Algebro-Geometric Aspects of Feynman Graphs

Joint work with H. Esnault and D. Kreimer

1. Rank 1 quadrics, ω(q), Schwinger period.

2. Renormalization Hopf algebra in the category of motives.

3. Limiting mixed Hodge structures and renormalization.

Rank 1 Quadrics

Γ connected graph, n loops, N edges. 0 → H_{1}(Γ) → ℚ[Edges] → ℚ[V ert] → ℚ → 0

0 → H → ℚ[Edge] → Mom → 0 H = H_{1}; Mom = ℚ[V ert]^{0} := Momentum .

H ⊂ ℚ[Edge] ⟶^{e^{∨}} ℚ; e ∈ Edge

∑_{1}^{N} A_{i}(e_{i}^{∨})^{2} linear series of rank 1 quadrics on ℙ(H). M_{H} = M_{H}(A) corresponding symmetric matrix. Graph polynomial Ψ_{H} = Ψ_{Γ} := det M.

Dualize

Mom^{∨} ⊂ ℚ[Edges]^{∨} = ℚ[Edges]

Same construction yields symmetric matrix M_{Mom^{∨}}(A), Ψ_{Mom^{∨}} = det M_{Mom^{∨}}(A)

Ψ_{H}(A) = (∏_{1}^{N} A_{i})Ψ_{Mom^{∨}}(A^{−1})

Q non-degenerate quadratic form on V ↔ symmetric matrix S. Then S^{−1} defines a quadratic from Q^{−1} on V^{∨} (gauge invariant)

R(A) := M_{Mom^{∨}}(A^{−1})^{−1} = quadratic form on (Mom^{∨})^{∨} = Mom. R(cA) = cR(A). Entries homogeneous of degree 1.

R_{ij} = Θ_{ij}/Ψ_{H}; deg Θ_{ij} = deg Ψ_{H} + 1 = n + 1

Differential Forms

Value of interest to physicists:

I_{Γ}(q) :=

\frac{1}{(4π)^{2n}} ∫_{A_{1},...,A_{N} =0}^{∞} \frac{exp(−qRq^{t})dA_{1} … dA_{N}}{Ψ_{Γ}^{2}}
q ∈ Mom

Motives?

Ω_{N−1} := ∑_{1}^{N}(−1)^{i}A_{i}dA_{1} … d̂A_{i} … dA_{n}

Substitute A_{i} = λB_{i}

(4π)^{2n}I_{Γ}(q) =

∫_{A_{i}≥0; ∑ A_{i}=1} \frac{Ω_{N−1}}{Ψ_{Γ}^{2}} ∫_{λ=0}^{∞} exp(λqRq^{t})λ^{N−1−2n}dλ

Substitute ν = λqRq^{t}, σ : A_{i} ≥ 0, ∀i

I_{Γ}(q) = \frac{Γ(2n − N)}{(4π)^{2n}} ∫_{σ} \frac{(qRq^{t})^{2n−N}Ω}{Ψ_{Γ}^{2}}

Periods and Motives

ℙ^{N−1}, coordinates A_{1}, …, A_{N}, X_{Γ} : Ψ_{Γ} = 0. Δ : A_{1}A_{2} ⋯ A_{N} = 0 coordinate simplex. σ real N − 1-chain with ∂σ ⊂ Δ. Assume 2n − N ≥ 0. (Divergent case) [σ] ∈ H_{N−1}(ℙ^{N−1}, Δ; ℤ);

ω(q) := \frac{(qRq^{t})^{2n−N}Ω}{Ψ_{Γ}^{2}},

[ω(q)] ∈ H_{DR}^{N−1}(ℙ^{N−1} − X_{Γ}) ∫_{σ} ω(q) =^{?} period (depending on q) associated to motive

H^{N−1}(ℙ^{N−1} − X_{Γ}, Δ − Δ ∩ X_{Γ})

Log divergent case, N = 2n, no q dependence.

The Structure of X_{Γ}

Spanning tree T ⊂ Γ: h_{0}(T) = 1, h_{1}(T) = 0, T contains all vertices of Γ.

Ψ_{Γ}(A) = ∑_{T spn.tr.} ∏_{e∉T} A_{e}

Example: Γ “banana graph” N lines between 2 vertices. Spanning trees = single lines. Ψ_{banana} = ∑_{i} A_{1}A_{2} ⋯ Â_{i} ⋯ A_{N} Ψ_{Γ} sum of monomials with coefficient +1.

Singularities of ∫_{σ} ω(q): σ : A_{i} ≥ 0, σ ∩ X_{Γ} = ⋃ L_{γ} L_{γ} coordinate linear space, L_{γ} ⊂ X_{Γ}

L_{γ} ⊂ X_{Γ} ↔ {γ ⊂ Γ | h_{1}(γ) > 0}

L_{γ} : A_{e} = 0, e ∈ γ.

Blowing Up; Renormalization

L ⊂ X_{Γ}, L = L_{γ}, γ ⊂ Γ.

π^{−1}L = E ∪ Y ⟶ P = BL(L ⊂ ℙ^{N−1})
↓  ↓^{π}
L  ⟶  ℙ^{N−1}

#(Edge γ) = r, E ≅ L × ℙ^{r−1}. Homogeneous coordinates on ℙ^{r−1} ↔ e ∈ γ Normal cone of L ⊂ X_{Γ}:

Ψ_{Γ} = Ψ_{γ}Ψ_{Γ//γ} + R

deg_{A_{e},e∈γ} R > h_{1}(γ) = deg Ψ_{γ}.

Y ∩ E = (X_{Γ//γ} × ℙ^{r−1}) ∪ (L × X_{γ}) E − E ∩ Y = (L − X_{Γ//γ}) × (ℙ^{r−1} − X_{γ})

Motivic Realization of the Renormalization Hopf Algebra

Δ ⊂ ℙ^{N−1}; Δ′ ⊂ P strict transform of Δ ⊃ L Δ′ ∩ E = (Δ_{L} × ℙ^{r−1}) ∪ (L × Δ_{ℙ^{r−1}})

Mot(Γ) := H^{N−1}(ℙ^{N−1}−X_{Γ}, Δ−Δ∩X_{Γ})(N−1) = H^{N−1}(P − (Y ∪ E), Δ′ − Δ′ ∩ (Y ∪ E))(N − 1)

⟶^{Res} H^{N−2}(E − E ∩ Y, Δ′ ∩ E)(N − 2) ≅

Mot(γ) ⊗ Mot(Γ//γ)

Also

Mot(Γ_{1} ⨿ Γ_{2}) = Mot(Γ_{1}) ⊗ Mot(Γ_{2})

This is because Ψ_{Γ_{1}⨿Γ_{2}}(A, B) = Ψ_{Γ_{1}}(A)Ψ_{Γ_{1}}(B), and

ℙ^{N_{1}+N_{2}−1} − Cone(X_{Γ_{1}}) − Cone(X_{Γ_{2}})

⟶^{𝔾_{m}-bundle} (ℙ^{N_{1}−1} − X_{Γ_{1}}) × (ℙ^{N_{2}−1} − X_{Γ_{2}})

Renormalization Hopf Algebra

H = ⨁ ℚ · Γ; Γ · Γ′ = Γ ⨿ Γ′

Δ(Γ) = ∑_{\begin{matrix}γ⊂Γ \\ diverg. \end{matrix}} γ ⊗ Γ/γ

Motivic version

H_{Mot} := ⨁ Mot(Γ),

Hopf algebra in the tannakian category of motives.

Periods - ω

Γ connected graph, log divergent (2n edges, n loops)

H := H^{2n−1}(ℙ^{2n−1} − X_{Γ}, Δ − X ∩ Δ) = Mot(Γ).

Rational structures: H_{Betti} and H_{DR}.

H_{Betti} ⊗_{ℚ} ℂ ≅ H_{DR} ⊗_{ℚ} ℂ

Recall ω := Ω/Ψ_{Γ}^{2} ∈ H_{DR}^{2n−1}(ℙ^{2n−1} − X_{Γ}). ω top degree form on affine ℙ^{2n−1} − X_{Γ} implies ω lifts canonically to H_{DR}.

Periods - σ

Assume Γ log divergent, n loops, 2n edges. Recall

σ : A_{i} ≥ 0, σ ∩ X_{Γ} = ⋃ L_{γ}

π : P = BL(L_{γ} ⊂ ℙ^{2n−1}) → ℙ^{2n−1}

E ⊂ P exceptional divisor

Proposition 1 Subgraph γ ⊂ Γ divergent iff π^{∗}ω has a pole along E.

Definition 2 Γ (assumed log-divergent) is primitive if it has no divergent subgraphs γ ⊊ Γ.

Proposition 3 Assume Γ primitive. Then there exists a sequence of blowups π : P → ℙ^{2n−1} of (strict transforms of) L_{γ} such that the strict transform of σ doesn’t meet the strict transform Y of X_{Γ}. In particular, ∫_{σ} ω becomes a well-defined period for H^{2n−1}(P − Y, Δ_{P} − Y ∩ Δ_{P}). Here Δ_{P} = π^{−1}(Δ).

Renormalization (Work in Progress)

Γ log divergent, n loops, 2n edges. Basic problem: If Γ has divergent subgraphs, then ∫_{σ} ω diverges. ℓ_{i}(A_{1}, …, A_{2n}) independent linear forms with coefs in ℝ. Δ_{ℓ} : ℓ_{1}ℓ_{2} ⋯ ℓ_{2n} = 0; σ_{ℓ} : ℓ_{i} ≥ 0. Example 4 ℓ_{i} : A_{i} − tA_{2n} = 0, ℓ_{2n} : A_{2n} − ∑_{1}^{2n−1} A_{i} = 0

0 < t < \frac{1}{2n − 1} ⇒ σ_{ℓ} ⊂ \{A_{i} > 0 | 1 ≤ i ≤ 2n\}

⇒ σ_{ℓ} ∩ X_{Γ} = ∅.

∫_{σ_{ℓ}} ω period for H^{2n−1}(ℙ^{2n−1} − X_{Γ}, Δ_{ℓ} − X ∩ Δ_{ℓ})

Limiting Mixed Hodge Structures

H_{t} = H^{2n−1}(ℙ^{2n−1} − X_{Γ}, Δ_{t} − X_{Γ} ∩ Δ_{t}), t ∈ D^{∗} Suitable variation of mixed Hodge structure over punctured disk D^{∗}. (Think of Δ_{t} general for t ≠ 0.) Limiting mixed Hodge structure H_{lim}. Monodromy of H_{t} quasi-unipotent. (Ramifying in t, we may assume unipotent.) Monodromy matrix exp(N log t) Period matrix P_{t} = (∫_{c_{i,t}} ω_{j,t}) c_{i,t} horizontal basis of H_{t}^{∨}; ω_{j,t} suitable algebraic basis of H_{t}. Basic result(Deligne): exp(−N log t)P_{t} is single-valued in a neighborhood of t = 0, and

M_{0} := lim_{t→0} exp(−N log t)P_{t}

exists. H_{lim} defined using M_{0}.

Good Lattices

ω = \frac{Ω_{2n−1}}{Ψ_{Γ}^{2}} =^{?} ω_{1,t}

Need

| ∫_{c_{i,t}} ω| = O(| log(t)|^{N}).

(Good Lattice condition.) γ ⊂ Γ subgraph, edges e_{1}, …, e_{p}. Assume h_{1}(γ) > 0.

L : A_{1} = … = A_{p} = 0; L ⊂ X_{Γ} π : P = BL(L ⊂ ℙ^{2n−1}) → ℙ^{2n−1}; E ⊂ P

Conclusion

Conjecture 5 Assume Γ has only logarithmically divergent subgraphs. Then for a suitable variation Δ_{t} and chains c_{i,t}, the column vector

exp(−N log t)(∫_{σ_{t}} ω, ∫_{c_{2,t}} ω, …, ∫_{c_{N,t}} ω)^{t}

is single-valued in a neighborhood of t = 0 and admits a limit as t → 0.