[Paper Review] Irreducible factors of Weil representations and TQFT

Julien Korinman|arXiv (Cornell University)|Oct 1, 2013

Advanced Algebra and Geometry22 references3 citations

TL;DR

This paper provides a complete decomposition of Weil representations of $Sp_{2g}(ℤ)$ at even levels into irreducible factors, generalizing prior results for odd levels. It establishes that quantum representations in $SU(2)$ and $SO(3)$ TQFTs at level $p$ decompose into irreducible modules indexed by prime powers, and shows that when $p$ is not divisible by 4, the universal TQFT construction without framed links yields the same theory as the standard one.

ABSTRACT

We give the decomposition into irreducible factors of Weil representations of the symplectic groups at even levels, generalizing previous decompositions at odd levels. We then derive the decomposition of the quantum representations of SL2(Z) arising in the SU(2) and SO(3) TQFTs. As application we show that, when the level indexing the TQFT is not a multiple of 4, the universal construction applied to a cobordism category without framed links leads to the same TQFT.

Motivation & Objective

• To extend the decomposition of Weil representations from odd to even levels.
• To derive the structure of quantum representations in $SU(2)$ and $SO(3)$ TQFTs via Weil representation decomposition.
• To prove that the universal TQFT construction without framed links recovers the standard TQFT when the level $p$ is not divisible by 4.
• To establish a precise isomorphism between the odd submodule of the Weil representation and the Witten-Reshetikhin-Turaev representations.

Proposed method

• Uses a topological and algebraic approach based on the universal construction of TQFTs, focusing on cobordism categories without framed links.
• Applies ring isomorphisms $\mu: \mathbf{k}_{ab} \to \mathbf{k}_a \otimes \mathbf{k}_b$ to decompose tensor products of Weil modules over cyclotomic rings.
• Employs recursive decomposition rules for $U_{r^{n+2}}^\otimes g$ into $U_{r^n}^\otimes g \oplus W_{r^{n+2}}^\otimes g$ and further splits into even and odd submodules.
• Uses explicit basis constructions and inner product computations in $U_p^\otimes g$ and $W_{r^n}^\otimes g$ to analyze irreducibility and invariance under $SL_2(\mathbb{Z})$.
• Relies on the identification $V_p \cong U_p^-$ to translate TQFT vector space structures into Weil representation components.
• Applies induction on the number of framed link components in 3-manifolds to show that any TQFT vector can be expressed without framed links when $4 \nmid p$.

Experimental results

Research questions

• RQ1 How do Weil representations of $Sp_{2g}(\mathbb{Z})$ decompose at even levels, generalizing known results at odd levels?
• RQ2 What is the structure of the quantum representations of $SL_2(\mathbb{Z})$ in $SU(2)$ and $SO(3)$ TQFTs at even levels?
• RQ3 When does the universal TQFT construction without framed links yield the same theory as the standard construction?
• RQ4 How does the odd submodule $U_p^-$ of the Weil representation relate to the Witten-Reshetikhin-Turaev representations?
• RQ5 Under what conditions on the level $p$ is the $SL_2(\mathbb{Z})$-invariant subspace $X_p$ closed under the action of the modular group?

Key findings

• The Weil representation at level $p$ decomposes into $\sigma(p)$ irreducible factors when $p$ is odd, and $\sigma(p/2)$ when $p$ is even.
• For coprime $a,b \geq 2$, there is an isomorphism $U_a^\otimes g \otimes U_b^\otimes g \cong U_{ab}^\otimes g$ as $Sp_{2g}(\mathbb{Z})$-modules.
• For prime powers, $U_{r^{n+2}}^\otimes g \cong U_{r^n}^\otimes g \oplus W_{r^{n+2}}^\otimes g$, with $W_{r^{n+2}}$ a free submodule.
• The irreducible factors are of the form $B_{r_1} \otimes \cdots \otimes B_{r_k}$ with $B_{r_i} \in \{U_{r_i}^{g,\pm}, W_{r_i^n}^{g,\pm}\}$, all pairwise non-isomorphic and irreducible.
• The $SU(2)$ and $SO(3)$ quantum representations at level $p$ decompose as direct sums of such irreducible modules, with the odd submodule $U_p^-$ isomorphic to the Witten-Reshetikhin-Turaev representation $V_p$.
• When $4 \nmid p$, the universal TQFT construction without framed links produces the same invariants as the standard construction, due to the equality $\Lambda_0 = \Lambda_1$ in the $SL_2(\mathbb{Z})$-module $V_p$.