Quantum Geometry from Higher Gauge Theoryrelativity.phys.lsu.edu/ilqgs/dittrich091019.pdf · 2019. 9. 9. · Quantum Geometry from Higher Gauge Theory. Main Message: Prospect for

Bianca Dittrich (Perimeter Institute)

withSeth Asante, Florian Girelli,Aldo Riello,Panagiotis Tsimiklis.

arxiv: 1908.05970

ILQGS Sep 10, 2019

Quantum GeometryfromHigher Gauge Theory

Main Message:

Prospect for a new 4d quantum geometry• Includes tetrads as basic variables (as opposed to just bi-vectors)

• Conjugated momenta: 2-connection, which is a two-form

• 2-curvature is a three-form: (space-time) diffeomorphism generator

• 2-curvature generates vertex translations (as opposed to edge translations)

Offers many advantages on the kinematic and dynamic level.

Overview

1.Motivation: 3D path integral for quantum gravity:

Trick: Enlarge configuration space

2. 4D quantum gravity - does it work there?

3. We should enlarge even more: BFCG action

4. Quantization, spin bubble model and quantum geometry

5. Outlook

3

Almost no (2-category) abstract non-sense

Motivation: 3D gravity - path integral

l1<latexit sha1_base64="vHJlzNlNbmXQOyxFuDi0QsChg1Q=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0IPpev1xxq+4cZJV4OalAjka//NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQllYzrErqWSRqj9bH7qlJxZZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMTXvsZl0lqULLFojAVxMRk9jcZcIXMiIkllClubyVsRBVlxqZTsiF4yy+vklat6l1Ua/eXlfpNHkcRTuAUzsGDK6jDHTSgCQyG8Ayv8OYI58V5dz4WrQUnnzmGP3A+fwD6p42Y</latexit>

l2<latexit sha1_base64="cGrDOFyL5pGKO3TAZPX9es7d10c=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0IPq1frniVt05yCrxclKBHI1++as3iFkaoTRMUK27npsYP6PKcCZwWuqlGhPKxnSIXUsljVD72fzUKTmzyoCEsbIlDZmrvycyGmk9iQLbGVEz0sveTPzP66YmvPYzLpPUoGSLRWEqiInJ7G8y4AqZERNLKFPc3krYiCrKjE2nZEPwll9eJa1a1buo1u4vK/WbPI4inMApnIMHV1CHO2hAExgM4Rle4c0Rzovz7nwsWgtOPnMMf+B8/gD8K42Z</latexit> l3

<latexit sha1_base64="TdXgAHJpejrcKXi6+g1CF2A3sI8=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0laQY9FLx4r2lpoQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgbjm5n/+IRK81g+mEmCfkSHkoecUWOle9Gv98sVt+rOQVaJl5MK5Gj2y1+9QczSCKVhgmrd9dzE+BlVhjOB01Iv1ZhQNqZD7FoqaYTaz+anTsmZVQYkjJUtachc/T2R0UjrSRTYzoiakV72ZuJ/Xjc14ZWfcZmkBiVbLApTQUxMZn+TAVfIjJhYQpni9lbCRlRRZmw6JRuCt/zyKmnXql69Wru7qDSu8ziKcAKncA4eXEIDbqEJLWAwhGd4hTdHOC/Ou/OxaC04+cwx/IHz+QP9r42a</latexit>

Domain of integration very complicated to impose:

• lengths positive

• triangle inequalities

• tetrahedral inequalities

non-local

constraints}

<latexit sha1_base64="z9I03uDaA3bS8qEriCB/zLQ+3nM=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF49V7Ae0oWy2m3bpZhN2J0IJ/QdePCji1X/kzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRRI3g7GtzO//cS1EbF6xEnC/YgOlQgFo2ilh960X664VXcOskq8nFQgR6Nf/uoNYpZGXCGT1Jiu5yboZ1SjYJJPS73U8ISyMR3yrqWKRtz42fzSKTmzyoCEsbalkMzV3xMZjYyZRIHtjCiOzLI3E//zuimG134mVJIiV2yxKEwlwZjM3iYDoTlDObGEMi3srYSNqKYMbTglG4K3/PIqadWq3kW1dn9Zqd/kcRThBE7hHDy4gjrcQQOawCCEZ3iFN2fsvDjvzseiteDkM8fwB87nD5/ajWs=</latexit>

Z =

ZDl e

i~Sgrav(l)

<latexit sha1_base64="ImZbRuuwrvTRF8jpbc54E2pA8BQ=">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</latexit>

complicated

action+

Motivation: 3D gravity - path integral

l1<latexit sha1_base64="vHJlzNlNbmXQOyxFuDi0QsChg1Q=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0IPpev1xxq+4cZJV4OalAjka//NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQllYzrErqWSRqj9bH7qlJxZZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMTXvsZl0lqULLFojAVxMRk9jcZcIXMiIkllClubyVsRBVlxqZTsiF4yy+vklat6l1Ua/eXlfpNHkcRTuAUzsGDK6jDHTSgCQyG8Ayv8OYI58V5dz4WrQUnnzmGP3A+fwD6p42Y</latexit>

l2<latexit sha1_base64="cGrDOFyL5pGKO3TAZPX9es7d10c=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0IPq1frniVt05yCrxclKBHI1++as3iFkaoTRMUK27npsYP6PKcCZwWuqlGhPKxnSIXUsljVD72fzUKTmzyoCEsbIlDZmrvycyGmk9iQLbGVEz0sveTPzP66YmvPYzLpPUoGSLRWEqiInJ7G8y4AqZERNLKFPc3krYiCrKjE2nZEPwll9eJa1a1buo1u4vK/WbPI4inMApnIMHV1CHO2hAExgM4Rle4c0Rzovz7nwsWgtOPnMMf+B8/gD8K42Z</latexit> l3

<latexit sha1_base64="TdXgAHJpejrcKXi6+g1CF2A3sI8=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0laQY9FLx4r2lpoQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgbjm5n/+IRK81g+mEmCfkSHkoecUWOle9Gv98sVt+rOQVaJl5MK5Gj2y1+9QczSCKVhgmrd9dzE+BlVhjOB01Iv1ZhQNqZD7FoqaYTaz+anTsmZVQYkjJUtachc/T2R0UjrSRTYzoiakV72ZuJ/Xjc14ZWfcZmkBiVbLApTQUxMZn+TAVfIjJhYQpni9lbCRlRRZmw6JRuCt/zyKmnXql69Wru7qDSu8ziKcAKncA4eXEIDbqEJLWAwhGd4hTdHOC/Ou/OxaC04+cwx/IHz+QP9r42a</latexit>

Domain of integration very complicated to impose:

• lengths positive

• triangle inequalities

• tetrahedral inequalities

non-local

constraints}

<latexit sha1_base64="z9I03uDaA3bS8qEriCB/zLQ+3nM=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF49V7Ae0oWy2m3bpZhN2J0IJ/QdePCji1X/kzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRRI3g7GtzO//cS1EbF6xEnC/YgOlQgFo2ilh960X664VXcOskq8nFQgR6Nf/uoNYpZGXCGT1Jiu5yboZ1SjYJJPS73U8ISyMR3yrqWKRtz42fzSKTmzyoCEsbalkMzV3xMZjYyZRIHtjCiOzLI3E//zuimG134mVJIiV2yxKEwlwZjM3iYDoTlDObGEMi3srYSNqKYMbTglG4K3/PIqadWq3kW1dn9Zqd/kcRThBE7hHDy4gjrcQQOawCCEZ3iFN2fsvDjvzseiteDkM8fwB87nD5/ajWs=</latexit>

Trick: enlarge configuration space … (and constrain again)

Triads and spin connection:

Z =

ZDl e

i~Sgrav(l)

<latexit sha1_base64="ImZbRuuwrvTRF8jpbc54E2pA8BQ=">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</latexit>

g = e · e � 0 , !<latexit sha1_base64="VGwlrGaXK9BSKzIqVWOA45AUHTY=">AAACDnicbVA9SwNBEN2LXzF+RS1tFkPAIoS7KGgjBG0sI5hEyIWwtze5LNm7vezuCeHIL7Dxr9hYKGJrbee/cS9JoYkPhnm8N8PuPC/mTGnb/rZyK6tr6xv5zcLW9s7uXnH/oKVEIik0qeBC3ntEAWcRNDXTHO5jCST0OLS94XXmtx9AKiaiOz2OoRuSIGJ9Rok2Uq9YDi4xYJf6Qmc9gBG23YpbqWB3lBAfuyKEgPSKJbtqT4GXiTMnJTRHo1f8cn1BkxAiTTlRquPYse6mRGpGOUwKbqIgJnRIAugYGpEQVDednjPBZaP4uC+kqUjjqfp7IyWhUuPQM5Mh0QO16GXif14n0f2LbsqiONEQ0dlD/YRjLXCWDfaZBKr52BBCJTN/xXRAJKHaJFgwITiLJy+TVq3qnFZrt2el+tU8jjw6QsfoBDnoHNXRDWqgJqLoET2jV/RmPVkv1rv1MRvNWfOdQ/QH1ucPhQOZ5g==</latexit>

Palatini action:

(su(2)-valued one-forms)

S =

ZTr(e ^ F (!))

<latexit sha1_base64="M1RSQTWpLw9ZPhxR2+dDonIAylE=">AAACDXicbVA9SwNBEN3z2/h1ammzGIWkCXdR0EYQBbFUNEbIHWFvM4lL9j7YnVPDkT9g41+xsVDE1t7Of+MmXqGJDwYe780wMy9IpNDoOF/WxOTU9Mzs3HxhYXFpecVeXbvScao41HgsY3UdMA1SRFBDgRKuEwUsDCTUg+7xwK/fgtIiji6xl4Afsk4k2oIzNFLT3ro4oJ6IkHoI95hdqn4JvDtodYCelLw4hA4rl5t20ak4Q9Bx4uakSHKcNe1PrxXzNIQIuWRaN1wnQT9jCgWX0C94qYaE8S7rQMPQiIWg/Wz4TZ9uG6VF27EyZQ4bqr8nMhZq3QsD0xkyvNGj3kD8z2uk2N73MxElKULEfxa1U0kxpoNoaEso4Ch7hjCuhLmV8humGEcTYMGE4I6+PE6uqhV3p1I93y0eHuVxzJENsklKxCV75JCckjNSI5w8kCfyQl6tR+vZerPef1onrHxmnfyB9fENp1iapw==</latexit>

example for a BF action: S =

ZTr(B ^ F (A))

<latexit sha1_base64="/l3uLzvMM6ItBUIftYYkPsYVSQQ=">AAACCHicbVDLSgNBEJz1GeMr6tGDg0FILmE3CnoRYgTxGDFRIRvC7KSTDJl9MNOrhiVHL/6KFw+KePUTvPk3Th4HjRY0FFXddHd5kRQabfvLmpmdm19YTC2ll1dW19YzG5tXOowVhxoPZahuPKZBigBqKFDCTaSA+Z6Ea693OvSvb0FpEQZV7EfQ8FknEG3BGRqpmdm5PKauCJC6CPeYVNUgV3bvoNUBepY7yeebmaxdsEegf4kzIVkyQaWZ+XRbIY99CJBLpnXdsSNsJEyh4BIGaTfWEDHeYx2oGxowH3QjGT0yoHtGadF2qEyZm0bqz4mE+Vr3fc90+gy7etobiv959RjbR41EBFGMEPDxonYsKYZ0mAptCQUcZd8QxpUwt1LeZYpxNNmlTQjO9Mt/yVWx4OwXihcH2VJ5EkeKbJNdkiMOOSQlck4qpEY4eSBP5IW8Wo/Ws/VmvY9bZ6zJzBb5BevjGzwImC4=</latexit>

defined in dimensions >1(d� 2)

<latexit sha1_base64="u5A4pJgezph9ZGQs0XSfltVC0R4=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBahHixJFPRY9OKxgmkLbSibzaZdutmE3Y1QQn+DFw+KePUHefPfuG1z0NYHA4/3ZpiZF6ScKW3b31ZpbX1jc6u8XdnZ3ds/qB4etVWSSUI9kvBEdgOsKGeCepppTruppDgOOO0E47uZ33miUrFEPOpJSv0YDwWLGMHaSF49vHDPB9Wa3bDnQKvEKUgNCrQG1a9+mJAspkITjpXqOXaq/RxLzQin00o/UzTFZIyHtGeowDFVfj4/dorOjBKiKJGmhEZz9fdEjmOlJnFgOmOsR2rZm4n/eb1MRzd+zkSaaSrIYlGUcaQTNPschUxSovnEEEwkM7ciMsISE23yqZgQnOWXV0nbbTiXDffhqta8LeIowwmcQh0cuIYm3EMLPCDA4Ble4c0S1ov1bn0sWktWMXMMf2B9/gBqCY3E</latexit>

-form

SU(2) gauge freedom

complicated

action+

[Horowitz 89]

Path integral for first order 3D gravitypath integral for Palatini action path integral over flat connections

Closed manifold Einstein-Cartan QG gravity

Introduce a discretization of the manifold completely capturing its topology

To awitch to “metric” variables, use Peter-Weyl thm + SU(2) recoupling theory

11

Ponzano-Regge model in a sketch

A theory of Flat connections

The j’s are eigenvalues of

length operator

For large q-numbers, it reproduces discrete Einstein-Hilbert

FIGURE

Z =

ZD[!]

Y

x

�(Fx[!])<latexit sha1_base64="6i0+aUjhhb5Wf51ldPLfW+3k1dw=">AAACH3icbVBNS8NAEN34bf2qevSyWIQKUhIV9SKIinhUsFVsQphspnVxkw27G7GE/hMv/hUvHhQRb/4bt7WCXw8GHu/NMDMvygTXxnXfnaHhkdGx8YnJ0tT0zOxceX6hoWWuGNaZFFJdRKBR8BTrhhuBF5lCSCKB59H1Qc8/v0GluUzPTCfDIIF2ylucgbFSWN663PV5amjhMxD0sNv0ZYJtCKi/Rv1MyTi8pX6MwkD1KLz9clfDcsWtuX3Qv8QbkAoZ4CQsv/mxZHmCqWECtG56bmaCApThTGC35OcaM2DX0MampSkkqIOi/1+Xrlglpi2pbNlb++r3iQISrTtJZDsTMFf6t9cT//OauWntBAVPs9xgyj4XtXJBjaS9sGjMFTIjOpYAU9zeStkVKGDGRlqyIXi/X/5LGus1b6O2frpZ2dsfxDFBlsgyqRKPbJM9ckxOSJ0wckceyBN5du6dR+fFef1sHXIGM4vkB5z3D1oxoe8=</latexit>

Z =

ZD[e]D[!] e

i2`Pl

RTr(e^F (!))

<latexit sha1_base64="6likPpwiLoD0NWXlTJjQYWnQe6w=">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</latexit>




11




length operator


FIGURE

Path integral for first order 3D gravitypath integral for Palatini action

Regularize on a lattice (dual to a triangulation):

path integral over flat connections




11




length operator


FIGURE

20

The first order formulation of general relativity, also known as Einstein–Cartan(–Palatini) grav-ity, is written in terms of a spin connection !ab

µ dxµ and a vielbein eaµdxµ, such that gµ⌫ = ⌘abeaµe

b⌫

with ⌘ab the flat metric of appropriate signature. This two sets of variables are considered to beindependent. The equations of motion, for invertible vielbeins, will imply that ! is a torsion freeLevi-Civita connection and gµ⌫ satisfies Einstein equations. In 3d, and in absence of a cosmolog-ical constant, first order general relativity is just an instantiation of BF theory. The associatedpartition function Z(M) for a (closed) 3-manifold M is formally given by

Z(M) =

ZDeD! e�iSBF [e,!] where SBF =

1

2`Pl

Z

M

ea ^ F a[!], (III.1)

where F a = ✏abcF bc, and F [!] = d! + ! ^ ! the curvature of !.Using the linearity of the BF action in the triad field e, one can formally integrate it out,

obtaining

Z(M) =

ZD! �(F [!]). (III.2)

Thus, we see that Z(M) computes the volume of the moduli space of flat spin connections on themanifold M .

This form of the partition function can be easily regularized through a discretization.23

Let us first introduce some notation, that will useful in the following too. Denote by � ={v, e, t,�} a simplicial decomposition of the three manifold M , where (v, e, t,�) are labels for itsvertices, edges, triangles, and tetrahedra, respectively. Denote then by �⇤ the Poincare dualcellular complex. It is composed by nodes n = �⇤ (zero-dimensional dual tetrahedra), links l = t⇤

(dual to triangles), faces f = e⇤ (dual to triangulation edges), and “bubbles” b = v⇤ (dual totriangulation vertices). Faces and edges need to be considered as (arbitrarily) oriented objects.

The introduction of � allows us to replace the continuum spin connection ! in (III.2) by SU(2)holonomies gl associated to the dual links l. Thus the discretization of the partition function (III.2)is given by

ZPR-group =

"Y

l

Z

SU(2)

dgl

#Y

f

�

✓ �Y

l3fg✏(l,f)l

◆, (III.3)

where ✏(l, f) = ±1 is the relative orientation of the link l and the face f , dg is the Haar measureon SU(2), and �(·) the corresponding group delta-function,

Rdg�(h�1g)f(g) = f(h). Notice that

ZPR-group computes also the (Haar) volume of the moduli space of flat SU(2) connections, but nowon the discretized manifold.

As we will explain this form of the partition function gives already the group (or holonomy)representation of the Ponzano–Regge model, hence the label ‘PR-group’. We will arrive at the orig-inal form of the Ponzano–Regge model, or the so–called spin representation, after a group Fouriertransform. Before explaining this we make a couple of remarks about the group representation.

First of all, in order to write the amplitude in the group representation, any cellular decom-position of the manifold would work. The condition of � being a simplicial decomposition of Mcan hence be dropped. This generalized form of the partition function can also translated into thespin picture, thus yielding a generalization of the original PR model, which is conversely based

23 An alternative derivation of the Ponzano–Regge model discretizes the BF action SBF together with its dynamicalvariables. This makes the formal integration over the triad fields in the now discretized path integral well defined.One then arrives also at (III.3) – however with SO(3) delta–functions instead of SU(2) delta–functions [22, 112].

Z =

ZD[!]

Y

x


Z =

ZD[e]D[!] e

i2`Pl

RTr(e^F (!))





11




length operator


FIGURE

g1<latexit sha1_base64="tNG2Bn+fACMu25bNhVTjzxMqoo4=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3TpZhN2N0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O4W19Y3NreJ2aWd3b/+gfHjU0kmmGDZZIhLVCahGwSU2DTcCO6lCGgcC28Hodua3n1BpnshHM07Rj2kkecgZNVZ6iPpev1xxq+4cZJV4OalAjka//NUbJCyLURomqNZdz02NP6HKcCZwWuplGlPKRjTCrqWSxqj9yfzUKTmzyoCEibIlDZmrvycmNNZ6HAe2M6ZmqJe9mfif181MeO1PuEwzg5ItFoWZICYhs7/JgCtkRowtoUxxeythQ6ooMzadkg3BW355lbRqVe+iWru/rNRv8jiKcAKncA4eXEEd7qABTWAQwTO8wpsjnBfn3flYtBacfOYY/sD5/AHzCY2T</latexit> g2

<latexit sha1_base64="zS5ajz98bp6SfgkudQ3O35BwjW4=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3TpZhN2N0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O4W19Y3NreJ2aWd3b/+gfHjU0kmmGDZZIhLVCahGwSU2DTcCO6lCGgcC28Hodua3n1BpnshHM07Rj2kkecgZNVZ6iPq1frniVt05yCrxclKBHI1++as3SFgWozRMUK27npsaf0KV4UzgtNTLNKaUjWiEXUsljVH7k/mpU3JmlQEJE2VLGjJXf09MaKz1OA5sZ0zNUC97M/E/r5uZ8NqfcJlmBiVbLAozQUxCZn+TAVfIjBhbQpni9lbChlRRZmw6JRuCt/zyKmnVqt5FtXZ/Wanf5HEU4QRO4Rw8uII63EEDmsAggmd4hTdHOC/Ou/OxaC04+cwx/IHz+QP0jY2U</latexit>

g3<latexit sha1_base64="JG6QxsGuk6JS91LQp2rr9CAGHRc=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0laQY9FLx4r2lpoQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgbjm5n/+IRK81g+mEmCfkSHkoecUWOl+2G/3i9X3Ko7B1klXk4qkKPZL3/1BjFLI5SGCap113MT42dUGc4ETku9VGNC2ZgOsWuppBFqP5ufOiVnVhmQMFa2pCFz9fdERiOtJ1FgOyNqRnrZm4n/ed3UhFd+xmWSGpRssShMBTExmf1NBlwhM2JiCWWK21sJG1FFmbHplGwI3vLLq6Rdq3r1au3uotK4zuMowgmcwjl4cAkNuIUmtIDBEJ7hFd4c4bw4787HorXg5DPH8AfO5w/2EY2V</latexit>

g4<latexit sha1_base64="8I/WkfEhE/x6NahUpaTPSkWKTrA=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4Kkkt6LHoxWNF+wFtKJvtJF262YTdjVBKf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXpIJr47rfztr6xubWdmGnuLu3f3BYOjpu6SRTDJssEYnqBFSj4BKbhhuBnVQhjQOB7WB0O/PbT6g0T+SjGafoxzSSPOSMGis9RP1av1R2K+4cZJV4OSlDjka/9NUbJCyLURomqNZdz02NP6HKcCZwWuxlGlPKRjTCrqWSxqj9yfzUKTm3yoCEibIlDZmrvycmNNZ6HAe2M6ZmqJe9mfif181MeO1PuEwzg5ItFoWZICYhs7/JgCtkRowtoUxxeythQ6ooMzadog3BW355lbSqFe+yUr2vles3eRwFOIUzuAAPrqAOd9CAJjCI4Ble4c0Rzovz7nwsWtecfOYE/sD5/AH3lY2W</latexit>

Path integral for first order 3D gravitypath integral for Palatini action

Regularize on a lattice (dual to a triangulation):

path integral over flat connections




11




length operator


FIGURE

20

The first order formulation of general relativity, also known as Einstein–Cartan(–Palatini) grav-ity, is written in terms of a spin connection !ab

µ dxµ and a vielbein eaµdxµ, such that gµ⌫ = ⌘abeaµe

b⌫

with ⌘ab the flat metric of appropriate signature. This two sets of variables are considered to beindependent. The equations of motion, for invertible vielbeins, will imply that ! is a torsion freeLevi-Civita connection and gµ⌫ satisfies Einstein equations. In 3d, and in absence of a cosmolog-ical constant, first order general relativity is just an instantiation of BF theory. The associatedpartition function Z(M) for a (closed) 3-manifold M is formally given by

Z(M) =

ZDeD! e�iSBF [e,!] where SBF =

1

2`Pl

Z

M

ea ^ F a[!], (III.1)

where F a = ✏abcF bc, and F [!] = d! + ! ^ ! the curvature of !.Using the linearity of the BF action in the triad field e, one can formally integrate it out,

obtaining

Z(M) =

ZD! �(F [!]). (III.2)

Thus, we see that Z(M) computes the volume of the moduli space of flat spin connections on themanifold M .

This form of the partition function can be easily regularized through a discretization.23

Let us first introduce some notation, that will useful in the following too. Denote by � ={v, e, t,�} a simplicial decomposition of the three manifold M , where (v, e, t,�) are labels for itsvertices, edges, triangles, and tetrahedra, respectively. Denote then by �⇤ the Poincare dualcellular complex. It is composed by nodes n = �⇤ (zero-dimensional dual tetrahedra), links l = t⇤

(dual to triangles), faces f = e⇤ (dual to triangulation edges), and “bubbles” b = v⇤ (dual totriangulation vertices). Faces and edges need to be considered as (arbitrarily) oriented objects.

The introduction of � allows us to replace the continuum spin connection ! in (III.2) by SU(2)holonomies gl associated to the dual links l. Thus the discretization of the partition function (III.2)is given by

ZPR-group =

"Y

l

Z

SU(2)

dgl

#Y

f

�

✓ �Y

l3fg✏(l,f)l

◆, (III.3)

where ✏(l, f) = ±1 is the relative orientation of the link l and the face f , dg is the Haar measureon SU(2), and �(·) the corresponding group delta-function,

Rdg�(h�1g)f(g) = f(h). Notice that

ZPR-group computes also the (Haar) volume of the moduli space of flat SU(2) connections, but nowon the discretized manifold.

As we will explain this form of the partition function gives already the group (or holonomy)representation of the Ponzano–Regge model, hence the label ‘PR-group’. We will arrive at the orig-inal form of the Ponzano–Regge model, or the so–called spin representation, after a group Fouriertransform. Before explaining this we make a couple of remarks about the group representation.

First of all, in order to write the amplitude in the group representation, any cellular decom-position of the manifold would work. The condition of � being a simplicial decomposition of Mcan hence be dropped. This generalized form of the partition function can also translated into thespin picture, thus yielding a generalization of the original PR model, which is conversely based

23 An alternative derivation of the Ponzano–Regge model discretizes the BF action SBF together with its dynamicalvariables. This makes the formal integration over the triad fields in the now discretized path integral well defined.One then arrives also at (III.3) – however with SO(3) delta–functions instead of SU(2) delta–functions [22, 112].

Variable transformation via group Fourier transform

Z =

ZD[!]

Y

x


Z =

ZD[e]D[!] e

i2`Pl

RTr(e^F (!))





11




length operator


FIGURE

g1<latexit sha1_base64="tNG2Bn+fACMu25bNhVTjzxMqoo4=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3TpZhN2N0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O4W19Y3NreJ2aWd3b/+gfHjU0kmmGDZZIhLVCahGwSU2DTcCO6lCGgcC28Hodua3n1BpnshHM07Rj2kkecgZNVZ6iPpev1xxq+4cZJV4OalAjka//NUbJCyLURomqNZdz02NP6HKcCZwWuplGlPKRjTCrqWSxqj9yfzUKTmzyoCEibIlDZmrvycmNNZ6HAe2M6ZmqJe9mfif181MeO1PuEwzg5ItFoWZICYhs7/JgCtkRowtoUxxeythQ6ooMzadkg3BW355lbRqVe+iWru/rNRv8jiKcAKncA4eXEEd7qABTWAQwTO8wpsjnBfn3flYtBacfOYY/sD5/AHzCY2T</latexit> g2

<latexit sha1_base64="zS5ajz98bp6SfgkudQ3O35BwjW4=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3TpZhN2N0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAqujet+O4W19Y3NreJ2aWd3b/+gfHjU0kmmGDZZIhLVCahGwSU2DTcCO6lCGgcC28Hodua3n1BpnshHM07Rj2kkecgZNVZ6iPq1frniVt05yCrxclKBHI1++as3SFgWozRMUK27npsaf0KV4UzgtNTLNKaUjWiEXUsljVH7k/mpU3JmlQEJE2VLGjJXf09MaKz1OA5sZ0zNUC97M/E/r5uZ8NqfcJlmBiVbLAozQUxCZn+TAVfIjBhbQpni9lbChlRRZmw6JRuCt/zyKmnVqt5FtXZ/Wanf5HEU4QRO4Rw8uII63EEDmsAggmd4hTdHOC/Ou/OxaC04+cwx/IHz+QP0jY2U</latexit>

g3<latexit sha1_base64="JG6QxsGuk6JS91LQp2rr9CAGHRc=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0laQY9FLx4r2lpoQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgbjm5n/+IRK81g+mEmCfkSHkoecUWOl+2G/3i9X3Ko7B1klXk4qkKPZL3/1BjFLI5SGCap113MT42dUGc4ETku9VGNC2ZgOsWuppBFqP5ufOiVnVhmQMFa2pCFz9fdERiOtJ1FgOyNqRnrZm4n/ed3UhFd+xmWSGpRssShMBTExmf1NBlwhM2JiCWWK21sJG1FFmbHplGwI3vLLq6Rdq3r1au3uotK4zuMowgmcwjl4cAkNuIUmtIDBEJ7hFd4c4bw4787HorXg5DPH8AfO5w/2EY2V</latexit>

g4<latexit sha1_base64="8I/WkfEhE/x6NahUpaTPSkWKTrA=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4Kkkt6LHoxWNF+wFtKJvtJF262YTdjVBKf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXpIJr47rfztr6xubWdmGnuLu3f3BYOjpu6SRTDJssEYnqBFSj4BKbhhuBnVQhjQOB7WB0O/PbT6g0T+SjGafoxzSSPOSMGis9RP1av1R2K+4cZJV4OSlDjka/9NUbJCyLURomqNZdz02NP6HKcCZwWuxlGlPKRjTCrqWSxqj9yfzUKTm3yoCEibIlDZmrvycmNNZ6HAe2M6ZmqJe9mfif181MeO1PuEwzg5ItFoWZICYhs7/JgCtkRowtoUxxeythQ6ooMzadog3BW355lbSqFe+yUr2vles3eRwFOIUzuAAPrqAOd9CAJjCI4Ble4c0Rzovz7nwsWtecfOYE/sD5/AH3lY2W</latexit>

(g) =X

j,m,n

(j,m, n)Djmn(g)

<latexit sha1_base64="Tn0SiEGzi9D/xz53H8O0Sa3/2BQ=">AAACH3icbVDLSgMxFM3UV62vqks3wSK0UMpMFXUjFHXhsoJ9QKcOmTRt0yaZIckIZeifuPFX3LhQRNz1b0ynXWjrgcDhnHO5uccPGVXatidWamV1bX0jvZnZ2t7Z3cvuH9RVEElMajhggWz6SBFGBalpqhlphpIg7jPS8Ic3U7/xRKSigXjQo5C0OeoJ2qUYaSN52XM3VDTfK8Ar6KqIe/GgyItiDF1NWYfAxE2kAoS3jwMv5mJs4l42Z5fsBHCZOHOSA3NUvey32wlwxInQmCGlWo4d6naMpKaYkXHGjRQJER6iHmkZKhAnqh0n943hiVE6sBtI84SGifp7IkZcqRH3TZIj3VeL3lT8z2tFunvZjqkII00Eni3qRgzqAE7Lgh0qCdZsZAjCkpq/QtxHEmFtKs2YEpzFk5dJvVxyTkvl+7Nc5XpeRxocgWOQBw64ABVwB6qgBjB4Bq/gHXxYL9ab9Wl9zaIpaz5zCP7AmvwAgG2gxw==</latexit>

Wigner (representation) matrices into path integral

Path integral for first order 3D gravity

6j symbols (spin network evaluation) associated to tetrahedra j1

j2

j3

[Ponzano-Regge 1968; with positive CC: Turaev-Viro 90’s] Spins give length of edges.

All inequalities implemented.

“State sum” over spin configurations (representation labels):

ZPR�spin =X

{je}

Y

tetra

{6je2tetra}<latexit sha1_base64="oPae74Mzdm4wgmTI0hH9j85r+do=">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</latexit>



j2

j3

[Ponzano-Regge 1968; with positive CC: Turaev-Viro 90’s]


• Well defined (after gauge fixing) path integral• Diffeo-symmetry and consistent Hamiltonians / constraint algebra (translations of vertices) • canonical quantization = path integral quantization• equivalence to Chern-Simons quantization• …

A very reasonable 3D quantum gravity theory:

[Barrett, Bonzom, Crane, BD, Freidel, Livine, Louapre, Meusburger, Noui, Perez, Rovelli, …]

ZPR�spin =X

{je}

Y

tetra


Spins give length of edges.All inequalities implemented.



j2

j3





Holographic duals for finite boundaries. Reproduces BMS vacuum character in asymptotic limit.


[BD, Goeller, Livine, Riello ’17,‘18]

ZPR�spin =X

{je}

Y

tetra





j2

j3





Holographic duals for finite boundaries. Reproduces BMS vacuum character in asymptotic limit.


[BD, Goeller, Livine, Riello ’17,‘18]

Open issues: sum over orientations, winding numbers (from holonomies), …

ZPR�spin =X

{je}

Y

tetra



({gl}) =X

{jl},{◆⌫}

({jl}, {◆⌫})Y

l

Djlmlnl

(gl)Y

⌫

◆mm0n...⌫

<latexit sha1_base64="MNqIOaj5G31ij2uf4QjnZm0p1gk=">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</latexit>

(2+1)D quantum gravity

• Hilbert space spanned by spin network functions:

• Quantum geometry operators, that measure lengths and (extrinsic curvature) angles.

• Arises also from quantization of (lattice) phase space: ⌦lT⇤SU(2)

<latexit sha1_base64="NHZxuR3qMxK5nirZw5zunbrEdzg=">AAAB/HicbVBNT8JAEN36ifhV5ehlIzFBD6RFEz0SvXjESIEEarNdtrBhu9vsbk0agn/FiweN8eoP8ea/cYEeFHzJJC/vzWRmXpgwqrTjfFsrq2vrG5uFreL2zu7evn1w2FIilZh4WDAhOyFShFFOPE01I51EEhSHjLTD0c3Ubz8SqajgTZ0lxI/RgNOIYqSNFNilntA0JipgsPlwBu+9Su00sMtO1ZkBLhM3J2WQoxHYX72+wGlMuMYMKdV1nUT7YyQ1xYxMir1UkQThERqQrqEcmYX+eHb8BJ4YpQ8jIU1xDWfq74kxipXK4tB0xkgP1aI3Ff/zuqmOrvwx5UmqCcfzRVHKoBZwmgTsU0mwZpkhCEtqboV4iCTC2uRVNCG4iy8vk1at6p5Xa3cX5fp1HkcBHIFjUAEuuAR1cAsawAMYZOAZvII368l6sd6tj3nripXPlMAfWJ8/mQOTbA==</latexit>

edge of the triangulation

dual link gl<latexit sha1_base64="DFJ/gMhSkcscO4IOIxBXgVEntgc=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBFclaQKuiy6cVnBPqANYTK9aYdOJmFmIpaQX3HjQhG3/og7/8ZpmoW2HrhwOOfeuXNPkHCmtON8W2vrG5tb25Wd6u7e/sGhfVTrqjiVFDo05rHsB0QBZwI6mmkO/UQCiQIOvWB6O/d7jyAVi8WDniXgRWQsWMgo0Uby7Vo2LB7JAp5Cjsc+z3277jScAniVuCWpoxJt3/4ajmKaRiA05USpgesk2suI1IxyyKvDVEFC6JSMYWCoIBEoLyvW5vjMKCMcxtKU0LhQf09kJFJqFgWmMyJ6opa9ufifN0h1eO1lTCSpBkEXi8KUYx3jeRB4xCRQzWeGECqZ+SumEyIJ1SauqgnBXT55lXSbDfei0by/rLduyjgq6ASdonPkoivUQneojTqIoif0jF7Rm5VbL9a79bFoXbPKmWP0B9bnD5O0lMg=</latexit>

el<latexit sha1_base64="7bcWfBIEgOopWOMTbkrEx3dei7I=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0gH3RL1fcqjsHWSVeTiqQo9Evf/UGMUsjlIYJqnXXcxPjZ1QZzgROS71UY0LZmA6xa6mkEWo/m586JWdWGZAwVrakIXP190RGI60nUWA7I2pGetmbif953dSE137GZZIalGyxKEwFMTGZ/U0GXCEzYmIJZYrbWwkbUUWZsemUbAje8surpFWrehfV2v1lpX6Tx1GEEziFc/DgCupwBw1oAoMhPMMrvDnCeXHenY9Fa8HJZ47hD5zPH0l4jcw=</latexit> {eal , gl} = gl⌧

a<latexit sha1_base64="/PlJVOVZqVXvzGhNDhWx0jp8dYg=">AAACAHicbVDLSsNAFJ3UV62vqAsXboJFcCElqYJuhKIblxXsA5oYbqaTduhkEmYmQgnZ+CtuXCji1s9w5984abvQ1gP3cjjnXmbuCRJGpbLtb6O0tLyyulZer2xsbm3vmLt7bRmnApMWjlksugFIwignLUUVI91EEIgCRjrB6KbwO49ESBrzezVOiBfBgNOQYlBa8s0DNyMP4LPTgc/c/KroClKtmFW7Zk9gLRJnRqpohqZvfrn9GKcR4QozkLLn2InyMhCKYkbyiptKkgAewYD0NOUQEellkwNy61grfSuMhS6urIn6eyODSMpxFOjJCNRQznuF+J/XS1V46WWUJ6kiHE8fClNmqdgq0rD6VBCs2FgTwILqv1p4CAKw0plVdAjO/MmLpF2vOWe1+t15tXE9i6OMDtEROkEOukANdIuaqIUwytEzekVvxpPxYrwbH9PRkjHb2Ud/YHz+AJWvlmY=</latexit>

1 + 1 = 2<latexit sha1_base64="UXJ6Ky2E/FmhhiTDDUx6doYc0xU=">AAACAHicbVDLSsNAFL2pr1pfURcu3AwWQRBKUgXdCEU3LivYB7ShTKYTO3QyCTMToYRs/BU3LhRx62e482+cpllo64GBwzn3NcePOVPacb6t0tLyyupaeb2ysbm1vWPv7rVVlEhCWyTikez6WFHOBG1ppjntxpLi0Oe0449vpn7nkUrFInGvJzH1QvwgWMAI1kYa2AdpPx+SGp0KjTPknrpX9WxgV52akwMtErcgVSjQHNhf/WFEktAMIRwr1XOdWHsplpoRTrNKP1E0xmRs1vQMFTikykvz3Rk6NsoQBZE0T2iUq787UhwqNQl9UxliPVLz3lT8z+slOrj0UibiRFNBZouChCMdoWkaaMgkJZpPDMFEMnMrIiMsMdEms4oJwZ3/8iJp12vuWa1+d15tXBdxlOEQjuAEXLiABtxCE1pAIINneIU368l6sd6tj1lpySp69uEPrM8f9h6V+Q==</latexit>

Dimensions:

4D gravity?Can we do a similar trick?

Palatini action

non-linear in tetrads S =

ZTr ⇤ (e ^ e) ^ F (!)

<latexit sha1_base64="TqHsxm86vCCMI1gp2o1xI010DlI=">AAACGXicbZDLSgMxFIYz9VbrrerSTbAIrUiZqYJuhKIgLiv2Bp1SMulpG5rMDElGKUNfw42v4saFIi515duYtrPQ6g+Bj/+cQ875vZAzpW37y0otLC4tr6RXM2vrG5tb2e2dugoiSaFGAx7IpkcUcOZDTTPNoRlKIMLj0PCGl5N64w6kYoFf1aMQ2oL0fdZjlGhjdbL27bnLfI1jVwpclWP3COPDPLj30O0DhgJO6CrvBgL6pNDJ5uyiPRX+C04COZSo0sl+uN2ARgJ8TTlRquXYoW7HRGpGOYwzbqQgJHRI+tAy6BMBqh1PLxvjA+N0cS+Q5pktp+7PiZgIpUbCM52C6IGar03M/2qtSPfO2jHzw0iDT2cf9SKOdYAnMeEuk0A1HxkgVDKzK6YDIgnVJsyMCcGZP/kv1EtF57hYujnJlS+SONJoD+2jPHLQKSqja1RBNUTRA3pCL+jVerSerTfrfdaaspKZXfRL1uc3rGKeOA==</latexit>


Palatini action

non-linear in tetrads

BF action

(topological) two-formS =

ZTrB ^ F (A)

<latexit sha1_base64="JsF8lUeCON5wuPXDXy1pvWBgLRk=">AAACCHicbVDLSgNBEJyNrxhfUY8eHAxCBAm7UdCLECOIx4h5QTaE2dlOMmT2wcysEpYcvfgrXjwo4tVP8ObfOEn2oIkFDUVVN91dTsiZVKb5baQWFpeWV9KrmbX1jc2t7PZOXQaRoFCjAQ9E0yESOPOhppji0AwFEM/h0HAGV2O/cQ9CssCvqmEIbY/0fNZllCgtdbL7dxc28xWObeHhqhjZxxiX7Qdwe4Cv85dHnWzOLJgT4HliJSSHElQ62S/bDWjkga8oJ1K2LDNU7ZgIxSiHUcaOJISEDkgPWpr6xAPZjiePjPChVlzcDYQufdRE/T0RE0/KoefoTo+ovpz1xuJ/XitS3fN2zPwwUuDT6aJuxLEK8DgV7DIBVPGhJoQKpm/FtE8EoUpnl9EhWLMvz5N6sWCdFIq3p7lSOYkjjfbQAcojC52hErpBFVRDFD2iZ/SK3own48V4Nz6mrSkjmdlFf2B8/gCOdJe/</latexit>

S =

ZTr ⇤ (e ^ e) ^ F (!)



Palatini action

non-linear in tetrads

BF action

(topological) two-formS =

ZTrB ^ F (A)

<latexit sha1_base64="JsF8lUeCON5wuPXDXy1pvWBgLRk=">AAACCHicbVDLSgNBEJyNrxhfUY8eHAxCBAm7UdCLECOIx4h5QTaE2dlOMmT2wcysEpYcvfgrXjwo4tVP8ObfOEn2oIkFDUVVN91dTsiZVKb5baQWFpeWV9KrmbX1jc2t7PZOXQaRoFCjAQ9E0yESOPOhppji0AwFEM/h0HAGV2O/cQ9CssCvqmEIbY/0fNZllCgtdbL7dxc28xWObeHhqhjZxxiX7Qdwe4Cv85dHnWzOLJgT4HliJSSHElQ62S/bDWjkga8oJ1K2LDNU7ZgIxSiHUcaOJISEDkgPWpr6xAPZjiePjPChVlzcDYQufdRE/T0RE0/KoefoTo+ovpz1xuJ/XitS3fN2zPwwUuDT6aJuxLEK8DgV7DIBVPGhJoQKpm/FtE8EoUpnl9EhWLMvz5N6sWCdFIq3p7lSOYkjjfbQAcojC52hErpBFVRDFD2iZ/SK3own48V4Nz6mrSkjmdlFf2B8/gCOdJe/</latexit>

S =

ZTr ⇤ (e ^ e) ^ F (!)


Plebanski action S =

ZTrB ^ F (A) + Tr�B ^B

<latexit sha1_base64="GgqcUmjj4vYdTs9eyGVXExciz0Y=">AAACJHicbVDJSgNBEO2JW4zbqEcvjUGIKGEmCgoixAjiMWI2yITQ06kkTXoWunuUMORjvPgrXjy44MGL32JnAWP0QcHjvSqq6rkhZ1JZ1qeRmJtfWFxKLqdWVtfWN8zNrYoMIkGhTAMeiJpLJHDmQ1kxxaEWCiCey6Hq9i6HfvUOhGSBX1L9EBoe6fiszShRWmqaZ7fnDvMVjh3h4ZIYOIe4gJ17aHUAX2Uu9vHBlOWEXfZjF5pm2spaI+C/xJ6QNJqg2DTfnFZAIw98RTmRsm5boWrERChGOQxSTiQhJLRHOlDX1CceyEY8enKA97TSwu1A6NIHj9TpiZh4UvY9V3d6RHXlrDcU//PqkWqfNmLmh5ECn44XtSOOVYCHieEWE0AV72tCqGD6Vky7RBCqdK4pHYI9+/JfUsll7aNs7uY4nS9M4kiiHbSLMshGJyiPrlERlRFFD+gJvaBX49F4Nt6Nj3FrwpjMbKNfML6+AcMiocw=</latexit>

Simplicity constraints impose*:

B = ?e ^ e<latexit sha1_base64="9PNMRIlxtpaXFUiVZN4BDQHoNkw=">AAACDHicbVDLSsNAFJ3UV62vqks3g0VwVZIq6EYodeOygn1AE8pkctMOnTyYmSgl5APc+CtuXCji1g9w5984TbPQ1gMDh3Pua44bcyaVaX4bpZXVtfWN8mZla3tnd6+6f9CVUSIodGjEI9F3iQTOQugopjj0YwEkcDn03Mn1zO/dg5AsCu/UNAYnIKOQ+YwSpaVhtZba+ZBU6xAqkuHWlS0VERjsB/BGgCHTVWbdzIGXiVWQGirQHla/bC+iSaAHUk6kHFhmrJyUCMUoh6xiJxJiQid65UDTkAQgnTS/I8MnWvGwHwn9QoVz9XdHSgIpp4GrKwOixnLRm4n/eYNE+ZdOysI4URDS+SI/4VhFeJYM9pgAqvhUE0IF07diOiaCUKXzq+gQrMUvL5Nuo26d1Ru357Vmq4ijjI7QMTpFFrpATXSD2qiDKHpEz+gVvRlPxovxbnzMS0tG0XOI/sD4/AHgIpt8</latexit>

BF action, Plebanski action, Palatini (-Holst) action and Ashtekar (-Barbero) variables

lead to similar phase space and quantum geometry.

(3+1)D Phase space and quantum geometry

gl<latexit sha1_base64="DFJ/gMhSkcscO4IOIxBXgVEntgc=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBFclaQKuiy6cVnBPqANYTK9aYdOJmFmIpaQX3HjQhG3/og7/8ZpmoW2HrhwOOfeuXNPkHCmtON8W2vrG5tb25Wd6u7e/sGhfVTrqjiVFDo05rHsB0QBZwI6mmkO/UQCiQIOvWB6O/d7jyAVi8WDniXgRWQsWMgo0Uby7Vo2LB7JAp5Cjsc+z3277jScAniVuCWpoxJt3/4ajmKaRiA05USpgesk2suI1IxyyKvDVEFC6JSMYWCoIBEoLyvW5vjMKCMcxtKU0LhQf09kJFJqFgWmMyJ6opa9ufifN0h1eO1lTCSpBkEXi8KUYx3jeRB4xCRQzWeGECqZ+SumEyIJ1SauqgnBXT55lXSbDfei0by/rLduyjgq6ASdonPkoivUQneojTqIoif0jF7Rm5VbL9a79bFoXbPKmWP0B9bnD5O0lMg=</latexit>

Bl<latexit sha1_base64="sMbYfWNc0SnwNjOymdgiLLDZ+bs=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9FjqxWNF+wFtKJvtpF262YTdjVBCf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/PbT6g0j+WjmSToR3QoecgZNVZ6qPdFv1R2K+4cZJV4OSlDjka/9NUbxCyNUBomqNZdz02Mn1FlOBM4LfZSjQllYzrErqWSRqj9bH7qlJxbZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMT3vgZl0lqULLFojAVxMRk9jcZcIXMiIkllClubyVsRBVlxqZTtCF4yy+vkla14l1WqvdX5Vo9j6MAp3AGF+DBNdTgDhrQBAZDeIZXeHOE8+K8Ox+L1jUnnzmBP3A+fwAUJo2p</latexit>

⌦lT⇤SU(2)


⌦lT⇤SO(4)

<latexit sha1_base64="55iT4lXDHPQfIbT2FGwV0ea/siI=">AAAB+3icbVDLSgMxFM34rPU11qWbYBGqizJTC7osunFnxb6gHYdMmmlDM8mQZMQy9FfcuFDErT/izr8xbWehrQcuHM65l3vvCWJGlXacb2tldW19YzO3ld/e2d3btw8KLSUSiUkTCyZkJ0CKMMpJU1PNSCeWBEUBI+1gdD31249EKip4Q49j4kVowGlIMdJG8u1CT2gaEeUz2Hg4u78tVU99u+iUnRngMnEzUgQZ6r791esLnESEa8yQUl3XibWXIqkpZmSS7yWKxAiP0IB0DeXI7PPS2e0TeGKUPgyFNMU1nKm/J1IUKTWOAtMZIT1Ui95U/M/rJjq89FLK40QTjueLwoRBLeA0CNinkmDNxoYgLKm5FeIhkghrE1fehOAuvrxMWpWye16u3FWLtassjhw4AsegBFxwAWrgBtRBE2DwBJ7BK3izJtaL9W59zFtXrGzmEPyB9fkDN+qTPg==</latexit>

or


encodes normal to triangle


encodes extrinsic curvature angle( with caveat*)

Dimensions:2 + 1 = 3

<latexit sha1_base64="qJ2nFfwAVyTvNs3EDk9X6wn5Fk8=">AAACAHicbVDLSsNAFL3xWesr6sKFm8EiCEJJWkE3QtGNywr2AW0ok+nEDp1MwsxEKCEbf8WNC0Xc+hnu/BunaRbaemDgcM59zfFjzpR2nG9raXlldW29tFHe3Nre2bX39tsqSiShLRLxSHZ9rChngrY005x2Y0lx6HPa8cc3U7/zSKVikbjXk5h6IX4QLGAEayMN7MO0nw9JjU6FxhmqnblX9WxgV5yqkwMtErcgFSjQHNhf/WFEktAMIRwr1XOdWHsplpoRTrNyP1E0xmRs1vQMFTikykvz3Rk6McoQBZE0T2iUq787UhwqNQl9UxliPVLz3lT8z+slOrj0UibiRFNBZouChCMdoWkaaMgkJZpPDMFEMnMrIiMsMdEms7IJwZ3/8iJp16puvVq7O680ros4SnAEx3AKLlxAA26hCS0gkMEzvMKb9WS9WO/Wx6x0ySp6DuAPrM8f+SyV+w==</latexit>

(3+1)D Phase space and quantum geometry



⌦lT⇤SU(2)


⌦lT⇤SO(4)

<latexit sha1_base64="55iT4lXDHPQfIbT2FGwV0ea/siI=">AAAB+3icbVDLSgMxFM34rPU11qWbYBGqizJTC7osunFnxb6gHYdMmmlDM8mQZMQy9FfcuFDErT/izr8xbWehrQcuHM65l3vvCWJGlXacb2tldW19YzO3ld/e2d3btw8KLSUSiUkTCyZkJ0CKMMpJU1PNSCeWBEUBI+1gdD31249EKip4Q49j4kVowGlIMdJG8u1CT2gaEeUz2Hg4u78tVU99u+iUnRngMnEzUgQZ6r791esLnESEa8yQUl3XibWXIqkpZmSS7yWKxAiP0IB0DeXI7PPS2e0TeGKUPgyFNMU1nKm/J1IUKTWOAtMZIT1Ui95U/M/rJjq89FLK40QTjueLwoRBLeA0CNinkmDNxoYgLKm5FeIhkghrE1fehOAuvrxMWpWye16u3FWLtassjhw4AsegBFxwAWrgBtRBE2DwBJ7BK3izJtaL9W59zFtXrGzmEPyB9fkDN+qTPg==</latexit>

or


encodes normal to triangle


encodes extrinsic curvature angle( with caveat*)

Tetrads need to be reconstructed from normals. (Issue for matter couplings.)

Problem (leading to caveat*):

Reconstructions from different tetrahedra do not fit (in shape).

[BD, Speziale ’07; BD, Ryan 08, ]

Triangle areas match, but triangle shapes do not match.

Dimensions:2 + 1 = 3

<latexit sha1_base64="qJ2nFfwAVyTvNs3EDk9X6wn5Fk8=">AAACAHicbVDLSsNAFL3xWesr6sKFm8EiCEJJWkE3QtGNywr2AW0ok+nEDp1MwsxEKCEbf8WNC0Xc+hnu/BunaRbaemDgcM59zfFjzpR2nG9raXlldW29tFHe3Nre2bX39tsqSiShLRLxSHZ9rChngrY005x2Y0lx6HPa8cc3U7/zSKVikbjXk5h6IX4QLGAEayMN7MO0nw9JjU6FxhmqnblX9WxgV5yqkwMtErcgFSjQHNhf/WFEktAMIRwr1XOdWHsplpoRTrNyP1E0xmRs1vQMFTikykvz3Rk6McoQBZE0T2iUq787UhwqNQl9UxliPVLz3lT8z+slOrj0UibiRFNBZouChCMdoWkaaMgkJZpPDMFEMnMrIiMsMdEms7IJwZ3/8iJp16puvVq7O680ros4SnAEx3AKLlxAA26hCS0gkMEzvMKb9WS9WO/Wx6x0ySp6DuAPrM8f+SyV+w==</latexit>

Simplicity constraints

[BD, Speziale ’07; BD, Ryan ’08, ]

Triangle areas match, but triangle shapes do not match.

Reconstructions from different tetrahedra do not fit (in shape).

[Freidel, Speziale ’10] Twisted Geometries

[BD, Ryan ’07 ] Due to: Edge simplicity constraints are not imposed.

holedge . e(B) = e(B)<latexit sha1_base64="o8NuhbyI4AEd0wWtYqcQLBquNAQ=">AAACG3icbZDLSgMxFIYz9VbrrerSTbAIClJmqqAbodSNywpWhU4pmfR0GprJDMkZoQx9Dze+ihsXirgSXPg2ptMuvB0I+fj/c0jOHyRSGHTdT6cwN7+wuFRcLq2srq1vlDe3rk2cag4tHstY3wbMgBQKWihQwm2igUWBhJtgeD7xb+5AGxGrKxwl0IlYqERfcIZW6pZrma8jOojluJsT9EIY+4eU+qgFU6EELcIBUthvHNCz/OqWK27VzYv+BW8GFTKrZrf87vdinkagkEtmTNtzE+xkTKPgEsYlPzWQMD5kIbQtKhaB6WT5bmO6Z5Ue7cfaHoU0V79PZCwyZhQFtjNiODC/vYn4n9dOsX/ayYRKUgTFpw/1U0kxppOgaE9o4ChHFhjXwv6V8gHTjKONs2RD8H6v/Beua1XvqFq7PK7UG7M4imSH7JJ94pETUicXpElahJN78kieyYvz4Dw5r87btLXgzGa2yY9yPr4AVWSfpQ==</latexit>

[BD, Ryan ’10 ] Are equivalent to secondary simplicity constraints (and to reality conditions on Ashtekar connection). Needed to reconstruct Levi-Civita connection.

Issue for spin foam/LQG- dynamics? [Alexandrov, Anza, Bonzom, Han, Hellmann, Kaminski, Oliveira, Speziale, …]

Summary: Phase space from BF theory

Basic variables: A<latexit sha1_base64="Lihtv2jYSe0RaYbwwPdS8141boc=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI+oF4+QyCOBDZkdemFkdnYzM2tCCF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR3cxvPaHSPJYPZpygH9GB5CFn1FipftMrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1dssjjycwCmcgwdXUIV7qEEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+AJQrjMk=</latexit>

B<latexit sha1_base64="wW6OXYFoJg3RvlrYIdlxqPzQzSE=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI8ELx4hkUcCGzI79MLI7OxmZtaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfju7nffkKleSwfzCRBP6JDyUPOqLFSo9YvltyyuwBZJ15GSpCh3i9+9QYxSyOUhgmqdddzE+NPqTKcCZwVeqnGhLIxHWLXUkkj1P50ceiMXFhlQMJY2ZKGLNTfE1MaaT2JAtsZUTPSq95c/M/rpia89adcJqlByZaLwlQQE5P512TAFTIjJpZQpri9lbARVZQZm03BhuCtvrxOWpWyd1WuNK5L1VoWRx7O4BwuwYMbqMI91KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AJWvjMo=</latexit>

so(4) valued

connection one-form

so(4) valued

two-form

1 + 2 = 3<latexit sha1_base64="aijunzAko7ujMDHRcvRPS83gJsI=">AAAB7HicbVBNSwMxEJ2tX7V+VT16CRZBEMpuK+hFKHrxWMFtC+1Ssmm2DU2yS5IVytLf4MWDIl79Qd78N6btHrT1wcDjvRlm5oUJZ9q47rdTWFvf2Nwqbpd2dvf2D8qHRy0dp4pQn8Q8Vp0Qa8qZpL5hhtNOoigWIaftcHw389tPVGkWy0czSWgg8FCyiBFsrOR7F7Wber9ccavuHGiVeDmpQI5mv/zVG8QkFVQawrHWXc9NTJBhZRjhdFrqpZommIzxkHYtlVhQHWTzY6fozCoDFMXKljRorv6eyLDQeiJC2ymwGellbyb+53VTE10HGZNJaqgki0VRypGJ0exzNGCKEsMnlmCimL0VkRFWmBibT8mG4C2/vEpatapXr9YeLiuN2zyOIpzAKZyDB1fQgHtogg8EGDzDK7w50nlx3p2PRWvByWeO4Q+czx9IP42u</latexit>

conjugated to each other

Geometry: (extrinsic)

curvature*

Areas

(normals to triangles)




so(4) valued

connection one-form

so(4) valued

two-form



Geometry: (extrinsic)

curvature*

Areas

(normals to triangles)

Hamiltonian: F (A) = 0<latexit sha1_base64="Vq5MBxy1yAsYq4oNor47MKTHAno=">AAAB7XicbVBNSwMxEJ31s9avqkcvwSLUS9mtgl6EqiAeK9gPaJeSTbNtbDZZkqxQlv4HLx4U8er/8ea/MW33oK0PBh7vzTAzL4g508Z1v52l5ZXVtfXcRn5za3tnt7C339AyUYTWieRStQKsKWeC1g0znLZiRXEUcNoMhjcTv/lElWZSPJhRTP0I9wULGcHGSo3b0tXJpdstFN2yOwVaJF5GipCh1i18dXqSJBEVhnCsddtzY+OnWBlGOB3nO4mmMSZD3KdtSwWOqPbT6bVjdGyVHgqlsiUMmqq/J1IcaT2KAtsZYTPQ895E/M9rJya88FMm4sRQQWaLwoQjI9HkddRjihLDR5Zgopi9FZEBVpgYG1DehuDNv7xIGpWyd1qu3J8Vq9dZHDk4hCMogQfnUIU7qEEdCDzCM7zCmyOdF+fd+Zi1LjnZzAH8gfP5A9oxjf8=</latexit>

Problem: Generates edge translations (instead of vertex translations). Reason: is a two-form.

[Waelbroeck, Zapata 94 ]: Propose to break symmetry down to vertex translations. Would result in non-local expressions.

[Thiemann 96]: Action based on tetrahedra. Geometric interpretation lost.




so(4) valued

connection one-form

so(4) valued

two-form



Would like

a (tetrad) one-form.

But then miss

a dimension.




so(4) valued

two-form



Trick: enlarge phase space … (and constrain again)

so(4) valued

connection one-formWould like

a (tetrad) one-form.

But then miss

a dimension.



so(4) valued

connection one-form

so(4) valued

two-form


valued two-form

(2-connection)

⌃<latexit sha1_base64="BPJq+0/bflPPUVaVxn+aiuvMPUE=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWNE84BkCbOT2WTMPJaZWSEs+QcvHhTx6v9482+cJHvQxIKGoqqb7q4o4cxY3//2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCKK92OsKGcSdqwzHLaTjTFIuK0FY1upn7riWrDlHyw44SGAg8kixnB1knN7j0bCNwrlf2KPwNaJkFOypCj3it9dfuKpIJKSzg2phP4iQ0zrC0jnE6K3dTQBJMRHtCOoxILasJsdu0EnTqlj2KlXUmLZurviQwLY8Yicp0C26FZ9Kbif14ntfFVmDGZpJZKMl8UpxxZhaavoz7TlFg+dgQTzdytiAyxxsS6gIouhGDx5WXSrFaC80r17qJcu87jKMAxnMAZBHAJNbiFOjSAwCM8wyu8ecp78d69j3nripfPHMEfeJ8/bK2PBw==</latexit>

R4<latexit sha1_base64="rppRLY1UTgW1Fa7J/HFzMkitLB0=">AAAB83icbVDLSsNAFL2pr1pfUZduBovgqiS1oMuiG5dV7AOaWCbTSTt0MgkzE6GE/oYbF4q49Wfc+TdO2iy09cDA4Zx7uWdOkHCmtON8W6W19Y3NrfJ2ZWd3b//APjzqqDiVhLZJzGPZC7CinAna1kxz2kskxVHAaTeY3OR+94lKxWLxoKcJ9SM8EixkBGsjeV6E9TgIsvvZY2NgV52aMwdaJW5BqlCgNbC/vGFM0ogKTThWqu86ifYzLDUjnM4qXqpogskEj2jfUIEjqvxsnnmGzowyRGEszRMazdXfGxmOlJpGgZnMM6plLxf/8/qpDq/8jIkk1VSQxaEw5UjHKC8ADZmkRPOpIZhIZrIiMsYSE21qqpgS3OUvr5JOveZe1Op3jWrzuqijDCdwCufgwiU04RZa0AYCCTzDK7xZqfVivVsfi9GSVewcwx9Ynz/ueJGc</latexit>

E<latexit sha1_base64="oW7OyFl8fZOuY5qYrNXg+guGckE=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BETwmYB6QLGF20puMmZ1dZmaFEPIFXjwo4tVP8ubfOEn2oIkFDUVVN91dQSK4Nq777eTW1jc2t/LbhZ3dvf2D4uFRU8epYthgsYhVO6AaBZfYMNwIbCcKaRQIbAWj25nfekKleSwfzDhBP6IDyUPOqLFS/a5XLLlldw6ySryMlCBDrVf86vZjlkYoDRNU647nJsafUGU4EzgtdFONCWUjOsCOpZJGqP3J/NApObNKn4SxsiUNmau/JyY00nocBbYzomaol72Z+J/XSU147U+4TFKDki0WhakgJiazr0mfK2RGjC2hTHF7K2FDqigzNpuCDcFbfnmVNCtl76JcqV+WqjdZHHk4gVM4Bw+uoAr3UIMGMEB4hld4cx6dF+fd+Vi05pxs5hj+wPn8AZo7jM0=</latexit>

valued one-form

(tetrads)





2 + 1 = 3<latexit sha1_base64="1Qd54H7gk1+CIzCp63Fga1BE24I=">AAAB7HicbVBNSwMxEJ2tX7V+VT16CRZBEMpuK+hFKHrxWMFtC+1Ssmm2DU2yS5IVytLf4MWDIl79Qd78N6btHrT1wcDjvRlm5oUJZ9q47rdTWFvf2Nwqbpd2dvf2D8qHRy0dp4pQn8Q8Vp0Qa8qZpL5hhtNOoigWIaftcHw389tPVGkWy0czSWgg8FCyiBFsrOTXLryber9ccavuHGiVeDmpQI5mv/zVG8QkFVQawrHWXc9NTJBhZRjhdFrqpZommIzxkHYtlVhQHWTzY6fozCoDFMXKljRorv6eyLDQeiJC2ymwGellbyb+53VTE10HGZNJaqgki0VRypGJ0exzNGCKEsMnlmCimL0VkRFWmBibT8mG4C2/vEpatapXr9YeLiuN2zyOIpzAKZyDB1fQgHtogg8EGDzDK7w50nlx3p2PRWvByWeO4Q+czx9IQY2u</latexit>

Enlarge phase space



so(4) valued

connection one-form

so(4) valued

two-form


valued two-form

(2-connection)




valued one-form

(tetrads)






Basic geometric variables.

Enlarge phase space

Enlarge phase space



so(4) valued

connection one-form

so(4) valued

two-form


valued two-form

(2-connection)




valued one-form

(tetrads)






2-curvature: G = dA⌃<latexit sha1_base64="FvFXrBW6+hBifRIhms31mT2Yq/c=">AAAB83icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6EWIetBjRPOA7BJmZ2eTITOzy8ysEJb8hhcPinj1Z7z5N04eB00saCiquunuClPOtHHdb2dpeWV1bb2wUdzc2t7ZLe3tN3WSKUIbJOGJaodYU84kbRhmOG2nimIRctoKBzdjv/VElWaJfDTDlAYC9ySLGcHGSv7tZdS9Qv4D6wncLZXdijsBWiTejJRhhnq39OVHCckElYZwrHXHc1MT5FgZRjgdFf1M0xSTAe7RjqUSC6qDfHLzCB1bJUJxomxJgybq74kcC62HIrSdApu+nvfG4n9eJzPxRZAzmWaGSjJdFGccmQSNA0ARU5QYPrQEE8XsrYj0scLE2JiKNgRv/uVF0qxWvNNK9f6sXLuexVGAQziCE/DgHGpwB3VoAIEUnuEV3pzMeXHenY9p65IzmzmAP3A+fwDbnpDr</latexit>

is a three-form. Will be important.

The BFCG action [Girelli, Pfeiffer, Popescu ’07]

S =

ZTrso(4)B ^ F (A) + TrR4C ^G(A,⌃)

<latexit sha1_base64="/+Rbt25WDX7EL4JURrFR54jh0mo=">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</latexit>


S =

ZTrso(4)B ^ F (A) + TrR4E ^G(A,⌃)

<latexit sha1_base64="q0YwDXztP9FbnpJHceXTBNj7nQI=">AAACP3icbVDLahsxFNW4r8R9ZJIsuxE1BZsaM+MYmk0gD5p0mTbxAyzXaOQ7trA0M0iaBjPMn2XTX+gu22y6aCjZZlfZnkVq94DgcM656N4TJIJr43k3TunJ02fPX2xsll++ev1my93e6eg4VQzaLBax6gVUg+ARtA03AnqJAioDAd1gejL3u99BaR5Hl2aWwEDSccRDzqix0tDtXBwQHhmcESXxpcqHmY6rrVqOj8kVjMaAT6tHNUzqH0j9UYZIaiZBkH3Nv7Vy/KmInlWP6uSCjyWtDd2K1/AWwOvEL0gFFTgfuj/JKGaphMgwQbXu+15iBhlVhjMBeZmkGhLKpnQMfUsjKkEPssX9OX5vlREOY2WfvWWhPp7IqNR6JgObnC+uV725+D+vn5pwf5DxKEkNRGz5UZgKbGI8LxOPuAJmxMwSyhS3u2I2oYoyYysv2xL81ZPXSafZ8PcazS+tyuFxUccGeoveoSry0Ud0iD6jc9RGDF2jW/Qb3Tk/nF/OH+d+GS05xcwu+gfOw19vKKzL</latexit> [Mikovic, Vojinovic ’11]


S =


<latexit sha1_base64="q0YwDXztP9FbnpJHceXTBNj7nQI=">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</latexit>

=

ZTrso(4)B ^ F (A) � TrR4T (A,E) ^ ⌃

<latexit sha1_base64="uBNYGuj2Tg6/CEjPx2OHk5FEIsk=">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</latexit>

T (A,E) = dAE<latexit sha1_base64="ViXRNSeceu1qe6Kgj4QSzTfZ00Y=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahgpSkCnoRWqXgsUK/oA1ls9m0SzebsLsRSunf8OJBEa/+GW/+G7dtDtr6YODx3gwz87yYM6Vt+9vKrK1vbG5lt3M7u3v7B/nDo5aKEklok0Q8kh0PK8qZoE3NNKedWFIcepy2vdH9zG8/UalYJBp6HFM3xAPBAkawNlKvUaxe1M5v/X4V1fr5gl2y50CrxElJAVLU+/mvnh+RJKRCE46V6jp2rN0JlpoRTqe5XqJojMkID2jXUIFDqtzJ/OYpOjOKj4JImhIazdXfExMcKjUOPdMZYj1Uy95M/M/rJjq4cSdMxImmgiwWBQlHOkKzAJDPJCWajw3BRDJzKyJDLDHRJqacCcFZfnmVtMol57JUfrwqVO7SOLJwAqdQBAeuoQIPUIcmEIjhGV7hzUqsF+vd+li0Zqx05hj+wPr8AV/dj/M=</latexit>

Torsion

Torsion freeness and vanishing curvature imposed by Lagrange multiplyers.

[Mikovic, Vojinovic ’11]


S =



=




Torsion


EOM: F (A) = 0<latexit sha1_base64="Vq5MBxy1yAsYq4oNor47MKTHAno=">AAAB7XicbVBNSwMxEJ31s9avqkcvwSLUS9mtgl6EqiAeK9gPaJeSTbNtbDZZkqxQlv4HLx4U8er/8ea/MW33oK0PBh7vzTAzL4g508Z1v52l5ZXVtfXcRn5za3tnt7C339AyUYTWieRStQKsKWeC1g0znLZiRXEUcNoMhjcTv/lElWZSPJhRTP0I9wULGcHGSo3b0tXJpdstFN2yOwVaJF5GipCh1i18dXqSJBEVhnCsddtzY+OnWBlGOB3nO4mmMSZD3KdtSwWOqPbT6bVjdGyVHgqlsiUMmqq/J1IcaT2KAtsZYTPQ895E/M9rJya88FMm4sRQQWaLwoQjI9HkddRjihLDR5Zgopi9FZEBVpgYG1DehuDNv7xIGpWyd1qu3J8Vq9dZHDk4hCMogQfnUIU7qEEdCDzCM7zCmyOdF+fd+Zi1LjnZzAH8gfP5A9oxjf8=</latexit>

dAB � 2E ^ ⌃ = 0<latexit sha1_base64="RpTaxW7vrBX+25cm+9K2iUnt9xU=">AAACAnicbVBNS8NAEN34WetX1JN4WSyCF0tSBb0ItSJ4rGg/oC1hs5m2S3eTsLtRSile/CtePCji1V/hzX/jts1BWx8MPN6bYWaeH3OmtON8W3PzC4tLy5mV7Ora+samvbVdVVEiKVRoxCNZ94kCzkKoaKY51GMJRPgcan7vcuTX7kEqFoV3uh9DS5BOyNqMEm0kz94NvAtcOipc4eYDBB3AzVvWEQSfO56dc/LOGHiWuCnJoRRlz/5qBhFNBISacqJUw3Vi3RoQqRnlMMw2EwUxoT3SgYahIRGgWoPxC0N8YJQAtyNpKtR4rP6eGBChVF/4plMQ3VXT3kj8z2skun3WGrAwTjSEdLKonXCsIzzKAwdMAtW8bwihkplbMe0SSag2qWVNCO70y7OkWsi7x/nCzUmuWErjyKA9tI8OkYtOURFdozKqIIoe0TN6RW/Wk/VivVsfk9Y5K53ZQX9gff4A7AqVNQ==</latexit>

G(A,⌃) = 0<latexit sha1_base64="3Y1dXCgStvAr0XXhEmyuXi3q1kM=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBahgpSkCnoRqh70WNF+QBvKZrtpl+4mcXdTKKG/w4sHRbz6Y7z5b9y2OWjrg4HHezPMzPMizpS27W8rs7S8srqWXc9tbG5t7+R39+oqjCWhNRLyUDY9rChnAa1ppjltRpJi4XHa8AY3E78xpFKxMHjUo4i6AvcC5jOCtZHc2+LVSfuB9QQ+vrQ7+YJdsqdAi8RJSQFSVDv5r3Y3JLGggSYcK9Vy7Ei7CZaaEU7HuXasaITJAPdoy9AAC6rcZHr0GB0ZpYv8UJoKNJqqvycSLJQaCc90Cqz7at6biP95rVj7F27CgijWNCCzRX7MkQ7RJAHUZZISzUeGYCKZuRWRPpaYaJNTzoTgzL+8SOrlknNaKt+fFSrXaRxZOIBDKIID51CBO6hCDQg8wTO8wps1tF6sd+tj1pqx0pl9+APr8wew5pC/</latexit>

T (A,E) = 0<latexit sha1_base64="Ldw80BAgLpAeb7Uk5GX3TNSUtAI=">AAAB73icbVBNS8NAEJ3Ur1q/qh69LBahgpSkFfQiVEXwWKFf0Iay2W7apZtN3N0IJfRPePGgiFf/jjf/jds2B219MPB4b4aZeV7EmdK2/W1lVlbX1jeym7mt7Z3dvfz+QVOFsSS0QUIeyraHFeVM0IZmmtN2JCkOPE5b3uh26reeqFQsFHU9jqgb4IFgPiNYG6ldL16f3Z1e2b18wS7ZM6Bl4qSkAClqvfxXtx+SOKBCE46V6jh2pN0ES80Ip5NcN1Y0wmSEB7RjqMABVW4yu3eCTozSR34oTQmNZurviQQHSo0Dz3QGWA/VojcV//M6sfYv3YSJKNZUkPkiP+ZIh2j6POozSYnmY0MwkczcisgQS0y0iShnQnAWX14mzXLJqZTKD+eF6k0aRxaO4BiK4MAFVOEeatAAAhye4RXerEfrxXq3PuatGSudOYQ/sD5/AOfZjpI=</latexit>

Topological theory.


S =



=




Torsion


S =



+Trso(4) � (B � ⇤(E ^ E))<latexit sha1_base64="2Zb/cQwXSktJx9JkLR3QvIAlQ88=">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</latexit>

Constrain:

[Mikovic, Vojinovic ’11; Mikovic, Oliveira, Vojinovic ’15-'18] constraint analysis of (continuum) actions

� = F<latexit sha1_base64="pvWqBWmTlb/671jiUG56K3kWsuc=">AAAB7XicbVBNSwMxEJ2tX7V+rXr0EiyCp7JbBb0IRUE8VrAf0C4lm2bb2GyyJFmhLP0PXjwo4tX/481/Y9ruQVsfDDzem2FmXphwpo3nfTuFldW19Y3iZmlre2d3z90/aGqZKkIbRHKp2iHWlDNBG4YZTtuJojgOOW2Fo5up33qiSjMpHsw4oUGMB4JFjGBjpWY3GbKr255b9ireDGiZ+DkpQ456z/3q9iVJYyoM4Vjrju8lJsiwMoxwOil1U00TTEZ4QDuWChxTHWSzayfoxCp9FEllSxg0U39PZDjWehyHtjPGZqgXvan4n9dJTXQZZEwkqaGCzBdFKUdGounrqM8UJYaPLcFEMXsrIkOsMDE2oJINwV98eZk0qxX/rFK9Py/XrvM4inAEx3AKPlxADe6gDg0g8AjP8ApvjnRenHfnY95acPKZQ/gD5/MHKL6O2g==</latexit>

G = 2 ⇤ (F ^ E)<latexit sha1_base64="7RO5+n25oVgkeozwgkSwT8SEPCA=">AAAB+nicbVDJSgNBEO2JW4zbRI9eGoMQPYSZKOhFCIrLMYJZIBlCT09N0qRnobvHEMZ8ihcPinj1S7z5N3aWgyY+KHi8V0VVPTfmTCrL+jYyS8srq2vZ9dzG5tb2jpnfrcsoERRqNOKRaLpEAmch1BRTHJqxABK4HBpu/2rsNx5BSBaFD2oYgxOQbsh8RonSUsfM316U8TEu3rQH4HUBXx91zIJVsibAi8SekQKaodoxv9peRJMAQkU5kbJlW7FyUiIUoxxGuXYiISa0T7rQ0jQkAUgnnZw+woda8bAfCV2hwhP190RKAimHgas7A6J6ct4bi/95rUT5507KwjhRENLpIj/hWEV4nAP2mACq+FATQgXTt2LaI4JQpdPK6RDs+ZcXSb1csk9K5fvTQuVyFkcW7aMDVEQ2OkMVdIeqqIYoGqBn9IrejCfjxXg3PqatGWM2s4f+wPj8AfLhkd0=</latexit>

Some EOM:

Key problem•Construction of spin/ representation pictures (“spin bubble model”) was so far not available

•Conjectured to be given by a model of quantum flat space (KBF model)

Key problem•Construction of spin/ representation pictures (“spin bubble model”) was so far not available

•Conjectured to be given by a model of quantum flat space (KBF model)

•Quantization of the BFCG action and correspondence with KBF model

•Key: solving part of the equations of motions (constraints), including edge simplicity constraints

•Construction of boundary Hilbert space:

•with consistent constraints

•G-network functions

Plan: [Asante, BD, Girelli, Riello, Tsimiklis ‘19]

Path integral=



Torsion freeness and vanishing curvature imposed by Lagrange multiplyers

S<latexit sha1_base64="vFchJGr6Z+gyWiveB04FdIL/myI=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELx4hyiOBDZkdemFkdnYzM2tCCF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaHSPJYPZpygH9GB5CFn1Fipft8rltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+AK9zjNs=</latexit>

Z =

ZD[A]D[E]

Y

x

�(Fx(A))Y

x

�(Tx(A,E))<latexit sha1_base64="7i3tmjPh6L/OQrObyxwyvseA/OE=">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</latexit>

Path integral=



Torsion freeness and vanishing curvature imposed by Lagrange multiplyers

S<latexit sha1_base64="vFchJGr6Z+gyWiveB04FdIL/myI=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELx4hyiOBDZkdemFkdnYzM2tCCF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaHSPJYPZpygH9GB5CFn1Fipft8rltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+AK9zjNs=</latexit>

Z =

ZD[A]D[E]

Y

x

�(Fx(A))Y

x

�(Tx(A,E))<latexit sha1_base64="7i3tmjPh6L/OQrObyxwyvseA/OE=">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</latexit>

Regularize on a lattice: holonomies associated to dual links, four-vectors to edges:

Zdiscr =

Z Y

l

dglY

e

dEe

Y

f

�SO(4) (holf (g))Y

t

�R4

�X

e2t

±E0e

�

<latexit sha1_base64="4cV0FLAX+7CtTxoPrlif3JyE32U=">AAACkXicbVFbb9MwFHbCbRQYZXvkxaJCpBKqklEJ9jBUDU1C4oFx6TZRl8hxTlJrdhLZJ5OqKP+H38Mb/wanjRBsHMlHn79zPvtckkpJi2H4y/Nv3b5z997O/cGDh492Hw+f7J3ZsjYC5qJUpblIuAUlC5ijRAUXlQGuEwXnyeW7Ln5+BcbKsviK6wqWmueFzKTg6Kh4+ONb3DCjaSqtMO0RkwVSVpkyjRVNc+e2F6DpiXPsJa U9k1GWgkIeN18+BtNxyxRkGNDNY6tStXEW5GNmZL7CMe2EWxn+kTHNcZUkzef2+7RlicwDl2ZrHTfgqqDYOoWmJy9i6ILjeDgKJ+HG6E0Q9WBEejuNhz9ZWopaQ4FCcWsXUVjhsuEGpVDQDlhtoeLikuewcLDgGuyy2Uy0pc8dk9KsNO64iWzYvxUN19audeIyuzbs9VhH/i+2qDF7s2xkUdUIhdh+lNWKYkm79bg9GBCo1g5wYaSrlYoVN1ygW+LADSG63vJNcHYwiV5NDj5NR7Pjfhw75Cl5RgISkddkRt6TUzInwtv1pt6R99bf9w/9md/n+l6v2Sf/mP/hN7dhxUE=</latexit> 1-Flatness constraintTriangle closure constraint

Parallel transport

Hypersurface

triangulation:

three tetrahedra meeting at an edge

Edge vector needs to be defined in one of the three adjacent tetrahedral reference

systems; parallel transported to the others.

Towards a spin bubble modelOne would now (2-)group Fourier-transform:

• SO(4)-group Fourier transform: not useful, as there are parallel transport matrices appearing in triangle closure constraints

• 2-group Fourier-transform: not known (no Peter-Weyl theorem)




[Asante, BD, Girelli, Riello, Tsimiklis ‘19] Way forward:

• solve part of the delta-functions: triangle closures

• Key insight: includes edge simplicity constraints (half of 1-flatness constraints)




[Asante, BD, Girelli, Riello, Tsimiklis ‘19] Way forward:

• solve part of the delta-functions: triangle closures

• Key insight: includes edge simplicity constraints (half of 1-flatness constraints)

hole(g) . Ee = Ee<latexit sha1_base64="IHNVadq1wT4VdOwm5NphYWK39q4=">AAACDnicbVBNS8NAEN3Ur1q/qh69LBahXkqigl6EoggeFawVmhA222m6uNmE3YlQQn+BF/+KFw+KePXszX/jtvag1gcDj/dmmJkXZVIYdN1PpzQzOze/UF6sLC2vrK5V1zeuTZprDi2eylTfRMyAFApaKFDCTaaBJZGEdnR7OvLbd6CNSNUVDjIIEhYr0ROcoZXC6k7h64T2UxlCPd4dUh+1YCqWoEXcR3oWwrGtsFpzG+4YdJp4E1IjE1yE1Q+/m/I8AYVcMmM6npthUDCNgksYVvzcQMb4LYuhY6liCZigGL8zpDtW6dJeqm0ppGP150TBEmMGSWQ7E4Z989cbif95nRx7R0EhVJYjKP69qJdLiikdZUO7QgNHObCEcS3srZT3mWYcbYIVG4L39+Vpcr3X8PYbe5cHtebJJI4y2SLbpE48ckia5JxckBbh5J48kmfy4jw4T86r8/bdWnImM5vkF5z3LzdMm5I=</latexit>

(on hypersurface)

Continuum:

• make use of 1-gauge invariance of path integral

dAE = 0 ) dAdAE = 0 ) F ^ E = 0<latexit sha1_base64="9pj7NBuLjeN/MYXUSiIcv+PuOUc=">AAACNHicdVDLSgMxFM34rPVVdekmWARXZaYKuhGqoghuqtgHdIYhk7ltQzMPk4yllH6UGz/EjQguFHHrN5hOu9BWDwROzjmX5B4v5kwq03wxZmbn5hcWM0vZ5ZXVtfXcxmZVRomgUKERj0TdIxI4C6GimOJQjwWQwONQ8zpnQ792D0KyKLxVvRicgLRC1mSUKC25uSvfPTk/NrF9lxDfvmGttiJCRN30jrF28T+JkYAv7C74LcA64ubyZsFMgaeJNSZ5NEbZzT3ZfkSTAEJFOZGyYZmxcvpEKEY5DLJ2IiEmtENa0NA0JAFIp58uPcC7WvFxMxL6hAqn6s+JPgmk7AWeTgZEteWkNxT/8hqJah45fRbGiYKQjh5qJhyrCA8bxD4TQBXvaUKoYPqvmLaJIFTpnrO6BGty5WlSLRas/ULx+iBfOh3XkUHbaAftIQsdohK6RGVUQRQ9oGf0ht6NR+PV+DA+R9EZYzyzhX7B+PoGkT+pkw==</latexit>

Implementing torsion-freeness(equivalent to primary and secondary simplicity constraints)

• triangle closure constraint

• edge simplicity constraint

proof

[ BD, Ryan ’10;Asante, BD, Girelli, Riello, Tsimiklis ‘19]

Holonomies are Levi-Civita:

g = Boost(✓)Rot(E)<latexit sha1_base64="7qCSSslE+KSYAwi8FFRxoCPRk5Q=">AAACDHicbVDLSgMxFM3UV62vqks3wSK0mzJTBd0IpSK4rGIf0BlKJk3b0MxkSO4IZegHuPFX3LhQxK0f4M6/MW1noa0HAodzzuXmHj8SXINtf1uZldW19Y3sZm5re2d3L79/0NQyVpQ1qBRStX2imeAhawAHwdqRYiTwBWv5o6up33pgSnMZ3sM4Yl5ABiHvc0rASN18YXCJE1cFuCalhknRhSEDUpprd9Io1yWTssv2DHiZOCkpoBT1bv7L7UkaBywEKojWHceOwEuIAk4Fm+TcWLOI0BEZsI6hIQmY9pLZMRN8YpQe7ktlXgh4pv6eSEig9TjwTTIgMNSL3lT8z+vE0L/wEh5GMbCQzhf1Y4FB4mkzuMcVoyDGhhCquPkrpkOiCAXTX86U4CyevEyalbJzWq7cnhWqtbSOLDpCx6iIHHSOqugG1VEDUfSIntErerOerBfr3fqYRzNWOnOI/sD6/AHI5pos</latexit>

Step 1:

Only free parameter:

extrinsic curvature




proof




Length=norm of E

Step 1:

Step 2: Use this in the 1-flatness constraints. This is an algebraic calculation and fixes:

Just left with variables. (`, ✓)<latexit sha1_base64="nAkV4S94t3VCmkxc1IsWoxAG+KU=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoMQQcJuFPQY9OIxgnlAdg2zk04yZPbBTK8SlvyHFw+KePVfvPk3TpI9aGJBQ1HVTXeXH0uh0ba/rdzK6tr6Rn6zsLW9s7tX3D9o6ihRHBo8kpFq+0yDFCE0UKCEdqyABb6Elj+6mfqtR1BaROE9jmPwAjYIRV9whkZ6KLsg5Rl1cQjITrvFkl2xZ6DLxMlIiWSod4tfbi/iSQAhcsm07jh2jF7KFAouYVJwEw0x4yM2gI6hIQtAe+ns6gk9MUqP9iNlKkQ6U39PpCzQehz4pjNgONSL3lT8z+sk2L/yUhHGCULI54v6iaQY0WkEtCcUcJRjQxhXwtxK+ZApxtEEVTAhOIsvL5NmteKcV6p3F6XadRZHnhyRY1ImDrkkNXJL6qRBOFHkmbySN+vJerHerY95a87KZg7JH1ifPz51kbI=</latexit>

✓ = ±⇥(E) = ±⇥(`)<latexit sha1_base64="Qj480HskT0BIJHXD0rp/Js5tZUo=">AAACDnicbZBNS0JBFIbn9mn2ZbVsMySCbuReC2oTSBG0NPALvCJzx6MOzv1g5txAxF/Qpr/SpkURbVu36980V12Y9sLAM+85h5nzepEUGm37x1pb39jc2k7tpHf39g8OM0fHdR3GikONhzJUTY9pkCKAGgqU0IwUMN+T0PCGt0m98QhKizCo4iiCts/6gegJztBYnUzOxQEgu3Yjn7rVBPN3hcWbC1IWOpmsXbSnoqvgzCFL5qp0Mt9uN+SxDwFyybRuOXaE7TFTKLiESdqNNUSMD1kfWgYD5oNuj6frTGjOOF3aC5U5AdKpuzgxZr7WI98znT7DgV6uJeZ/tVaMvav2WARRjBDw2UO9WFIMaZIN7QoFHOXIAONKmL9SPmCKcTQJpk0IzvLKq1AvFZ3zYunhIlu+mceRIqfkjOSJQy5JmdyTCqkRTp7IC3kj79az9Wp9WJ+z1jVrPnNC/sj6+gUkDJrj</latexit>


extrinsic curvature




proof




Length=norm of E

Step 1:

Step 2: Use this in the 1-flatness constraints. This is an algebraic calculation and fixes:

Step 3:

Just left with variables. (`, ✓)<latexit sha1_base64="nAkV4S94t3VCmkxc1IsWoxAG+KU=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoMQQcJuFPQY9OIxgnlAdg2zk04yZPbBTK8SlvyHFw+KePVfvPk3TpI9aGJBQ1HVTXeXH0uh0ba/rdzK6tr6Rn6zsLW9s7tX3D9o6ihRHBo8kpFq+0yDFCE0UKCEdqyABb6Elj+6mfqtR1BaROE9jmPwAjYIRV9whkZ6KLsg5Rl1cQjITrvFkl2xZ6DLxMlIiWSod4tfbi/iSQAhcsm07jh2jF7KFAouYVJwEw0x4yM2gI6hIQtAe+ns6gk9MUqP9iNlKkQ6U39PpCzQehz4pjNgONSL3lT8z+sk2L/yUhHGCULI54v6iaQY0WkEtCcUcJRjQxhXwtxK+ZApxtEEVTAhOIsvL5NmteKcV6p3F6XadRZHnhyRY1ImDrkkNXJL6qRBOFHkmbySN+vJerHerY95a87KZg7JH1ifPz51kbI=</latexit>

✓ = ±⇥(E) = ±⇥(`)<latexit sha1_base64="Qj480HskT0BIJHXD0rp/Js5tZUo=">AAACDnicbZBNS0JBFIbn9mn2ZbVsMySCbuReC2oTSBG0NPALvCJzx6MOzv1g5txAxF/Qpr/SpkURbVu36980V12Y9sLAM+85h5nzepEUGm37x1pb39jc2k7tpHf39g8OM0fHdR3GikONhzJUTY9pkCKAGgqU0IwUMN+T0PCGt0m98QhKizCo4iiCts/6gegJztBYnUzOxQEgu3Yjn7rVBPN3hcWbC1IWOpmsXbSnoqvgzCFL5qp0Mt9uN+SxDwFyybRuOXaE7TFTKLiESdqNNUSMD1kfWgYD5oNuj6frTGjOOF3aC5U5AdKpuzgxZr7WI98znT7DgV6uJeZ/tVaMvav2WARRjBDw2UO9WFIMaZIN7QoFHOXIAONKmL9SPmCKcTQJpk0IzvLKq1AvFZ3zYunhIlu+mceRIqfkjOSJQy5JmdyTCqkRTp7IC3kj79az9Wp9WJ+z1jVrPnNC/sj6+gUkDJrj</latexit>

U(1) Fourier transform

�(✓ ±⇥(l)) =X

s2Zeis(✓±⇥(l))

<latexit sha1_base64="sjIMhgt0i6oh6cC15Mt143G6VsY=">AAACN3icbVBNSyNBFOxRd1fj7hr16KUxCPESZlRwLwuiF0+iYFQ2nQ09nRfTpLtn6H6zEIb5V178G9704kERr/4De+KAX1vQUFS9ot+rOFXSYRheB1PTM1++fpudq81///Fzob64dOKSzApoi0Ql9izmDpQ00EaJCs5SC1zHCk7j0V7pn/4D62RijnGcQlfzcyMHUnD0Uq9+wPqgkDcZDgE5SzU7LklTra//Zi7TvdwxaSjTHIdxnP8pCgp/c+mqAPUJ+hopevVG2AonoJ9JVJEGqXDYq1+xfiIyDQaF4s51ojDFbs4tSqGgqLHMQcrFiJ9Dx1PDNbhuPrm7oGte6dNBYv0zSCfq20TOtXNjHfvJcn/30SvF/3mdDAe/urk0aYZgxMtHg0xRTGhZIu1LCwLV2BMurPS7UjHklgv0Vdd8CdHHkz+Tk41WtNnaONpq7OxWdcySFbJKmiQi22SH7JND0iaCXJAbckfug8vgNngIHl9Gp4Iqs0zeIXh6Bp8KrFU=</latexit>


extrinsic curvature

Spin bubble model for quantum flat space-time

Obtain amplitudes of the KBF-model:[ Korepanov `02; Baratin, Freidel `07] turn it into well-defined model via gauge fixing

Z =

ZD[è]

X

{st2Z},{✏�=±1}

µ(è, st)Y

�

exp�i✏�

X

t2�

st⇥t(`)�

<latexit sha1_base64="vuj3HepDFo5k1WkR3StY26iD4wE=">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</latexit>

Sum over enforces flatness. st<latexit sha1_base64="DS3a4VGNb2ceztQ1s+VmssFx7ZA=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2m3bpZhN2J0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRRI3g7GtzO//cS1EbF6xEnC/YgOlQgFo2ilB9PHfrniVt05yCrxclKBHI1++as3iFkacYVMUmO6npugn1GNgkk+LfVSwxPKxnTIu5YqGnHjZ/NTp+TMKgMSxtqWQjJXf09kNDJmEgW2M6I4MsveTPzP66YYXvuZUEmKXLHFojCVBGMy+5sMhOYM5cQSyrSwtxI2opoytOmUbAje8surpFWrehfV2v1lpX6Tx1GEEziFc/DgCupwBw1oAoMhPMMrvDnSeXHenY9Fa8HJZ47hD5zPH2rsjeI=</latexit>

Spin bubble model for quantum flat space-time

Obtain amplitudes of the KBF-model:[ Korepanov `02; Baratin, Freidel `07] turn it into well-defined model via gauge fixing

Z =

ZD[è]

X

{st2Z},{✏�=±1}

µ(è, st)Y

�

exp�i✏�

X

t2�

st⇥t(`)�

<latexit sha1_base64="vuj3HepDFo5k1WkR3StY26iD4wE=">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</latexit>

Sum over enforces flatness. st<latexit sha1_base64="DS3a4VGNb2ceztQ1s+VmssFx7ZA=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2m3bpZhN2J0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRRI3g7GtzO//cS1EbF6xEnC/YgOlQgFo2ilB9PHfrniVt05yCrxclKBHI1++as3iFkacYVMUmO6npugn1GNgkk+LfVSwxPKxnTIu5YqGnHjZ/NTp+TMKgMSxtqWQjJXf09kNDJmEgW2M6I4MsveTPzP66YYXvuZUEmKXLHFojCVBGMy+5sMhOYM5cQSyrSwtxI2opoytOmUbAje8surpFWrehfV2v1lpX6Tx1GEEziFc/DgCupwBw1oAoMhPMMrvDnSeXHenY9Fa8HJZ47hD5zPH2rsjeI=</latexit>

st = at(le)<latexit sha1_base64="J3I/rRyG1oQoA9whmH2IMdoFz80=">AAAB83icbVBNS8NAEN3Ur1q/qh69BItQLyWpgl6EohePFewHtCFstpN26WYTdidCCf0bXjwo4tU/481/47bNQVsfDDzem2FmXpAIrtFxvq3C2vrG5lZxu7Szu7d/UD48aus4VQxaLBax6gZUg+ASWshRQDdRQKNAQCcY3838zhMozWP5iJMEvIgOJQ85o2ikvvbxhvpYFT6c++WKU3PmsFeJm5MKydH0y1/9QczSCCQyQbXuuU6CXkYVciZgWuqnGhLKxnQIPUMljUB72fzmqX1mlIEdxsqURHuu/p7IaKT1JApMZ0RxpJe9mfif10sxvPYyLpMUQbLFojAVNsb2LAB7wBUwFBNDKFPc3GqzEVWUoYmpZEJwl19eJe16zb2o1R8uK43bPI4iOSGnpEpcckUa5J40SYswkpBn8krerNR6sd6tj0VrwcpnjskfWJ8/RgiRLg==</latexit>

[Baratin, Freidel `14]

This state sum model can be reconstructed from 2-group representation theory.

(Simplex amplitude = Contraction of 2-intertwiners)

and are labels of representations and intertwiners of the Euclidean 2-group.

le<latexit sha1_base64="90sTdXQyj+5nkX+DEcH8DaopNOk=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0IPrYL1fcqjsHWSVeTiqQo9Evf/UGMUsjlIYJqnXXcxPjZ1QZzgROS71UY0LZmA6xa6mkEWo/m586JWdWGZAwVrakIXP190RGI60nUWA7I2pGetmbif953dSE137GZZIalGyxKEwFMTGZ/U0GXCEzYmIJZYrbWwkbUUWZsemUbAje8surpFWrehfV2v1lpX6Tx1GEEziFc/DgCupwBw1oAoMhPMMrvDnCeXHenY9Fa8HJZ47hD5zPH0mGjcw=</latexit>

st<latexit sha1_base64="DS3a4VGNb2ceztQ1s+VmssFx7ZA=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2m3bpZhN2J0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRRI3g7GtzO//cS1EbF6xEnC/YgOlQgFo2ilB9PHfrniVt05yCrxclKBHI1++as3iFkacYVMUmO6npugn1GNgkk+LfVSwxPKxnTIu5YqGnHjZ/NTp+TMKgMSxtqWQjJXf09kNDJmEgW2M6I4MsveTPzP66YYXvuZUEmKXLHFojCVBGMy+5sMhOYM5cQSyrSwtxI2opoytOmUbAje8surpFWrehfV2v1lpX6Tx1GEEziFc/DgCupwBw1oAoMhPMMrvDnSeXHenY9Fa8HJZ47hD5zPH2rsjeI=</latexit>

Obtain (Regge) gravity by constraining .

Quantum geometry

[Relation to Freidel, Livine, Pranzetti ’19]

Boundary Hilbert space:

• quantization of all variables:

• consistent quantization of all constraints: 1-Flatness, 2-Flatness, 1-Gauss, 2-Gauss

• (first class) subalgebra: 1-Gauss, 2-Gauss and Edge Simplicity

• reduced wave functions depend only on (`, ✓)<latexit sha1_base64="IX2NMjhfvfWybPunzJ7iX7cABRY=">AAAB9HicbVBNS8NAEN3Ur1q/qh69BItQQUpSBT0WvXisYD+gCWWznbRLN5u4OymU0t/hxYMiXv0x3vw3btsctPXBwOO9GWbmBYngGh3n28qtrW9sbuW3Czu7e/sHxcOjpo5TxaDBYhGrdkA1CC6hgRwFtBMFNAoEtILh3cxvjUBpHstHHCfgR7QvecgZRSP5ZQ+EuPBwAEjPu8WSU3HmsFeJm5ESyVDvFr+8XszSCCQyQbXuuE6C/oQq5EzAtOClGhLKhrQPHUMljUD7k/nRU/vMKD07jJUpifZc/T0xoZHW4ygwnRHFgV72ZuJ/XifF8MafcJmkCJItFoWpsDG2ZwnYPa6AoRgbQpni5labDaiiDE1OBROCu/zyKmlWK+5lpfpwVardZnHkyQk5JWXikmtSI/ekThqEkSfyTF7JmzWyXqx362PRmrOymWPyB9bnD+YdkYg=</latexit>

G-network functionsFuture work: relate explicitly to 2-group representation theory.

• Hamiltonian: 2-Flatness constraint

Generates vertex translations (diffeos) for the tetrad sector.

[Asante, BD, Girelli, Riello, Tsimiklis ‘19]

Gc =X

f2c

±⌃0f

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Summary and OutlookHigher gauge theory:An alternative approach to quantum geometry:

• Includes tetrads. • Geometric transparent Hamiltonian and Diffeomorphism constraints.

Constructed relation between 2-group and 2-representation picture of quantum flat space model.•Boundary Hilbert space with all constraints. •G-network functions. [Asante, BD, Girelli, Riello, Tsimiklis '19]

Infinite many things to do, both on the mathematical (TQFT) and physical (QG) side:

• (3+1)D flat space holography [Asante, BD, Haggard 18]

• construct explicit 2-group Fourier transform (for more general 2-groups)• study defects: coupling to strings and particles• generalize to homogeneously curved space-times (McDowell-Mansouri action)• impose simplicity constraints: in phase space• quantization: which spectra do survive?• improve Hamiltonian constraints• …

Summary and OutlookHigher gauge theory:An alternative approach to quantum geometry:

• Includes tetrads. • Geometric transparent Hamiltonian and Diffeomorphism constraints.

Constructed relation between 2-group and 2-representation picture of quantum flat space model.•Boundary Hilbert space with all constraints. •G-network functions. [Asante, BD, Girelli, Riello, Tsimiklis '19]

Infinite many things to do, both on the mathematical (TQFT) and physical (QG) side:

• (3+1)D flat space holography [Asante, BD, Haggard 18]

• construct explicit 2-group Fourier transform (for more general 2-groups)• study defects: coupling to strings and particles• connection to tele-parallel gravity• generalize to homogeneously curved space-times (McDowell-Mansouri action)• impose simplicity constraints: in phase space• quantization: which spectra do survive?• improve Hamiltonian constraints• …

Documents

Quantum Geometry from Higher Gauge Theoryrelativity.phys.lsu.edu/ilqgs/dittrich091019.pdf · 2019. 9. 9. · Quantum Geometry from Higher Gauge Theory. Main Message: Prospect for