Modular Constraints on Conformal Field Theories with Currents

We study constraints coming from the modular invariance of the partition function of two-dimensional conformal field theories. We constrain the spectrum of CFTs in the presence of holomorphic and anti-holomorphic currents using the semi-definite programming. In particular, we find the bounds on the twist gap for the non-current primaries depend dramatically on the presence of holomorphic currents, showing numerous kinks and peaks. Various rational CFTs are realized at the numerical boundary of the twist gap, saturating the upper limits on the degeneracies. Such theories include Wess-Zumino-Witten models for the Deligne's exceptional series, the Monster CFT and the Baby Monster CFT. We also study modular constraints imposed by $\mathcal{W}$-algebras of various type and observe that the bounds on the gap depend on the choice of $\mathcal{W}$-algebra in the small central charge region.


The Character Decomposition
• The (Virasoro) vacuum characters and primary characters are defined by The torus partition function of unitary CFT admit the character decomposition, where the degeneracies d (h,h), d(j) andd(j) are positive integers.
The constraints from SL(2, Z) • T -transformation : All states should have integer spin.
where the function Z λ (τ,τ ) is defined as

Modular Bootstrap -Basic Strategy
• In the computation, we mainly use the reduced character for convenience.
• Find αm,n such that, If we find such αm,n, then we conclude that no modular invariant partition function can be exist. This problem can be converted to the semi-definite programming.

Scalar Gap Problem
In this problem, we impose a gap ∆ s only to the scalar operator. ∆ ≥ ∆ s for j = 0, ∆ ≥ j for j ̸ = 0.

Overall Gap Problem
In this problem, we impose a gap ∆ o to the certain low-spin operators.

Twist Gap Problem
In this problem, we impose a gap ∆ t to the twist, defined as

Expected CFTs on the numerical bound (Twist Gap)
• For the Wess-Zumino-Witten model with affine Lie algebraĝ and level-k, • The above formulae suggest that the twist gap problem realize level-1 WZW models on the numerical boundary!(c ≤ 8)

The Modular Differential Equation(MDE)
• Idea : n characters of rational conformal field theory(RCFT) are the solutions to the n-th order modular differential equation,
• The coefficients {a0, a2, a3, · · · } are positive integer only for • The primary characters have the form of .
• The coefficients in the primary characters are not completely fixed from the modular differential equation.
and solve the following problem via the semi-definite programming.

Extremal Functional Method
• Suppose the degeneracies of all primaries saturated the maximum bound. Then, the modular bootstrap equation is reduced to the below form.
• The solutions to the second order MDE with c = 26 5 gives : • The relation between partition function and reduced partition function is given by, • The partition function of (F4)1 WZW model is known : This perfectly agree with the numerical result.

The Result Summary (c ≤ 8)
• In case of (G2)1, (F4)1 and (E7)1 WZW model, its modular invariant partition function is known. In terms of the solutions to the second order MDE, they are written as and in case of (E6)1 WZW model, For them, we checked the spectral analysis successfully reproduce the known partition function.

Examine ECFTs via the modular bootstrap
• CLAIM : Twist gap problem realize the ECFTs with c = 24, 48 on the boundary.
• The partition function of c = 24 ECFT is obatined by the solutions to the third order MDE, while the c = 48 partition function is realized by the fourth order MDE.
• The EFM analysis suggests that all of them have the states with integer ∆.

CFTs without Kac-Moody symmetry
• In the mathematics, the corresponding vertex operator algebra was constructed.

Bootstrapping with W-algebra
• In case of the W(2, 3)-algebra, we have spin-3 generator Wn. The corresponding fugacity p = e 2πiz should be introduced in the character.
The modular transformation property is only known up to W 2 0 order. We will focus on the unrefined character which means W0-zeroth order character.
• Assuming the non-vacuum module is non-degenerate, the unrefined character of rank-r W(d1, d2, · · · dr )-algebra is given by,  • (c = 3, ∆ = 3 4 ) sits on the numerical boundary that obtained using the unrefined character of rank-3 W(2, 3, 4) algebra. Note that c = 3 is not in the list of the two-character RCFTs.
• The hypothetical CFT with c = 3 can be identified to the (A3)1 WZW model. This theory is realized by third order MDE, with Kac-Moody symmtery.

Conclusion and Outlook
• The twist gap problem with holomorphic currents (j ≥ 1) successfully realize two-character RCFTs and three-character RCFTs on the numerical bound. The various RCFTS include level-one WZW models and extremal conformal field theories.
• When the holomorphic currents are included from j = 2, the CFTs without Kac-Moody symmetry are realized on the numerical boundary. It include c = 8, c = 16 CFTs and baby monster CFT. We suggest the modular invariant partition function of those theories based on the numerical results.
The coefficients in partition function can be decomposed by the dimension of irrpes of the O + 10 (2) or baby monster group. They are expected to be a underlying symmetry of the three special theories.
• The numerical analysis extended to the W-algebra cases, using the unrefined character. The numerical bounds suggest the absence of the degenerate states in unitary irreducible representation when c ≥ r .
• Application to the supersymmetric cases : Can we examine the super WZW models, super extremal conformal field theory? Unexpected super-RCFTs?