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John Baez (Jul 04 2020 at 16:42):If you want to do work on higher gauge theory using just 2-categories, you can try to apply higher gauge theory to condensed matter physics. That's a fairly active subject now. People would like to find actual materials described by 2-group gauge theories.
ADITTYA CHAUDHURI (Jul 04 2020 at 17:04):John Baez said:
If you want to do work on higher gauge theory using just 2-categories, you can try to apply higher gauge theory to condensed matter physics. That's a fairly active subject now. People would like to find actual materials described by 2-group gauge theories.
Thank you very much Sir for the reply. My Background in Physics is very poor.(Almost zero). But still I am very much enthusiastic about learning how to apply 2-group gauge theory in condensed matter physics. Though I do not know How and Where tho begin with!!
Sir , I have seen in your paper An invitation to higher gauge theory you mentioned that there is some relation between Higher Gauge theory(Higher Parallel Transports) and Quantum Gravity. So it would be really great if you kindly suggest some literatures on both connection between condensed matter physics and 2-group Gauge theories and connection between Higher Gauge theory and Quantum Gravity. I am highly interested in understanding and later working on both the topics.
John Baez (Jul 04 2020 at 17:28):I think research in quantum gravity is doing very poorly right now, and throwing more mathematics at it will not help much: we need people to think hard about the underlying physics. So, I don't really want to help any young people work on quantum gravity. But I encourage you to learn general relativity and quantum field theory, since not only are these prerequisites for work on quantum gravity, more importantly they are theories that make verified predictions about our world - so by learning them, you're learning real physics.
Condensed matter physics is doing much better these days, so it's possible for someone who knows a lot of math and a little physics to make a useful contribution.
I don't know the best literature on 2-group gauge theories in condensed matter physics, so these are just some random selections:
https://link.springer.com/article/10.1007/JHEP10(2018)049
https://arxiv.org/abs/1907.01608
https://arxiv.org/abs/1711.04186
You should probably look around the arXiv for better papers!
John Baez (Jul 04 2020 at 17:29):I would also recommend learning condensed matter physics, if that's what you want to work on.
John Baez (Jul 04 2020 at 17:30):I find R. Shankar's book Quantum Field Theory and Condensed Matter to be very good.
John Baez (Jul 04 2020 at 17:31):In general, learning physics is a prerequisite for applying math to physics, unless you have a physicist friend to collaborate with.
ADITTYA CHAUDHURI (Jul 04 2020 at 18:25):John Baez said:
I think research in quantum gravity is doing very poorly right now, and throwing more mathematics at it will not help much: we need people to think hard about the underlying physics. So, I don't really want to help any young people work on quantum gravity. But I encourage you to learn general relativity and quantum field theory, since not only are these prerequisites for work on quantum gravity, more importantly they are theories that make verified predictions about our world - so by learning them, you're learning real physics.
Condensed matter physics is doing much better these days, so it's possible for someone who knows a lot of math and a little physics to make a useful contribution.
I don't know the best literature on 2-group gauge theories in condensed matter physics, so these are just some random selections:
https://link.springer.com/article/10.1007/JHEP10(2018)049
https://arxiv.org/abs/1907.01608
https://arxiv.org/abs/1711.04186
You should probably look around the arXiv for better papers!
Thank You Sir very much for the references. Also thanks for explaining about the current state or status of Mathematical aspects of Quantum Gravity. I did not have any idea about its current state. As you suggested I will definitely try to learn Quantum field theory and General relativity.
ADITTYA CHAUDHURI (Jul 04 2020 at 18:28):John Baez said:
I would also recommend learning condensed matter physics, if that's what you want to work on.
Yes Sir, I want to work on the application of 2-Group Gauge Theory in Condensed Matter Physics.
ADITTYA CHAUDHURI (Jul 04 2020 at 18:29):John Baez said:
I find R. Shankar's book Quantum Field Theory and Condensed Matter to be very good.
Thank you Sir.
ADITTYA CHAUDHURI (Jul 04 2020 at 18:30):John Baez said:
In general, learning physics is a prerequisite for applying math to physics, unless you have a physicist friend to collaborate with.
I can understand Sir!
ADITTYA CHAUDHURI (Jul 04 2020 at 18:48):@John Baez I am reading the references you suggested. I think it will take some time for me to digest everything.
John Baez (Jul 04 2020 at 19:07):Of course! By the way, have you read this?
Condensed matter theorists like to use a "discretized" approach to gauge theory where spacetime is chopped up into cube or simplexes; this paper tries to work out 2-group gauge theory in that context, so it might be good before reading other papers on higher gauge theory in condensed matter physics.
Other people have written papers complaining about this paper, so it might have some problems - I'm not sure - but not actual mistakes.
ADITTYA CHAUDHURI (Jul 04 2020 at 19:11):John Baez said:
Of course! By the way, have you read this?
- Girelli and Pfeiffer, Higher gauge theory - differential versus integral formulation.
Condensed matter theorists like to use a "discretized" approach to gauge theory where spacetime is chopped up into cube or simplexes; this paper tries to work out 2-group gauge theory in that context, so it might be good before reading other papers on higher gauge theory in condensed matter physics.
Other people have written papers complaining about this paper, so it might have some problems - I'm not sure - but not actual mistakes.
Thank you Sir for the reference. No Sir I did not read this paper before. So as you suggested I am trying to get the ideas of this paper first before reading the other references you provided in your previous comment.
Arthur Parzygnat (Jul 04 2020 at 20:26):@ADITTYA CHAUDHURI There is a TON of higher gauge theory being done in condensed matter at the moment. I noticed @John Baez already mentioned several authors such as Apoorv Tiwari. There are also papers by Ryan Thorngren, Anton Kapustin, and Sergei Gukov for example. You might also want to take a look at some of Dan Freed and Greg Moore's work in condensed matter (https://arxiv.org/abs/1208.5055 is quite influential, and even understanding the appendix will get you very far into the mathematical aspects of topological condensed matter theory). These are more mathematically-minded writers. From a high energy physicist's perspective, check out https://arxiv.org/pdf/1412.5148.pdf and people who cite this work. For more condensed matter focus, I'd say it's paramount you learn the Levin--Wen model pretty well (https://arxiv.org/abs/cond-mat/0404617), which is a foundational paper in topological condensed matter. The old papers of Kohmoto (http://physics.gu.se/~tfkhj/Kohmoto.pdf) and and TKNN https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.49.405. I also think it's important to have a solid foundation of the ordinary gauge theoretic aspects of condensed matter before trying to go for the "higher" gauge theoretic aspects. For example, the Berry phase is nothing but the holonomy along a subbundle of a trivial bundle, which is then equipped with the induced connection via projections. If any of those words seem unfamiliar, it's probably important to learn those as well (check out Barry Simon's paper on the subject). There's so much out there, I hope it's not overwhelming. (I actually also have two papers on the subject if you want to check them out btw. they're more on the introductory and mathematical side though...)
John Baez (Jul 04 2020 at 20:29):Thanks, Arthur. I haven't been following the literature so my recommendations were nearly random. I sometimes feel like getting back into higher gauge theory, with much more of a focus on condensed matter physics, but I don't know enough to have any idea of what I would do.
John Baez (Jul 04 2020 at 20:30):Is there any sort of "big problem" that people in this subject are trying to solve?
(I don't usually try to solve "big problems", I try to create new problems - but it's always good to know what people consider important.)
Arthur Parzygnat (Jul 04 2020 at 20:49):@John Baez Unfortunately, I stopped working in this area, too, but I think I'll find out more in August since I will be attending a conference to check out the current status. Though I could also probably ask Apoorv or Ryan... From what I remember, our understanding of quantum field theory, and the notion of symmetry, seems to be changing quite a lot (even standard things like Noether's theorem---see the first few minutes of Zohar Komargodski's talk https://www.youtube.com/watch?v=FkMkCKb1nTw&list=PLx5f8IelFRgGfQMuGGOuqHtRaelyJFj0u&index=14&t=0s). There are many kinds of "higher-dimensional" operators, generalizing things like Wilson loop operators (such as t'Hooft operators), and I don't think they're all fully understood yet. Don't take my word for it though, I can barely follow the literature.
Arthur Parzygnat (Jul 04 2020 at 21:03):@ADITTYA CHAUDHURI you might also be interested in this long workshop on higher structures in field theory: https://www.esi.ac.at/events/e299/
John Baez (Jul 04 2020 at 21:08):There are many kinds of "higher-dimensional" operators, generalizing things like Wilson loop operators (such as t'Hooft operators), and I don't think they're all fully understood yet.
That sounds fun!
ADITTYA CHAUDHURI (Jul 05 2020 at 01:49):Arthur Parzygnat said:
ADITTYA CHAUDHURI There is a TON of higher gauge theory being done in condensed matter at the moment. I noticed John Baez already mentioned several authors such as Apoorv Tiwari. There are also papers by Ryan Thorngren, Anton Kapustin, and Sergei Gukov for example. You might also want to take a look at some of Dan Freed and Greg Moore's work in condensed matter (https://arxiv.org/abs/1208.5055 is quite influential, and even understanding the appendix will get you very far into the mathematical aspects of topological condensed matter theory). These are more mathematically-minded writers. From a high energy physicist's perspective, check out https://arxiv.org/pdf/1412.5148.pdf and people who cite this work. For more condensed matter focus, I'd say it's paramount you learn the Levin--Wen model pretty well (https://arxiv.org/abs/cond-mat/0404617), which is a foundational paper in topological condensed matter. The old papers of Kohmoto (http://physics.gu.se/~tfkhj/Kohmoto.pdf) and and TKNN https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.49.405. I also think it's important to have a solid foundation of the ordinary gauge theoretic aspects of condensed matter before trying to go for the "higher" gauge theoretic aspects. For example, the Berry phase is nothing but the holonomy along a subbundle of a trivial bundle, which is then equipped with the induced connection via projections. If any of those words seem unfamiliar, it's probably important to learn those as well (check out Barry Simon's paper on the subject). There's so much out there, I hope it's not overwhelming. (I actually also have two papers on the subject if you want to check them out btw. they're more on the introductory and mathematical side though...)
Thank you Sir very much for the references and the information about the subject. I felt I have to learn a quite a lot of stuff before contributing something to the field. But it seems the journey will be exciting. I am also excited to know about your two papers on the subject. So can you please share the links of those two?
ADITTYA CHAUDHURI (Jul 05 2020 at 01:56):Arthur Parzygnat said:
ADITTYA CHAUDHURI you might also be interested in this long workshop on higher structures in field theory: https://www.esi.ac.at/events/e299/
Thank you very much for the link. But I could not find the option to register for the workshop. Also it seems that the participants list is already published. I am curious to know.. is there a way to attend this workshop?
ADITTYA CHAUDHURI (Jul 05 2020 at 02:01):@John Baez Sir, although it seems a "long way to go" for me according to my current state of knowledge in this area before contributing something to the field but "I feel the journey would be super exciting!"
John Baez (Jul 05 2020 at 05:16):You can do interesting things en route.
ADITTYA CHAUDHURI (Jul 05 2020 at 06:34):John Baez said:
You can do interesting things en route.
I am also looking forward to that!!
ADITTYA CHAUDHURI (Jul 05 2020 at 06:43):@John Baez Sir, in The n-Category Cafe page in one of your article on 2-groups in condensed matter physics https://golem.ph.utexas.edu/category/2018/11/2groups_in_condensed_matter_ph.html I came across these 2 papers https://arxiv.org/abs/1810.12965 and https://arxiv.org/abs/1808.09394. Among which in the first one at least many of the terms seem familiar to me . Also I saw these papers are quite recent publications . Are these papers relevant to me if I want to work in the application of 2 Group Gauge theory in Condensed Matter Physics?
Arthur Parzygnat (Jul 05 2020 at 06:57):ADITTYA CHAUDHURI said:
Thank you Sir very much for the references and the information about the subject. I felt I have to learn a quite a lot of stuff before contributing something to the field. But it seems the journey will be exciting. I am also excited to know about your two papers on the subject. So can you please share the links of those two?
Sure, they are https://arxiv.org/abs/1802.01139 and https://arxiv.org/abs/1410.6938. The first paper does a lot of higher dimensional non-abelian calculus and provides a lattice picture for the surface-ordered exponential. The second paper (written earlier) proves some theorems on gauge invariance/covariance for surface holonomy and computes some examples.
I should also mention Saikat Chatterjee and Amitabha Lahiri, who are both currently in India. Saikat is a mathematician who knows a lot about higher categorical structures in general (latest paper: https://arxiv.org/pdf/1907.00375.pdf) and Amitabha is a physicist who knows a lot about GR and field theory. If you happen to live near either of them, I would definitely be in contact with them. I'm curious, are you learning these things on your own or do you have an advisor who works in the field?
ADITTYA CHAUDHURI (Jul 05 2020 at 07:00):Arthur Parzygnat said:
ADITTYA CHAUDHURI said:
Thank you Sir very much for the references and the information about the subject. I felt I have to learn a quite a lot of stuff before contributing something to the field. But it seems the journey will be exciting. I am also excited to know about your two papers on the subject. So can you please share the links of those two?
Sure, they are https://arxiv.org/abs/1802.01139 and https://arxiv.org/abs/1410.6938. The first paper does a lot of higher dimensional non-abelian calculus and provides a lattice picture for the surface-ordered exponential. The second paper (written earlier) proves some theorems on gauge invariance/covariance for surface holonomy and computes some examples.
I should also mention Saikat Chatterjee and Amitabha Lahiri, who are both currently in India. Saikat is a mathematician who knows a lot about higher categorical structures in general (latest paper: https://arxiv.org/pdf/1907.00375.pdf) and Amitabha is a physicist who knows a lot about GR and field theory. If you happen to live near either of them, I would definitely be in contact with them. I'm curious, are you learning these things on your own or do you have an advisor who works in the field?
Thank you very much for the references.. What a coincidence!! I am pursuing my PhD under the supervision of Saikat Chatterjee only!!
ADITTYA CHAUDHURI (Jul 05 2020 at 07:06):@Arthur Parzygnat It will take some time for me to digest the materials in your papers.
Arthur Parzygnat (Jul 05 2020 at 07:06):Okay, great! I figured! Please tell him I said hi! That's good news then. You'll learn a lot! Also follow the work of Kiyonori Gomi in Japan who transitioned from higher gauge theory to condensed matter.
Arthur Parzygnat (Jul 05 2020 at 07:08):ADITTYA CHAUDHURI said:
Arthur Parzygnat It will take some time for me to digest the materials in your papers.
Yes, there's a lot of information on the subject. I did try to write my papers in more of an expository manner (which is a reason why they're long), so I hope they're at least somewhat accessible. Please feel free to ask questions if you have them (but probably not in this "introduce yourself" stream).
ADITTYA CHAUDHURI (Jul 05 2020 at 07:08):Arthur Parzygnat said:
Okay, great! I figured! Please tell him I said hi! That's good news then. You'll learn a lot! Also follow the work of Kiyonori Gomi in Japan who transitioned from higher gauge theory to condensed matter.
Thanks!! sure I will tell him.
ADITTYA CHAUDHURI (Jul 05 2020 at 07:10):Arthur Parzygnat said:
ADITTYA CHAUDHURI said:
Arthur Parzygnat It will take some time for me to digest the materials in your papers.
Yes, there's a lot of information on the subject. I did try to write my papers in more of an expository manner (which is a reason why they're long), so I hope they're at least somewhat accessible. Please feel free to ask questions if you have them (but probably not in this "introduce yourself" stream).
Thanks a lot.. Can I please get your email Id? Or should I message here in personal messages?
Morgan Rogers (he/him) (Jul 05 2020 at 10:06):@ADITTYA CHAUDHURI If this develops into a significant branch of ACT, we can reopen a dedicated stream to it later :)
ADITTYA CHAUDHURI (Jul 05 2020 at 14:10):[Mod] Morgan Rogers said:
ADITTYA CHAUDHURI If this develops into a significant branch of ACT, we can reopen a dedicated stream to it later :)
Ok Sir.
John Baez (Jul 05 2020 at 18:44):Rongmin Lu said:
John Baez said:
Is there any sort of "big problem" that people in this subject are trying to solve?
(I don't usually try to solve "big problems", I try to create new problems - but it's always good to know what people consider important.)
You could ask your good friend from your MIT days. Just saying... :smiley:
Which "good friend" of mine is an expert on higher gauge theory in condensed matter physics?
John Baez (Jul 05 2020 at 21:18):You mean Mathai? Yeah, if I wanted to work on the quantum Hall effect I'd ask him, but I don't feel any special competence in that area. I think I might still be better at higher gauge theory than most condensed matter physicists.
Philip Zucker (Jul 06 2020 at 04:04):Oh it's quite the effect though. I've been hearing that there's been interesting experimental progress lately
ADITTYA CHAUDHURI (Jul 06 2020 at 09:00):In the paper 2 Bundles https://arxiv.org/pdf/math/0410328.pdf by Bartels it was mentioned that in future it is hoped that
there is a possibility that Gerbes will be generalised to stack of coherent 2-groups . We know that there is a cocycle description of Non abelian Gerbes over a manifold using Crossed modules (which are equivalent to Strict 2-Groups).
I have two Questions in this regard:
1. What is the current status of the generalisation of gerbes as a stack of Coherent 2 groups?
2 If we call such gerbes "Weak Gerbes" (as we are using coherent 2-groups) then is there any corresponding cocycle description of Weak gerbes over a manifold?
jjjjjj.png
Jules Hedges (Jul 06 2020 at 09:22):[Mod] Morgan Rogers said:
ADITTYA CHAUDHURI If this develops into a significant branch of ACT, we can reopen a dedicated stream to it later :)
Perhaps I'm wrong, but I was under the impression that this (higher categories in theoretical physics) was a much bigger field than the whole of ACT (and possibly that most people doing it don't know or care what ACT is because they already have their own identity, same as applied categories in computer science)
Morgan Rogers (he/him) (Jul 06 2020 at 09:47):If so, I should modify my comment to "If this develops into a significant branch of the ACT discussion on Zulip, we can reopen a dedicated stream to it later :)"
Konstantinos Meichanetzidis (Jul 06 2020 at 13:21): ADITTYA CHAUDHURI (Jul 06 2020 at 13:23):Rongmin Lu said:
ADITTYA CHAUDHURI said:
I have two Questions in this regard:
1. What is the current status of the generalisation of gerbes as a stack of Coherent 2 groups?
2 If we call such gerbes "Weak Gerbes" (as we are using coherent 2-groups) then is there any corresponding cocycle description of Weak gerbes over a manifold?I think Nikolaus and Waldorf have an answer in this 2013 paper:
Four equivalent versions of nonabelian gerbes
Thomas Nikolaus and Konrad Waldorf
Pacific J. Math. Vol. 264 (2013), No. 2, 355–420Abstract:
We recall and partially improve four versions of smooth, nonabelian gerbes: Čech cocycles, classifying maps, bundle gerbes, and principal 2-bundles. We prove that all four versions are equivalent, and so establish new relations between interesting recent developments. Prominent partial results that we prove are a bijection between the continuous and smooth nonabelian cohomology, and an explicit equivalence between bundle gerbes and principal 2-bundles as 2-stacks.
I am just a beginner in the subject so I might be wrong but it seems to me that they only considered Strict Lie 2 groups in the above paper. Am I making mistakes or interpreting in a wrong way? ttttttt.png
ADITTYA CHAUDHURI (Jul 06 2020 at 13:26):Konstantinos Meichanetzidis said:
Thanks. It looks interesting . I am attending it.
Arthur Parzygnat (Jul 06 2020 at 18:22):This doesn't fully answer your question, but Baez and Lauda in their 2-groups paper show that weak 2-groups are equivalent to coherent ones (in Section 5) and coherent ones are equivalent to special ones (Section 8.3), so at the very least there may be a way of carrying the extra data given by the cohomology class (though I'm not sure how to do this). But if you wanted a cocycle description of such weak 2-group 2-bundles, I would naively think you can use the same idea in Section 3.2 https://arxiv.org/pdf/1410.6938.pdf (which I originally learned about from Christoph Wockel's work https://arxiv.org/abs/0803.3692) by replacing the target category by a weak 2-group and use the definition of a weak 2-functor of 2-categories to define the local cocycle data. I'm not sure if this has been done (it's possible that this has been addressed in the -category language, but I can't follow that work).
ADITTYA CHAUDHURI (Jul 06 2020 at 19:44):@Arthur Parzygnat Thank you very much for link of your paper. I also thought once before about using the ideas of section 8.3 of Baez Lauda paper but could not proceed much that time. I will try once more. Thanks again!
ADITTYA CHAUDHURI (Jul 06 2020 at 19:56):@Arthur Parzygnat " I'm not sure if this has been done (it's possible that this has been addressed in the \infty∞-category language,"
Some time back I saw a similar kind of idea in the paper Semistrict Higher Gauge Theory by Jurco, Samann and Wolf in the (section 3.2) https://arxiv.org/pdf/1403.7185.pdf. But I am not sure whether they are completely similar.
ADITTYA CHAUDHURI (Jul 06 2020 at 20:04):@Arthur Parzygnat Do you know whether there is any published work about Weak notion of Gerbes ? I am asking can"Exploring the notion of Gerbes over a manifold using coeherent 2 group or equivalently special 2 groups(as mentioned in Baez -Lauda paper) be a good problem to work on?
John Baez (Jul 06 2020 at 20:17):I imagine Urs Schreiber's work involves a lot of weak gerbes, formulated using -categorical language.
ADITTYA CHAUDHURI (Jul 06 2020 at 20:25):@John Baez Thank you Sir. Any specific reference you would like to mention? I have seen some papers on Principal infinity bundles by Schreiber but right now I do not have sufficient background of infinity category to understand the material of those papers! So is there any reference which uses only 2 category theory? (I am currently learning infinity category theory but still my background is not sufficient)
John Baez (Jul 06 2020 at 20:27):No, he doesn't use just 2-category theory.
ADITTYA CHAUDHURI (Jul 06 2020 at 20:28):Ok.
John Baez (Jul 06 2020 at 20:28):In string theory you need not just 2-groups but also 3-groups, 5-groups etc., so it becomes cleaner to work more generally.
ADITTYA CHAUDHURI (Jul 06 2020 at 20:30):@John Baez I see! Thank you. So I have to first learn infinity categories. Then only I can understand about Weak Gerbes.
John Baez (Jul 06 2020 at 20:43):If you want to use just 2-groups in gauge theory, the most exciting area of study right now is probably applying them to condensed matter physics.
John Baez (Jul 06 2020 at 20:43):If you want to do everything as "weakly" as possible, it's good to learn about -categories.
John Baez (Jul 06 2020 at 20:44):I think weak gerbes are already pretty well understood in that approach, at least by a few people like Urs.
ADITTYA CHAUDHURI (Jul 06 2020 at 20:46):@John Baez Thank you Sir.
Nivedita (Jul 06 2020 at 20:47):John Baez said:
If you want to use just 2-groups in gauge theory, the most exciting area of study right now is probably applying them to condensed matter physics.
Can you give the most recent reference you came across in this regard. I recently came across the use of 2-groups in taking forward Mermin's classification of textures and defects. In there any scope of newer developments in this direction?
John Baez (Jul 06 2020 at 20:51):I listed 3 references earlier in this discussion, and @Arthur Parzygnat listed more. I'm not really paying much attention to this subject. I think Arthur been paying attention more recently. Here's what he said:
There is a TON of higher gauge theory being done in condensed matter at the moment. I noticed John Baez already mentioned several authors such as Apoorv Tiwari. There are also papers by Ryan Thorngren, Anton Kapustin, and Sergei Gukov for example. You might also want to take a look at some of Dan Freed and Greg Moore's work in condensed matter (https://arxiv.org/abs/1208.5055 is quite influential, and even understanding the appendix will get you very far into the mathematical aspects of topological condensed matter theory). These are more mathematically-minded writers. From a high energy physicist's perspective, check out https://arxiv.org/pdf/1412.5148.pdf and people who cite this work. For more condensed matter focus, I'd say it's paramount you learn the Levin--Wen model pretty well (https://arxiv.org/abs/cond-mat/0404617), which is a foundational paper in topological condensed matter. The old papers of Kohmoto (http://physics.gu.se/~tfkhj/Kohmoto.pdf) and and TKNN https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.49.405. I also think it's important to have a solid foundation of the ordinary gauge theoretic aspects of condensed matter before trying to go for the "higher" gauge theoretic aspects. For example, the Berry phase is nothing but the holonomy along a subbundle of a trivial bundle, which is then equipped with the induced connection via projections. If any of those words seem unfamiliar, it's probably important to learn those as well (check out Barry Simon's paper on the subject). There's so much out there, I hope it's not overwhelming. (I actually also have two papers on the subject if you want to check them out btw. they're more on the introductory and mathematical side though...)
John Baez (Jul 06 2020 at 20:55):Nivekida:
I recently came across the use of 2-groups in taking forward Mermin's classification of textures and defects.
Can you point me to that paper? That might help me see what has been done and what's left to do.
Nivedita (Jul 06 2020 at 20:59):John Baez said:
Nivekida:
I recently came across the use of 2-groups in taking forward Mermin's classification of textures and defects.
Can you point me to that paper? That might help me see what has been done and what's left to do.
I am not sure if there's a published version.
https://arxiv.org/abs/1810.12965
https://inspirehep.net/literature/1701181
Arthur Parzygnat (Jul 07 2020 at 06:20):ADITTYA CHAUDHURI said:
Arthur Parzygnat Do you know whether there is any published work about Weak notion of Gerbes ? I am asking can"Exploring the notion of Gerbes over a manifold using coeherent 2 group or equivalently special 2 groups(as mentioned in Baez -Lauda paper) be a good problem to work on?
As for the specific question if I know there is published work on it, no, I'm not sure. But that's not because it doesn't exist, more so because I was never curious. The paper you mentioned by Jurco et. al. seems to address the weak 2-group 2-bundle situation, so I would look at all papers that cited this and check those out if that's the kind of question that interests you. I honestly never understood any of the papers by Samann and company---I honestly found it easier reading Urs Schreiber's work (like, I remember reading his paper "Higher prequantum geometry" https://ncatlab.org/schreiber/files/hgp.pdf and watching his related videos on the topic and being completely fascinated---I always told myself if I ever went back into this field, I would devote some time in understanding his work). As for applications of higher gauge theory, I see two directions. On the one hand you have string theory and M-theory. On the other hand, you have condensed matter physics and low-energy field theories, which can have extended object excitations. In either case, it seems one needs a reasonable understanding of QFT and renormalization.
If you want my personal opinion, I would recommend you work on what interests you the most, and at the same time something you think might be a useful contribution. If the topic you want to work on most is currently too far out of reach, have those motivating questions/goals in the back of your mind, and work on things that will help you achieve those goals some time in the future.
ADITTYA CHAUDHURI (Jul 07 2020 at 07:56):Arthur Parzygnat said:
ADITTYA CHAUDHURI said:
Arthur Parzygnat Do you know whether there is any published work about Weak notion of Gerbes ? I am asking can"Exploring the notion of Gerbes over a manifold using coeherent 2 group or equivalently special 2 groups(as mentioned in Baez -Lauda paper) be a good problem to work on?
As for the specific question if I know there is published work on it, no, I'm not sure. But that's not because it doesn't exist, more so because I was never curious. The paper you mentioned by Jurco et. al. seems to address the weak 2-group 2-bundle situation, so I would look at all papers that cited this and check those out if that's the kind of question that interests you. I honestly never understood any of the papers by Samann and company---I honestly found it easier reading Urs Schreiber's work (like, I remember reading his paper "Higher prequantum geometry" https://ncatlab.org/schreiber/files/hgp.pdf and watching his related videos on the topic and being completely fascinated---I always told myself if I ever went back into this field, I would devote some time in understanding his work). As for applications of higher gauge theory, I see two directions. On the one hand you have string theory and M-theory. On the other hand, you have condensed matter physics and low-energy field theories, which can have extended object excitations. In either case, it seems one needs a reasonable understanding of QFT and renormalization.
If you want my personal opinion, I would recommend you work on what interests you the most, and at the same time something you think might be a useful contribution. If the topic you want to work on most is currently too far out of reach, have those motivating questions/goals in the back of your mind, and work on things that will help you achieve those goals some time in the future.
Thanks a lot!! As you mentioned "In either case, it seems one needs a reasonable understanding of QFT and renormalization" I have a very poor Physics background at the current stage. So are you suggesting that to apply Higher Gauge theory(Or understanding Higher Gauge Theory properly) the Knowledge of QFT will be very important?
ADITTYA CHAUDHURI (Jul 07 2020 at 07:58):@Arthur Parzygnat If you want my personal opinion, I would recommend you work on what interests you the most, and at the same time something you think might be a useful contribution. If the topic you want to work on most is currently too far out of reach, have those motivating questions/goals in the back of your mind, and work on things that will help you achieve those goals some time in the future. :+1: Thank you very much for the advice!
Arthur Parzygnat (Jul 07 2020 at 10:47):ADITTYA CHAUDHURI said:
Thanks a lot!! As you mentioned "In either case, it seems one needs a reasonable understanding of QFT and renormalization" I have a very poor Physics background at the current stage. So are you suggesting that to apply Higher Gauge theory(Or understanding Higher Gauge Theory properly) the Knowledge of QFT will be very important?
I would say if you're merely interested in the purely mathematical aspects, there are many interesting things one can do without knowing any QFT whatsoever. But as far as I am aware, the main point of higher gauge theory was to provide a formulation for the gauge theory for extended objects, and in both of the examples I am aware of where such objects occur, knowledge of QFT seems to be needed. That's not to say it has no applicability outside these two areas, I just don't know about them (actually, there has also been some work in teleparallel gravity https://arxiv.org/abs/1204.4339, which only requires some knowledge of GR, but I don't know how far this has gone).
Outside of physics, one can also view higher bundles as additional structures one can place on manifolds. Is there a K-theory associated with them? Do they have a sensible theory of characteristic classes? Are there differential refinements? I don't know if there are answers to any of these questions (other than the theory of Dixmier--Douady classes), though I suspect Schreiber and others work is exactly meant to address at least some of them.
ADITTYA CHAUDHURI (Jul 07 2020 at 11:38):Arthur Parzygnat said:
ADITTYA CHAUDHURI said:
Thanks a lot!! As you mentioned "In either case, it seems one needs a reasonable understanding of QFT and renormalization" I have a very poor Physics background at the current stage. So are you suggesting that to apply Higher Gauge theory(Or understanding Higher Gauge Theory properly) the Knowledge of QFT will be very important?
I would say if you're merely interested in the purely mathematical aspects, there are many interesting things one can do without knowing any QFT whatsoever. But as far as I am aware, the main point of higher gauge theory was to provide a formulation for the gauge theory for extended objects, and in both of the examples I am aware of where such objects occur, knowledge of QFT seems to be needed. That's not to say it has no applicability outside these two areas, I just don't know about them (actually, there has also been some work in teleparallel gravity https://arxiv.org/abs/1204.4339, which only requires some knowledge of GR, but I don't know how far this has gone).
Outside of physics, one can also view higher bundles as additional structures one can place on manifolds. Is there a K-theory associated with them? Do they have a sensible theory of characteristic classes? Are there differential refinements? I don't know if there are answers to any of these questions (other than the theory of Dixmier--Douady classes), though I suspect Schreiber and others work is exactly meant to address at least some of them.
Thank you very much for the detailed explanation.