A QFT for non-semisimple TQFT
We construct a family of 3d quantum field theories $\mathcal T_{n,k}^A$ that conjecturally provide a physical realization -- and derived generalization -- of non-semisimple mathematical TQFT's based on the modules for the quantum group $U_q(\mathfrak{sl}_n)$ at an even root of unity $q=\text{ex...
| Main Authors: | , , , |
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| Format: | Journal article |
| Language: | English |
| Published: |
International Press
2024
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| _version_ | 1826316095884099584 |
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| author | Creutzig, T Dimofte, T Garner, N Geer, N |
| author_facet | Creutzig, T Dimofte, T Garner, N Geer, N |
| author_sort | Creutzig, T |
| collection | OXFORD |
| description | We construct a family of 3d quantum field theories $\mathcal T_{n,k}^A$ that
conjecturally provide a physical realization -- and derived generalization --
of non-semisimple mathematical TQFT's based on the modules for the quantum
group $U_q(\mathfrak{sl}_n)$ at an even root of unity $q=\text{exp}(i\pi/k)$.
The theories $\mathcal T_{n,k}^A$ are defined as topological twists of certain
3d $\mathcal N=4$ Chern-Simons-matter theories, which also admit
string/M-theory realizations. They may be thought of as $SU(n)_{k-n}$
Chern-Simons theories, coupled to a twisted $\mathcal N=4$ matter sector (the
source of non-semisimplicity). We show that $\mathcal T_{n,k}^A$ admits
holomorphic boundary conditions supporting two different logarithmic vertex
operator algebras, one of which is an $\mathfrak{sl}_n$-type Feigin-Tipunin
algebra; and we conjecture that these two vertex operator algebras are related
by a novel logarithmic level-rank duality. (We perform detailed computations to
support the conjecture.) We thus relate the category of line operators in
$\mathcal T_{n,k}^A$ to the derived category of modules for a boundary
Feigin-Tipunin algebra, and -- using a logarithmic Kazhdan-Lusztig-like
correspondence that has been established for $n=2$ and expected for general $n$
-- to the derived category of $U_q(\mathfrak{sl}_n)$ modules. We analyze many
other key features of $\mathcal T_{n,k}^A$ and match them from quantum-group
and VOA perspectives, including deformations by flat $PSL(n,\mathbb C)$
connections, one-form symmetries, and indices of (derived) genus-$g$ state
spaces. |
| first_indexed | 2024-12-09T03:39:43Z |
| format | Journal article |
| id | oxford-uuid:b0a583e5-bf6c-48b0-8028-0b2a38dbc184 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-12-09T03:39:43Z |
| publishDate | 2024 |
| publisher | International Press |
| record_format | dspace |
| spelling | oxford-uuid:b0a583e5-bf6c-48b0-8028-0b2a38dbc1842024-12-06T17:42:34ZA QFT for non-semisimple TQFTJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:b0a583e5-bf6c-48b0-8028-0b2a38dbc184EnglishSymplectic ElementsInternational Press2024Creutzig, TDimofte, TGarner, NGeer, NWe construct a family of 3d quantum field theories $\mathcal T_{n,k}^A$ that conjecturally provide a physical realization -- and derived generalization -- of non-semisimple mathematical TQFT's based on the modules for the quantum group $U_q(\mathfrak{sl}_n)$ at an even root of unity $q=\text{exp}(i\pi/k)$. The theories $\mathcal T_{n,k}^A$ are defined as topological twists of certain 3d $\mathcal N=4$ Chern-Simons-matter theories, which also admit string/M-theory realizations. They may be thought of as $SU(n)_{k-n}$ Chern-Simons theories, coupled to a twisted $\mathcal N=4$ matter sector (the source of non-semisimplicity). We show that $\mathcal T_{n,k}^A$ admits holomorphic boundary conditions supporting two different logarithmic vertex operator algebras, one of which is an $\mathfrak{sl}_n$-type Feigin-Tipunin algebra; and we conjecture that these two vertex operator algebras are related by a novel logarithmic level-rank duality. (We perform detailed computations to support the conjecture.) We thus relate the category of line operators in $\mathcal T_{n,k}^A$ to the derived category of modules for a boundary Feigin-Tipunin algebra, and -- using a logarithmic Kazhdan-Lusztig-like correspondence that has been established for $n=2$ and expected for general $n$ -- to the derived category of $U_q(\mathfrak{sl}_n)$ modules. We analyze many other key features of $\mathcal T_{n,k}^A$ and match them from quantum-group and VOA perspectives, including deformations by flat $PSL(n,\mathbb C)$ connections, one-form symmetries, and indices of (derived) genus-$g$ state spaces. |
| spellingShingle | Creutzig, T Dimofte, T Garner, N Geer, N A QFT for non-semisimple TQFT |
| title | A QFT for non-semisimple TQFT |
| title_full | A QFT for non-semisimple TQFT |
| title_fullStr | A QFT for non-semisimple TQFT |
| title_full_unstemmed | A QFT for non-semisimple TQFT |
| title_short | A QFT for non-semisimple TQFT |
| title_sort | qft for non semisimple tqft |
| work_keys_str_mv | AT creutzigt aqftfornonsemisimpletqft AT dimoftet aqftfornonsemisimpletqft AT garnern aqftfornonsemisimpletqft AT geern aqftfornonsemisimpletqft AT creutzigt qftfornonsemisimpletqft AT dimoftet qftfornonsemisimpletqft AT garnern qftfornonsemisimpletqft AT geern qftfornonsemisimpletqft |