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Non-Chiral Vertex Operator Algebra Associated To Lorentzian Lattices And Narain CFTs
by Ranveer Kumar Singh, Madhav Sinha
Submission summary
| Authors (as registered SciPost users): | Ranveer Singh |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202401_00008v1 (pdf) |
| Date submitted: | 2024-01-10 18:47 |
| Submitted by: | Singh, Ranveer |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Frenkel, Lepowsky, and Meurman constructed a vertex operator algebra (VOA) associated to any even, integral, Euclidean lattice. In the language of physics, these are examples of chiral conformal field theories. In this paper, we define non-chiral vertex operator algebra and some associated notions. We then give a construction of a non-chiral VOA associated to an even, integral, Lorentzian lattice and construct their irreducible modules. We obtain the moduli space of such modular invariant non-chiral VOAs based on self-dual Lorentzian lattices of signature (m,n) assuming the validity of a technical result about automorphisms of the lattice. We finally show that Narain conformal field theories in physics are examples of non-chiral VOA. Our formalism helps us to identify the chiral algebra of Narain CFTs in terms of a particular sublattice and break its partition function into sum of characters.
Current status:
Reports on this Submission
Strengths
1. This paper gives mathematically rigorous and axiomatic definitions to various physical notions in 2-dimensional conformal field theory.
2. This paper presents a new interesting construction of an example from a certain Lorentzian lattice.
3. This paper formulats a precise mathematical conjecture, and presents a description of a certain moduli space based on this conjecture.
Weaknesses
1. I wonder whether certain terminology is appropriate.
2. Relations to preceding works should be clearly given.
3. I have a doubt about validity of one remark.
Report
The authors formulate a non-chiral vertex operator algebra, its modules and intertwiners, and a non-chiral conformal field theory in terms of mathematical axioms, and construct an example for each even integral Lorentzian lattice. This construction extends a well-known construction of a vertex operator algebra for each even integral Euclidean lattice. They study the moduli space of non-chiral CFTs over certain Lorentzian lattices and identify this space based on a conjecture given in this paper. Then they show that Narain CFT's in physics literature fall within this framework. This is a nice contribution to mathematical physics of conformal field theory, and I recommend publication of this paper.
Requested changes
(1) The name "non-chiral vertex operator algebra" sounds unnatural to me and "full vertex operator algebra" sounds much more natural. If the authors insists on this name, one needs some justification.
(2) Relations of the formulation of the authors to that in Moriwaki [8] should be clarified.
(3) page 4:
Hence. N_{i,0}=\delta_{i,0}, N_{0,\bar i}=\delta_{0,\bar i}.
I don't understand this claim. There are type I modular invariants corresponding to D_{2n}, E_6, and E_8 as in the following paper. For them, N_{i,0} can be 1 for nonzero i, where the algebra A is given by the SU(2)_k WZW-model. There are many such examples.
Cappelli, A.; Itzykson, C.; Zuber, J.-B.
The A-D-E classification of minimal and A^{(1)}_1 conformal invariant theories
Comm. Math. Phys. 113 (1987), no. 1, 1-26.
(4) page 9:
Infact --> In fact
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