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 jAut γj e e v e γ2G1; e2Eγ v2Vγ 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 2 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 2 G2 4 6 1 2 7 3 5 Z γ X 1 Y Y Y ~ G (p1; :::;ppVγ ) = iλγv dqf P (qi) jAut Γj d int Γ2G2; v2VΓ f2F R fi;jg2EΓ @Γ=γ Γ Feynman rules: 1 coupling iλγv for each vertex v 2 VΓ with vertex graph γv 2 propagator P~(qi) for each internal edge e = fi; jg 2E Γ, 3 R int a Lebesgue integral d dqf for internal face f 2 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 8s 2S : ν ◦ µ ◦ σ1(s) = ν ◦ µ(s) an involution σ2 : S!S pairing strands at edges: 8s 2S : 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 2 6 / 28 Vertex-graph representation −1 −1 Vertex graph gv = (Vv; Hv; νv;ι v) := ν (v); (ν ◦ µ) (v); µjHv ;σ 1jHv −! gv = Represent 2-graphs via vertex graphs G πvg :(V; H; ν; ι; S; µ,σ 1; σ2) 7! gv; ι, σ2 v2V (i) Not bijective! In general gv = tigv , vertex belonging information lost... βvg :(V; H; ν; ι; S; µ,σ 1; σ2) 7! fgvgv2V ; 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) = (f1; 2; 3; 4; 5; 6; 7g; (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 = (fh1; h2; :::; hng; fs1n; s12; s21; s23; :::sn n−1; sn1g; 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 = ffs1; s2ngj(s1:::s2n) 2 FH g, external faces are shrunken to the strands at the new contracted vertices. e e e EG=H = EG nE H , SG=H = SG nS 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)) & : G ! G ; Γ 7! &Γ := π (skl(Γ)) = F γ 2 1 vg v2VΓ 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 := f@Γigi = βvg(res(Γ)) & : G !P(G ) ; Γ 7! &Γ := fγ g = β (skl(Γ)) e 2 1 e v v2VΓ 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 2 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 := fΓ = Γ0=ΘjΘ ⊂ Γ0 2 K; Θ 2 Pg contraction closure K := G2 K 2-graph subbialgebra 2-graphs of restricted vertex types V: G2(V) := fΓ 2 G2 j &eΓ 2 P(V)g Prop: hG2(V)i is a subbialgebra of G. for field theory with interactions V 2 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, : G!A wrt convolution product: φ ∗ := mA ◦ (φ ⊗ ) ◦ ∆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 2 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.