[Paper Review] Higher Rank TQFT Representations of SL(2,Z) are Reducible

Qi Chen, Thomas Kerler|ArXiv.org|Jun 26, 2007

Advanced Algebra and Geometry10 references3 citations

TL;DR

This paper demonstrates that higher-rank TQFT representations of SL(2,ℤ) derived from quantum PSU(N) for N > 2 are generically reducible, with explicit decomposition via a ℤ/2 symmetry exchanging representation labels with their conjugates. For quantum PSU(3) at prime order r ≡ 2 mod 3, it identifies an irreducible summand isomorphic to the PSU(2) representation, proving reducibility beyond the initial parity splitting.

ABSTRACT

In this article we give examples which show that the TQFT representations of the mapping class groups derived from quantum SU(N) for N>2 are generically decomposable. One general decomposition of the representations is induced by the symmetry which exchanges SU(N) representation labels by their conjugates. The respective summands of a given parity are typically still reducible into many further components. Specifically, we give an explicit basis for an irreducible direct summand in the SL(2,Z) representation obtained from quantum PSU(3) when the order of the root of unity is a prime r=2 mod 3. We show that this summand is isomorphic to the respective PSU(2) representation.

Motivation & Objective

• To investigate the decomposition structure of TQFT representations of mapping class groups when moving from sl₂ to higher-rank Lie algebras.
• To determine whether TQFT representations from quantum PSU(N) for N > 2 remain irreducible, as they do in the sl₂ case.
• To identify explicit summands in the SL(2,ℤ) representation from quantum PSU(3) when r ≡ 2 mod 3.
• To show that the irreducible summands from PSU(2) appear as subrepresentations within the PSU(3) TQFT at specific root-of-unity orders.
• To establish that the initial ℤ/2-grading is only the first step in a deeper reducibility structure.

Proposed method

• Utilizes the ℤ/2-action induced by conjugating representation labels in the Weyl alcove of quantum groups.
• Applies the TQFT construction from quantum groups U_ζ(𝔰𝔩_N)′, restricting to PSU(N) representations via the Y-subcategory.
• Employs skein-theoretic techniques and modular S-matrices to compare representations of SL(2,ℤ) from PSU(3) and PSU(2).
• Uses character sums and Gauss sums over finite fields to prove proportionality of S-matrices between PSU(3) and PSU(2) representations.
• Analyzes the action of generators S and T of SL(2,ℤ) on vector spaces labeled by weights in the Weyl alcove.
• Applies number-theoretic identities involving Legendre symbols and quadratic Gauss sums to verify orthogonality and proportionality of matrix entries.

Experimental results

Research questions

• RQ1 Are TQFT representations of SL(2,ℤ) derived from quantum PSU(N) for N > 2 irreducible, as in the sl₂ case?
• RQ2 Does the ℤ/2-grading induced by conjugate symmetry on representation labels lead to a non-trivial decomposition of the SL(2,ℤ) representation?
• RQ3 Can an irreducible PSU(2) representation be embedded as a subrepresentation within the PSU(3) TQFT at prime order r ≡ 2 mod 3?
• RQ4 What is the precise structure of the decomposition of the SL(2,ℤ) representation in the quantum PSU(3) TQFT when r ≡ 2 mod 3?
• RQ5 To what extent does the reducibility observed at genus one extend to higher genus mapping class group representations?

Key findings

• For N > 2 and ζ of order k > 2N with gcd(k, N) = 1, the quantum PSU(N) TQFT is ℤ/2-graded via conjugation symmetry on representation labels.
• The SL(2,ℤ) representation from quantum PSU(3) at prime order r ≡ 2 mod 3 contains an irreducible summand isomorphic to the quantum PSU(2) representation.
• The S-matrices of the PSU(3) and PSU(2) representations are proportional, confirming that the PSU(2) representation is a subrepresentation.
• The summand corresponding to the PSU(2) representation is irreducible when r is an odd prime, as established by the same argument used in the sl₂ case.
• The decomposition is not exhausted by the ℤ/2-grading; further reducibility occurs within each graded component.
• For r ≡ 1 mod 3, such as r = 7, the PSU(3) representation decomposes into summands of dimensions 1 and 4, while the PSU(2) representation (dimension 3) is not a subrepresentation, showing the condition r ≡ 2 mod 3 is necessary.