Wonderful Renormalization Marko Berghoff, Humboldt Universit¨ at zu Berlin Potsdam, February 2016
Introduction QFT in position space / Causal perturbation theory ◮ Stueckelberg, Bogoliubov, Shirkov (late 50’s): Axiomatic approach to S -matrix, � S = 1 + T n . n > 1 ◮ Epstein and Glaser (’73): Renormalization of S translates into an extension (splitting) problem for distributions. ◮ Simplified version by Stora (ca.’00), used in QFT on curved spacetimes.
Introduction ◮ Bergbauer, Brunetti, Kreimer (’10): Version for single graphs. Example (Euclidean φ 4 4 -theory) x 0 G y z 1 � � Feynman rules Φ : G �− → ω G = dxdydz ( x − y ) 4 ( y − z ) 2 z 4 x 2 � What is ω G ? ◮ Easy answer: ∞ . ◮ Tricky answer: Find renormalized value ...
Introduction Idea (Atiyah; Axelrod, Singer): Use a smooth model to arrange the divergences in a ”nice” way, renormalize on this model, then push the result back to original spacetime. Definition Let A = { A 1 , . . . , A k } be a family of smooth subvarities in an algebraic variety X . A smooth model is a smooth variety Y together with a proper, surjective map β : Y → X , such that E := β − 1 ( ∪ A ∈A A ) is a normal crossing divisor and β | Y \E a diffeomorphism.
Wonderful models Such smooth models are given by the wonderful model construction by DeConcini and Procesi. Idea is based on Fulton and MacPherson’s “Compactification of Configuration Spaces”: The configuration space of n -points in an algebraic variety X is C n ( X ) = { ( x 1 , . . . , x n ) ∈ X n | x i � = x j for all i � = j } . Fulton and MacPherson construct its compactification X [ n ] by a sequence of blow-ups along the (strict transforms) of diagonals of increasing dimension. A limiting point in X [ n ] \ C n ( X ) is encoded by a nested set of diagonals.
Wonderful models Definition Let A be a linear arrangement in a vector space X . The wonderful model ( Y A , β ) is defined as follows: The graph of the map � � π A : X \ A − → P ( X / A ) A ∈A A ∈A is locally closed in X × � A ∈A P ( X / A ). Define Y A as its closure and β : Y A → X as the projection onto the first factor. ◮ An explicit construction is given by a sequence of blowups along (strict transforms of) elements of a building set B ⊆ A , giving local charts ( U i , κ i ), i = ( N , B ), where N is a nested set of elements of B and B an adapted, marked basis of X . ◮ B controls the number of irreducible components of E ⊆ Y B , while the B -nested sets describe a stratification of E .
Graphs and arrangements Feichtner (’05): These notions can all be defined combinatorially! Either in terms of the intersection lattice of A , � � L A := { A 1 ∩ · · · ∩ A k | A i ∈ A} , ⊇ , or, in our case, using the poset of divergent subgraphs of G . Definition Let G = ( V , E ) be a graph. ◮ Its superficial degree of divergence is defined by s ( G ) = dh 1 ( G ) − 2 | E | ( d = dim. of spacetime). G is called at most logarithmic if s ( g ) ≤ 0 holds for all g ⊆ G . ◮ The divergent poset of G is defined as � � D G := { g ⊆ G | s ( g ) ≤ 0 } , ⊆ .
Graphs and arrangements Now consider the following Feynman rules: Let G be a connected graph. Orient G and choose a spanning tree t ⊆ G . The Feynman rules map Φ sends G to the pair ( X G , ˜ v G ) of a chain X G = ( R d ) E ( t ) and a form defined by the rational function � x e if e ∈ E ( t ) − d � v G : x �− → y 2 , y e = e e ′ ∈ E ( t e ) σ t ( e ′ ) x e ′ � else. e ∈ E ( G ) Here t e is the unique path in t connecting the source and target vertices of e and σ t : E ( t ) → {± 1 } given by the orientation on G .
Graphs and arrangements ∈ L 1 ( X G ) - by viewing v G as We avoid the infrared problem - v G / (the kernel of) a distribution on X G . On the other hand, the ∈ L 1 ultraviolet problem - v G / loc ( X G ) - is characterized by the following Proposition Let G be at most logarithmic. ◮ v G defines a distribution on X G \ � g ∈D G A g , where A g := { y e = 0 | e ∈ E ( g ) } ⊆ X G . ◮ D G is a graded (distributive) lattice with join and meet operations given by g ∨ h := g ∪ h g ∧ h := g ∩ h .
Wonderful combinatorics Definition Let L be a lattice. B ⊆ L is a building set for L if ◮ for all p ∈ L > ˆ 0 and { q 1 , . . . , q k } = max B ≤ p there is an isomorphism of posets k � [ˆ → [ˆ ϕ A : 0 , q i ] − 0 , p ] i =1 with ϕ p (ˆ 0 , . . . , q j , . . . , ˆ 0) = q j for j = 1 , . . . , k . ◮ the ranking function on L satisfies k � r ( p ) = r ( q i ) . i =1 In our case r is given by codim( A g ) = d ( | E ( g ) | − h 1 ( g )).
Wonderful combinatorics Definition Let B be a building set in L . A subset N ⊆ B is B -nested if for all subsets { p 1 , . . . , p k } ⊆ N of pairwise incomparable elements their join p 1 ∨ · · · ∨ p k exists (in L ) and does not belong to B . Definition ◮ A basis b of X is adapted to N if for all A ∈ N the set b ∩ A generates A ⇐ ⇒ b is given by the edges of an adapted spanning tree, i.e. t ⊆ G such that t ∩ g is spanning for all g ∈ N . ◮ A marking of b is for every A ∈ N the choice of an element b A ∈ b ∩ A ⇐ ⇒ for every g ∈ N a choice b g ∈ { b e } e ∈ E ( t ∩ g ) .
Wonderful renormalization Let ( Y B , β ) be a wonderful model for a building set B ⊆ D = D G and v = v G the Feynman distribution associated to a graph G . Proposition v s = v s | dx | In local coordinates on U i , i = ( N , B ) , the pullback of ˜ (s = regularization parameter) along β is a density on Y given by � ( ω s ) i := ( β ∗ ˜ v s ) i = f s u g ( s , · ) | dx | , i g ∈N u g ( s , x ) = | x g | − 1+ r ( g )( s − 1) , x g marked . → R is in L 1 The map f i : κ i ( U i ) − loc ( κ i ( U i )) and smooth in the marked variables x g , g ∈ N .
Wonderful renormalization The next step is to study the Laurent expansion of ω s . To formulate this we need a local version of graph contraction. Definition Let g ⊆ G and N be nested. The contraction relative to N is defined as � g / ( � γ ∈N < g γ ) if g ∈ N , g / / N := g / ( g ∩ � γ ∈N ,γ ∩ g < g γ ) else . For J ⊆ N the poset ( N / / J , ⊑ ) is given by the underlying set N / / J := { g / / J | g ∈ N} , partially ordered by inclusion (in general ⊑� = ⊆ !).
Wonderful renormalization Theorem ◮ The Laurent expansion of ω s at s = 1 has a pole of order N where N is the cardinality of the largest B -nested set. ◮ The coefficients ˜ a k in the principal part of the Laurent expansion ω s = � a k ( s − 1) k ˜ − N ≤ k are densities with supp ˜ a k = � |N| = − k E N for k < 0 . ◮ Consider the minimal building set I ( D ) ⊆ D . Assume G ∈ I ( D ) . Let N be the cardinality of a maximal nested set and denote by χ the constant function on the wonderful model Y I ( D ) . Then � � � ˜ a − N | χ � = P ( γ/ / M ) . γ ∈M |M| = N
Wonderful renormalization Definition (“Local subtraction at fixed conditions”) In every chart U i let ν = { ν i g } g ∈N denote a collection of smooth functions on κ ( U i ), each ν i g depending only on the coordinates x e with e ∈ E ( t ) ∩ E ( g \ N < g ), satisfying ν i g | x g =0 = 1 and compactly supported in all other directions. For u ∈ D ′ ( R \ { 0 } ) and µ ∈ D ([ − 1 , 1]) let r µ [ u ] ∈ D ′ ( R ) denote the extended distribution r µ [ u ] : ϕ �→ � u | ϕ � − � u | ϕ (0) µ � . The extension of ω s is defined by R ν [ ω s ] loc . = R i ν [ f s � u g ( s ) | dx | ] := f s � r ν i g [ u g ( s )] | dx | i i g ∈N g ∈N � ( − 1) |J | ν i J · ( ω s =: i ) E J . J ⊆N
Wonderful renormalization Theorem ◮ R ν [ ω s ] defines a density-valued holomorphic function in a neighborhood of s = 1 . ◮ Define the renormalized Feynman rules by the map Φ R : G �− → ( X G , R [ v G ]) with R [ v G ] := β ∗ R ν [ ω s ] | s =1 and evaluation on ϕ ∈ D ( X G ) given by � R [ v G ] | ϕ � = � β ∗ R ν [ ω s ] | s =1 | ϕ � = � R ν [ ω s ] | s =1 | β ∗ ϕ � . Then R satisfies the Epstein-Glaser locality principle.
Renormalization group What happens if the renormalization point ν is changed? Theorem Consider ( R ν ′ − R ν )[ ω s ] for two choices of function families ν ′ and ν . Locally in U i , applied on a test function ϕ = β ∗ ψ for ψ ∈ D ( β ( U i ) ∩ κ i ( U i )) we have j � ( R i � ν ′ − R i ν )[ ω s ν [( ω s i ] | ϕ � = c J � R / J ) j ] | δ J [ ϕ ] � G / ∅� = J ⊆N with � � R k ν [( ω s / J ) j ] | ν ′ c J := γ � . γ/ γ ∈J The indices j , k correspond to ( N / / J ) ⊑ G / / J and ( N / / J ) ⊑ γ/ / J , respectively.
Conclusions & Outlook ◮ Geometric ansatz put in combinatorial language ◮ Simplifies the ”wonderful” construction and adds discrete toolbox ◮ Reconstruction of Epstein-Glaser method via models for K n → Fulton-MacPherson compactification ◮ Dyson-Schwinger equations? ◮ Renormalization group equation / flow? ◮ Renormalization Hopf algebra? It encodes the stratification of the exceptional divisor E ...
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