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Parity and the modular bootstrap

by Tarek Anous, Raghu Mahajan, Edgar Shaghoulian

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Authors (as registered SciPost users): Tarek Anous · Raghu Mahajan
Submission information
Preprint Link:  (pdf)
Date submitted: 2018-04-12 02:00
Submitted by: Anous, Tarek
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We consider unitary, modular invariant, two-dimensional CFTs which are invariant under the parity transformation $P$. Combining $P$ with modular inversion $S$ leads to a continuous family of fixed points of the $SP$ transformation. A particular subset of this locus of fixed points exists along the line of positive left- and right-moving temperatures satisfying $\beta_L \beta_R = 4\pi^2$. We use this fixed locus to prove a conjecture of Hartman, Keller, and Stoica that the free energy of a large-$c$ CFT$_2$ with a suitably sparse low-lying spectrum matches that of AdS$_3$ gravity at all temperatures and all angular potentials. We also use the fixed locus to generalize the modular bootstrap equations, obtaining novel constraints on the operator spectrum and providing a new proof of the statement that the twist gap is smaller than $(c-1)/12$ when $c>1$. At large $c$ we show that the operator dimension of the first excited primary lies in a region in the $(h,\overline{h})$-plane that is significantly smaller than $h+\overline{h}<c/6$. All our results extend to theories without parity symmetry through the construction of an auxiliary parity-invariant partition function.

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Reports on this Submission

Anonymous Report 2 on 2018-7-10 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1803.04938v2, delivered 2018-07-10, doi: 10.21468/SciPost.Report.530


1. The authors used clear and concise arguments to argue qualitative properties of 2d CFT. $\\$

2. It is a nice idea to utilize a continuous line of fixed points to bound theories. $\\$

3. The paper is well-written and clarifies several potential confusions along the way.


1. Parts of the paper do not feel complete, even for the purpose of demonstrating qualitative features. For example, the right side of Figure 2 shows two sharp kinks at 3 derivative order. One may thus wonder about the shape of the region at higher orders, and about whether such kinks persist. Such a qualitative question can be answered without going to too high an order. $\\$

2. Almost all argued properties were already known in the literature. The reduction of the allowed region in Figure 2 was phrased as a "novel constraint" in the abstract; however, the reduction had not been explicitly mapped out in the past mainly due to the lack of a clear target statement worth pursuing. A new revealing statement is also lacking in the present paper.


This paper explores how a continuous line of fixed points in parity-invariant CFT$_2$ may be used to generate bounds and demonstrate universal features. Their major achievement is to prove a previous conjecture about the universality of the spectrum in large c theories. Other utilizations of the continuous line of fixed points are also discussed, and some are further generalized to theories without parity symmetry.

Requested changes

1. Above (3.8), I suppose the authors want to keep $\beta_L$ and $\beta_R$ BOTH fixed, with $\beta_L \beta_R \neq 4\pi^2$. $\\$

2. I think the statement "This shows..." below (3.8) follows more from (3.7) than (3.8). Suppose $Z_H = 100 Z_{H^c}$, one would not say that $Z_{H^c}$ is dominating, but (3.8) would remain true. $\\$

3. Below (3.12), it is commented that "If $\sqrt{E_L E_R} > c/24$, we can do the integral by saddle point." Isn't one still able to perform a saddle point approximation even when $\sqrt{E_L E_R} > c/24$, except that now the saddle value depends on the details of $Z$? $\\$

4. I think it should be made clearer that Section 3.2 is just a review. $\\$

5. Does the proof in Section 4.3 only apply to parity-invariant theories? Relatedly, does the last sentence in the abstract need certain qualifiers? $\\$

6. I personally find Figure 3 not very illuminating. I would rather they write up their analytic argument following (4.12) in more explicit detail.

  • validity: top
  • significance: good
  • originality: good
  • clarity: high
  • formatting: perfect
  • grammar: perfect

Author:  Tarek Anous  on 2018-08-02  [id 302]

(in reply to Report 2 on 2018-07-10)

Dear Referee, thank you for taking the time to assess our paper. We will now try to address your comments.


In reference to weakness 1) we plan to clarify figure 2. To do so, we will modify it from simply showing the upper bound on the operator dimension to showing the entire region (including now a lower bound), where an operator must exist as dictated by modular invariance. We have changed the caption accordingly and have added a parenthetical in the ensuing description to indicate that the sharp features are only a result of the large-c limit and are not related to the low derivative order.

It is incumbent upon us to address weakness 2) as it would seem that the referee is suggesting that most of our results are not novel. However, the only previously known result from our paper is the twist gap, for which we provide an alternate derivation. The continuous fixed line as a practical tool for the numerical/analytic modular bootstrap is a new idea. The proof of the HKS conjecture and the reduction of the allowed region in Figure 2, both of which form the primary results of the paper, are also new. In summary, we respectfully disagree with the referee's commentary and contend that the majority of our results are indeed novel.

Requested changes:

1) We will include the word ``both'' in the line above equation (3.8) in order to clarify that we must keep both temperatures fixed.

2) If $Z_H = 100 Z_{H^c}$, we would indeed still say that the non-heavy states dominate the free energy, because considering their contribution alone gives the correct large-$c$ answer. In other words, $\log(100) = O(1)$ in large-$c$ counting, while $\log Z_{H^c} = O(c)$. We therefore intend to leave this part unchanged.

3) In section 3.2 we will reword the discussion to indicate that we can always do the integral by saddle point.

4) It is our opinion that it would be a mischaracterization to label section 3.2 as review. The precise reason one might be interested in proving that CFTs have a universal free energy at large $c$ is that this property explains why the Cardy formula extends to states beyond its naive regime of validity. Since our paper manages to finally show that CFTs have a universal free energy at large $c$, for arbitrary angular potentials, section 3.2 stands as a direct consequence of this result.

5) The proof in section 4.3 extends to parity non-invariant theories. This is because it is a statement about the twist, defined as the $2\text{min}(h,\overline{h})$ and is not a bound on the individual $(h,\overline{h})$. Phrasing it as a bound on the twist makes the result hold for theories without parity symmetry, despite the fact that figure 3 would seem to suggest that we are bounding $h$.

We will change the last line of the abstract to emphasize this point.

6) While we agree with the referee that the description after (4.12) is not as explicit as it could be, this was done in order to preserve clarity rather than the opposite. The reason is that applying $\mathcal{B}$ to a generic partition function results in a nasty and cumbersome set of equations involving Dedekind eta functions and derivatives thereof. We quote the limits of these functions, which the reader is free to check, but we feel that writing down explicit equations would more likely bog the reader down given their sheer length.

We would again like to thank you for taking the time to assess and improve our paper.

Tarek Anous, Raghu Mahajan, and Edgar Shaghoulian

Anonymous Report 1 on 2018-6-25 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1803.04938v2, delivered 2018-06-25, doi: 10.21468/SciPost.Report.514


1-Elegant argument that allows to extend previous results.
2-Results fill in gaps in previous papers.
3-Short and clearly written.


1-The methods used are not genuinely new, but rather an application of known methods to a combined new symmetry.


This paper considers a combined parity and modular transformation of 2d CFTs. This symmetry allows to apply bootstrap techniques to obtain new results. Namely, the universality of the phase diagram of 2d CFTs with sparse spectrum is established for general spin potentials. It also improves the bound on the lightest operator in a large $c$ theory. These are both nice additions to previous results in the literature.

Requested changes


  • validity: top
  • significance: high
  • originality: good
  • clarity: high
  • formatting: perfect
  • grammar: perfect

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