Algebraic Structure in Renormalization of Combinatorially Non-Local Field

Algebraic structure in renormalization of combinatorially non-local ﬁeld theories

Johannes Th¨urigen,

based on arXiv:2102.12453 (MPAG to appear), 2103.01136

Local QFT Seminar, HU Berlin – May 10, 2021

1 / 28 Perturbative ﬁeld theory

D 0 R 0 Fields φ : R → R with covariance/propagator P (xx,xx ) = dµ[φ]φ(x)φ(x ): Z n (n) iSia[φ] Y G (x1, ...,xxn) = dµ[φ]e φ(xi) i=1 Z Z k k k k Y X Y ˜ Sia[φ] = dx λkφ(x) = λk dqi δ qi φ(qi) D D R R i=1 i=1 i=1

Point-like interactions, e.g. quartic k = 4 : =∼

l iSia[φ] P (iSia) Perturbative exp. e = l l! ⇒ formal power series over graphs γ:

X 1 Y Z Y X G(n)(p , ...,pp ) = dq P˜(q ) iλ δ q 1 n |Aut γ| e e v e γ∈G1, e∈Eγ v∈Vγ e@v e Nγ =n

2 / 28 Combinatorially non-local ﬁeld theory (cNLFT)

Combinatorially non-local interactions for ﬁelds φ :(Rd)r → R:

Z k k Y Y a b Y ˜ Sia[φ] = λγ dqi δ(qi − qj ) φ(qi) i=1 (ia,jb) i=1

pairwise convolution of individual entries qa ∈ Rd, a = 1, ..., r

=∼

Combinatorics of interaction: vertex graph γ = c , not just k = Vγ

3 / 28 Perturbation theory: 2-graphs

Perturbative series over “ribbon graphs”, “stranded diagramms” ... here general concept: 2-graphs Γ ∈ G2

4 6 1 2 7

3 5

Z γ X 1 Y Y Y ˜ G (p1, ...,ppVγ ) = iλγv dqf P (qi) |Aut Γ| d int Γ∈G2, v∈VΓ f∈F R {i,j}∈EΓ ∂Γ=γ Γ Feynman rules:

1 coupling iλγv for each vertex v ∈ VΓ with vertex graph γv

2 propagator P˜(qi) for each internal edge e = {i, j} ∈E Γ,

3 R int a Lebesgue integral d dqf for internal face f ∈ F (qf = q identiﬁed) R Γ i

4 / 28 Renormalization

Integrals might not converge → renormalization needed various prescriptions how to remove inﬁnite part of the integral always necessary: forest formula to subtract subdivergences universally described by the Connes-Kreimer Hopf algebra principle of locality needed for this

Hopf algebra of Feynman graphs generalizes to 2-graphs in cNLFT Locality is captured by vertex graphs Applications of cNLFT: Probability & combinatorics: random matrices → random tensors Quantum ﬁeld theory: non-trivial solvable models, e.g. Grosse-Wulkenhaar model (topology the missing ingredient in QFT?) Quantum gravity: Generating function of lattice gauge theories (“group ﬁeld theory”)

5 / 28 Outline

1 Non-local ﬁeld theory Combinatorial non-locality

2 2-graphs From 1-graphs to 2-graphs Contraction and boundary Algebra

3 Renormalization in cNLFT Renormalizable ﬁeld theories The BPHZ momentum scheme Outlook: combinatorial DSE Half-edge graphs + strands

A 1-graph is a tuple g = (V, H, ν, ι) with a set of vertices V ι a set of half-edges H an adjacency map ν : H→V an involution ι : H→H pairing edges (ﬁxed points are external edges)

A 2-graph is G = (V, H, ν, ι; S, µ,σ 1, σ2) with a set of strand sections S ι, σ2 an adjacency map µ : S→H

a fixed-point free involution σ1 : S→S with ∀s ∈S : ν ◦ µ ◦ σ1(s) = ν ◦ µ(s) an involution σ2 : S→S pairing strands at edges: ∀s ∈S : ι ◦ µ(s) = µ ◦ σ2(s) and s is fixed point of σ2 iff µ(s) is fixed point of ι.

H v e S Involutions ι,σ 1, σ2 are equivalent to edge sets E⊂ 2 and S , S ∈ 2

6 / 28 Vertex-graph representation

−1 −1 Vertex graph gv = (Vv, Hv, νv,ι v) := ν (v), (ν ◦ µ) (v), µ|Hv ,σ 1|Hv

−→ gv =

Represent 2-graphs via vertex graphs G πvg :(V, H, ν, ι; S, µ,σ 1, σ2) 7→ gv, ι, σ2 v∈V

(i) Not bijective! In general gv = tigv , vertex belonging information lost... βvg :(V, H, ν, ι; S, µ,σ 1, σ2) 7→ {gv}v∈V , ι, σ2 is bijection

7 / 28 Example: combinatorial maps

A combinatorial map is a triple (H, σ, ι) with permutation σ : H→H whose cycles deﬁne (oriented) vertices

Example: 1 2 4 6 7 (H, σ, ι) = ({1, 2, 3, 4, 5, 6, 7}, (1)(234)(576), (12)(35)(46)(7)) = 3 5 Maps are 2-graphs with 2 strands per edge

Strands encoded in oriented vertices, e.i. σ-cycles v = (h1, h2, ..., hn) deﬁne

gv = ({h1, h2, ..., hn}, {s1n, s12, s21, s23, ...sn n−1, sn1}, νv, ιv)

and ιv pairs sij with sji iﬀ σ(hi) = hj.

4 6 4 6 1 2 1 2 7 7 =∼

3 5 3 5 8 / 28 Example: edge-coloured graphs

Feynman diagrams of rank-r tensor theories: regular edge-coloured graphs (r + 1)-coloured graphs are 2-graphs with r strands per edge colour c = 0 edges → 2-graph edges colour c 6= 0 subgraph components → vertex graphs

stranding of edges σ2 ﬁxed by colour preservation

c1 c2 c1 c2

∼=

Bijective only for connected vertex graphs (usual case in tensor theories)

Topology of edge-coloured graphs

(r + 1)-coloured graphs ⇐⇒ r-dimensional pseudo manifolds [Gurau ’11] (abstract simplicial complexes)

9 / 28 Outline

1 Non-local ﬁeld theory Combinatorial non-locality

2 2-graphs From 1-graphs to 2-graphs Contraction and boundary Algebra

3 Renormalization in cNLFT Renormalizable ﬁeld theories The BPHZ momentum scheme Outlook: combinatorial DSE Subgraphs

For a 2-graph G, a subgraph H is a 2-graph diﬀering from G in EH ⊂E G and e e SH ⊂S G. Then one writes H ⊂ G.

Example

E 2 G subgraphs per 2-graph G, H0 = for example for H1 = 1 c1 4 5 c2 8

G = : H2 = 2 3 6 7 H3 =

10 / 28 Contraction

Contraction of H ⊂ G: shrinking all stranded edges of H:

VG/H = KH the set of connected components of H ext ext HG/H = HH , SG/H = SH , only external half-edges of H remain v ext SG/H = {{s1, s2n}|(s1...s2n) ∈ FH }, external faces are shrunken to the strands at the new contracted vertices. e e e EG/H = EG \E H , SG/H = SG \S H (deleting stranded edges of H)

Example: G/H for H =

1 c 4 5 c 8 1 c 8 1 c 8 4 5 c1 = c2 = c : 1 8 2 7 3 6 2 3 6 7 2 c 7 2 7 4 3 1 c1 4 5 c2 8 c1 c2 c1 c2 1 8 1 7 1 7 c1 6= c2 : 2 8 2 8 2 3 6 7 5 6 2 7

11 / 28 Labelled vs. Unlabelled

Unlabelled 2-graphs

Isomorphism j : G1 → G2 is a triple of bijections j = (jV , jH, jS ) s.t.: −1 −1 νG2 = jV ◦ νG1 ◦ jH and µG2 = jH ◦ µG1 ◦ jS −1 ιG2 = jH ◦ ιG1 ◦ jH −1 −1 σ1G2 = jS ◦ σ1G1 ◦ jS and σ2G2 = jS ◦ σ2G1 ◦ jS ∼ Then equivalence G1 = G2, unlabelled 2-graph, Γ = [G1]∼= = [G2]∼=. Compatible with contractions. Example: ∼ H1 = = H2 =

1 c 8 4 5 ⇒ [G/H1]∼= = 1 8 = [G/H2]∼= = 2 7 ∼= 3 6 ∼= 2 c 7

12 / 28 Boundary and external structure

Residue and skeleton ∗ 2-graph has two characteristic 2-graphs without edges R ⊂ G2: ∗ res : G2 → R , Γ 7→ Γ/Γ , the “external structure” ∗ skl : G2 → R , Γ 7→ Θ0 , the subgraph without edges

Boundary and vertex graphs Can be used to deﬁne the boundary 1-graph of a 2-graph:

∂ : G2 → G1, Γ 7→ ∂Γ := πvg(res(Γ)) ς : G → G , Γ 7→ ςΓ := π (skl(Γ)) = F γ 2 1 vg v∈VΓ v For r-coloured 2-graphs: indeed (r − 1)-dimensional boundary ps. manifolds

External structure must be sensitive to con. comp. (e.g. t ): F ∂e : G2 → P(G1), Γ = i Γi 7→ ∂eΓ := {∂Γi}i = βvg(res(Γ)) ς : G → P(G ) , Γ 7→ ςΓ := {γ } = β (skl(Γ)) e 2 1 e v v∈VΓ vg

13 / 28 Outline

1 Non-local ﬁeld theory Combinatorial non-locality

2 2-graphs From 1-graphs to 2-graphs Contraction and boundary Algebra

3 Renormalization in cNLFT Renormalizable ﬁeld theories The BPHZ momentum scheme Outlook: combinatorial DSE Coalgebra

Algebra

Let G := hG2i be the Q-algebra generated by all 2-graphs Γ ∈ G2 with

m : G ⊗ G → G , Γ1 ⊗ Γ2 7→ Γ1 t Γ2

Unital commutative algebra with u : Q → G, q 7→ q1 (1 empty 2-graph) Coalgebra X ∆ : G → G ⊗ G, Γ 7→ Θ ⊗ Γ/Θ Θ⊂Γ

Associative counital coalgebra with counit = χR∗ : G → Q In fact, also bialgebra (all proofs completely parallel to 1-graphs)

Example: ∆Γ = ⊗

+ ⊗ + ⊗

14 / 28 Subalgebras

Contraction closure

Let P, K ⊂ G2. P-contraction closure PK := {Γ = Γ0/Θ|Θ ⊂ Γ0 ∈ K, Θ ∈ P} contraction closure K := G2 K

2-graph subbialgebra

2-graphs of restricted vertex types V: G2(V) := {Γ ∈ G2 | ςeΓ ∈ P(V)} Prop: hG2(V)i is a subbialgebra of G.

for ﬁeld theory with interactions V ∈ G1: “theory space” hG2(V)i

Example: Matrix/Tensor ﬁeld theory 2-graphs characterized by ﬁxed # of strands at edges = tensor rank r

for rank-r interactions Vr: G2(Vr) = G2(Vr) contraction closed

r-coloured diagrams generate subbialgebra hG2(Vr)i

15 / 28 Hopf algebra of 2-graphs

interest: group structure on algebra homomorphisms φ, ψ : G → A wrt

convolution product: φ ∗ ψ := mA ◦ (φ ⊗ ψ) ◦ ∆G

Hopf algebra of 2-graphs The bialgebra of 2-graphs G is a Hopf algebra, i.e. there is a coinverse S:

S ∗ id = id ∗ S = u ◦ .

G The set ΦA of algebra homomorpisms from G to a unital commutative φ G algebra A is a group with inverse S = φ ◦ S for every φ ∈ ΦA,

φ φ S ∗ φ = φ ∗ S = uA ◦ G.

The subbialgebra hG2(V)i for speciﬁc vertex graphs V ⊂ G1 is a Hopf subalgebra of G.

16 / 28 Outline

1 Non-local ﬁeld theory Combinatorial non-locality

2 2-graphs From 1-graphs to 2-graphs Contraction and boundary Algebra

3 Renormalization in cNLFT Renormalizable ﬁeld theories The BPHZ momentum scheme Outlook: combinatorial DSE Renormalizability

cNLFT T = (E, V, ω, d) given by dimension d ∈ N, E, V ⊂ G1, weights

ω : E ∪ V → Z

T T Feynman diagrams G2 := G2(V) generate a Hopf algebra GT := hG2 i Hopf algebra of divergent Feynman 2-graphs sd P P Superﬁcial degree of divergence ω (Γ) = ω(γv) − ω(γe) + d · FΓ v∈VΓ e∈EΓ T is renormalizable iﬀ ωsd(Γ) = ω(∂Γ) for all Γ with ωsd(Γ) > 0

s.d. n G T sd o PT := Γ = i∈I Γi ∈ G2 1PI |∀i ∈ I :Γi 6∈ R ⇒ ω (Γi) ≥ 0

f2g s.d. HT = hPT i is the Hopf algebra of divergent 2-graphs of T Hopf subalgebra of GT since contraction closed due to renormalizablity.

17 / 28 Example: Matrix ﬁeld theory

n φD matrix ﬁeld theory [Wulkenhaar’19][Hock’20]: dimension d = D/2 (D spacetime dim. of related non-commutative QFT)

interactions V are polygons up to n vertices, ω|V = 0 Feynman diagrams Γ are 2-graphs bijective to combinatorial maps

sd ω (Γ) = d · FΓ − EΓ n ! d − 1 X (k) = −d(V − 1) + kV − V − d (2g + K − 1) Γ 2 Γ ∂Γ Γ ∂Γ k=1

4 Grosse-Wulkenhaar model ≡ φD matrix ﬁeld theory D − 2 2ωsd(Γ) = D − V + (D − 4)V − D(2g + K − 1) . 2 ∂Γ Γ Γ ∂Γ

just renormalizable in D = 4

only planar maps (gΓ = 0) with single boundary (K∂Γ = 1) divergent

18 / 28 Tensorial ﬁeld theory

n φd,r tensorial ﬁeld theory [BenGeloun’14]: similar to dr = d(r − 1) dimensional local ﬁeld theory dr −2ζ interactions V are r-coloured graphs, ω(γv) = dr − 2 Vγv Feynman diagrams Γ are 2-graphs bijective to (r + 1)-coloured graphs

Divergence degree (for general propagator ω(γe) = 2ζ):

d − 2ζ 2ωg − 2ωg ωsd(Γ) = d − r V − d Γ ∂Γ + K − 1 . r 2 ∂Γ (r − 1)! ∂Γ

g P Gurau degree ω = J gJ generalizes genus g (J generalize Heegaard surfaces)

theories renormalizable for interactions up to n = b 2dr c dr −2ζ 4 1 4 4 just-renormalizable φd,r theories: dr = 4ζ (e.g. ζ = 2 : φ2,2, φ1,3) g g coproduct preserves ω [Raasakka/Tanasa’13] ⇒ renormalizability for ωΓ > 0 6 K∂Γ > 1 possible: e.g. φ1,4 theory [BenGeloun/Rivasseau’13] needs ∈ V

19 / 28 Outline

1 Non-local ﬁeld theory Combinatorial non-locality

2 2-graphs From 1-graphs to 2-graphs Contraction and boundary Algebra

3 Renormalization in cNLFT Renormalizable ﬁeld theories The BPHZ momentum scheme Outlook: combinatorial DSE Momemtum scheme in cNLFT

algebra homomorphism A : G → A to the algebra A of integrals with rational integrands Z Y Y Y ˜ AΓ = A(Γ) : {pf }f∈F ext, 7→ AΓ({pf }) := λγv dqf P (qi) eΓ d v∈V int R {i,j}∈E Γ f∈FΓ Γ

sd Superﬁcial degree of divergence: ω (Γ) = d · FΓ − 2ζEΓ Momentum subtraction operator

|~k| Λ ω P 1 ∂ AΓ Q kf R[AΓ]({pf }) := T AΓ ({pf }) = k 0 p {pf } ~k! Q f f ~ sd f ∂pf ext |k|≤ω (Γ) f∈FeΓ Renormalized amplitude for primitive divergent 2-graphs (no subdivergences):

Ar(Γ) := (A − R ◦ A)(Γ)

20 / 28 Example: Tadpole diagrams in tensorial theories

4 1 φ theory with P˜(p) = (i.e. ω(γe) = 1): =∼ d=2,r=2 |p1|+|p2|+1

q2 Z 1 1 Ar (p1) ≡ Ar p1 = λ 1 − Tp1 dq2 2 |p1| + |q2| + 1 R = 2πλ (|p1| + 1) log (|p1| + 1) − |p1|

φ4 theory with P˜(p) = 1 : two tadpoles d=1,r=3 |p1|+|p2|+|p3|+1 Z Z 1 dq2dq3 p1 Ar = λ c 1 − Tp1 |p1| + |q2| + |q3| + 1 R R = 4λ c (|p1| + 1) log (|p1| + 1) − |p1| q1 Z dq A = λ 1 − T 0 1 r p , p c p2,p3 2 3 |q1| + |p2| + |p3| + 1 R

= −2λ c log(|p2| + |p3| + 1)

21 / 28 Subdivergences

In a renormalizable local ﬁeld theory T : sd BPHZ: ∀Γ with ω (Γ) ≥ 0 there is a counter term s.t. Ar(Γ) converges Zimmermann: forest formula for counter term of nested subdivergences a fg fg Kreimer: counter term Sr : H → A from antipode S in Hopf alg. H :

a Ar = Sr ∗ A a a X h a i Sr (Γ) = −R [(Sr ∗ A ◦ P )(Γ)] = − R Sr (Θ)A(Γ/Θ) Θ∈Hfg Θ(Γ Renormalization in cNLFT a counter term Sr in the same way on the Hopf algebra of 2-graphs a f2g if cNLFT T is renormalizable, Ar = Sr ∗ A on HT gives ren. amplitudes a BPHZ momentum scheme: Sr is algebra homomorphism since R is a Rota-Baxter operator (R[AB] + R[A]R[B] = R[R[A]B + AR[B]]) as in local QFT

22 / 28 4 Example: sunrise diagram in φ2,2 theory

c c Sunrise 2-graph Γ = =∼ :

p p1 1 q2 a q p Ar(Γ)(p1, p2) = A + Sr 2 A 2 q1 p2 q1 q1 q 2 q2 p1 q2 a q a + Sr 1 A p1 + Sr q1 p p2 2

Last counter term calculated recursively:

p " p1 1 q2 a q p Sr (Γ) = −R A − R A 2 A 2 q1 p2 q1 q1 q 2 q2 # − R A q1 A p1

23 / 28 4 Example: sunrise diagram in φ2,2 theory

p1 q2 Ar q1 p2 Z Z 2 1 1 1 1 = λ 1 − Tp1,p2 dq1 dq2 2 2 |p1| + |q2| + 1 |q1| + |q2| + 1 |q1| + |p2| + 1 R R 1 0 1 1 + −Tp1,q1 |q1| + |p2| + 1 |p1| + |q2| + 1 |q1| + |q2| + 1 1 0 1 1 + −Tq2,p2 |p1| + |q1| + 1 |q1| + |q2| + 1 |q2| + |p2| + 1 2 2 4π X = λ |p1||p2|ζ2 + (|p1| + |p2| + 1) (|pi| + 1) log(|pi| + 1) − |pi| |p1| + |p2| + 1 i=1,2 Y X − (|pi| + 1) log(|pi| + 1) + |pi|(|pi| + 1)Li2(−|pi|) i=1,2 i=1,2

in agreement with [Hock2020] 2 multiple polylogarithms as in local QFT, but ζ2 = π /6 is peculiar

24 / 28 4 Example: sunrise diagram in φ1,3 theory

p1 q2, q3 Sunrise diagram has q1 p2, p3 only logarithmic divergence ωsd(Γ) = 3 − 3 = 0

only one proper divergent 1PI subgraph q2, q3

⇒ no overlapping divergence ⇒ factorizing Ar Z 2 0 1 Ar(p1, p2, p3) = λ c 1 − Tp1,p2,p3 dq1 1 |q1| + |p2| + |p3| + 1 R Z Z 0 1 1 × 1 − Tp1,q1 dq2 dq3 |q1| + |q2| + |q3| + 1 |p1| + |q2| + |q3| + 1 R R

more restricted set of LO diagrams (“melonic”) in tensorial theories conjecture: all amplitudes are multiple polylogarithms

25 / 28 Outline

1 Non-local ﬁeld theory Combinatorial non-locality

2 2-graphs From 1-graphs to 2-graphs Contraction and boundary Algebra

3 Renormalization in cNLFT Renormalizable ﬁeld theories The BPHZ momentum scheme Outlook: combinatorial DSE Combinatorial Dyson-Schwinger equations

Comb. Green’s fct. expanded in # faces F (=# loops for planar maps in MFT):

∞ X Γ X Xγ = r ± αFΓ = r ± αjcγ γ |Aut Γ| γ j f2g j=1 Γ∈HT ∂Γ=γ

Γ Comb. Dyson-Schwinger eq. hold with usual comb. factors in B+ [Kreimer ’08]: γ X k X Γ γ X = rγ ± α B+ (X Qγ ) k≥1 Γ prim. FΓ=k ∂Γ=γ

by construction Hochschild 1-cocycles

26 / 28 Example: cDSE in quartic matrix ﬁeld theory

simpliﬁcations: |Aut Γ| = 1 for any map with boundary only two primitives for the 2-point function (due to planarity): X = − α B+ + B+ (X )

cDSE hold!! (no mixing with planar irregular sector necessary [Tanasa/Kreimer ’13])

Challenges: still inﬁnitely many primitives for X

factor # maximal forests in B+ non-trivial: practical use of cDSE? possible to identify the reason for solvability here??? necessary to modify the cDSE framework?

27 / 28 Outlook

Result: algebraic structure of renormalization generalizes to cNLFT Random geometry/quantum gravity occurs at criticality → understand non-perturbative cNLFT! identify algebraic structure underlying solvability of matrix ﬁeld theory apply to tensors of rank r > 2

Thanks for your attention!

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