Okay, then let's start. Hello everyone, to The Curious all scheduled today's speakers younger DT with quantum geometry from higher gauge theories
Thank you everyone for tuning in. So I will speak about this part of geometry form if he and I should think my key and so on and we'll
For actually organizing a quantum repeater last window.
And they put together some people into reading books. And so this is one of these open loops this this people which appeared in August.
And it's basically weapon by my muscles open book, and as I will explain. It's about
Extending he retired he read and getting some very interesting lessons for point of view on it and it candidate for quantum geometry out of it.
So the main message will be that this
prospect for new for the quantum geometry which will include
Us.
Listen, okay.
Hello.
Can anybody hear me.
Yes, we can hear you fine.
Yes.
Okay so includes Edwards as as basic many units are supposed to just be vectors and the recon you get the momentum. And that's the two connection which
If the need to the higher THC. So it's a two page Siri, and there's two connection will be a to phone so if you form the two curvature. It really ski form.
And it turns out that the sleep on us as a spacetime different often generator for for the tech bytes, particularly, it really does immediately generate the Nordics translations.
And to Siri, as opposed to consideration of the edges which is done by the just the user curvature and be absolutely
So, it offers many advantages on the kinematic and dynamic level and we will see that we already learn one lesson about
The dynamic content of quantum gravity.
So here's an overview of my talk, a try to give a very fast.
If you have sweetie quantum reality and see how nicely things look to see maybe them to compare to 40 quantum gravity.
And this will be based on like to acknowledge the configuration space. And I would argue that maybe 40 we should actually a notch more than we usually do.
And this would be basically based on the BSC G action.
And then I represent a bit what we actually did. And the people that would be the quantization also section. The I think the spin bobin order and quantum geography and the closest an Outlook. So for all people expected
And introduction for two categories and release every almost no to category.
Descriptions appearing here. So it's mostly based on on geometrical insights, which actually indeed help us to solve a problem in society. He
Context.
So let's start with the CV gravity. Imagine we do a positive good see veggie I close
The water. Start with a triangulation attach some legs very evil to the edges and try to define or to compute as possible.
Now the actually be a very complicated function on polynomial legs for earbuds. Additionally, you have to worry about the domain of integration because they will be
tying it in the qualities and inequality.
And so on. So you get some complicated constraints.
The trick here is to enlarge configuration Spain's space and then to contain again.
And particular to introduce clients.
So that when since I'm a clickable have some positivity properties. And in this case, Google, it was a first order formulation and introduce an independent spin connection.
So we were introduced to gauge freedom and just needs to support any action.
And one except also be affection and as you can see the Palestinian action is just Ninian's tired fields.
So if you'd like to ask them to go to the actual listen I we can integrate out I feared and VCs are basically this Thai food. Excellent. Thanks, Mike. Much don't have to impose anything curvature. So what we end this is
Also be going for for quantum gravity and really it's just the positive
Actions and this we should idolize Francis some lettuce and so he gets a presentation of
Authenticity authentically in
Any event on hollowness
Now, what we use on a technical level is appropriate one form. So we expand functions in this case a data function over group element.
To be a representation in business and use various states and so on. And then end up Miss upon sandwiches order which he is just within the tetrahedron people left out signs of dementia factors and so on.
So we get to this loop a positive. So we just assign it to eat all get to six seasons, which automatically satisfy all this time and any qualities
Associated with that.
And, you know, was actually invented in 68 based on a completely different arguments.
And this gives you a very reasonable TV quantum gravity Siri, which unfortunately is still a bit
unappreciated to can make it very defined of the vaccine.
There's lots of work which teaches that you can even understand if you do symmetry and consistent Hamiltonian aijalon could say algebra. In this model, and this is me audience and the different multiple constantly translates vertices
Into hyper surface triangulation, and this case.
We know that canonical quantization.
Can choose canonical quantization. So that's experiment positively quantization. One can also show equivalents to transfer payments and so on.
So the recent calculation, which we have done is to define all the classic toys or find the boundaries.
And if you take the limit and as opposed to kind of limit you actually even was be speaking collectors for this petition function.
And this was a calculation, which has been done first and affirmative duty, so you can immediately make you can actually make contact with petabyte of pitino calculations. This is Facebook, of course, he has open fashions these questions.
They put into two sessions about what on what does one understand our theory of functionality. So make sure you could ask this, this model invoice some orientations. There was a widening numbers, which also show up in this
In this example of the BMS we can come into should say busy no should say not busy and so on, but otherwise it's a completely consistent quantization of sitting
On the canonical side us one we can lie to Connecticut a bit space and place the network functions and you can define all this quantum jump up are believers.
And this also part of an ISIS formal quantization of lettuce a space which is given by associating
He started as a tool which comes with some slight exception to each link. So one copy of this is equal to each link now becomes a very important point. So this link actually represents that instruction, the sense that
We, the subject expansion sounds like the title to get to the enemies and to tie it is me it over the edges of the emulation something one dimension.
So, hello. Nice. Nice. Almost meeting or connection over one dimension and actually is doing. So it's also something one dimensional
So here we have a situation where, basically, these two variables us me the way each one dimension and one plus one is two. It gives us the space, the dimension of space. And that's important to get us a subject exception to get the big mountain, you can
Sometimes matrix or measure independent syntactic structure.
And as we'll see in four dimensions and we need to get see on the right hand side and then we need to decide how to split this into two plus one one plus two.
So let's look at 4070. Can we do a symmetric. So there's a first order formulation based on the quality action which is nonlinear and templates.
We have to be affection, which is topological
And maybe it's a true form.
And of course we know that to get really gravity would have to add constraints to the SPF action and if it's a pretty ballsy action resist constraints and for simplicity of the two
So to form field to lies on a very
Of course we know also that, in this context, the simplicity constraints aquatic oneself more cycles. I don't comment on that. So one of these sectors is this one here is a geometric sector.
Interestingly, these different formulations. So as a BF actions that have been ski action then Apollo teeny action if you add to hold stone to it and the Africa by be able formulation already to similar space and to open up a similar kind of quantum geometry.
So you
Go now to the CD hyper surface and sip as one and you triangulate and again to each when
You associate a copy for instance of these ties it all if you're already in time gauge and so on, or if you are not scared.
Of the style. So, for
In this case,
The variables which are connected to each other one I quality a bee bee. So before, so he feared and he encodes knowledge was a triangle. So it's based on devices form.
This meeting was applying and so hold on me. It's coming again from the integral was a link. So again, they have a conjugate appeal.
But this time, we need really to integrate one of these barriers and really be in this case was something two dimensional because in the end be meter that is dimension at up to see
So from from this be like door to funkier so from be we
Usually would deconstruct the pilot
And then you can complete a new template.
Was this problem.
On issue if you want. And because if one the concert series hide from one template and the tides for another
One notices that shape sheep have two triangles, which one would like to identify actually do not match the time as much but the kinds of shapes, do not match.
And
I think the notice it.
In the canonical formalism first though with James I know 2008
And these constraints actually appeared before on the similar in 2007
So be there, but the supplies exam because we expected. That'd be fine. Really, Reggie, some tissue geometry.
Um, but now it's a very known fact and effect was one motivation for calling the local remedy geometries with the geometries.
Later we found out that that will be important point is
That this is an argument of office space of geometry advisors to due to the fact that
Certain height of constraints which we need edge simplicity constraints are not imposed.
So these edge Simplicity Simplicity constraints, the month following. If you will reconstruct
For instance, a tired from your beef fields and to transport this diet, which is associated to get an edge and I want to actually say if you wanted this tight Victoria stable analysis on me.
So actually this this constraint.
Is equivalent to the secondary simplicity constrained and the secondary simplicity constraints I proven to the reality conditions on the African connection and effect. It's also a property of a display which will be a connection.
And actually later we will see that this constraint is actually considered as a disk utilization also offers a first BMP identity.
Which comes from basically that the conditions that portion vanishes.
Um,
There was some discussion about the interpretation of the ICO later. Again, we will really see that you can identify that this some kind of fortunate degrees of freedom is still ongoing discussion with this.
This issue.
Isn't it's a problem for the spin form and a pudgy dynamics. So for me that was always a question, what do we do with this and notched face face of geometries.
I can also ask this question, it's been foam and one person which which pointed out that order forms the beginning, maybe also the Alexander
Was also built by Simone and answer. How about ocean. How should be defined the dynamics on this extended face face.
So basically, for me, it was, it goes back to suspected want us to see
Because if you want to gauge formulation. So you really want is a whole other means, which was near the world war one I mentioned things that will links and then
You get us going to get it very that's something which is defined as a to form. And so this as a director feuds
And
We say import users extend the curvature. There's a star. It's actually the curvature, because, in fact, it says, which has anybody find if you have actually uniquely construction of the pilots.
Which allows you then to reconstruct the lever to be the connection effect as we have seen the edge simplicity constraints is about having a very defined he beat up on action. And so if that's not possible, you accept the curvature is a biggest
And the beef years unquote areas moments to find it.
And furthermore, if you if you really look at BF
Dynamics. This is based on
On the consented to banish it's a to form and as a to form a basically can see only edges can see one dimensional objects. And in fact, it generates edge translations instead of vertical stations.
And there's an auto proposal by Republicans upon which proposals to break this excellence Asian cemetery don't have an explanation.
If you think about it, what was I didn't know nope expressions nonetheless it's, it's, it's an interesting proposal which, to my knowledge, unfortunately, nobody really followed up this
And the more modern incarnation of the Hamiltonian contains actions sections actually based on paper Edo over to my opinion has a disadvantage because you lose of it's a geometric interpretation.
Say virtue and virtue paperless
Hamiltonian constraints generate erotics translation.
If you have an action based on paper, you know, I'm Penn State my heat and your hypothesis, it should actually also like a long object. So you would expect the kind of fun. Okay.
So what one actually what like this to a place to be fields which are based on time. Is this
One forms.
To really get
The Matrix geometry.
But since the titles are one forms, it would miss one dimension.
In fact, if you look at the canonical analysis of the votes, the monkey action we do saw this one phones but then
We find that
It's very connected to the
Connection is actually
So energized and most
Of these one forms and that leads us to the to form.
So here what what this proposes to again applies a strict and a notch. The space and then constraining in
This announcement proposes to introduce actually to form.
Which is a connection.
And since the now. Oh, from one connection form poor to connection for it, we are free to choose to either templates.
As the quantity editable.
And so we use it. Not only is one plus to see, but also the two plus one is St. So it's a very natural.
Allotment
So notice that I chose here the connection and the be fields. So for that, you'd be choose the to form and pilots templates to be are for valued and that will have a specific reason. And the reason is that they want to fit into this to gauge theory, which I will explain on the next slide.
So later we can imagine that we actually want Spain's of leaders and the connection one form and have nots will be a will basically
Be feared and the connections will be expressed in terms of establishments.
Let me introduce one quantity which is to curvature
To cook curvature is defined as a call and
Also to have some connection.
And this would be a sequel.
So you can ask no Uzi study this come from some actual question. Yeah.
Is churchy variable really something like a proxy for the tourism Stevie wedgie me
Given up. He is a portion of the video almost see it on the next slide and two seconds.
So, but it comes from an action and that is a BSc G action which was defined by what we own
And like 500,000 in preventable and 2007
And you can all contemplate why it's called a BSc G action, given that existed a b f x
Y you complete it.
Contemplated it will renamed the HCG action to a BF eg action. So, this we will actually identifying the CPS with atheists.
Now remember that G was Jessica derivative of this to connection.
I think so. It depends on the one connection and on the connection.
And if you do an integration by parts.
In fact what appears is an older when the limit of x on the
Fields and that'd be fine to the torsion.
So, yet the action of basically BF C D can understand us
An action for enforcing to constraints here.
One is a one curvature constraints, until eventually the player is be
On the second part, we have a choice. We choose here that the single to torsion as a constraint. So we have supported on saying answer to connection very acute just as an advanced microphone.
Okay, so this turns out to be a topological action.
You have
Efficient of motions all the certifications of motions for for the fourth years which appear or this equation of motions also lead to constraints.
So you have so flatness constrained F is equal to zero, you have the towels container which appears in the user form, plus a term which is a bit unusual, but which is needed to take to to generate
As a form HTTPS. So for me, if somebody
Has a torsion on St.
And if you have to, to curvature constraint which is a G.
So this is a topological theory, it's not maybe key, but now you have actually directly in there is a field and the field. So, it is electric field and
Pepper fields and what you can now easily put at least as silly.
As a simplicity constraints which know just comes into forums that he has to be equal to start either he
And
I can find these papers which will save it here as a creation of motion for this form of action and the constraint on analysis of the continuum action.
And you just mentioned that some observations. So the luggage might forgets to be fixed to be if it was a curvature was one curvature. So, this indicates us. It's also the case and BF series of simplicity constraints.
Maybe second Johnson. Thank you, but also get secondary simplicity constraints.
And geez to curvature consent evaluates to
Some secular star f g
So this if you translate to standard look on gravity variables. Actually, we
Kind of look like San Antonio St. So it's actually the invasion of motion and policy formulation, as I mentioned,
Until you see already here that this that G will be created to the
Ancient times are expressed in the one in terms of one connection and the pilots.
So G is a function of the one Connection. Connection you already see some constraints between the two connection and one connection.
Okay, so, and puts it looks very promising. So you could also avoid people play around with this more and keep hold on, or one key automation. Was it
Was
Nothing available, which amounts to.
To group for you transform. So you could ask, you know, what is the spin or the representation picture of this model and
The connection between the for the spin Bobby model and a BSc G action or is a cool picture because it didn't exist yet.
Most conjectured said it was given by model of quantum fleet quantum flat space, which we call it KBS also was defined by button, based on the call.
So the rest of my talk.
It's a plan to be to be presented with what's actually in our people. So it's a quantization or to be HCG action and the correspondence
Will give it a bit postponements as a PDF motor
As a key of offices correspondence that being not to construct a to group returns home. But actually, and solving odd also equations of motion or the constraints and
He his idea was actually that this part of the constraints of foods, he
Will be constructed. The only in that space of the ceiling.
And you can say Otter also constraints and they have a consistent angel.
And he defined support G network functions which you can send us a generalization of the sprint network function. So it's been a look functions in my house constraints and the G network functions environment has one of those constraints.
And the two goals can send us a portion. And so one of those constraints. So usually those constraints.
Okay, so are there any questions here.
Okay, right, and I continue
This a possible
So, we have seen that action we can understand just also enforcing to constraints.
So once that is complete and total constraint.
And so that means it will sink. I can integrate out a good answer to connections.
Either alive at the partition function nature just integrated was remaining feeds the connection and
One forms.
Lissa condition is the constraints and this. These are the data functions that the one coverage of anxious and
Now you just need to pay again NATO allies. So extraction
And so for the first one flightless we get to usually expansion. This is given by a portable global economies associated to the edges was an inside of it will face.
So that's, that's the usual one flatness constraints and these will also
Usually one anonymous
Function contains actually. It's also quite straightforward contain
It translates into the
The associated vector
To each edge.
And so you get the X Factor and to extract all associated to the edges of the financial close
So this gives us actually playing a total constraints.
Now there's one subcommittee.
Which I will explain going to see the torso two types of his family strangulation and this is know that they need to to worry about edge vectors which also see two edges.
And which that you need to worry about in between you define these ethnic ethnic dogs.
It's not a problem, an equity value of the victors associate to tiny let's hear you and I could you just choose one of the enabling technologies.
Are some reference fame for for for the speed like to, and then you just have the consistency condition. And how's it be like door looks like a scene from the other free
So here you have actors and separate all sharing it gives match. So you have to choose one or two step by eat all
I said reference system for you actually go and, furthermore, you need to define how to call a transport you expect or who the other tip I, you know, which has this edge and this case if you're on CDs or two choices because you can always
Cyclical cyclical on your edge.
And so also the time at Oracle saying
This not only involved the edge make those but also won Emmys, because these are needed to transport vehicles.
And so in effect.
If you think about know, doing, doing good for you transform if you just would do so for good for you transform
So this it would do if you would have just be f series. So just one flatness constrains then you can do and so forth. We transform you would get the user
It's a 10 or 15 Jason birds.
But this piece we can't do that because actually is a
One on one is also appeared in the portion
And on the other hand, if you really wanted to, uh, to go for it and so on, they unfortunately as magicians do not know what it
Is so there's no Peter YC available.
What, what we did is actually to see that we can go forward by serving part of the data functions.
And part and actually puts a little piece of time. You told us. In fact, we already
Did a partial fleet as form in some sense.
This part of it for you transform. We didn't do it as a states, some, but we did it by doing the integration my parts and the action so you can also start from from these constraints that connection to curvature is finishing. And then if we transform
Our form and then if they actually finds a tiny closure as a contact form is a triangle closure constraints.
So, so I think for the time enclosure constraints we would actually impose also took those constraints.
And here he is again, the point is is partly transports appealing and the time constraints matter because this we have we have two choices pilot transports. And there are several choices on Channel two, one flight is concerns all these choices Uno momento.
But the principles define us different versions of Pluto consent and a key inside is that being not even We not only have to take some time constraints, but we should actually add
The edge simplicity constraints.
And the actual physical constraints are basically what we had before impose it an edge backdoor pilot has floated around the edge itself. And it's stable.
If you remember what I said on the previous slide. This ensures
That the public transport which Troy see
That's not met on a few things. And if you count. This gives you sleep conditions so that imposes extra exactly half of the one flight is constrained to the full flattens contango visit the whole only own G on the only on every edge would be vanishing
Um,
And it also has a continuum analogy.
So was a time constraints come from the condition that the potion manages if you take again.
To have certain condition you get f g is SEO and so that's actually the first can kayden at the edge simplicity constraints. I actually just a discrete those offices identity.
So we will also imposes excellent the sequence things which you can understand as harmful to one.
And at the end of it also make use of one gauge in my hands of the Pacific.
Okay.
So yeah, just to give a sketch sketch of support.
Of the mechanism.
We have obviously steps.
Step one is
This couple that important, the time constraints and the actual physical same
As everybody basically a reduction and the degrees of freedom which certain salamis
Effect. You can use these two constraints and you can prove that salamis a deuce to two cards and one part is completely determined by your adverts
And if you go to time gauge it will be actually a rotation it as I thought only has one free parameter and this one free power meter mine was actually
Curvature and if, again, if you go to have it set to be boost.
Because it's an so for so for rotation.
Which will be perpendicular to the to the time because the plane of Steiner
Um,
So yeah, it is. It all the G's.
The free power meter left or if you discounters touchpads it just given my one. How many this English
Now you can put this form of the GS back into the one flatness of St.
It's an entrepreneur population and this case which we considered, which was a surface office complex it fixes modal one sign this expensive furniture cubbies eccentric curvature of a flatly embedded simplex. So you fix it and the curvature as a function of
Subsets of your office complex and all what you need from the template. It's actually the length
And so we just fix the seat us
To be this explicit functions.
Which are given by this isn't even an angle of reflection flattering. But it's complex.
In terms of its edge edge links.
So actually this year will be addressed left
This to kind of enables excellent and CC to ingots so in this that the mid also use officer. So for each alliance or c x
Last, so now we have
This thing is, and what would you say you want for each one form for the data functions, which is the impulsive, meaning one flatness
So we get what you what you need to remember here is that we get an amplitude patrols looks like exponential is so the variable also see to triangle.
Sita is
Valuable and you have asked us functions of capital city as he studied with angles of a flight simplex actual Piccoli
So what you if you put that back into into the partition function.
As this stone exponential is see Tom has a problem. He said concerts versus actual definition of suffering.
And so if you're just left with these terms which I won't mention is times the healing is of a flat simplex
And so this
It to to get for each simplex and a new triangulation.
So what you see here is basically the action.
You have also this epsilon, epsilon is just giving you the orientation of the simplex as it follows home the fact that flatness constraint have two solutions one give us the hidden Inca if you would sit inside this template someone if using a heater ingots if you sit outside simplex
S.
O. 's that value because it comes home.
So free transform what compared word Inger
Living in this one the legs are continuous variables.
And doesn't measure to
Measure tome. Oh, you can determine Boyd demonic triangulation in this order and has been shown
Lots of
By our speech no longer need it be fights you and population in my own model.
So yeah, you have to partition function and Ibiza basically the spin Bobby version.
Of also be a CG
based production function.
I'm the sense that
In fact, it turns out that
The next
Thing you labeled AND APPLAUSE And this all the time, the inequalities and post together. This is live, it's St.
Can also be labels of representations and intertwine us also feeding tubes.
And you can click on stop this action causes me to have this model believe directly from to walk representation silly as a sensor to simply simply totally arises from a collection of support to intertwine us
So this is equivalent of office, beautiful picture of what we have seen in CD.
Again here if you if you take the action to
Hear you see more or less direct is a rejection, if you would really have the rejection st would not be a fee that either, but it would be to have a tiger or six expressed also function or two legs.
So here's the point is, it s is a free valuable it's constrained on deck, because that's what you need to get to a data function.
As a deficit ingots um and if you just start from this as a political action and so constrained. Again, this integration Magnolia as t minus at you, again, immediately see that solving for Michael O'Neill and solving for the constraints you get back to the match action.
So this basically also proves that
They can be F modesty related comes from the BSC G action.
And it connects and also a picture and representation picture for this for beer for the BSC GT
What we also provided in the paper is the differentiation of vulnerable space and the quantum geometry up operators which you can define on the spot here but space.
So you basically is very much you have again with this ability to form one connection the connection.
And the template many events.
So these are represented by again beat
Me that was associated to triangles.
seasick boy leader relatives on so for
Me.
I couldn't get it to the whole mystique which are
associated to the length of the bold letters.
You
Have
Really been associated to the edges of the nation. So they present the edge towards and additionally you have many events which conforms to connection. So this will be to colonies.
And they'll associate to the
New faces.
And to do it manually do it was a triangulation.
And the timing. It's like the pine nuts.
I do it through the lens and the edges on board to the faces. So you see that it always come on autologous eclectic selection.
Be also provided
A democratization and quantization of all the constraints.
So in fact, in this topological theory that you don't even have any kind of factor ordering issues.
And it's consistent in the sense that, though. This is like structure defined or the constraints constraints. I actually feel scans. So there's one flatness to flip this one goes into ghost good sense.
And you can define a sub edge about constraints which is also first class in itself, which which is defined by one goes to goes into he implicitly constrained. So these are the constraints we have
Imposed and capacity and this if you impose them. This leads to reduce the functions, all the functions which and only on the edge. Later on two legs labels and on the
Healing is eco on just doing a power meter which you can define from
Holland. Nice.
So such functions would define what we call it g for Gauss network functions.
And future. What will we be to delete these functions explicitly to like objects and
Which come from to hope representations of his presentation matrices.
And
So the flatness constraints. So two flavors constraints.
I would he explicitly. They have a very simple form. So the flavors constraints are defined for each said what you see so complex and
The definition is similar to what we are us from the wall, one goes constrained
And the demands that the connection many beds, which are suited to the two faces. If you some this connection believers.
Associates associated to the faces which on a given. So this combination for Spanish
So as a plus minus sign come from choices of orientation. So one has to set all
The time again is one possibly has actually on a transport for these events or also one connection was a wonderful enemies appearances constraint.
But given this right this constraint is the stress given also some over these events sigma
It's easy to see. And I was going to give it to the templates, it's actually easy to see that these
Two flavors constraint generator vertical translation for the template sector.
So it says says this sentence Celsius. So it's a dual two vertices. And so give me two letters constraint would act on all the edges which are decent to given vertex and
Since you have the same given each parameter in front of the goals constraint of this tool also to flatness constraint. This would lead to the just to adding a given vector to all the inspectors adjacent to this edge.
Okay, so there has been also recent paper by by long for the living on safety and one of the points is also that the define
That
Include pirates actually
I was very the tides are defined on boundaries and they have conjugated momentum from this point they can define the often constrained
And so this is obligation because here we have also templates so to flatness constitute connection is quite a bit to the side. That's the two flatness constraints act as a center tune in and different office and constraints.
Right. Let me summarize so
This is more
Advertisements at higher dt. It's an alternative approach to quantum geometry and look on gravity users. He itself.
And US US poach to the quantum geometry.
So here we generalize it to me. And I think has a particular advantages and sleep as one dimensions.
So, we have seen that includes the templates and we get also geometric transplant Hamiltonian, a different model constraints and this type of sexual
So what we put in particular did and our people is to construct the relation between the two group and representation picture of this quantum flat space model, and that includes a bona yes basis or constraints and the G network functions, the definition of the genetic functions.
Rather, many things to do.
On the on the mathematical side and on the precipice on it.
This this project actually to look into to this model of quantum flat space time. It was actually motivated to understand calligraphy and slippers one dimensions.
And it's easier to understand, philosophy and see first one dimensions, if you, if you go to quantum flat space. So here you can do that explicitly. I'm using both the picture and
Put the presentation picture.
Again, this is motivated by our broken and purpose one the enemy who local fee busy. The solution within this picture was essential to divide all these mice. Besides, we had
So he can do you can think about similar setup to define two theories on and on the CD bonding.
Here we use dramatic insight to construct basically to replace it totally transform, but we think that this actually can be used to define it to hopefully transform also for more general talks so that would be something on the mathematical side.
Um,
Another advantage is now we have be BSC G.
As an action and
It has been go by is and an openness, for instance, and coupling metal boy your defects to the St. Louis, and since f is a tool for what you need to couple
Things.
But now we have also the CG Terminator took a picture. So you can also come here, our tickets
And in fact I'm highly engaged theory and higher categories are usually used to introduce boundaries of law and order I mentioned and your finger and so strings, basically, if you will.
Because when I mentioned on here and articles, if you lower dimension one plus one dimension boundary. So that's
All consistent with this idea of
Now, being able to accomplish things in particular.
But also relates to the caveat model because the PDF model was busy invented as a kind of the to copy Fineman items on offer.
Particles propagating articles to some kind of lead quantum slide deck on
Various, it could be also some connections with Telefonica key for the simple reason that they have a portion explicitly appearing in this window.
For one interesting observation is to generalize too much of space time in particular to the place of war by, say something.
Then, so this, this will be
More
Mathematical side.
Of imposing simplicity concerns and face face into quantum so the fan voted interesting functions and this case I fled building blocks. We have teams at the SPS
Commerce discrete spectrum because he's forgiven by these S3 buckets, which we have that valued as the length of this continuous vector was eventually these continuous spectral conforms affected be used for
So if you if you're imposing constraints at NSF to be equal to the lens. You can also which which spectrum to actually survive in this case I'm most icing is actually to
Look, it's a geometric
Geometry, which is encoded in the screen opens
To fly things could sense which generate nicely vertex translation and to use it to teach a better understanding and possibly to import them, it only sense
Okay, so I would close here.
I have only two extra slides. If anyone has a question about what the water to group and water to connection actually is. If not,
We open for questions.
If
You see the definition of that is actually related to responding to form that appears in the definitional possible
In the definition of what version of their
Concert concert concert content is such as a relation to such as a look so
Because it seems to me that it's to connection cheese. Who's exterior derivative was combined exterior is the
Answer is not
Okay, I would have to explicitly what's us wanting to form is
I mentioned this.
Possible relation to too.
Long and yellow.
And they define this quantity. In fact, in terms of cost of charges.
Which come from integrations over regions or word boundaries of these features so
And in fact, what what seems so
I could be related to that. So the two flattens concerned, you can infect understand also kind of conversation conservation law.
In fact, it's a it's a one bank identity, which then you can also understand that.
And also to one.
So you can you can understand also also
As convert conversation.
What an economist and us also equation of motion. If you import a Sophistic on St. This to connection.
Seems to
Claim is that it's finishing
And if that's the case, it makes a dramatic interpretation somewhat difficult
So,
Its purpose.
And the observation.
On cello some physical constraint is related to the
Matter.
Is is cheap, is the script values to to flatness becomes one of the flux of his press ENTER depends for across the building.
So I don't know what this platform in se, but it sounds awesome.
Okay.
Can you
Know, see what's possible brackets.
Between alanna Musa Fluxus
And
Needs background independence independent this meeting because two plus one. And yeah, you are this one plus two, can you can you explain was structural comes out and does a barbarian Mitzi parameter appear on the right hand
Side so so to answer the last question we have not yet available in music parameter. But you think it's possible but
I'm
The beta quantization of these of these websites.
So again, as you to select one
Which is very similar to my which which is a b f which comes from to the afternoon. So this sector will look exactly like and look on
So P is conjugated to G is a piece to commute and so on.
Then only we have also put a subject exceptional on the Air Force ECHO and this is that
10 megabits a canonical on you get it to the connection.
And both are for. So, that makes sense. And the templates in this for so they actually mute.
And just selecting like it's building the template feeds vanish. And so we still plan to the iPad with all these modern methods of driving subtracting
Structures for silly only owners and so on. But that's just like instruction.
And there you have said basically beefiest to be few there's two dimension to dimension things G sweet or one dimension, things that I asked me it over the edges.
And so the conjugated valuable connection.
Is defined on the faces.
So awesome standard gravity.
All these structures are basically consistent in the sense that the dimension it up those variables limits which are connected to each other, always defined on on
An object like a link and the tool object. In this case, a triangle or an edge or face and so on.
And so we could also buy down or constraints and even into discrete with all kinds of choices of public transport, you need to make
And then you can compute that all these constraints effect on all to be first class. We have a consistent percentage, wha, and this constraint algebra and it reflects upon center support you.
Thank you.
Okay, any other questions.
Yeah, it's how can you hear me.
Yeah.
I was when you spoke about the two representation labels that
Was like
Okay.
And the fact that the SLA bows and integer clearly imposes some sort of constraints on the elves of that triangle.
And
Oh you disappeared.
Oh, no. I was just wondering if the the constraints imposed on the elves have some interpretation in terms of the representation theory.
By Tuesday yesterday was being imposed on St. John's day
Well, because there's only some else that will satisfy that the the area.
Was every else. Yeah, so
If you import kind of simplicity
That's what you're asking. Yes.
Yeah, in fact,
If you, if you do it. And this, let me see in a fee.
Which I'm seeing naive because
There will be also secondary constraints you
A picture.
Which I have not looking. And actually, if you, if you look at this, if your answer is contained in the actual meal. You see that lunch, right. You play against actually fixed to be to be the deficit in so
Which means you can expect that this constraint. The second house, but also is always the case with this client. So now you can choose
It's sort of a textbook style you actually that you have to go back to classic a silly.
And go to the advocates and so on.
It can also choose any kind of
Version, which is just a spectacle quantization.
And what I want to say is it's not very clear.
If you will really go back to formulate formalism that opposing this constraints that the SEC between and one is this
Very good insight that you should. That is actually survives assume this is what survives and effect. It's an element, which has been made by Voinovich just that he claims.
Or Zelda slick that solutions on the air is one of the solutions is at all edits, or just equal to one, for instance, or equal to some global scale.
Which I don't think it's a nice solution.
Because I usually if you go back, if you go to larger, larger
Next variables that you will find
More solutions so that
So the question is, is it likely to do if you do it, then you will find stone, stone constraints on on the it's it's maybe an interesting question. How these constraints behave
If you go to larger links for me.
And all this events to find a dense set of solutions or not.
Yeah.
Good. I
Like it just comes creation. He
Said,
Before we have. So to see order as one
Yeah.
And then
Which is the contract.
Yeah, that'd be continuous partially
Series.
Continues
So,
So partially or two. So to see or the scope and you would expect that some of these SS
Have to be continuous another interesting question. Just go to curve building blocks. If you go to craft a TD and maybe the editor become the seat, but it's when somebody to me.
So certainly there's an interesting question register which kind of spectrum might survive if you try to also success equals. Thanks.
Thank you.
Are there further questions.
Does not appear to be the case. So that's Bianca again.
Thank you all.
Right.