Hopf algebras and factorial divergent power series: Algebraic tools for graphical enumeration Michael Borinsky 1 Humboldt-University Berlin Departments of Physics and Mathematics Workshop on Enumerative Combinatorics, ESI, October 2017 - Renormalized asymptotic enumeration of Feynman diagrams, Annals of Physics, October 2017 arxiv:1703.00840 1 borinsky@physik.hu-berlin.de M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 1
Overview 1 Motivation 2 Asymptotics of multigraph enumeration 3 Ring of factorially divergent power series 4 Counting subgraph-restricted graphs 5 Conclusions M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 2
Motivation Perturbation expansions in quantum field theory may be organized as sums of integrals. Each integral maybe represented as a graph. Usually these expansions have vanishing radius of convergence. The coefficients behave as f n CA n Γ(n + β) for large n. Divergence is believed to be caused by proliferation of graphs. M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 3
In zero dimensions the integration becomes trivial. Observables are generating functions of certain classes of graphs [Hurst, 1952, Cvitanović et al., 1978, Argyres et al., 2001, Molinari and Manini, 2006]. For instance, the partition function in ϕ 3 theory is formally, ) dx ( Z ϕ3 ( ) = e 1 x2 2 + x3 3!. 2π R Formal expand under the integral sign and integrate over Gaussian term by term, Z ϕ3 ( ) := n=0 n (2n 1)!![x 2n ]e x3 3! M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 4
Z ϕ3 is the exponential generating function of 3-valent multigraphs, Z ϕ3 ( ) = n=0 n (2n 1)!![x 2n ]e x3 3! = cubic graphs Γ The parameter counts the excess of the graph. E(Γ) V (Γ) Aut Γ Z ϕ3 ( ) = φ ( 1 + 1 8 + 1 48 + 1 16 = 1 + + 1 12 + 1 16 ( 1 8 + 1 12 + 1 128 + 1 288 + 1 + 1 8 24 ) + 385 1152 2 +... + 1 96 +... ) where φ(γ) = E(Γ) V (Γ) maps a graph with excess n to n. M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 5
Arbitrary degree distribution More general with arbitrary degree distribution, Z( ) = n (2n 1)!![x 2n ]e n=0 = graphs Γ k 3 λ k k! x k E(Γ) V (Γ) v V (Γ) λ v Aut Γ where the sum is over all multigraphs Γ and v is the degree of vertex v. This corresponds to the pairing model of multigraphs [Bender and Canfield, 1978]. M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 6
We obtain a series of polynomials n (2n 1)!![x 2n ]e n=0 + 1 128 + 1 16 k 3 + 1 288 + 1 16 λ k k! x k + 1 96 + 1 8 ( = φ 1 + 1 8 + 1 48 + 1 24 + 1 12 ) + + 1 8 where φ(γ) = E(Γ) V (Γ) v V (Γ) λ v (( 1 = 1 + 8 + 1 ) λ 2 3 + 1 ) 12 8 λ 4 ( 385 + 1152 λ4 3 + 35 64 λ2 3λ 4 + 35 384 λ2 4 + 7 48 λ 3λ 5 + 1 ) 48 λ 6 2 + = P n (λ 3, λ 4,...) n n=0 M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 7
Example Take the degree distribution λ k = 1 for all k 3. Z stir ( ) := n (2n 1)!![x 2n ]e n=0 ( = φ 1 + 1 8 + 1 12 k 3 1 k! xk + 1 8 ) +... where φ(γ) = ( 1) V (Γ) E(Γ) V (Γ) ( 1 = 1 + 8 ( 1)2 + 1 12 ( 1)2 + 1 ) 8 ( 1)1 +... = 1 + 1 12 + 1 288 2 139 51840 3 +... = e Stirling series as a signed sum over multigraphs. n=1 B n+1 n(n+1) n M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 8
Define a functional F : R[[x]] R[[ ]] F : S(x) n (2n 1)!![x 2n ]e n=0 x 2 2 +S(x) where S(x) = x2 2 + λ k k 3 k! x k. F maps a degree sequence to the corresponding generating function of multigraphs. M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 9
Expressions as F[S(x)] = n (2n 1)!![x 2n ]e n=0 x 2 2 +S(x) are inconvenient to expand, because of the and the 1 exponent we have to sum over a diagonal. in the Theorem This can be expressed without a diagonal summation [MB, 2017]: F[S(x)] = n (2n + 1)!![y 2n+1 ]x(y), n=0 where x(y) is the (power series) solution of y 2 2 = S(x(y)). M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 10
Asymptotics Calculate asymptotics of F[S(x)] by singularity analysis of x(y): Locate the dominant singularity of x(y) by analysis of the (generalized) hyperelliptic curve, y 2 2 = S(x). The dominant singularity of x(y) coincides with a branch-cut of the local parametrization of the curve. M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 11
4 2 y 0 2 4 4 2 0 2 4 x Figure: Example: The curve y 2 2 = x 2 2 x 3 3! associated to Z ϕ3. x(y) has a (dominant) branch-cut singularity at y = ρ = 2 3, where x(ρ) = τ = 2. M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 12
Near the dominant singularity: x(y) τ = d 1 1 y ρ + ( d j 1 y ) j 2 ρ j=2 ( ) [y n ]x(y) Cρ n n 3 e k 2 1 + n k Therefore, k=1 [ n ]F[S(x)] = (2n + 1)!![y 2n+1 ]x(y) = 2n Γ(n + 3 2 ) π [y 2n+1 ]x(y) ( C 2 n ρ 2n Γ(n) 1 + What is the value of C and the e k? k=1 e k n k ) M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 13
Theorem ([MB, 2017]) Let (x, y) = (τ, ρ) be the location of the dominant branch-cut singularity of y 2 2 = S(x). Then R 1 [ n ]F[S(x)]( ) = c k A (n k) Γ(n k) + O ( A n Γ(n R) ), k=0 where A = S(τ) and c k = 1 2π [ k ]F[S(τ) S(x + τ)]( ). The asymptotic expansion can be expressed as a generating function of graphs. M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 14
Example For cubic graphs or equivalently ϕ 3 theory, we are interested in S(x) = x2 2 + x3 3!, F [S(x)] ( ) = φ ( 1 + 1 + 1 + 1 8 12 128 1 + 5 385 + 24 1152 2 + 85085 82944 3 + +... ) We find τ = 2, A = 2 3 and the coefficients of the asymptotic expansion c k k = 1 1 2 F[S(τ) S(τ + x)]( ) = 2π 2π F[ x 2 + x 3 3! ]( ) k=0 = 1 ( 1 5 385 + 2π 24 1152 2 85085 ) 82944 3 +... The asymptotic expansion is [ n ]F [S(x)] ( ) = R 1 k=0 c ka n+k Γ(n k) + O(A n+r Γ(n R)). M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 15
Ring of factorially divergent power series Power series which have Poincaré asymptotic expansion of the form R 1 f n = c k A n+k Γ(n k) + O(A n Γ(n R)) R 0, k=0 form a subring of R[[x]] which is closed under composition and inversion of power series [MB, 2016]. For instance, this allows us to calculate the complete asymptotic expansion of constructions such as in closed form. log F [S(x)] ( ) M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 16
Renormalization Renormalization Restriction on the allowed bridgeless subgraphs. P is a set of forbidden subgraphs. We wish to have a map { E(Γ) V (Γ) if Γ has no subgraph in P φ P (Γ) = 0 else which is compatible with our generating function and asymptotic techniques. M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 17
G is the Q-algebra generated by multigraphs. Split G = G G + such that G + is the set of graphs without subgraphs in P. G is the set of graphs with subgraphs in P. We know a map φ : G R[[ ]]. Construct φ P such that φ P G + = φ and φ P G = 0. This is a Riemann-Hilbert problem. M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 18
Hopf algebra of graphs Pick a set of bridgeless graphs P, such that if γ 1 γ 2 then γ 1, γ 2 P iff γ 1, γ 2 /γ 1 P (1) if γ 1, γ 2 P then γ 1 γ 2 P (2) P H is the Q-algebra of graphs in P. Define a coaction on G: : G H G Γ γ Γ/γ γ Γ s.t.γ P H is a left-comodule over G. can be set up on H. : H H H. H is a Hopf algebra Connes and Kreimer [2001]. (1) implies is coassociative (id ) = ( id). M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 19 (3)
The coaction shall keep information about the number of edges it was connected in the original graph. The graphs in P have legs or hairs. Example: Suppose,, P, then = γ /γ = 1 + 3 + γ s.t.γ P where we had to consider the subgraphs,,, the complete and the empty subgraph. M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 20
Gives us an action on algebra morphisms G R[[ ]]. For ψ : H R[[ ]] and φ : G R[[ ]], ψ φ : G R[[ ]] ψ φ = m (ψ φ) and a product of algebra morphisms H R[[ ]]. For ξ : H R[[ ]] and ψ : H R[[ ]], ξ ψ : H R[[ ]] ξ ψ = m (ξ ψ) Coassociativity of implies associativity of : (ξ ψ) φ = ξ (ψ φ) M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 21
The set of all algebra morphisms H R[[ ]] with the -product forms a group. The identity ɛ : H R[[ ]] maps the empty graph to 1 and all other graphs to 0. The inverse of ψ : H R[[ ]] maybe calculated recursively: ψ 1 (Γ) = ψ(γ) γ Γ s.t.γ P ψ 1 (γ)ψ(γ/γ) Corresponds to Moebius-Inversion on the subgraph poset. Simplifies on many (physical) cases to a (functional) inversion problem on power series. Solves the Riemann-Hilbert problem: Invert φ : Γ E(Γ) V (Γ) restricted to H. Then { (φ 1 H φ)(γ) = E(Γ) V (Γ) if Γ G + 0 else M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 22
The identity van Suijlekom [2007], Yeats [2008], X = graphs Γ where X = graphs Γ v V (Γ) Γ Aut Γ and X ( v ) P X (k) P = Γ Aut Γ. Γ s.t.γ P and Γ has k legs Γ Aut Γ, can be used to make this accessible for asymptotic analysis: φ 1 H φ (X ) = m ( φ 1 H φ) X = ( ) φ 1 H X ( v ) φ(γ) P Aut Γ graphs Γ v V (Γ) M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 23
φ 1 H φ (X ) = graphs Γ v V (Γ) ( φ 1 H X ( v ) P ) φ(γ) Aut Γ The generating function of all graphs with arbitrary weight for each vertex degree is Therefore, φ 1 H n (2n 1)!![x 2n ]e n=0 = graphs Γ v V (Γ) λ v k 0 φ (X ) = n (2n 1)!![x 2n ]e n=0 λ k k! x k E(Γ) V (Γ) Aut Γ ( k 0 xk k! φ 1 H X (k) P ) M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 24
φ 1 H φ (X ) = n (2n 1)!![x 2n ]e n=0 ( k 0 xk k! φ 1 H In quantum field theories we want the set P to be all graphs with a bounded number of legs. This corresponds to restrictions on the edge-connectivity of the graphs. In renormalizable quantum field theories the expressions φ 1 H ( X (k) P ) are relatively easy to expand. ( ) are called The generating functions φ 1 H X (k) P counterterms in this context. They are (almost) the generating functions of the number of primitive elements of H: Γ is primitive iff Γ = Γ 1 + 1 Γ. We can obtain the full asymptotic expansions in these cases. M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 25 X (k) P )
Examples Expressed as densities over all multigraphs: In ϕ 3 -theory (i.e. three-valent multigraphs): P(Γ is 2-edge-connected) = e 1 ( 1 23 1 21 n +...). P(Γ is cyclically 4-edge-connected) = e ( 10 3 1 133 1 3 n + ) In ϕ 4 -theory (i.e. four-valent multigraphs): P(Γ is cyclically 6-edge-connected) = e ( 15 4 1 126 1 n + ). Arbitrary high order terms can be obtained by iteratively solving implicit equations. M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 26
Conclusions The zero-dimensional path integral is a convenient tool to enumerate multigraphs by excess. Asymptotics are easily accessible: The asymptotic expansion also enumerates graphs. With Hopf algebra techniques restrictions on the set of enumerated graphs can be imposed. M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 27
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MB. Renormalized asymptotic enumeration of feynman diagrams. arxiv preprint arxiv:1703.00840, 2017. LG Molinari and N Manini. Enumeration of many-body skeleton diagrams. The European Physical Journal B-Condensed Matter and Complex Systems, 51(3):331 336, 2006. Walter D van Suijlekom. Renormalization of gauge fields: A hopf algebra approach. Communications in Mathematical Physics, 276(3):773 798, 2007. Karen Yeats. Growth estimates for dyson-schwinger equations. arxiv preprint arxiv:0810.2249, 2008. M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 27