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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 27th 2024
   • (edited Aug 27th 2024)
   • PermaLink
   Author: Urs
   Format: MarkdownItexToday I started having a closer look at * Chen: *Instanton Density Operator in Lattice QCD from Higher Category Theory* [[arXiv:2406.06673](https://arxiv.org/abs/2406.06673)] I find the author makes a convincing and fascinating proposal. In my paraphrase: > Lattice gauge theory cannot tell how the gauge field behaves "in between lattice cells", and this is on purpose, in order to reduce the configuration space to be finite-dimensional. But it also means that one cannot meaningfully determine topological (instanton) sectors from a conventional lattice gauge configuration, since one does not know the "winding" of the gauge field in between lattice cells. > So in order to still control the global topological sectors of the fields, one must remember, if not the full field configuration so at least its local topological winding "in between lattice cells". Since this is essentially discrete information, adding it in is possible in that it does retain the nature of lattice gauge theory while providing access to well-defined topological observables. > In low dimensions/degrees this is actually well-understood and known (for better or worse) as *Villainization*. > But in in order to make this work for the practically most important topological observables of instanton numbers in 4d YM theory, the local topological information to be retained must, so the argument, take cell-wise values not in groups but in stacky 2-groups modeled by judiciously chosen simplicial models. In contrast to "Villainization" in lower dimensions/degress, the prescription for how to do this for instantons in 4d is no longer easy or even possible to guess from a traditional lattice QFT point of view. But it can be read off from the math literature of models for, essentially, the String 2-group aka the WZW bundle gerbe. > The proposal is hence to enrich the usual lattice configurations for $SU(N)$ YM fields by certain stacky but nevertheless suitably discrete data on plaquettes and cubes, to thereby record the local contributions to the global instanton number. This is the idea. I find this intriguing, as far as it goes as a motivation, and it would seem to have enormous potential if successful. But after a first read through the (long) article, I am still shaky on what precisely the proposed improved lattice model for 4d YM with instanton observables is. It first appears on p. 53, and then more details seem to be provided on p. 106 around (126). I sure recognize many familiar ingredients, but I may need a second read to digest how exactly the proposed lattice model is defined. Has anyone delved further into this proposal?

   Today I started having a closer look at

   • Chen: Instanton Density Operator in Lattice QCD from Higher Category Theory [arXiv:2406.06673]

   I find the author makes a convincing and fascinating proposal. In my paraphrase:

   Lattice gauge theory cannot tell how the gauge field behaves “in between lattice cells”, and this is on purpose, in order to reduce the configuration space to be finite-dimensional. But it also means that one cannot meaningfully determine topological (instanton) sectors from a conventional lattice gauge configuration, since one does not know the “winding” of the gauge field in between lattice cells.

   So in order to still control the global topological sectors of the fields, one must remember, if not the full field configuration so at least its local topological winding “in between lattice cells”. Since this is essentially discrete information, adding it in is possible in that it does retain the nature of lattice gauge theory while providing access to well-defined topological observables.

   In low dimensions/degrees this is actually well-understood and known (for better or worse) as Villainization.

   But in in order to make this work for the practically most important topological observables of instanton numbers in 4d YM theory, the local topological information to be retained must, so the argument, take cell-wise values not in groups but in stacky 2-groups modeled by judiciously chosen simplicial models. In contrast to “Villainization” in lower dimensions/degress, the prescription for how to do this for instantons in 4d is no longer easy or even possible to guess from a traditional lattice QFT point of view. But it can be read off from the math literature of models for, essentially, the String 2-group aka the WZW bundle gerbe.

   The proposal is hence to enrich the usual lattice configurations for $SU(N)$ YM fields by certain stacky but nevertheless suitably discrete data on plaquettes and cubes, to thereby record the local contributions to the global instanton number.

   This is the idea. I find this intriguing, as far as it goes as a motivation, and it would seem to have enormous potential if successful.

   But after a first read through the (long) article, I am still shaky on what precisely the proposed improved lattice model for 4d YM with instanton observables is.

   It first appears on p. 53, and then more details seem to be provided on p. 106 around (126). I sure recognize many familiar ingredients, but I may need a second read to digest how exactly the proposed lattice model is defined.

   Has anyone delved further into this proposal?

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 28th 2024
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexLooks interesting. > The categories used in TQFT are equipped with--more precisely, enriched by--complex linear spaces, which will play the role of the quantum Hilbert spaces. In comparison, the categories we used in this paper have no built-in linear structure, but we know our constructions do have the quantum mechanical linearity, simply because, in the end, we are constructing well-defined path integrals. (p. 118) Is this related to the thrust behind [[groupoidification]] and that work you did on Burnside rings a few years ago, that the combinatorics is enough?

   Looks interesting.

   The categories used in TQFT are equipped with–more precisely, enriched by–complex linear spaces, which will play the role of the quantum Hilbert spaces. In comparison, the categories we used in this paper have no built-in linear structure, but we know our constructions do have the quantum mechanical linearity, simply because, in the end, we are constructing well-defined path integrals. (p. 118)

   Is this related to the thrust behind groupoidification and that work you did on Burnside rings a few years ago, that the combinatorics is enough?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeAug 28th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexWhat you quote appears to be just a version of the statement that in the (lattice-)path integral formulation no quantum states ever appear, only the (non-linear) field configurations themselves. $\,$ Meanwhile, I have further spent time with the main proposed definitions (64) and (71), and I find myself currently unable to follow what the definition is of the crucial terms $e^{i \mathcal{W}_p}\, \mu_{g_{v \in \partial p}, m_{l \in \partial p}, \widehat{n}_{l \in \partial p}}$ and $e^{i \mathcal{C}_c} \, \nu_{g_{l \in \partial c}, m_{p \in \partial c}, \widehat{n}_{p \in \partial c}}$, respectively. Also, I am missing discussion of the consistency requirement that these new partition functions locally reduce to the desired ordinary ones. For the moment I give up on following this. But if anyone has more insights to offer, I'd be interested.

   What you quote appears to be just a version of the statement that in the (lattice-)path integral formulation no quantum states ever appear, only the (non-linear) field configurations themselves.

   $\,$

   Meanwhile, I have further spent time with the main proposed definitions (64) and (71), and I find myself currently unable to follow what the definition is of the crucial terms $e^{i \mathcal{W}_p}\, \mu_{g_{v \in \partial p}, m_{l \in \partial p}, \widehat{n}_{l \in \partial p}}$ and $e^{i \mathcal{C}_c} \, \nu_{g_{l \in \partial c}, m_{p \in \partial c}, \widehat{n}_{p \in \partial c}}$, respectively. Also, I am missing discussion of the consistency requirement that these new partition functions locally reduce to the desired ordinary ones.

   For the moment I give up on following this. But if anyone has more insights to offer, I’d be interested.