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QG from SymQRG: AdS$_3$/CFT$_2$ Correspondence as Topological Symmetry-Preserving Quantum RG Flow
by Ning Bao, Ling-Yan Hung, Yikun Jiang, Zhihan Liu
Submission summary
| Authors (as registered SciPost users): | Yikun Jiang |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2412.12045v2 (pdf) |
| Date submitted: | March 18, 2025, 5:07 p.m. |
| Submitted by: | Yikun Jiang |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
By analyzing the non-perturbative RG flows that explicitly preserve given symmetries, we demonstrate that they can be expressed as quantum path integrals of the $\textit{SymTFT}$ in one higher dimension. When the symmetries involved include Virasoro defect lines, such as in the case of $T\bar{T}$ deformations, the RG flow corresponds to the 3D quantum gravitational path integral. For each 2D CFT, we identify a corresponding ground state of the SymTFT, from which the Wheeler-DeWitt equation naturally emerges as a non-perturbative constraint. These observations are summarized in the slogan: $\textbf{SymQRG = QG}$. The recently proposed exact discrete formulation of Liouville theory in [1] allows us to identify a universal SymQRG kernel, constructed from quantum $6j$ symbols associated with $U_q(SL(2,\mathbb{R}))$. This kernel is directly related to the quantum path integral of the Virasoro TQFT, and manifests itself as an exact and analytical 3D background-independent MERA-type holographic tensor network. Many aspects of the AdS/CFT correspondence, including the factorization puzzle, admit a natural interpretation within this framework. This provides the first evidence suggesting that there is a universal holographic principle encompassing AdS/CFT and topological holography. We propose that the non-perturbative AdS/CFT correspondence is a $\textit{maximal}$ form of topological holography.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
Weaknesses
Report
This paper involves lots of interesting ideas, perhaps a large fraction of the ideas appearing on the hep-th arxiv in recent years. It is very long and my feeling is that it is not optimally presented.
The goal seems to be to make a connection between the recent developments of "SymTFT" or "topological holography" and the usual holography between asymptotically-AdS gravity and quantum field theories, i.e. the AdS/CFT correspondence. On the one hand, that such a connection exists is implicit in the name "topological holography", and indeed many examples of "topological holography" are related to the topological sectors of explicit AdS/CFT dual pairs realized in string theory. On the other hand, the goal of "topological holography", at least as most clearly expressed in the well-known work of Moore, Freed and Teleman, is rather different. That is, it is {\it not} a duality, but rather an attempt to abstract the symmetry structure from the dynamical information, generalizing the strategy of group theory. It is not clear to me that mixing these ideas up is a helpful thing to do.
The authors attempt to make these vague ideas more concrete with two "examples". One is a discussion of TTbar deformations, attempting to rewrite various integrals, following previous work of S.S. Lee, in the form of a higher-dimensional path integral. It is not clear to me what we learn by writing down this path integral. The other example involves tensor network representations of critical 2d partition functions. I am confused by the discussion of RG flow in that context. If we are really talking about the CFT partition function, there is no RG flow. If we are talking about a discretization of the CFT partition function, then there is some flow to the continuum theory. I therefore found the statement that these ideas ``can be applied to continuum field theory" very confusing -- if we are already in the continuum in a CFT, what RG flow are we talking about? Perhaps the authors are instead talking about the groundstate wavefunction of the CFT, and a construction of a tensor network representation of it along the lines of a MERA, and interpreting the layers of this network as an RG flow?
One very confusing thing about this discussion regards the distinction between topological field theory and quantum gravity. I think the discussion in the last section of Witten's paper called "Topological Quantum Field Theory" is quite relevant here. There he argues that there are two kinds of metric-independent theories -- those where we sum over geometries (quantum gravity) and those where we never introduce a metric in the first place (topological field theory), and that they should be regarded as two phases of one kind of system. That is, both classes of systems are topological in the sense that they are independent of a choice of metric. With this in mind, when the authors say that ``the bulk is not topological" I am not sure what it means.
Section 1.1 attempts to summarize the concrete results of the paper.
They seem to be in the realm of quantum gravity and more specifically AdS/CFT correspondence. I do not understand them.
I was particularly perplexed by the sentence
"In general, the bulk theory coming from a given CFT does not sum over all geometries." but this is just one small part of my problem.
The authors claim in various places, and I think this is a motivating idea in this work, that the way to understand AdS/CFT is that it should be the SymTFT that "makes explicit all the symmetries". I think this is incorrect. We know that AdS/CFT is a special case of a much more general correspondence involving non-conformal field theories which have much less symmetry.
I have tried several times in good faith to read this paper. I can see that it contains many interesting ideas and some correct detailed statements. However I must admit that I have failed to understand the concrete outcome of all these words and symbols. I see three (not mutually exclusive) possibilities:
1) I am just not smart enough to understand it and this is a failure on my part. 2) The outcomes do not merit all the pages and effort involved.
3) The paper really does contain incisive and understandable results, but they are presented in a way that makes them difficult to discern.
If option 1 is the case, I should not stand in the way of the paper being published. If option 2 is the case, the paper should not be published. If option 3 is the case, the authors should revise the paper to make it easier for the poor reader to discern a clear and well-defined payoff. Since it would be the happiest circumstance for all involved, I believe we should work under the assumption that option 3 is the case, and ask the authors to try again to clearly explain what they are doing, in a smaller number of pages, and without using sentences that cannot be assigned meaning.
A good example of the kind of meaning-free statement that is preventing me from understanding the paper is on page 6: "the non-topological boundary condition ... carries most of the dynamical information of the QFT" where the key problematic word is "most".
Here are some more examples that confused me:
-- on page 4: the authors discuss the ``data of RG flow in SymTFT bulk". I really do not understand what this means. The SymTFT is a topological field theory where there is no possible notion of RG flow. I see that the authors later try to complicate this notion, but at least at this point in the paper it is very difficult to assign meaning to these words.
-- on page 4: in the footnote, the authors use the phrase "topological symmetry". I do not understand the meaning of this term. What is a symmetry that is not topological?
-- on page 5: "In this paper we demonstrate ..." I do not understand
-- on page 5: does "irrational" just mean "infinite"
-- I do not understand the meaning of Figure 2.
-- page 37 "In the seminar work of Moore and Seiberg" should be "In the seminal work of Moore and Seiberg"
Requested changes
Please see the report
Recommendation
Ask for major revision
Strengths
Concrete Example: The Liouville tensor network is a concrete instantiation that illustrates the general framework.
Rigorous Symmetry Analysis: The treatment of Virasoro lines and quantum group structures is detailed and mathematically sound.
Weaknesses
RG Flow Details: In Section 3, the derivation of the discrete RG step is algebraically intricate. Could the authors clarify the choice of blocking transformation and justify its uniqueness?
Bulk Interpretation: The mapping between SymTFT and the 3D gravity path integral relies on certain regularization schemes. Can the authors compare different regularizations and comment on scheme‑dependence?
TT̄ Deformation Regime: The connection to TT̄ is introduced but not fully explored. Up to which order in the deformation parameter λ is the topological interpretation valid?
Quantum Group Data: The role of U_q(SL(2,ℝ)) representations in the Hilbert space construction could be elaborated, particularly regarding boundary conditions and edge modes.
Report
This manuscript presents a significant theoretical advance in our understanding of holographic RG flows by framing them as topological field theories. With clarifications on the questions above and minor structural edits, it will be a valuable contribution to the AdS/CFT literature.
Requested changes
In Equation (2.15), you introduce a topological defect line implementing Virasoro symmetry. Can you provide an explicit example of how this line acts on primary operators?
On page 14, the saddle‑point evaluation of the 3D path integral is sketched. Could you include a more detailed derivation or an appendix with intermediate steps?
How does your discrete tensor network reduce to the continuous Liouville path integral in the continuum limit? Are there subtleties in taking this limit?
Could you clarify whether the SymQRG framework preserves modular invariance at each RG step?
Have you checked whether anomalies (e.g., gravitational anomaly in CFT) affect the topological symmetry‑preserving property?
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Strengths
Report
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
We thank the referee very much for the encouraging feedback and thoughtful comments on our draft. Below, we address the specific points raised:
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The discrete tensor network formulation does not completely coincide with the field theory TTbar deformation at the non-perturbative level (for example, there are higher order terms in the Ishibashi states that will become important when the hole size is not small). These two constructions really correspond to different ways of turning on irrelevant deformations. In our opinion, the lattice realization we study is better defined in some respects, as it avoids the issue of complex energies often encountered in the TTbar-deformed theories.
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In fact, any combination coming from T and Tbar operators will do the job, since by definition they commute with the topological defects.
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This is an excellent question question that we currently don't have a complete answer, some possible deformations come from imposing the boundary with irregular shapes, though a systematic classification is currently lacking. For the second question, in our perspective, they are actually the same in the following sense: we encode the symmetry preserving deformation as acting on the physical boundary, rather than on the topological boundary. In the triangulation picture, the deformation corresponds to the gluing of tetrahedra, which is entirely determined by the F symbols associated with the symmetry. Of course, when we compute the result form the sandwich including the topological boundary, the explicit answer will depend on the detailed data of the boundary theory. This is analogous to the case of TTbar deformation, which has a universal form across different CFTs, though the precise answer definitely depend on model-specific details such as the operator spectrum.
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This is a question we cannot yet answer with complete certainty, as no explicit examples are known. Our current belief is that the bulk theory realizing generalized symmetries of the boundary need not always be a TQFT, particularly in cases involving an infinite number of symmetry defects. Constructing or ruling out such an example would indeed be very interesting.
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We distinguish two types of “non-geometrical” contributions. The first arises from singular geometries that do not correspond to saddle points in the semiclassical approximation. The second stems from the fact that in the presence of non-gravitational bulk matter, some contributions may not admit a geometric interpretation.
Author: Yikun Jiang on 2025-07-22 [id 5664]
(in reply to Report 2 on 2025-07-12)We thank the referee for the comments and suggestions. Below are our responces to the questions in this report: 1. The cases regarding the rational CFTs and other chiral algebras were actually analyzed in detail in some other earlier papers by one of the author Ling-Yan Hung, and we included the reference in the paper. Since the main focus of this paper is to connect to gravity, we do not put much emphasize on these cases. 2. Our block spin transform is one that preserves the topological symmetry by construction. We can choose other transformations on the \Omega state such that the overlap with the \Psi state is invariant, but the part that breaks the symmetry will always be projected out when we take overlap with the \Psi state at a different scale. 3. In our construction actually the bulk theory does not depend on the scheme, the choice of scheme is reflected in the choice of the boundary condition instead. There is an analagous way of seeing this in the context of some rational cases . In those examples, the bulk can be either described by two copies of the continuum Chern-Simons theory, or by the discrete Turaev-Viro theory. These two are mathematically equivalent. The difference however will come into play when we set boundary conditions for these two formulations, since they are the ones that bring "scales" into the boundary field theory. 4. TTbar deformation corresponds to a specific deformation and RG trajectory. In the language of SymTFT, they correspond to a particular set of choices for the boundary conditions. The topological holography interpretation is always there. The difference is that in the continuum case and discrete case, we have chosen different 3D boundary conditions and corresponding RG trajectories with the same IR fixed point. To be more specfic, the discrete case first introduces higher orders in T terms (from higher orders in Ishibashi), and the fact that the operators are inserted on discrete set of points instead of the whole 2D spacetime. 5. Thanks for the suggestion. We do not include them in the paper since in this paper we do not discuss cutting the bulk open, introducing the edge modes. However, we will consider adding some comments and references. 6. We will consider adding references on how these line operators on local operators in Liouville theory. 7. 14 probably is not the right page, maybe there is a typo? 8. There are indeed subtleties in this procedure (mostly associated to normalization and divergences) and more details were in our previous paper discussing Liouville theory. 9. This is a very interesting question. In the continuum, this was explored in the context of TTbar deformation where it was shown that modular invariance is preserved in an intricate way (see https://arxiv.org/pdf/1808.02492). We have not explored in full detail of our lattice construction, but we believe similar results shall hold, although more needs to be done. 10. In the current formulation, the CFT are well-defined 2D theories without gravitational anomalies (c_L=c_R and modular invariance), and are exactly equivalent to the 3D sandwiches. We have not explored much on how anomalous theories fall into this paradigm.