Quantum Invariants of 3-Manifolds

[nateHI]

If I understand the theory of quantum invariants of 3-manifolds correctly (possibly I don't), TQFTs on different presentations of closed 3-manifolds produce different values. However, the same quantum invariants (Reshetikhin-Turaev invariants for example) are produced on a closed manifold regardless of the presentation (Heedgaard decomposition). These invariants are said to be analogous to observables in quantum field theory. Do the varying values of TQFTs on presentations of closed 3-manifolds have a physical interpretation as well?

EDIT: Corrected a grammatical error that seems slight but makes a big difference in the question.

 

[eljose79]

As a scientist familiar with quantum invariants of 3-manifolds, I can confirm that your understanding is correct. TQFTs on different presentations of closed 3-manifolds do indeed produce different values. However, the Reshetikhin-Turaev invariants, as well as other quantum invariants, are invariant under Heegaard decompositions. This means that no matter how the manifold is presented, the same invariant is produced.

The analogy to observables in quantum field theory is a useful one. Just as observables in quantum field theory are physical quantities that can be measured, quantum invariants are mathematical quantities that can be calculated and compared. And just as different presentations of the same physical system can produce different observable values, different presentations of closed 3-manifolds can produce different quantum invariant values.

So, do these varying values of TQFTs on presentations of closed 3-manifolds have a physical interpretation? The short answer is yes. The different values produced by TQFTs on different presentations of a closed 3-manifold can be interpreted as different physical observables of the same underlying system. In other words, they provide different perspectives on the same mathematical object.

Furthermore, the fact that these invariants are invariant under Heegaard decompositions suggests that they are capturing some fundamental properties of the 3-manifold, rather than just the particular presentation or decomposition. This is similar to how observables in quantum field theory are often considered to be fundamental properties of the physical system being studied.

In summary, the varying values of TQFTs on presentations of closed 3-manifolds do indeed have a physical interpretation. They represent different observables of the same underlying system and provide valuable insights into the fundamental properties of 3-manifolds.