Loops in AdS and the Wilson-Fisher BCFT

Simone Giombi

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QFT in AdS, EPFL, Feb. 26 2026

Based mainly on: arXiv:2506.14699 SG, Z. Sun

arXiv: 2007.04955 SG, H. Khanchandani

CFT in AdS = BCFT

• Consider a boundary conformal field theory (BCFT) in flat half-space R^d-1 x R₊, with coordinates (z,x), z>0
• By a Weyl transformation, we can map the problem to hyperbolic space, or Euclidean AdS_d

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• Similarly, a BCFT on disk/hemisphere can be mapped to the hyperbolic ball

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• Thus, placing a CFT in AdS gives a natural approach to study the BCFT

Paulos, Penedones, Toledo, van Rees, Vieira ’16

Carmi, Di Pietro, Komatsu ‘18 SG, Khanchandani ‘20 …

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CFT in AdS = BCFT

• The SO(d,1) symmetry of the BCFT is manifest in the AdS approach. SO(d,1) remains even away from fixed point.
• Correlation functions take simple form due to maximal symmetry
• One-point functions are constant
• Two-point functions are function of the geodesic distance
• Correlation functions of boundary operators can be computed borrowing the technology developed in AdS/CFT (Witten diagrams)

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• The AdS approach is especially useful to compute quantities like the boundary central charge, which can be obtained from the free energy F_AdS of the CFT in AdS

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AdS approach to defects

• A similar AdS approach can be used to study defect CFTs. Starting with a CFT_d with a p-dimensional defect, one can map to AdS_p+1 x S^d-p-1. Defect is at the boundary of AdS_p+1.

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• Examples
• Monodromy defects from AdS_d-1 x S¹ SG, Helfenberger, Ji, Khanchandani ’21
• Line defects from AdS₂ x S^d-2

Kapustin ‘05; Cuomo, Komargodski, Mezei ’21; SG, Helfenberger, Khanchandani ’21

• Surface defects from AdS₃ x S^d-3 SG, Liu ‘23
• RG interfaces from AdS_d in Janus coords SG, Helfenberger, Khanchandani ‘24

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Example: free scalar BCFT

• As a simple example, consider free massless scalar on half space

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• Defines two BCFTs, corresponding to the two conformally invariant boundary conditions
• Neumann Boundary scalar operator has dimension d/2-1
• Dirichlet Boundary scalar operator has dimension d/2

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• The Neumann and Dirichlet BCFTs are connected by an RG flow Neumann-->Dirichlet triggered by a boundary mass term

Free conformal scalar in AdS

• Mapping the free massless scalar theory to AdS

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• The dimension of the boundary scalar operator can be obtained by the familiar mass/dimension relation

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• The RG flow between Neumann and Dirichlet BCFTs can be seen essentially as a special case of the general “double-trace flows” in AdS/CFT corresponding to alternate b.c. in AdS

Boundary central charge

Boundary central charge from free energy

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• The boundary central charge c can be extracted from the free energy on a hemisphere

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• Equivalently, it can be extracted from the free energy on Euclidean AdS₃

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where the logarithmic divergence now comes from the regularized volume of AdS₃ , vol(AdS₃) =-2πlog(Rμ)+…For the free scalar, one finds c^Neum= -c^Dirich =1/16

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• The AdS approach is in practice more convenient for explicit calculations, especially in interacting theories

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Dimensional continuation of the free energy

• We will be interested in the epsilon-expansion, so a natural question is how to define a suitable dimensional continuation of F in general d
• For a CFT on the round sphere, a natural quantity is SG, Klebanov ‘14

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which smoothly interpolates between anomaly coefficients in even d, and F- coefficients in odd d. Satisfies a generalized F-theorem interpolating between a/c-theorems and F-theorem

• Its epsilon-expansion was used to obtain estimates for F of 3d Ising and O(N) model CFTs, e.g. predicting SG, Klebanov ’14; Fei, SG, Klebanov, Tarnopolsky ‘15
• In very good agreement with recent fuzzy sphere result

Hu, Zhu, He ‘24

Dimensional continuation of the AdS free energy

• For the CFT in AdS, it is natural to define the analogous quantity

SG, Khanchandani ‘20

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• In d=3 (and other odd d), the sine factor cancels the pole coming from AdS regularized volume, leaving the boundary central charge. Explicitly:

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• Generalizing the boundary c-theorem in d=3, a natural conjecture is that in general d, under boundary RG flow it satisfies

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Example: Neumann-Dirichlet flow for free scalar

• For the free scalar in AdS, it is straightforward to compute the difference between Neumann and Dirichlet free energies adapting known formulas for double-trace flows. One has SG, Khanchandani ‘20

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• In d=3 one finds , in agreement with the known results for the boundary central charges c^N=-c^D=1/16.

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• For all continuous d, it satisfies as expected

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A better quantity

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Hence this quantity smoothly interpolates between boundary central charge and g-function (and their higher dimensional analogs), and should satisfy

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• This can also be understood as a special case of a “generalized g-theorem” for p-dimensional defects

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Kobayashi et al ’18

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For line defects (p=1), this is the g-theorem proved in Cuomo, Komargodski, Raviv-Moshe ‘22

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O(N) model with a boundary

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K_ij=K in bulk, K_ij=K₁ on surface

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Bray, Moore, Burkhardt, Cardy, Diehl,…

Liendo, Rastelli, van Rees, …

Metlitski ‘22

O(N) model with a boundary

O(N) model with a boundary

• The O(N) φ⁴ theory in flat half-space
• Two O(N) breaking boundary universality classes
• “Normal”: Explicit O(N) symmetry breaking. Corresponds to adding boundary magnetic field φ^I n^I , with n^I a fixed N-dimensional unit vector.

Non-vanishing one-point function

• “Extraordinary”: Spontaneous symmetry breaking. Can realize by the same boundary coupling, but with n^I dynamical
• For N=1 (Ising), the normal and extraordinary classes are equivalent. For d>3 and general N they are also believed to be essentially equivalent Bray, Moore, Burkhardt, Cardy, Diehl,…
• In d=3 for 1<N<N_c the extraordinary class takes a modified “extraordinary-log” form Metlitski ’22
• The “ordinary” and “normal” classes exist for all N and all 2<d<4. We will focus on these two cases

Available methods

Wilson-Fisher BCFT in AdS

• We now map the problem to AdS

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• The action in AdS is

Wilson-Fisher BCFT in AdS

• The action in AdS is

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• The ordinary and special BCFTs are easy to understand: We just do perturbation theory around the two possible boundary conditions

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Ordinary Special

Wilson-Fisher BCFT in AdS

The O(N) breaking “Normal” BCFT

Free Energy of “normal” BCFT: Tree level and one-loop

• There is a tree-level contribution from classical action

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• At one-loop

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• The one-loop determinants can be computed explicitly as …SG, Khanchandani…

“Normal” Free Energy: Two loops SG, Sun ‘25

“Normal” Free Energy: Two loops SG, Sun ‘25

“Normal” Free Energy: Final result

Pade resummation

• We can also compute the one-point function to one-loop

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• After suitable normalization so that in flat space , we find

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• Our Pade resummation compared to available results

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(For N=1, we can again apply “two-sided” Pade, using exact result in 2d Ising)

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Free Energy of the “Ordinary” BCFT

• In this case, we have standard perturbation theory with Dirichlet boundary conditions. Propagators are simpler

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• We computed the free energy to 4-loop order

SG, Khanchandani ‘20

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SG, Sun ’25

• A byproduct of the free energy calculation is that we can get the one-point function to order λ² without computing any new diagrams
• Diagrams for one-point function

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• Since one-point function are constant in AdS, we can just integrate over insertion point and divide by the AdS volume. Then the diagrams are the same as the ones contributing to the free-energy (up to stripping off tadpole factors)
• We find the result

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• Here is the renormalized operator obtained by multiplying by the Z-factor fixed by flat space renormalization

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Comparison to known result from analytic bootstrap

Mixing with curvature

Back to the Free energy

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• Putting together all diagrams and going to the fixed point, we find the final result for

Pade estimate for N=1

A surface defect in the O(N) model

Krishnan, Metliski, ‘23

A surface defect in the O(N) model

Large N approach

Large N approach

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• Assuming O(N) invariance, we can integrate out the scalars to obtain

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• Looking for saddle points with constant σ,

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• This gives the two O(N) invariant solutions

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• Similarly, one can recover the symmetry broken phase with O(N) -> O(N-1) and

Large N check of the free energy

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• The quadratic action for the σ fluctuations is

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• We need to compute the determinant of G_φ(x,y)² in AdS. This can be done by expanding G_φ(x,y)² into the harmonic functions

σ fluctuations determinant

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• In more detail, one finds Dyatlik, SG, Sun in progress

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• Then the one-loop determinant is given by

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σ fluctuations determinant

σ fluctuations determinant

Conclusion

A curious observation

Carmi, Di Pietro, Komatsu ’18; SG, Helfenberger, Khanchandani ‘22

• Just to mention a curious observation, possibly related to the 3d bose/fermi duality
• In the critical O(N) model in d=3 with a boundary, the large N expansion gives the dimension of the boundary scalar at the “ordinary” boundary condition as

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• This happens to be the same as the dimension of the boundary fermion operator in a d=3 free massless fermion theory
• Similarly, large N analysis of the critical Gross-Neveu model shows that there are two possible boundary conditions, with boundary fermion dimensions

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• These are the same as the boundary scalar dimensions for a free 3d scalar with Dirichlet or Neumann b.c., respectively
• Seems natural to expect that this should be related to the 3d bose/fermi duality?

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Thank You!