On c-theorems in arbitrary dimensions

Aninda Sinha; Arpan Bhattacharyya; Kallol Sen; Ling-Yan Hung

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On c-theorems in arbitrary dimensions

Arpan Bhattacharyya, Ling-Yan Hung, Kallol Sen, Aninda Sinha

High Energy Physics - Theory · Submitted 10 Jul 2012

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In three dimensions, we observe that by finding an exact Hamilton-Jacobi counterterm, one can find a non-polynomial action which is invariant under global Weyl rescalings.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.75 · mentions code or data availability 0.02.

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The dilaton action in 3+1 dimensions plays a crucial role in the proof of the a-theorem. This action arises using Wess-Zumino consistency conditions and crucially relies on the existence of the trace anomaly. Since there are no anomalies in odd dimensions, it is interesting to ask how such an action could arise otherwise. Motivated by this we use the AdS/CFT correspondence to examine both even and odd dimensional CFTs. We find that in even dimensions, by promoting the cut-off to a field, one can get an action for this field which coincides with the WZ action in flat space. In three dimensions, we observe that by finding an exact Hamilton-Jacobi counterterm, one can find a non-polynomial action which is invariant under global Weyl rescalings. We comment on how this finding is tied up with the F-theorem conjectures.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The dilaton action in 3+1 dimensions plays a crucial role in the proof of the a-theorem.	background 1.00	0.09	0.05	asserted 1.00
s1	This action arises using Wess-Zumino consistency conditions and crucially relies on the existence of the trace anomaly.	background 1.00	0.11	0.04	asserted 1.00
s2	Since there are no anomalies in odd dimensions, it is interesting to ask how such an action could arise otherwise.	gap 0.68	0.05	0.03	asserted 0.80
s3	Motivated by this we use the AdS/CFT correspondence to examine both even and odd dimensional CFTs.	method 0.71	0.78	0.05	asserted 0.90
s4	We find that in even dimensions, by promoting the cut-off to a field, one can get an action for this field which coincides with the WZ action in flat space.	result 0.99	0.94	0.19	asserted 0.91
s5	In three dimensions, we observe that by finding an exact Hamilton-Jacobi counterterm, one can find a non-polynomial action which is invariant under global Weyl rescalings.	result 1.00	0.94	0.23	asserted 0.98
s6	We comment on how this finding is tied up with the F-theorem conjectures.	interpretation 1.00	0.54	0.09	unclear 0.76

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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Original abstract

The dilaton action in 3+1 dimensions plays a crucial role in the proof of the a-theorem. This action arises using Wess-Zumino consistency conditions and crucially relies on the existence of the trace anomaly. Since there are no anomalies in odd dimensions, it is interesting to ask how such an action could arise otherwise. Motivated by this we use the AdS/CFT correspondence to examine both even and odd dimensional CFTs. We find that in even dimensions, by promoting the cut-off to a field, one can get an action for this field which coincides with the WZ action in flat space. In three dimensions, we observe that by finding an exact Hamilton-Jacobi counterterm, one can find a non-polynomial action which is invariant under global Weyl rescalings. We comment on how this finding is tied up with the F-theorem conjectures.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Entanglement entropy in higher derivative holography
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Applying the Lewkowycz-Maldacena generalized gravitational entropy method to four-derivative gravity, the authors show Gauss-Bonnet gravity admits an area functional matching prior proposals, while general four-deriva...
On generalized gravitational entropy, squashed cones and holography
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The Fursaev-Patrushev-Solodukhin squashed cone regularization, combined with the Lewkowycz-Maldacena generalized gravitational entropy procedure, reproduces the known universal entanglement entropy terms for spherical...
Entanglement Entropy for $TT$ deformed CFT in general dimensions
hep-th 2019-04 conditional novelty 4.0 of 10

The exact entanglement entropy of a large-N CFT deformed by a stress-tensor-bilinear operator matches the Ryu-Takayanagi prediction in cutoff AdS for dimensions 2 through 6, up to a renormalization-scheme area term.

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9 arXiv papers cite this in 19 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 10 sentences without a confident label

Builds on it 1 paper

In this section, we are going to generalized the idea in [ref] to find the dilaton actions in GQC gravity.

A Holographic Study of the $a$-theorem and RG Flow in General Quadratic Curvature Gravity · Dilaton action in GQC gravity · 1.00

Uses its method or tool 1 paper

After finding all the necessary Lagrangians, following to [ref] , we introduce a radial cut-off as a scalar function of the boundary variables.

A Holographic Study of the $a$-theorem and RG Flow in General Quadratic Curvature Gravity · Summary and Discussion · 0.92

Compares with it 1 paper

In all the above calculations in even $d≤6$ dimension, the $a$-anomaly coefficients reduce to the GB results in [ref] when $a_1=a_2=0$.

A Holographic Study of the $a$-theorem and RG Flow in General Quadratic Curvature Gravity · TEXT · 0.99

Cites it as background 5 papers

In [ref] the same idea is investigated in the context of the AdS/CFT.

A Holographic Study of the $a$-theorem and RG Flow in General Quadratic Curvature Gravity · Introduction · 0.85
For a more detailed exposition to the structure of these counter terms, the reader could refer to [ref] .

Entanglement Entropy for $TT$ deformed CFT in general dimensions · On Holographic dictionary · 0.97
The trace-anomaly exists only for even $d$, and therefore it is a priori clear that for $d$ odd, the low-energy dilaton effective action simply consists of a derivative-expansion of Weyl-invariants.[A holographic approach to the dilaton effective action in $d$-dimensions was recently discussed in [ref] .] Such Weyl-invariants must also be included when writing down the dilaton effective action in even $d$, in addition to the Wess-Zumino action whose Weyl-variation produces the integral of the trace anomaly polynomial traceT.

RG flows in d dimensions, the dilaton effective action, and the a-theorem · Introduction & Summary · 0.91

What it cites · 45 sentences without a confident label

Uses the method or tool of 2 papers

For these theories, the generalized Gibbons-Hawking term is known [ref] which facilitates our calculation.

Hamilton-Jacobi Counterterms for Einstein-Gauss-Bonnet Gravity · Introduction · 0.89
May be the methods based on symmetry for e.g., in [ref] will be handy here.

Galileons as Wess-Zumino Terms · Discussion · 0.81

Compares with 2 papers

One can easily verify that (upto total derivatives) the above recursion relations reproduce the results of [ref] One obtains at six derivative orderΠ_ab[S_2()]Π^ab[S_1()]=L^4/(d-4)(d-2)^3[2_3+d^2-9d+6/4(d-1)_2+d(d+2)/16(d-1)^3] +L^4/(d-4)(d-2)^3[-2^c_a∇_b∇_cG^ab-d/4(d-1)+^ab_ab],which can be put into the form using the definition of the Cotton tensor$C_abc$asS_3()=L^5√()/(d-6)(d-4)(d-2)^3[d-4/d-2_3-3d(d-4)/4(d-1)(d-2)_2+d^3-20d+16/16(d-1)^2(d-2)^3 -1/2C^2-^ik^mnW_mikn].Using the above form for the counterterm in$d=3$we observe that it coincides with the expansion of the DBI counterterm[Similarly for the $d=5$ case we have from the above counterterm $_̱7=-7/5184$ which coincides with the coefficients that we derive from the simple c-theorem constraint [ref] .].

Counterterms, critical gravity and holography · Counterterms from the Hamiltonian constraint · 0.96
L_pole2=6τ(∇τ)^2+4(∇τ)^4+2(d-2)G^ab∇_aτ∇_bτwhere we have used the following relations obtained by integrating by parts, [equation] Combining everything we get, [equation] If we compare with [ref] then (after setting $τ_here=-τ_there$), we can identify $I_pole=-S_anomaly$.

On Renormalization Group Flows in Four Dimensions · Weyl transformation of pole terms · 1.00

Cites as support 1 paper

We note that this property of scale invariance dominated by $√(hℛ^d)$ when expanded in the interior of the AdS space is a general property of the re-summed counter terms in arbitrary odd dimensions, at least in the case of Einstein gravity as demonstrated in [ref] .

Counterterms, critical gravity and holography · Discussion · 0.97

Cites as background 5 papers

On the holographic side, this has been studied originally in [ref] and in recent times for example in [ref] .

On higher derivative gravity, c-theorems and cosmology · Introduction · 0.96
On the holographic side, this has been studied originally in [ref] and in recent times for example in [ref] .

Holographic c-theorems in arbitrary dimensions · Introduction · 0.96
On the holographic side, this has been studied originally in [ref] and in recent times for example in [ref] .

Towards the F-Theorem: N=2 Field Theories on the Three-Sphere · Introduction · 0.96

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