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Parity and the modular bootstrap

by Tarek Anous, Raghu Mahajan, Edgar Shaghoulian

Submission summary

As Contributors: Tarek Anous
Arxiv Link: (pdf)
Date accepted: 2018-09-10
Date submitted: 2018-08-02 02:00
Submitted by: Anous, Tarek
Submitted to: SciPost Physics
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 $h+\overline{h}<c/6$. Our results for the free energy and constraints on the operator spectrum extend to theories without parity symmetry through the construction of an auxiliary parity-invariant partition function.

Ontology / Topics

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AdS$_3$ gravity Conformal bootstrap Conformal field theory (CFT) Parity symmetry

Published as SciPost Phys. 5, 022 (2018)

Author comments upon resubmission

We would like to thank the referees for their kind assessment of our paper. We hope that we have addressed their concerns in our replies and in the list of changes describing this resubmission.

List of changes

1) In order to clarify which results are independent of the parity symmetry, we have changed the last line of the abstract to: ``Our results for the free energy and constraints on the operator spectrum extend to theories without parity symmetry through the construction of an auxiliary parity-invariant partition function.''

2) We have included the word ``both'' in the line above equation (3.8) in order to clarify that we must keep both temperatures fixed.

3) In section 3.2 we have reworded the discussion to indicate that we can always do the integral by saddle point. We have added the sentence ``If $\sqrt{E_{L}\,E_R}>c/24$, this saddle point is at a value of $\beta_L$ and $\beta_R$ such that $\beta_L \beta_R < 4\pi^2$.'' to explain the significance of that condition in the final equation for the Cardy formula.

4) We have modified figure 2 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 modified the relevant section and 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.

5) We have fixed a typo in the original submitted version of our paper. Previously it was written that $\tilde{\rho}\leq 2\rho(h,\overline{h})$. We have replaced this with the correct expression (3.12):$\tilde{rho} \leq 2\, \text{max}\left[\rho(h, \overline{h}),\rho(\overline{h},h)\right]$.

Reports on this Submission

Anonymous Report 1 on 2018-8-16 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1803.04938v3, delivered 2018-08-16, doi: 10.21468/SciPost.Report.557


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-Most techniques are existing ones, and some results are re-derivations of old results.


This paper explores how a continuous line of fixed points in parity-invariant CFT2 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-Following up on Weakness 1 in my earlier report, I find the comment below (4.9) about the kink in the infinite c limit slightly misleading to the reader who is less familiar with bootstrap. Here there are two limits involved, infinite derivative order and infinite c. The physical order of limit is first infinite derivative order, and then infinite c. So the physical meaning of a kink that develops in the infinite c limit at 3 derivative order is unclear. I suggest that the authors add a footnote explaining this point, saying that the kink may be an artifact of finite derivative order. Better yet, if they wish, the authors may go to 5 or 7 derivatives to check the fate of the kink, and make a more definitive statement.$\\$

Note: There has also been no change or response regarding Changes 2, 4 and 6 in my earlier report, but since they were subjective comments, I leave those points to the authors' liberty.

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

Author:  Tarek Anous  on 2018-08-30  [id 311]

(in reply to Report 1 on 2018-08-16)
answer to question

Dear Referee,

Thank you again for your report. We would like to stress that the plots we have shown do not come from an exhaustive optimization up to a certain order of derivatives, as is usually the case in the bootstrap. They simply present the constraint that arises from a single combination of derivatives, with a judiciously chosen set of coefficients. In particular, the meaning of "the shape at higher orders" is, in not a well-defined notion in our presentation.

If we had chosen to perform a systematic optimization, one should expect a shape similar to the one that appears in our paper, but then the question of how the shape changes as we increase derivative order can be addressed, in that context. We instead judged that such a systematic optimization was outside the scope of our paper, opting instead for a qualitative illustration.

Without undertaking a systematic optimization, the best one can do to address your comment is to keep the combination of derivatives fixed and vary the central charge. As seen by the plots, the kinks only develop at large central charge.

As a result we believe it is best to leave the section unchanged.

Thank you for taking the time to assess our paper.

Tarek Anous, Raghu Mahajan, Edgar Shaghoulian

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