WE ARE NOT STUCK WITH GLUING a response to a note of A. Ocneanu
이들은 표준 15J 기호를 정의하고, 그것을 이용하여 네트워크 상에 할당 된 스피너를 기반으로 토폴로지 인바리에이션 함수를 정의합니다.
그들의 논문에서 Crane과 Yetter는 A. Ocneanu가 제안한 구성법이 실제로는 15J 기호와 관련된 4차원 토폴로지 양자장론에 대해 1을 항상 얻는다는 것을 주장합니다.
그러나 Crane과 Yetter는 Ocneanu의 결과를 부정하고, 실제로는 15J 기호와 3차원 토폴로지 양자장론을 사용하여 생성되는 4차원 토폴로지 인바리에이션 함수가 다르다고 주장합니다.
그들은 그들의 계산에서 15J 기호와 관련된 토폴로지 인바리에이션 함수는 네트워크의 크기나 형태에 따라 달라질 수 있으며, Ocneanu의 결과가 틀렸다는 것을 보입니다.
마지막으로, Crane과 Yetter는 그들의 구성법이 4차원 토폴로지 양자장론에 더 일반적인 방법을 제공할 수 있고, 그들이 생성한 인바리에이션 함수를 사용하여 homeomorphic 한 4 차원 다양체를 구별하는가 하는 문제가 여전히 불명의 상태라고 말합니다.
영어 요약 시작:
L. Crane and D. Yetter's paper from 1993 describes a construction method for generating 4-dimensional topological quantum field theories (TQFT) using modular tensor categories (MTC).
They define the standard 15J symbols and use them to define a topology invariant function based on spin assignments to networks.
Crane and Yetter argue that A. Ocneanu's proposed construction method actually yields 1 for any closed 4-manifold, which they dispute.
The authors claim that their calculations show that the invariant of S^4 is not constant and depends on the network size and topology, contradicting Ocneanu's results.
They also argue that their construction method provides a more general approach to generating invariants for compact 4-manifolds and that it may be possible to use their invariants to distinguish between homeomorphic 4-manifolds.
Crane and Yetter conclude that Ocneanu's assumption is not supported by the facts and that their own construction method remains an open problem.
WE ARE NOT STUCK WITH GLUING a response to a note of A. Ocneanu
arXiv:hep-th/9302118v1 24 Feb 1993WE ARE NOT STUCK WITH GLUING a response to a note of A. Ocneanuby D. Yetter and L. CraneIn [1], we outlined a procedure for constructing a 4D topological QuantumField Theory(TQFT) from a modular tensor category (MTC).The construction is related to the well known construction of a 3d tqft . Inour announcement we gave the formula for the invariant as follows:XN #vertices−#edges Yfacesdimq(j)Ytetrahedradim−1q (p)Y4−simplexes15Jq(∗)where the sum ranges over all assignments of spins to the faces and tetrahedraof the triangulation and j represents the spin labelling a face, p represents thespin labelling the cut interior to a tetrahedron, dimq is the quantum dimension,and N is the sum of the squares of the quantum dimensions.
Here by spins, wemean irreducible representations of quantized sl2 at a root of unity.We have two different ways of thinking of our quantum 15J symbols. One,which really plays a heuristic role for us, is as an invariant of a labelled surfaceembedded in S3.
The other, which we use directly in our proof, is as a recombi-nation diagram in a braided tensor category. Perhaps we have been a little toocavalier in using the first picture, since the connection between the two involvessome subtleties.In [2], A. Ocneanu announced the result that the invariant we define is always1, and asserted that our procedure is equivalent to one he examined earlier, ina different context.Although we think that professor Ocneanu’s argument is interesting, and infact that the construction he suggests is of interest even if it does give 1 for anyclosed 4 manifold, we do not believe that the two constructions are the same.In particular, we know by direct calculation that our invariant is not constant,nor is it 1 for all simple cases.The point of departure in [2] is the assertion that the formula above is thesame as gluing of the 3 manifolds with boundary which are related to the 15J-qsymbols defined in [1].
We do not see how this could be the case. Note that inour formula the internal and external spins do not enter in the same way.
Gluingwould be regarding the 15J symbol as coming from a manifold with boundary,in which the external and internal spins play identical roles. Thus our formuladoes not appear to have the proper symmetry to express gluing.There seems to be no way to make professor Ocneanu’s results agree withour calculations.
In the first place, he does not get a result which is independantof the triangulation of the 4 manifold. The formula he computed reduces theinvariant of a 4 manifold to one for a connected sum of copies of S3 × S1 [in a3D TQFT], where the number of copies depends on the triangulation chosen.1
If we are to take it that our formula, as normalized, is equivalent to gluing,then we are being told that a topological invariant is equal to a non-invariant.If we are to believe that our formula without the normalization is equal togluing, we run into the immediate problem that when we join 3 15J symbolsaround a common face then we do not get the result corresponding to gluingtopologically, but rather an extra factor corresponding to a loop which is splitoff, which corresponds to our factors of N.If our process were in fact gluing, then the same arguments which allow usto join together parts of the boundary surfaces corresponding to disjoint 15J’swould also allow us to join separate segments of a connected boundary surfaceto itself. In fact, the combination rules in the category which allow us to joindisjoint categorical diagrams do not extend to that case.
Indeed, before realizingthis, we briefly thought that our invariant would be quite simple [althoughcertainly not constant].Furthermore, if we take it that the invariant of a connected sum of S1 ×S1’sis always 1, we would be led to the conclusion that our invariant was always 1. ( Probably any constant could be absorbed as a normalization).
This, however,contradicts the calculations we have been able to do by hand.Our calculations show that the invariant of S4 is N (= 2 for r=3, =4 for r=4). Calculating the invariant of S3 × S1 is more complicated , but yields 1 forr=3,4, not agreeing with the number for S4.
The case of S2 × S2 is much morecomplicated. Our initial calculuations, which we have not thoroughly checkedyield an expression involving the braiding.We are left with the problem of how often we obtain a power of N as invariant,and whether some modification of Ocneanu’s argument could tell us that.The interest in the theory we construct does not reduce to the invariantsof compact 4 manifolds.
In fact for possible applications to quantum gravity,compact 4 manifolds are irrelevant, since compact spacetimes are not causal.It is also possible that the relative form of our construction for manifolds withboundary could give invariants of embedded surfaces which are richer than theinvariants of closed 4 manifolds. For these reasons, we think that Ocneanu’sconstruction may also be of considerable interest, regardless of its triviality onclosed 4 manifolds.Finally, we still do not know if our invariants can distinguish homeomorphic4 manifolds.
We can see no solution to this problem, except to compute themfor some examples like Dolgachev surfaces.In summary, we believe that professor Ocneanu’s assumption that our for-mula is equivalent to gluing, although a natural hypothesis, is not supported bythe facts.REFERENCES:1.L. Crane and D. Yetter A Categorical Construction of 4D TopologocalQuantum Field Theories ksu preprint2
2.A. Ocneanu A Note on Simplicial Dimension Shifting preprint3
출처: arXiv:9302.118 • 원문 보기