2ngravity = V2n(4π)n(−2)n+1(n − 1)!∫ m√dμ⎧⎨⎩2n−2∏j=0(n + μ − 1 − j)⎫⎬⎭ψ(n + μ) + · · · . (15)After including the volume factor and using again the duplication formula for the gammafunction, we finally obtain [3]S2ngravity = 2n · ln 2n(n +m√), (16)which ought to be compared with the gauge result (6).Odd dimensions: d = 2n + 1This case is qualitatively different from the above; now the volume gets mixed with thesubleading term in the −expansion of the effective Lagrangian. The volume asymptoticsbehaves asvol(HD) = 1L2n+1 + V2n+1 + o(1). (17)The precise value of the renormalized volume V2n+1 will not be relevant for it only contributesa polynomial in the mass. The important term is the ‘integrated holographic trace anomaly’[8–10] which is unambiguously given by L2n+1 = (−1)n 2πn(n+1) R2n+1.The effective Lagrangian is now finite as D → 2n + 1, and polynomial in m. Thenontrivial part, which eventually leads to Barnes’ multiple gamma function, comes from the4J. Phys. A: Math. Theor. 43 (2010) 295401 R Aros et alcancelation of the pole in the volume against the linear-in- term in the expansion of theeffective Lagrangian2S2n+1gravity =( 12 − n)(−2)nπ 12 (1 + n)∫ m√dμ(n + 12 + μ)( 12 − n + μ)×(ψ(n +12+ μ)+ ψ(12− n + μ))+ · · · (18)= 21−nπ12( 12 + n)(1 + n)∫ m√dμ⎧⎨⎩2n−1∏j=0(n − 12+ μ − j)⎫⎬⎭ψ(n +12+ μ)+ · · · .A little algebraic manipulation yieldsS2n+1gravity = 2n · ln 2n+1(n +12+m√). (19)In all, on the gravity side the one-loop effective action after inclusion of the volume termin d dimensions, even and odd, readsSdgravity = 2[d/2] · ln d(d2+m√)(20)in remarkable correspondence with the gauge one-loop effective action (6) and with d = oddand d = even on equal footing.AcknowledgmentsWe kindly acknowledge E Friedman and S Ruijsenaars for a useful conversation on Barnes’zeta and gamma functions. DED wishes to thank H Dorn for valuable discussions. Thiswork was partially funded through Fondecyt-Chile 3090012, UNAB AR-01-09 and UNABAR-04-09.Appendix. Useful identities• Integral representation of Barnes’ multiple gamma, which effectively performs theWeierstrass regularization:ln N(w) =∫ ∞0dtt⎛⎝e−wtN∏j=111 − e−aj t− tNN−1∑k=0(−t)kk!BN,k(w) − (−1)NN !e−tBN,N(w)⎞⎠ , (A.1)with BN,k(w) being the Bernoulli-type polynomials [6, 7].• Gamma function, duplication formula(z) (z +12)= 21−2z (12)(2z). (A.2)• Gamma function, reflection formula(12+ z)(12− z)= πcos πz. (A.3)2 An analogous ‘conspiracy’ was also crucial in establishing a holographic formula for functional determinant inAdS [5].5J. Phys. A: Math. Theor. 43 (2010) 295401 R Aros et alReferences[1] Branson T 1993 The Functional Determinant (Lecture Note Series vol 4) (Seoul: Global Analysis ResearchCenter, Seoul National University)[2] Dunne G V 2008 Functional determinants in quantum field theory J. Phys. A: Math. Theor. 41 304006(arXiv:0711.1178 [hep-th])[3] Basar G and Dunne G V 2010 A Gauge-gravity relation in the one-loop effective action J. Phys. A: Math.Theor. 43 072002 (arXiv:0912.1260 [hep-th])[4] Camporesi R 1992 The Spinor heat kernel in maximally symmetric spaces Commun. Math. Phys. 148 283[5] Diaz D E and Dorn H 2007 Partition functions and double-trace deformations in AdS/CFT J. High EnergyPhys. JHEP05(2007)046 (arXiv:hep-th/0702163)[6] Ruijsenaars S N M 2000 On Barnes’ multiple zeta and gamma functions Adv. Math. 156 107–32[7] Friedman E and Ruijsenaars S N M 2004 Shintani–Barnes zeta and gamma functions Adv. Math. 187 362–95[8] Henningson M and Skenderis K 1998 The holographic Weyl anomaly J. High Energy Phys. JHEP07(1998)023(arXiv:hep-th/9806087)Henningson M and Skenderis K 2000 Holography and the Weyl anomaly Fortsch. Phys. 48 125(arXiv:hep-th/9812032)[9] Graham C R 2000 Volume and area renormalizations for conformally compact Einstein metrics Rend. Circ.Mat. Palermo (2) Suppl. no. 63 31–42 (arXiv:math.DG/9909042)[10] Albin P 2005 Renormalizing Curvature Integrals on Poincare–einstein Manifolds arXiv:math.DG/05041616

A note on a gauge–gravity relation and functional determinants

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A note on a gauge-gravity relation and functional determinants

A note on a gauge-gravity relation and functional determinants

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