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edited Apr 13, 2017 at 12:58

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I have an incomplete understanding of this, but I will try to say what I know.

For each $c \in \mathbb{C}$, we define $Verma_c$ to be the category whose objects are Verma modules $V_{h,c}$ of central charge $c$ and highest weight $h$ for some $h \in \mathbb{C}$, and whose maps are Virasoro-module maps. Feigin and Fuks worked out the morphisms in all of these categories (they are embeddings), and they found that the categories $Verma_c$ and $Verma_{26-c}$ are antiequivalent. In particular, when $V_{h,c}$ has an embedded Verma submodule $V_{h+x,c}$, the corresponding module $V_{-1-h,26-c}$ embeds into $V_{-1-h-x,26-c}$.

When you consider representations of Virasoro with central charge $0 < c < 1$ and positive $h$, the corresponding objects will have central charge $25 < 26-c < 26$, and highest weight that is negative. In particular, unitary representations don't seem to make an appearance anywhere in the complementary picture.

According to this MathOverflow question this MathOverflow question, Positselski has developed the antiequivalence into a derived category statement, where modules are resolved by complexes of Vermas. However, I have been unable to extract an explicit theorem about Virasoro from his monograph.

I have an incomplete understanding of this, but I will try to say what I know.

For each $c \in \mathbb{C}$, we define $Verma_c$ to be the category whose objects are Verma modules $V_{h,c}$ of central charge $c$ and highest weight $h$ for some $h \in \mathbb{C}$, and whose maps are Virasoro-module maps. Feigin and Fuks worked out the morphisms in all of these categories (they are embeddings), and they found that the categories $Verma_c$ and $Verma_{26-c}$ are antiequivalent. In particular, when $V_{h,c}$ has an embedded Verma submodule $V_{h+x,c}$, the corresponding module $V_{-1-h,26-c}$ embeds into $V_{-1-h-x,26-c}$.

When you consider representations of Virasoro with central charge $0 < c < 1$ and positive $h$, the corresponding objects will have central charge $25 < 26-c < 26$, and highest weight that is negative. In particular, unitary representations don't seem to make an appearance anywhere in the complementary picture.

According to this MathOverflow question, Positselski has developed the antiequivalence into a derived category statement, where modules are resolved by complexes of Vermas. However, I have been unable to extract an explicit theorem about Virasoro from his monograph.

I have an incomplete understanding of this, but I will try to say what I know.

For each $c \in \mathbb{C}$, we define $Verma_c$ to be the category whose objects are Verma modules $V_{h,c}$ of central charge $c$ and highest weight $h$ for some $h \in \mathbb{C}$, and whose maps are Virasoro-module maps. Feigin and Fuks worked out the morphisms in all of these categories (they are embeddings), and they found that the categories $Verma_c$ and $Verma_{26-c}$ are antiequivalent. In particular, when $V_{h,c}$ has an embedded Verma submodule $V_{h+x,c}$, the corresponding module $V_{-1-h,26-c}$ embeds into $V_{-1-h-x,26-c}$.

When you consider representations of Virasoro with central charge $0 < c < 1$ and positive $h$, the corresponding objects will have central charge $25 < 26-c < 26$, and highest weight that is negative. In particular, unitary representations don't seem to make an appearance anywhere in the complementary picture.

According to this MathOverflow question, Positselski has developed the antiequivalence into a derived category statement, where modules are resolved by complexes of Vermas. However, I have been unable to extract an explicit theorem about Virasoro from his monograph.

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created May 14, 2013 at 3:13

S. Carnahan ♦

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I have an incomplete understanding of this, but I will try to say what I know.

For each $c \in \mathbb{C}$, we define $Verma_c$ to be the category whose objects are Verma modules $V_{h,c}$ of central charge $c$ and highest weight $h$ for some $h \in \mathbb{C}$, and whose maps are Virasoro-module maps. Feigin and Fuks worked out the morphisms in all of these categories (they are embeddings), and they found that the categories $Verma_c$ and $Verma_{26-c}$ are antiequivalent. In particular, when $V_{h,c}$ has an embedded Verma submodule $V_{h+x,c}$, the corresponding module $V_{-1-h,26-c}$ embeds into $V_{-1-h-x,26-c}$.

When you consider representations of Virasoro with central charge $0 < c < 1$ and positive $h$, the corresponding objects will have central charge $25 < 26-c < 26$, and highest weight that is negative. In particular, unitary representations don't seem to make an appearance anywhere in the complementary picture.

According to this MathOverflow question, Positselski has developed the antiequivalence into a derived category statement, where modules are resolved by complexes of Vermas. However, I have been unable to extract an explicit theorem about Virasoro from his monograph.