Modular invariance of (logarithmic) intertwining operators

Let $V$ be a $C_2$-cofinite vertex operator algebra without nonzero elements of negative weights. We prove the conjecture that the spaces spanned by analytic extensions of pseudo-$q$-traces ($q=e^{2\pi i\tau}$) shifted by $-\frac{c}{24}$ of products of geometrically-modified (logarithmic) intertwining operators among grading-restricted generalized $V$-modules are invariant under modular transformations. The convergence and analytic extension result needed to formulate this conjecture and some consequences on such shifted pseudo-$q$-traces were proved by Fiordalisi in [F1] and [F2] using the method developed in [H2]. The method that we use to prove this conjecture is based on the theory of the associative algebras $A^{N}(V)$ for $N\in \mathbb{N}$, their graded modules and their bimodules introduced and studied by the author in [H8] and [H9]. This modular invariance result gives a construction of $C_2$-cofinite genus-one logarithmic conformal field theories from the corresponding genus-zero logarithmic conformal field theories.

The modular invariance conjecture of Moore and Seiberg Let V be a vertex operator algebra and W 1 , W 2 and W 3 lower-bounded generalized V -modules. A (non-logarithmic) intertwining operator is a linear map satisfying the lower-truncation property, the Jacobi identity and the L(−1)-derivative property. In 1988, Moore and Seiberg conjectured that for rational conformal field theories, the space spanned by q-traces (shifted by − c 24 ) of products of chiral vertex operators (intertwining operators) is invariant under modular transformations. Based on these two conjectures, they derived the Verlinde formula and discovered modular tensor category structures.
The modular invariance conjecture of Moore and Seiberg Let V be a vertex operator algebra and W 1 , W 2 and W 3 lower-bounded generalized V -modules. A (non-logarithmic) intertwining operator is a linear map satisfying the lower-truncation property, the Jacobi identity and the L(−1)-derivative property. In 1988, Moore and Seiberg conjectured that for rational conformal field theories, the space spanned by q-traces (shifted by − c 24 ) of products of chiral vertex operators (intertwining operators) is invariant under modular transformations. Based on these two conjectures, they derived the Verlinde formula and discovered modular tensor category structures.
The modular invariance conjecture of Moore and Seiberg Let V be a vertex operator algebra and W 1 , W 2 and W 3 lower-bounded generalized V -modules. A (non-logarithmic) intertwining operator is a linear map satisfying the lower-truncation property, the Jacobi identity and the L(−1)-derivative property. In 1988, Moore and Seiberg conjectured that for rational conformal field theories, the space spanned by q-traces (shifted by − c 24 ) of products of chiral vertex operators (intertwining operators) is invariant under modular transformations. Based on these two conjectures, they derived the Verlinde formula and discovered modular tensor category structures.
The modular invariance conjecture of Moore and Seiberg Let V be a vertex operator algebra and W 1 , W 2 and W 3 lower-bounded generalized V -modules. A (non-logarithmic) intertwining operator is a linear map satisfying the lower-truncation property, the Jacobi identity and the L(−1)-derivative property. In 1988, Moore and Seiberg conjectured that for rational conformal field theories, the space spanned by q-traces (shifted by − c 24 ) of products of chiral vertex operators (intertwining operators) is invariant under modular transformations. Based on these two conjectures, they derived the Verlinde formula and discovered modular tensor category structures.
Zhu's theorem: A special case of the conjecture of Moore and Seiberg In 1990, Zhu formulated precisely and proved a special case of the modular invariance conjecture of Moore and Seiberg. Let V be a vertex operator algebra V satisfying the following conditions: 1 V has no nonzero elements of negative weights, Every lower-bounded generalized V -module is completely reducible.
Zhu proved that the space spanned by q-traces (shifted by − c 24 ) of products of vertex operators for V -modules is invariant under modular transformations.
In particular, Zhu proved that the characters of irreucible V -modules span a module for the modular group SL(2, Z).
Zhu's theorem: A special case of the conjecture of Moore and Seiberg In 1990, Zhu formulated precisely and proved a special case of the modular invariance conjecture of Moore and Seiberg. Let V be a vertex operator algebra V satisfying the following conditions: 1 V has no nonzero elements of negative weights, Every lower-bounded generalized V -module is completely reducible.
Zhu proved that the space spanned by q-traces (shifted by − c 24 ) of products of vertex operators for V -modules is invariant under modular transformations.
In particular, Zhu proved that the characters of irreucible V -modules span a module for the modular group SL(2, Z).
Zhu's theorem: A special case of the conjecture of Moore and Seiberg In 1990, Zhu formulated precisely and proved a special case of the modular invariance conjecture of Moore and Seiberg. Let V be a vertex operator algebra V satisfying the following conditions: 1 V has no nonzero elements of negative weights, Every lower-bounded generalized V -module is completely reducible.
Zhu proved that the space spanned by q-traces (shifted by − c 24 ) of products of vertex operators for V -modules is invariant under modular transformations.
In particular, Zhu proved that the characters of irreucible V -modules span a module for the modular group SL(2, Z).
Zhu's theorem: A special case of the conjecture of Moore and Seiberg In 1990, Zhu formulated precisely and proved a special case of the modular invariance conjecture of Moore and Seiberg. Let V be a vertex operator algebra V satisfying the following conditions: 1 V has no nonzero elements of negative weights, Every lower-bounded generalized V -module is completely reducible.
Zhu proved that the space spanned by q-traces (shifted by − c 24 ) of products of vertex operators for V -modules is invariant under modular transformations.
In particular, Zhu proved that the characters of irreucible V -modules span a module for the modular group SL(2, Z).
Zhu's theorem: A special case of the conjecture of Moore and Seiberg In 1990, Zhu formulated precisely and proved a special case of the modular invariance conjecture of Moore and Seiberg. Let V be a vertex operator algebra V satisfying the following conditions: 1 V has no nonzero elements of negative weights, Every lower-bounded generalized V -module is completely reducible.
Zhu proved that the space spanned by q-traces (shifted by − c 24 ) of products of vertex operators for V -modules is invariant under modular transformations.
In particular, Zhu proved that the characters of irreucible V -modules span a module for the modular group SL(2, Z).
Zhu's theorem: A special case of the conjecture of Moore and Seiberg In 1990, Zhu formulated precisely and proved a special case of the modular invariance conjecture of Moore and Seiberg. Let V be a vertex operator algebra V satisfying the following conditions: 1 V has no nonzero elements of negative weights, Every lower-bounded generalized V -module is completely reducible.
Zhu proved that the space spanned by q-traces (shifted by − c 24 ) of products of vertex operators for V -modules is invariant under modular transformations.
In particular, Zhu proved that the characters of irreucible V -modules span a module for the modular group SL(2, Z).
Zhu's theorem: A special case of the conjecture of Moore and Seiberg In 1990, Zhu formulated precisely and proved a special case of the modular invariance conjecture of Moore and Seiberg. Let V be a vertex operator algebra V satisfying the following conditions: 1 V has no nonzero elements of negative weights, Every lower-bounded generalized V -module is completely reducible.
Zhu proved that the space spanned by q-traces (shifted by − c 24 ) of products of vertex operators for V -modules is invariant under modular transformations.
In particular, Zhu proved that the characters of irreucible V -modules span a module for the modular group SL(2, Z).
Zhu's theorem: A special case of the conjecture of Moore and Seiberg In 1990, Zhu formulated precisely and proved a special case of the modular invariance conjecture of Moore and Seiberg. Let V be a vertex operator algebra V satisfying the following conditions: 1 V has no nonzero elements of negative weights, Every lower-bounded generalized V -module is completely reducible.
Zhu proved that the space spanned by q-traces (shifted by − c 24 ) of products of vertex operators for V -modules is invariant under modular transformations.
In particular, Zhu proved that the characters of irreucible V -modules span a module for the modular group SL(2, Z).
Miyamoto's pseudo-q-traces and nonsemisimple generalization of Zhu's theorem In 2002, Miyamoto proved a nonsemisimple generalization of Zhu's theorem. In the nonsemisimple case, q-traces are not enough. Miyamoto introduced pseudo-q-traces of operators on grading-restricted generalized V -modules. Let V be a vertex operator algebra V satisfying the conditions: 1 V has no nonzero elements of negative weights.
There are no finite-dimensional irreducible V -modules.
Miyamoto proved that the space spanned by pseudo-q-traces (shifted by − c 24 ) of products of vertex operators for grading-restricted generalized V -modules is invariant under modular transformations. Arike and Nagatomo first noticed that Condition 3 is needed in Miyamoto's paper. There are examples of vertex operators satisfying Conditions 1 and 2 but not Condition 3.
Miyamoto's pseudo-q-traces and nonsemisimple generalization of Zhu's theorem In 2002, Miyamoto proved a nonsemisimple generalization of Zhu's theorem. In the nonsemisimple case, q-traces are not enough. Miyamoto introduced pseudo-q-traces of operators on grading-restricted generalized V -modules. Let V be a vertex operator algebra V satisfying the conditions: 1 V has no nonzero elements of negative weights.
There are no finite-dimensional irreducible V -modules.
Miyamoto proved that the space spanned by pseudo-q-traces (shifted by − c 24 ) of products of vertex operators for grading-restricted generalized V -modules is invariant under modular transformations. Arike and Nagatomo first noticed that Condition 3 is needed in Miyamoto's paper. There are examples of vertex operators satisfying Conditions 1 and 2 but not Condition 3.
Miyamoto's pseudo-q-traces and nonsemisimple generalization of Zhu's theorem In 2002, Miyamoto proved a nonsemisimple generalization of Zhu's theorem. In the nonsemisimple case, q-traces are not enough. Miyamoto introduced pseudo-q-traces of operators on grading-restricted generalized V -modules. Let V be a vertex operator algebra V satisfying the conditions: 1 V has no nonzero elements of negative weights.
There are no finite-dimensional irreducible V -modules.
Miyamoto proved that the space spanned by pseudo-q-traces (shifted by − c 24 ) of products of vertex operators for grading-restricted generalized V -modules is invariant under modular transformations. Arike and Nagatomo first noticed that Condition 3 is needed in Miyamoto's paper. There are examples of vertex operators satisfying Conditions 1 and 2 but not Condition 3.
Miyamoto's pseudo-q-traces and nonsemisimple generalization of Zhu's theorem In 2002, Miyamoto proved a nonsemisimple generalization of Zhu's theorem. In the nonsemisimple case, q-traces are not enough. Miyamoto introduced pseudo-q-traces of operators on grading-restricted generalized V -modules. Let V be a vertex operator algebra V satisfying the conditions: 1 V has no nonzero elements of negative weights.
There are no finite-dimensional irreducible V -modules.
Miyamoto proved that the space spanned by pseudo-q-traces (shifted by − c 24 ) of products of vertex operators for grading-restricted generalized V -modules is invariant under modular transformations. Arike and Nagatomo first noticed that Condition 3 is needed in Miyamoto's paper. There are examples of vertex operators satisfying Conditions 1 and 2 but not Condition 3.
Miyamoto's pseudo-q-traces and nonsemisimple generalization of Zhu's theorem In 2002, Miyamoto proved a nonsemisimple generalization of Zhu's theorem. In the nonsemisimple case, q-traces are not enough. Miyamoto introduced pseudo-q-traces of operators on grading-restricted generalized V -modules. Let V be a vertex operator algebra V satisfying the conditions: 1 V has no nonzero elements of negative weights.
There are no finite-dimensional irreducible V -modules.
Miyamoto proved that the space spanned by pseudo-q-traces (shifted by − c 24 ) of products of vertex operators for grading-restricted generalized V -modules is invariant under modular transformations. Arike and Nagatomo first noticed that Condition 3 is needed in Miyamoto's paper. There are examples of vertex operators satisfying Conditions 1 and 2 but not Condition 3.
Miyamoto's pseudo-q-traces and nonsemisimple generalization of Zhu's theorem In 2002, Miyamoto proved a nonsemisimple generalization of Zhu's theorem. In the nonsemisimple case, q-traces are not enough. Miyamoto introduced pseudo-q-traces of operators on grading-restricted generalized V -modules. Let V be a vertex operator algebra V satisfying the conditions: 1 V has no nonzero elements of negative weights.
There are no finite-dimensional irreducible V -modules.
Miyamoto proved that the space spanned by pseudo-q-traces (shifted by − c 24 ) of products of vertex operators for grading-restricted generalized V -modules is invariant under modular transformations. Arike and Nagatomo first noticed that Condition 3 is needed in Miyamoto's paper. There are examples of vertex operators satisfying Conditions 1 and 2 but not Condition 3.
Miyamoto's pseudo-q-traces and nonsemisimple generalization of Zhu's theorem In 2002, Miyamoto proved a nonsemisimple generalization of Zhu's theorem. In the nonsemisimple case, q-traces are not enough. Miyamoto introduced pseudo-q-traces of operators on grading-restricted generalized V -modules. Let V be a vertex operator algebra V satisfying the conditions: 1 V has no nonzero elements of negative weights.
There are no finite-dimensional irreducible V -modules.
Miyamoto proved that the space spanned by pseudo-q-traces (shifted by − c 24 ) of products of vertex operators for grading-restricted generalized V -modules is invariant under modular transformations. Arike and Nagatomo first noticed that Condition 3 is needed in Miyamoto's paper. There are examples of vertex operators satisfying Conditions 1 and 2 but not Condition 3.
Miyamoto's pseudo-q-traces and nonsemisimple generalization of Zhu's theorem In 2002, Miyamoto proved a nonsemisimple generalization of Zhu's theorem. In the nonsemisimple case, q-traces are not enough. Miyamoto introduced pseudo-q-traces of operators on grading-restricted generalized V -modules. Let V be a vertex operator algebra V satisfying the conditions: 1 V has no nonzero elements of negative weights.
There are no finite-dimensional irreducible V -modules.
Miyamoto proved that the space spanned by pseudo-q-traces (shifted by − c 24 ) of products of vertex operators for grading-restricted generalized V -modules is invariant under modular transformations. Arike and Nagatomo first noticed that Condition 3 is needed in Miyamoto's paper. There are examples of vertex operators satisfying Conditions 1 and 2 but not Condition 3.
The precise formulation and proof of the modular invariance conjecture of Moore-Seiberg In 2003, I formulated precisely and proved the modular invariance conjecture of Moore and Seiberg.
Let V be a vertex operator algebra satisfying the three conditions needed in Zhu's theorem. Then the modular invariance conjecture of Moore and Seiberg is true. That is, the space spanned by q-traces (shifted by − c 24 ) of products of intertwining operators is invariant under modular transformations.
The method used by Zhu (which is also used by Miyamoto) cannot be used to prove this conjecture of Moore and Seiberg for the q-traces shifted by c 24 of products of more than one intertwining operators.
A completely different method is developed to prove this conjecture. This is the reason why it took 13 years after Zhu's theorem to prove this conjecture.
The precise formulation and proof of the modular invariance conjecture of Moore-Seiberg In 2003, I formulated precisely and proved the modular invariance conjecture of Moore and Seiberg.
Let V be a vertex operator algebra satisfying the three conditions needed in Zhu's theorem. Then the modular invariance conjecture of Moore and Seiberg is true. That is, the space spanned by q-traces (shifted by − c 24 ) of products of intertwining operators is invariant under modular transformations.
The method used by Zhu (which is also used by Miyamoto) cannot be used to prove this conjecture of Moore and Seiberg for the q-traces shifted by c 24 of products of more than one intertwining operators.
A completely different method is developed to prove this conjecture. This is the reason why it took 13 years after Zhu's theorem to prove this conjecture.
The precise formulation and proof of the modular invariance conjecture of Moore-Seiberg In 2003, I formulated precisely and proved the modular invariance conjecture of Moore and Seiberg.
Let V be a vertex operator algebra satisfying the three conditions needed in Zhu's theorem. Then the modular invariance conjecture of Moore and Seiberg is true. That is, the space spanned by q-traces (shifted by − c 24 ) of products of intertwining operators is invariant under modular transformations.
The method used by Zhu (which is also used by Miyamoto) cannot be used to prove this conjecture of Moore and Seiberg for the q-traces shifted by c 24 of products of more than one intertwining operators.
A completely different method is developed to prove this conjecture. This is the reason why it took 13 years after Zhu's theorem to prove this conjecture.
The precise formulation and proof of the modular invariance conjecture of Moore-Seiberg In 2003, I formulated precisely and proved the modular invariance conjecture of Moore and Seiberg.
Let V be a vertex operator algebra satisfying the three conditions needed in Zhu's theorem. Then the modular invariance conjecture of Moore and Seiberg is true. That is, the space spanned by q-traces (shifted by − c 24 ) of products of intertwining operators is invariant under modular transformations.
The method used by Zhu (which is also used by Miyamoto) cannot be used to prove this conjecture of Moore and Seiberg for the q-traces shifted by c 24 of products of more than one intertwining operators.
A completely different method is developed to prove this conjecture. This is the reason why it took 13 years after Zhu's theorem to prove this conjecture.
The precise formulation and proof of the modular invariance conjecture of Moore-Seiberg In 2003, I formulated precisely and proved the modular invariance conjecture of Moore and Seiberg.
Let V be a vertex operator algebra satisfying the three conditions needed in Zhu's theorem. Then the modular invariance conjecture of Moore and Seiberg is true. That is, the space spanned by q-traces (shifted by − c 24 ) of products of intertwining operators is invariant under modular transformations.
The method used by Zhu (which is also used by Miyamoto) cannot be used to prove this conjecture of Moore and Seiberg for the q-traces shifted by c 24 of products of more than one intertwining operators.
A completely different method is developed to prove this conjecture. This is the reason why it took 13 years after Zhu's theorem to prove this conjecture.
Conjecture on the modular invariance of (logarithmic) intertwining operators In the nonsemisimple case, we have to consider more general intertwining operators: A (logarithmic) intertwining operator is a linear map satisfying the same conditions above for intertwining operators. We will sometimes call a (logarithmic) intertwining operator simply an intertwining operator.

Conjecture (2003):
For a C 2 -cofinite vertex operator algebra V without nonzero elements of negative weights, the space of pseudo-q-traces (shifted by − c 24 ) of products of (logarithmic) intertwining operators is invariant under modular transformations. Conjecture on the modular invariance of (logarithmic) intertwining operators In the nonsemisimple case, we have to consider more general intertwining operators: A (logarithmic) intertwining operator is a linear map satisfying the same conditions above for intertwining operators. We will sometimes call a (logarithmic) intertwining operator simply an intertwining operator.

Conjecture (2003):
For a C 2 -cofinite vertex operator algebra V without nonzero elements of negative weights, the space of pseudo-q-traces (shifted by − c 24 ) of products of (logarithmic) intertwining operators is invariant under modular transformations. Conjecture on the modular invariance of (logarithmic) intertwining operators In the nonsemisimple case, we have to consider more general intertwining operators: A (logarithmic) intertwining operator is a linear map satisfying the same conditions above for intertwining operators. We will sometimes call a (logarithmic) intertwining operator simply an intertwining operator.

Conjecture (2003):
For a C 2 -cofinite vertex operator algebra V without nonzero elements of negative weights, the space of pseudo-q-traces (shifted by − c 24 ) of products of (logarithmic) intertwining operators is invariant under modular transformations.

Pseudo-traces
Let P be a finite-dimensional associative algebra and M a finitely generated right projective P-module. Then M has a pair of sets Let φ : P → C be a symmetric linear function on P, that is, a linear map satisfying φ(p 1 p 2 ) = φ(p 2 p 1 ) for all p 1 , p 2 ∈ P. The pseudo-trace of A ∈ End P M associated to φ is defined to be The pseudo-trace Tr φ M is a symmetric linear function on End P M.

Pseudo-traces
Let P be a finite-dimensional associative algebra and M a finitely generated right projective P-module. Then M has a pair of sets Let φ : P → C be a symmetric linear function on P, that is, a linear map satisfying φ(p 1 p 2 ) = φ(p 2 p 1 ) for all p 1 , p 2 ∈ P. The pseudo-trace of A ∈ End P M associated to φ is defined to be The pseudo-trace Tr φ M is a symmetric linear function on End P M.

Pseudo-traces
Let P be a finite-dimensional associative algebra and M a finitely generated right projective P-module. Then M has a pair of sets Let φ : P → C be a symmetric linear function on P, that is, a linear map satisfying φ(p 1 p 2 ) = φ(p 2 p 1 ) for all p 1 , p 2 ∈ P. The pseudo-trace of A ∈ End P M associated to φ is defined to be The pseudo-trace Tr φ M is a symmetric linear function on End P M.

Pseudo-traces
Let P be a finite-dimensional associative algebra and M a finitely generated right projective P-module. Then M has a pair of sets Let φ : P → C be a symmetric linear function on P, that is, a linear map satisfying φ(p 1 p 2 ) = φ(p 2 p 1 ) for all p 1 , p 2 ∈ P. The pseudo-trace of A ∈ End P M associated to φ is defined to be The pseudo-trace Tr φ M is a symmetric linear function on End P M.

Pseudo-traces
Let P be a finite-dimensional associative algebra and M a finitely generated right projective P-module. Then M has a pair of sets Let φ : P → C be a symmetric linear function on P, that is, a linear map satisfying φ(p 1 p 2 ) = φ(p 2 p 1 ) for all p 1 , p 2 ∈ P. The pseudo-trace of A ∈ End P M associated to φ is defined to be The pseudo-trace Tr φ M is a symmetric linear function on End P M.

Pseudo-traces
Let P be a finite-dimensional associative algebra and M a finitely generated right projective P-module. Then M has a pair of sets Let φ : P → C be a symmetric linear function on P, that is, a linear map satisfying φ(p 1 p 2 ) = φ(p 2 p 1 ) for all p 1 , p 2 ∈ P. The pseudo-trace of A ∈ End P M associated to φ is defined to be The pseudo-trace Tr φ M is a symmetric linear function on End P M.

Pseudo-q-traces
Let V be a vertex operator algebra, P a finite-dimensional associative algebra and φ a symmetric linear function on P.
Let W be a grading-restricted generalized V -P-bimodule (a grading-restricted generalized V -modue which is also a right P-module such that the vertex operators and the action of P commutes), projective as a right P-module and L W (0) K +1 N W = 0 for some K ∈ Z + . In particular, the homogeneous subspaces of W are finitely generated projective P-modules.
The pseudo-q-trace (associated to φ and shifted by c 24 ) of A ∈ Hom P (W , W ) is defined to be where for n ∈ C, π n is the projection from W to W [n] .
Yi-Zhi Huang Modular invariance April 28, 2023 Pseudo-q-traces Let V be a vertex operator algebra, P a finite-dimensional associative algebra and φ a symmetric linear function on P.
Let W be a grading-restricted generalized V -P-bimodule (a grading-restricted generalized V -modue which is also a right P-module such that the vertex operators and the action of P commutes), projective as a right P-module and L W (0) K +1 N W = 0 for some K ∈ Z + . In particular, the homogeneous subspaces of W are finitely generated projective P-modules.
The pseudo-q-trace (associated to φ and shifted by c 24 ) of A ∈ Hom P (W , W ) is defined to be where for n ∈ C, π n is the projection from W to W [n] .
Yi-Zhi Huang Modular invariance April 28, 2023 Pseudo-q-traces Let V be a vertex operator algebra, P a finite-dimensional associative algebra and φ a symmetric linear function on P.
Let W be a grading-restricted generalized V -P-bimodule (a grading-restricted generalized V -modue which is also a right P-module such that the vertex operators and the action of P commutes), projective as a right P-module and L W (0) K +1 N W = 0 for some K ∈ Z + . In particular, the homogeneous subspaces of W are finitely generated projective P-modules.
The pseudo-q-trace (associated to φ and shifted by c 24 ) of A ∈ Hom P (W , W ) is defined to be where for n ∈ C, π n is the projection from W to W [n] .
Yi-Zhi Huang Modular invariance April 28, 2023 Pseudo-q-traces Let V be a vertex operator algebra, P a finite-dimensional associative algebra and φ a symmetric linear function on P.
Let W be a grading-restricted generalized V -P-bimodule (a grading-restricted generalized V -modue which is also a right P-module such that the vertex operators and the action of P commutes), projective as a right P-module and L W (0) K +1 N W = 0 for some K ∈ Z + . In particular, the homogeneous subspaces of W are finitely generated projective P-modules.
The pseudo-q-trace (associated to φ and shifted by c 24 ) of A ∈ Hom P (W , W ) is defined to be where for n ∈ C, π n is the projection from W to W [n] . Products of intertwining operators Let W 1 , . . . , W n , W 1 , . . . , W n−1 be grading-restricted generalized V -modules and let W be a grading-restricted generalized V -P-bimodule which is projective as a right P-module.
Let Y 1 , . . . , Y n be intertwining operators of types , . . . , W n−1 WnW , respectively. In the case that V is C 2 -cofinite, I had proved that for w 1 ∈ W 1 , . . . , w n ∈ W n , Y 1 (w 1 , z 1 ) · · · Y n (w n , z n ) ∈ Hom(W , W ) in the region |z 1 | > · · · > |z n > 0 and can be analytically extended to a multivalued analytic function valued in Hom(W , W ) on the region z i = 0 for i = 1, . . . , n, z i = z j for i = j.

Products of intertwining operators
Let W 1 , . . . , W n , W 1 , . . . , W n−1 be grading-restricted generalized V -modules and let W be a grading-restricted generalized V -P-bimodule which is projective as a right P-module.
Let Y 1 , . . . , Y n be intertwining operators of types WnW , respectively. In the case that V is C 2 -cofinite, I had proved that for w 1 ∈ W 1 , . . . , w n ∈ W n , Y 1 (w 1 , z 1 ) · · · Y n (w n , z n ) ∈ Hom(W , W ) in the region |z 1 | > · · · > |z n > 0 and can be analytically extended to a multivalued analytic function valued in Hom(W , W ) on the region z i = 0 for i = 1, . . . , n, z i = z j for i = j.

Products of intertwining operators
Let W 1 , . . . , W n , W 1 , . . . , W n−1 be grading-restricted generalized V -modules and let W be a grading-restricted generalized V -P-bimodule which is projective as a right P-module.
Let Y 1 , . . . , Y n be intertwining operators of types WnW , respectively. In the case that V is C 2 -cofinite, I had proved that for w 1 ∈ W 1 , . . . , w n ∈ W n , Y 1 (w 1 , z 1 ) · · · Y n (w n , z n ) ∈ Hom(W , W ) in the region |z 1 | > · · · > |z n > 0 and can be analytically extended to a multivalued analytic function valued in Hom(W , W ) on the region z i = 0 for i = 1, . . . , n, z i = z j for i = j.
Assume that Y 1 (w 1 , z 1 ) · · · Y n (w n , z n ) commutes with the action of P.

Products of intertwining operators
Let W 1 , . . . , W n , W 1 , . . . , W n−1 be grading-restricted generalized V -modules and let W be a grading-restricted generalized V -P-bimodule which is projective as a right P-module.
Let Y 1 , . . . , Y n be intertwining operators of types WnW , respectively. In the case that V is C 2 -cofinite, I had proved that for w 1 ∈ W 1 , . . . , w n ∈ W n , Y 1 (w 1 , z 1 ) · · · Y n (w n , z n ) ∈ Hom(W , W ) in the region |z 1 | > · · · > |z n > 0 and can be analytically extended to a multivalued analytic function valued in Hom(W , W ) on the region z i = 0 for i = 1, . . . , n, z i = z j for i = j.
Assume that Y 1 (w 1 , z 1 ) · · · Y n (w n , z n ) commutes with the action of P.

Products of intertwining operators
Let W 1 , . . . , W n , W 1 , . . . , W n−1 be grading-restricted generalized V -modules and let W be a grading-restricted generalized V -P-bimodule which is projective as a right P-module.
Let Y 1 , . . . , Y n be intertwining operators of types WnW , respectively. In the case that V is C 2 -cofinite, I had proved that for w 1 ∈ W 1 , . . . , w n ∈ W n , Y 1 (w 1 , z 1 ) · · · Y n (w n , z n ) ∈ Hom(W , W ) in the region |z 1 | > · · · > |z n > 0 and can be analytically extended to a multivalued analytic function valued in Hom(W , W ) on the region z i = 0 for i = 1, . . . , n, z i = z j for i = j.
Assume that Y 1 (w 1 , z 1 ) · · · Y n (w n , z n ) commutes with the action of P.
For w 1 ∈ W 1 , . . . , w n ∈ W n , let F w 1 ,...,wn be the space spanned by all such multivalued analytic functions.
Fiordalisi also proved in 2015 that these multivalued analytic functions satisfy the genus-one associativity and genus-one commutativity.
For w 1 ∈ W 1 , . . . , w n ∈ W n , let F w 1 ,...,wn be the space spanned by all such multivalued analytic functions.
Fiordalisi also proved in 2015 that these multivalued analytic functions satisfy the genus-one associativity and genus-one commutativity.
For w 1 ∈ W 1 , . . . , w n ∈ W n , let F w 1 ,...,wn be the space spanned by all such multivalued analytic functions.
Fiordalisi also proved in 2015 that these multivalued analytic functions satisfy the genus-one associativity and genus-one commutativity.
For w 1 ∈ W 1 , . . . , w n ∈ W n , let F w 1 ,...,wn be the space spanned by all such multivalued analytic functions.
Fiordalisi also proved in 2015 that these multivalued analytic functions satisfy the genus-one associativity and genus-one commutativity.
Here is the precise formulation of the theorem: Theorem (H., 2023) Let V be a C 2 -cofinite vertex operator algebra without nonzero elements of negative weights. Then the modular transformation of F φ Y 1 ,...,Yn (w 1 , . . . , w n ; z 1 , . . . , z n ; τ ) ∈ F w 1 ,...,wn under α β γ δ ∈ SL(2, Z) is in F w 1 ,...,wn . In particular, the space F w 1 ,...,wn is an SL(2, Z)-module. Reduce the proof to the case of one intertwining operator We have the genus-one associativity . . , w n ; z 1 , . . . , z k −1 , z k +1 , . . . , z n ; τ ) proved by Fiordalisi in his thesis. As in the proof of the modular invariance conjecture of Moore-Seiberg, using the genus-one associativity, the modular invariance of F w 1 ,...,wn for w 1 ∈ W 1 , . . . , w n ∈ W n is reduced to the modular invariance of F w for w ∈ W . This step was the main difficulty in the proof of the modular invariance conjecture of Moore-Seiberg. The same method works in the nonsemisimple case.
Reduce the proof to the case of one intertwining operator We have the genus-one associativity . . , w n ; z 1 , . . . , z k −1 , z k +1 , . . . , z n ; τ ) proved by Fiordalisi in his thesis. As in the proof of the modular invariance conjecture of Moore-Seiberg, using the genus-one associativity, the modular invariance of F w 1 ,...,wn for w 1 ∈ W 1 , . . . , w n ∈ W n is reduced to the modular invariance of F w for w ∈ W . This step was the main difficulty in the proof of the modular invariance conjecture of Moore-Seiberg. The same method works in the nonsemisimple case.
Reduce the proof to the case of one intertwining operator We have the genus-one associativity . . , w n ; z 1 , . . . , z k −1 , z k +1 , . . . , z n ; τ ) proved by Fiordalisi in his thesis. As in the proof of the modular invariance conjecture of Moore-Seiberg, using the genus-one associativity, the modular invariance of F w 1 ,...,wn for w 1 ∈ W 1 , . . . , w n ∈ W n is reduced to the modular invariance of F w for w ∈ W . This step was the main difficulty in the proof of the modular invariance conjecture of Moore-Seiberg. The same method works in the nonsemisimple case.
Reduce the proof to the case of one intertwining operator We have the genus-one associativity . . , w n ; z 1 , . . . , z k −1 , z k +1 , . . . , z n ; τ ) proved by Fiordalisi in his thesis. As in the proof of the modular invariance conjecture of Moore-Seiberg, using the genus-one associativity, the modular invariance of F w 1 ,...,wn for w 1 ∈ W 1 , . . . , w n ∈ W n is reduced to the modular invariance of F w for w ∈ W . This step was the main difficulty in the proof of the modular invariance conjecture of Moore-Seiberg. The same method works in the nonsemisimple case.
In the proof of the modular invariance conjecture of Moore and Seiberg, one needs to use the modular transformation of the pseudo-q τ -trace of an intertwining operator to obtain a symmetric linear function on a bimodule for Zhu's algebra A(V ).
In our nonsemisimple case, bimodules for Zhu's algebra does not work anymore. In Miyamoto's work, he used the generalizations A n (V ) of Zhu's algebra by Dong, Li and Mason. But this is in fact the reason why Miyamoto had to assume the additional condition that the vertex operator algebra does not have finite-dimensional irreducible modules. Also for modular invariance of (logarithmic) intertwining operators, A n (V )-bimodules are also not enough.
Later, I found that in order to prove the modular invariance of (logaritmic) intertwining operators, one needs to introduce much larger associative algebras. These are the associative algebras A ∞ (V ) and A N (V ) for N ∈ N introduced in 2020.

Associative algebras and modular invariance
In the proof of the modular invariance conjecture of Moore and Seiberg, one needs to use the modular transformation of the pseudo-q τ -trace of an intertwining operator to obtain a symmetric linear function on a bimodule for Zhu's algebra A(V ).
In our nonsemisimple case, bimodules for Zhu's algebra does not work anymore. In Miyamoto's work, he used the generalizations A n (V ) of Zhu's algebra by Dong, Li and Mason. But this is in fact the reason why Miyamoto had to assume the additional condition that the vertex operator algebra does not have finite-dimensional irreducible modules. Also for modular invariance of (logarithmic) intertwining operators, A n (V )-bimodules are also not enough.
Later, I found that in order to prove the modular invariance of (logaritmic) intertwining operators, one needs to introduce much larger associative algebras. These are the associative algebras A ∞ (V ) and A N (V ) for N ∈ N introduced in 2020.

Associative algebras and modular invariance
In the proof of the modular invariance conjecture of Moore and Seiberg, one needs to use the modular transformation of the pseudo-q τ -trace of an intertwining operator to obtain a symmetric linear function on a bimodule for Zhu's algebra A(V ).
In our nonsemisimple case, bimodules for Zhu's algebra does not work anymore. In Miyamoto's work, he used the generalizations A n (V ) of Zhu's algebra by Dong, Li and Mason. But this is in fact the reason why Miyamoto had to assume the additional condition that the vertex operator algebra does not have finite-dimensional irreducible modules. Also for modular invariance of (logarithmic) intertwining operators, A n (V )-bimodules are also not enough.
Later, I found that in order to prove the modular invariance of (logaritmic) intertwining operators, one needs to introduce much larger associative algebras. These are the associative algebras A ∞ (V ) and A N (V ) for N ∈ N introduced in 2020.

Associative algebras and modular invariance
In the proof of the modular invariance conjecture of Moore and Seiberg, one needs to use the modular transformation of the pseudo-q τ -trace of an intertwining operator to obtain a symmetric linear function on a bimodule for Zhu's algebra A(V ).
In our nonsemisimple case, bimodules for Zhu's algebra does not work anymore. In Miyamoto's work, he used the generalizations A n (V ) of Zhu's algebra by Dong, Li and Mason. But this is in fact the reason why Miyamoto had to assume the additional condition that the vertex operator algebra does not have finite-dimensional irreducible modules. Also for modular invariance of (logarithmic) intertwining operators, A n (V )-bimodules are also not enough.
Later, I found that in order to prove the modular invariance of (logaritmic) intertwining operators, one needs to introduce much larger associative algebras. These are the associative algebras A ∞ (V ) and A N (V ) for N ∈ N introduced in 2020.
Associative algebras A ∞ (V ) and A N (V ) There exists a product in is an associative algebra and ϑ W induces an A ∞ (V )-module structure on W .
For N ∈ N, let U N (V ) be the subspace of U ∞ (V ) consisting of matrices with (k , l)-entries being 0 for k , l ≥ N + 1. Let A N (V ) be the subset of A ∞ (V ) consisting of the cosets containing elements of U N (V ). Then A N (V ) is a subalgebra of A ∞ (V ). Let Then the restriction of ϑ W to A N (V ) gives an A N (V )-module structure on Ω 0 N (W ).

Associative algebras A ∞ (V ) and A N (V )
There exists a product in is an associative algebra and ϑ W induces an A ∞ (V )-module structure on W .
For N ∈ N, let U N (V ) be the subspace of U ∞ (V ) consisting of matrices with (k , l)-entries being 0 for k , l ≥ N + 1. Let A N (V ) be the subset of A ∞ (V ) consisting of the cosets containing elements of U N (V ). Then A N (V ) is a subalgebra of A ∞ (V ). Let Then the restriction of ϑ W to A N (V ) gives an A N (V )-module structure on Ω 0 N (W ).

Associative algebras A ∞ (V ) and A N (V )
There exists a product in is an associative algebra and ϑ W induces an A ∞ (V )-module structure on W .
For N ∈ N, let U N (V ) be the subspace of U ∞ (V ) consisting of matrices with (k , l)-entries being 0 for k , l ≥ N + 1. Let A N (V ) be the subset of A ∞ (V ) consisting of the cosets containing elements of U N (V ). Then A N (V ) is a subalgebra of A ∞ (V ). Let Then the restriction of ϑ W to A N (V ) gives an A N (V )-module structure on Ω 0 N (W ).

Associative algebras A ∞ (V ) and A N (V )
There exists a product in is an associative algebra and ϑ W induces an A ∞ (V )-module structure on W .
For N ∈ N, let U N (V ) be the subspace of U ∞ (V ) consisting of matrices with (k , l)-entries being 0 for k , l ≥ N + 1. Let A N (V ) be the subset of A ∞ (V ) consisting of the cosets containing elements of U N (V ). Then A N (V ) is a subalgebra of A ∞ (V ). Let Then the restriction of ϑ W to A N (V ) gives an A N (V )-module structure on Ω 0 N (W ).
Associative algebras A ∞ (V ) and A N (V ) There exists a product in is an associative algebra and ϑ W induces an A ∞ (V )-module structure on W .
For N ∈ N, let U N (V ) be the subspace of U ∞ (V ) consisting of matrices with (k , l)-entries being 0 for k , l ≥ N + 1. Let A N (V ) be the subset of A ∞ (V ) consisting of the cosets containing elements of U N (V ). Then A N (V ) is a subalgebra of A ∞ (V ). Let Then the restriction of ϑ W to A N (V ) gives an A N (V )-module structure on Ω 0 N (W ).
A ∞ (V )-bimodules, A N (V )-bimodules and intertwining operators For each lower-bounded generalized V -module W , we also construct an A ∞ (V )-bimodule A ∞ (W ) using the space of column-finite infinite matrices with entries in W . The space of intertwining operators of type W 3 W 1 W 2 is isomorphic to Given a graded A N (V )-module M, there is a lower-bounded generalized V -module S N (M) satisfying a universal property. In the case that N is sufficiently large and W 2 and W 3 are equivalent to S N (Ω 0 N (W 2 )) and S N (Ω 0 N (W 3 )), respectively, the space of intertwining operators of type W 3 W 1 W 2 is isomorphic to A ∞ (V )-bimodules, A N (V )-bimodules and intertwining operators For each lower-bounded generalized V -module W , we also construct an A ∞ (V )-bimodule A ∞ (W ) using the space of column-finite infinite matrices with entries in W . The space of intertwining operators of type W 3 W 1 W 2 is isomorphic to Given a graded A N (V )-module M, there is a lower-bounded generalized V -module S N (M) satisfying a universal property. In the case that N is sufficiently large and W 2 and W 3 are equivalent to S N (Ω 0 N (W 2 )) and S N (Ω 0 N (W 3 )), respectively, the space of intertwining operators of type W 3 W 1 W 2 is isomorphic to A ∞ (V )-bimodules, A N (V )-bimodules and intertwining operators For each lower-bounded generalized V -module W , we also construct an A ∞ (V )-bimodule A ∞ (W ) using the space of column-finite infinite matrices with entries in W . The space of intertwining operators of type W 3 W 1 W 2 is isomorphic to Given a graded A N (V )-module M, there is a lower-bounded generalized V -module S N (M) satisfying a universal property. In the case that N is sufficiently large and W 2 and W 3 are equivalent to S N (Ω 0 N (W 2 )) and S N (Ω 0 N (W 3 )), respectively, the space of intertwining operators of type W 3 W 1 W 2 is isomorphic to

1-point genus-one correlation functions and symmetric linear functions
The analytic extension F φ Y (w 1 ; z; τ ) of the pseudo-q τ -trace of an intertwining operator Y of type W 2 W 1 W 2 is in fact a 1-point genus-one correlation function. Then a modular transformation of F φ Y (w 1 ; z; τ ) is also a 1-point genus-one correlation function. Using the properties of 1-point genus-one correlation functions, we prove that for N ∈ N, a modular transformation of F φ Y (w 1 ; z; τ ) gives a symmetric linear function on A N (W 1 ). This step is the most difficult part of the proof of this conjecture.
By a theorem of Fiordalisi proved in his thesis using a theorem of Miyamoto and Arike on pseudo-traces and square-zero extension of associative algebras, a symmetric linear function on A N (W 1 ) is a sum of pseudo-traces of elements of

1-point genus-one correlation functions and symmetric linear functions
The analytic extension F φ Y (w 1 ; z; τ ) of the pseudo-q τ -trace of an intertwining operator Y of type W 2 W 1 W 2 is in fact a 1-point genus-one correlation function. Then a modular transformation of F φ Y (w 1 ; z; τ ) is also a 1-point genus-one correlation function. Using the properties of 1-point genus-one correlation functions, we prove that for N ∈ N, a modular transformation of F φ Y (w 1 ; z; τ ) gives a symmetric linear function on A N (W 1 ). This step is the most difficult part of the proof of this conjecture.
By a theorem of Fiordalisi proved in his thesis using a theorem of Miyamoto and Arike on pseudo-traces and square-zero extension of associative algebras, a symmetric linear function on A N (W 1 ) is a sum of pseudo-traces of elements of

1-point genus-one correlation functions and symmetric linear functions
The analytic extension F φ Y (w 1 ; z; τ ) of the pseudo-q τ -trace of an intertwining operator Y of type W 2 W 1 W 2 is in fact a 1-point genus-one correlation function. Then a modular transformation of F φ Y (w 1 ; z; τ ) is also a 1-point genus-one correlation function. Using the properties of 1-point genus-one correlation functions, we prove that for N ∈ N, a modular transformation of F φ Y (w 1 ; z; τ ) gives a symmetric linear function on A N (W 1 ). This step is the most difficult part of the proof of this conjecture.
By a theorem of Fiordalisi proved in his thesis using a theorem of Miyamoto and Arike on pseudo-traces and square-zero extension of associative algebras, a symmetric linear function on A N (W 1 ) is a sum of pseudo-traces of elements of

1-point genus-one correlation functions and symmetric linear functions
The analytic extension F φ Y (w 1 ; z; τ ) of the pseudo-q τ -trace of an intertwining operator Y of type W 2 W 1 W 2 is in fact a 1-point genus-one correlation function. Then a modular transformation of F φ Y (w 1 ; z; τ ) is also a 1-point genus-one correlation function. Using the properties of 1-point genus-one correlation functions, we prove that for N ∈ N, a modular transformation of F φ Y (w 1 ; z; τ ) gives a symmetric linear function on A N (W 1 ). This step is the most difficult part of the proof of this conjecture.
By a theorem of Fiordalisi proved in his thesis using a theorem of Miyamoto and Arike on pseudo-traces and square-zero extension of associative algebras, a symmetric linear function on A N (W 1 ) is a sum of pseudo-traces of elements of The last step of the proof For N sufficiently large, the C 2 -cofiniteness of V implies that W 2 and W 2 are equivalent to S N (Ω 0 N (W 2 )) and S N (Ω 0 N (W 2 )), respectively. Then In particular, we obtain intertwining operators of types W 2 W 1 W 2 . We take the difference of the modular transformation of F φ Y (w 1 ; z; τ ) and the sum of the analytic extensions of pseudo-q τ -traces of these intertwining operators. This is still a genus-one correlation function.
Repeat the step above. Finally we can find finitely many intertwining operators such that the sum of the analytic extensions of pseudo-q τ -traces of these intertwining operators is equal to the modular transformation of F φ Y (w 1 ; z; τ ), proving the modular invariance theorem. The last step of the proof For N sufficiently large, the C 2 -cofiniteness of V implies that W 2 and W 2 are equivalent to S N (Ω 0 N (W 2 )) and S N (Ω 0 N (W 2 )), respectively. Then In particular, we obtain intertwining operators of types W 2 W 1 W 2 . We take the difference of the modular transformation of F φ Y (w 1 ; z; τ ) and the sum of the analytic extensions of pseudo-q τ -traces of these intertwining operators. This is still a genus-one correlation function.
Repeat the step above. Finally we can find finitely many intertwining operators such that the sum of the analytic extensions of pseudo-q τ -traces of these intertwining operators is equal to the modular transformation of F φ Y (w 1 ; z; τ ), proving the modular invariance theorem. The last step of the proof For N sufficiently large, the C 2 -cofiniteness of V implies that W 2 and W 2 are equivalent to S N (Ω 0 N (W 2 )) and S N (Ω 0 N (W 2 )), respectively. Then In particular, we obtain intertwining operators of types W 2 W 1 W 2 . We take the difference of the modular transformation of F φ Y (w 1 ; z; τ ) and the sum of the analytic extensions of pseudo-q τ -traces of these intertwining operators. This is still a genus-one correlation function.
Repeat the step above. Finally we can find finitely many intertwining operators such that the sum of the analytic extensions of pseudo-q τ -traces of these intertwining operators is equal to the modular transformation of F φ Y (w 1 ; z; τ ), proving the modular invariance theorem. The last step of the proof For N sufficiently large, the C 2 -cofiniteness of V implies that W 2 and W 2 are equivalent to S N (Ω 0 N (W 2 )) and S N (Ω 0 N (W 2 )), respectively.
Then Hom A N (V ) (A N (W 1 ) ⊗ A N (V ) W 2 , W 2 ) is isomorphic to the space of intertwining operators of type W 2 W 1 W 2 . In particular, we obtain intertwining operators of types W 2 W 1 W 2 . We take the difference of the modular transformation of F φ Y (w 1 ; z; τ ) and the sum of the analytic extensions of pseudo-q τ -traces of these intertwining operators. This is still a genus-one correlation function.
Repeat the step above. Finally we can find finitely many intertwining operators such that the sum of the analytic extensions of pseudo-q τ -traces of these intertwining operators is equal to the modular transformation of F φ Y (w 1 ; z; τ ), proving the modular invariance theorem. The last step of the proof For N sufficiently large, the C 2 -cofiniteness of V implies that W 2 and W 2 are equivalent to S N (Ω 0 N (W 2 )) and S N (Ω 0 N (W 2 )), respectively.
Then Hom A N (V ) (A N (W 1 ) ⊗ A N (V ) W 2 , W 2 ) is isomorphic to the space of intertwining operators of type W 2 W 1 W 2 . In particular, we obtain intertwining operators of types W 2 W 1 W 2 . We take the difference of the modular transformation of F φ Y (w 1 ; z; τ ) and the sum of the analytic extensions of pseudo-q τ -traces of these intertwining operators. This is still a genus-one correlation function.
Repeat the step above. Finally we can find finitely many intertwining operators such that the sum of the analytic extensions of pseudo-q τ -traces of these intertwining operators is equal to the modular transformation of F φ Y (w 1 ; z; τ ), proving the modular invariance theorem.

A main possible application
The proof of the modular invariance conjecture of Moore and Seiberg led to the proof of the Verlinde formula and the proof of the rigidity and modularity of the braided tensor category of V -modules when V satisfies in addition a condition on the weight-one subspace of V , the existence of a nondegenerate bilinear invariant form on V and a complete reducibility condition.
In the case that complete reducibility condition is not satisfied, I also conjectured many years ago that the braided tensor category of grading-restricted generalized V -modules is rigidity. Later, it was also conjectured bu other people that this category should be modular in a suitable sense.
It is natural to expect that the modular invariance theorem we have discussed above be useful in the proof of these conjectures. A main possible application The proof of the modular invariance conjecture of Moore and Seiberg led to the proof of the Verlinde formula and the proof of the rigidity and modularity of the braided tensor category of V -modules when V satisfies in addition a condition on the weight-one subspace of V , the existence of a nondegenerate bilinear invariant form on V and a complete reducibility condition.
In the case that complete reducibility condition is not satisfied, I also conjectured many years ago that the braided tensor category of grading-restricted generalized V -modules is rigidity. Later, it was also conjectured bu other people that this category should be modular in a suitable sense.
It is natural to expect that the modular invariance theorem we have discussed above be useful in the proof of these conjectures.
Yi-Zhi Huang Modular invariance April 28, 2023 20 / 21 A main possible application The proof of the modular invariance conjecture of Moore and Seiberg led to the proof of the Verlinde formula and the proof of the rigidity and modularity of the braided tensor category of V -modules when V satisfies in addition a condition on the weight-one subspace of V , the existence of a nondegenerate bilinear invariant form on V and a complete reducibility condition.
In the case that complete reducibility condition is not satisfied, I also conjectured many years ago that the braided tensor category of grading-restricted generalized V -modules is rigidity. Later, it was also conjectured bu other people that this category should be modular in a suitable sense.
It is natural to expect that the modular invariance theorem we have discussed above be useful in the proof of these conjectures.
Yi-Zhi Huang Modular invariance April 28, 2023 20 / 21 A main possible application The proof of the modular invariance conjecture of Moore and Seiberg led to the proof of the Verlinde formula and the proof of the rigidity and modularity of the braided tensor category of V -modules when V satisfies in addition a condition on the weight-one subspace of V , the existence of a nondegenerate bilinear invariant form on V and a complete reducibility condition.
In the case that complete reducibility condition is not satisfied, I also conjectured many years ago that the braided tensor category of grading-restricted generalized V -modules is rigidity. Later, it was also conjectured bu other people that this category should be modular in a suitable sense.
It is natural to expect that the modular invariance theorem we have discussed above be useful in the proof of these conjectures.