Perturbative Quantum Field Theory and Homotopy Algebras

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PoS(CORFU2019)199
Perturbative Quantum Field Theory
and Homotopy Algebras
Branislav Jurˇ
co
Faculty of Mathematics and Physics, Mathematical Institute
Charles University Prague, Prague 186 75, Czech Republic
E-mail: branislav.jurco@gmail.com
Hyungrok Kim
Maxwell Institute and Department of Mathematics
Heriot–Watt University, Edinburgh EH14 4AS, United Kingdom
E-mail: hk55@hw.ac.uk
Tommaso Macrelli
Department of Mathematics, University of Surrey
Guildford GU2 7XH, United Kingdom
E-mail: t.macrelli@surrey.ac.uk
Christian Saemann∗†
Maxwell Institute and Department of Mathematics
Heriot–Watt University, Edinburgh EH14 4AS, United Kingdom
E-mail: c.saemann@hw.ac.uk
Martin Wolf
Department of Mathematics, University of Surrey
Guildford GU2 7XH, United Kingdom E-mail: m.wolf@surrey.ac.uk
We review the homotopy algebraic perspective on perturbative quantum field theory: classical
field theories correspond to homotopy algebras such as A∞- and L∞-algebras. Furthermore, their
scattering amplitudes are encoded in minimal models of these homotopy algebras at tree level
and their quantum relatives at loop level. The translation between Lagrangian field theories and
homotopy algebras is provided by the Batalin–Vilkovisky formalism. The minimal models are
computed recursively using the homological perturbation lemma, which induces useful recursion
relations for the computation of scattering amplitudes. After explaining how the homolcogical
perturbation lemma produces the usual Feynman diagram expansion, we use our techniques to
verify an identity for the Berends–Giele currents which implies the Kleiss–Kuijf relations.
Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity"
(CORFU2019)
31 August - 25 September 2019
Corfu, Greece
∗Speaker.
†Report numbers: DMUS–MP-20/01, EMPG–20–06
c
Copyright owned by the author(s) under the terms of the Creative Commons
Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/

PoS(CORFU2019)199
Perturbative Quantum Field Theory and Homotopy Algebras Christian Saemann
1. Introduction
Homotopy algebras such as A∞- or L∞-algebras arise very naturally in string theory. For exam-
ple, the B-field in string theory is part of the connective structure of a gerbe, a higher or categorified
versions of a principal circle bundle. Such a higher bundle comes naturally with higher versions
of symmetries, which are most conveniently encoded in L∞-algebras. Similarly, the intricate gauge
symmetries arising from the underlying diffeomorphism invariance of string theory fixes the ac-
tions of open and closed string field theories to be the homotopy Maurer–Cartan theories of A∞-
and L∞-algebras, respectively.
More surprising may be the fact that homotopy algebras arise equally naturally already in the
context of ordinary quantum field theory and, moreover, that they are very useful in the analysis of
correlation functions and scattering amplitudes. Indeed, they capture literally the usual description
of tree level amplitudes in perturbative quantum field theory. It is therefore not surprising that
theoretical physicists have already used some of these structures, albeit implicitly. For example,
the Berends–Giele recursion relation for currents [1] is nothing but a quasi-isomorphism of L∞-
algebras.1
The bridge between Lagrangian field theories and homotopy algebras is provided by the clas-
sical part of the Batalin–Vilkovisky (BV) formalism [3,4], which yields a differential graded com-
mutative algebra (dgca), usually called the BV complex, one of a number of equivalent descriptions
of an L∞-algebra. Any L∞-algebra comes with a homotopy Maurer–Cartan action as well as an
isomorphism class of minimal models. The homotopy Maurer–Cartan action reproduces the clas-
sical action from which the L∞-algebra was constructed, and thus any field theory is a homotopy
Maurer–Cartan theory. The minimal models encode the tree level scattering amplitudes.
Since the minimal models are computed recursively, most conveniently by using the homo-
logical perturbation lemma, there are useful recursion relations underlying the computation of tree
level scattering amplitudes. In the case of four-dimensional Yang–Mills theory, these are precisely
the aforementioned Berends–Giele recursion relations.
The full power of this homotopy algebraic perspective on field theory has not been exploited
yet, and it may lead to more direct (or, indeed, first) answers to many questions in quantum field
theory. It also provides a very concise explanation of the computation of scattering amplitudes
in perturbative quantum field theory2that should be more readily accessible to mathematicians.
We therefore believe that it is important to explain this perspective to a broader audience and to
promote its use.
We begin with a short introduction to homotopy algebras. We then explain how the BV for-
malism assigns a homotopy algebra to any field theory and how the original action is recovered by
the homotopy Maurer–Cartan theory of that homotopy algebra. Our key examples here are scalar
field theory and Yang–Mills theory. We then explain how minimal models are computed with
the homological perturbation lemma and how they lead to the Feynman diagrams of perturbative
1Interestingly, this paper was published in the same journal in which four years later the notion of L∞-algebra was
first defined [2]. It seems that the journal for groundbreaking research on strong homotopy Lie algebras is a journal for
theoretical physics.
2We hasten to add that the homotopy algebraic perspective given here does not address regularisation and renormal-
isation, but we intend to formulate these issues in the context of homotopy algebras soon.
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quantum field theory, both at tree and quantum level. We close with a short new proof for a combi-
natorial identity for the tree-level Berends–Giele currents in Yang–Mills theory, which implies the
Kleiss–Kuijf relations [5,6]
Pointers to some of the relevant original literature
The historical development of the homotopy algebraic perspective after the invention of the BV
formalism becomes quickly very involved and it would be impossible to give a complete account
of the literature. In the following, we point out some key references. The most important paper
is certainly [2], which introduced L∞-algebras, already together with their quantum variants, and
established the link to the BV formalism. The closely related A∞-algebras are much older [7,8].
Deeper explanations of the geometry and the meaning of the classical part of the BV formalism
were given by various authors in the 90ies, see e.g. [4,9,10]. Concrete differential complexes
underlying classical field theories were then calculated in the following decade, see e.g. [11,12,
13,14,15,16,17,18]. The fact that classical field theories have underlying L∞-algebras was later
rediscovered several times, see e.g. [19,20]. It also underlies much of the work of Costello, cf. [21]
The existence of minimal models of homotopy algebras, in particular A∞- and L∞-algebras,
was known for some time [22]. The decomposition theorem together with a useful perturbative
algorithm for the computation of the minimal model was given much later in [23]. In this reference,
it was also pointed out that minimal models are related to a Feynman diagram expansion in general,
following earlier suggestions going back to [24]. This fact was also used in [25] to derive Wick’s
theorem and Feynman rules for finite-dimensional integrals.
The homological perturbation lemma [26,27,28] then entered the discussion with [29] and in
particular with [30], but its relevance seems to have been clear to several people much earlier. The
extension of the homological perturbation lemma to computations of quantum minimal models is
due to [30].
The explicit application to computations of S-matrices at tree level was rather recent [31,32,
33,34,35] and the generalisation to loop level is found in [36].
Much more detailed explanations of our formalism and its mathematical background in the
conventions used here as well as a more complete list of pointers to the original literature are found
in the papers [37,38,39,33,36].
2. Homotopy algebras
Homotopy algebras are generalisations of classical algebras, such as associative, Leibniz or
Lie algebras, in which the relevant structural identities, that is associativity, the Leibniz identity
or the Jacobi identity, respectively, hold only up to homotopies. Usually, an additional qualifier
“strong” implies that the structural identities hold up to “nicely behaved” or coherent homotopies.
There are three different perspectives on homotopy algebras3that are relevant to the discussion
of perturbative quantum field theory:
1. higher products or higher brackets which give rise to the formulation of an action principle;
3A fourth one being that of algebras over operads in the category of chain complexes, which can also be useful.
Particular types of strong homotopy Lie algebras are also good descriptions of categorified Lie algebras.
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2. codifferential graded coalgebras, where all higher products are packaged in a single codif-
ferential, and which is best suited for performing perturbation theory;
3. differential graded algebras, which is the linear dual of the latter and which is the output of
the classical part of the BV formalism.
In the following, we take a brief tour through all of these descriptions and indicate how these
are linked to each other.
2.1 Higher products
Consider a unital associative algebra a. A good example to have in mind is a matrix algebra
which includes the identity matrix. Besides being a vector space, ais endowed with an associative
product m2:a×a→asatisfying
m2(m2(a1,a2),a3) = m2(a1,m2(a2,a3)) (2.1)
for all a1,2,3∈a. A simple generalisation of this is a strict A∞-algebra, which is a differential
Z
-graded associative algebra,
a=M
i∈
Z
ai, .. . m1
−−→ a−1
m1
−−→ a0
m1
−−→ a1
m1
−−→ .. . m2:ai×aj→ai+j,(2.2)
where m1is a differential which is compatible with the associative product m2. That is,
m1(m1(a1)) = 0,
m1(m2(a1,a2)) = m2(m1(a1),a2)+(−1)|a1|m2(a1,m1(a2)) ,
m2(m2(a1,a2),a3) = m2(a1,m2(a2,a3)) ,
(2.3)
where |a1|denotes the degree of a1.
To obtain a general A∞-algebra, we first have to lift associativity up to coherent homotopy by
introducing a trilinear map m3:a×a×a→aof degree −1,
m3:ai×aj×ak→ai+j+k−1,(2.4)
such that
m2(m2(a1,a2),a3)−m2(a1,m2(a2,a3))
=m1(m3(a1,a2,a3)) + m3(m1(a1),a2,a3)+
+ (−1)|a1|m3(a1,m1(a2),a3)+(−1)|a1|+|a2|m3(a1,a2,m1(a3)) .
(2.5)
More succinctly, we can write
m1m3−m2(m2⊗id) + m2(id ⊗m2) + m3(m1⊗id ⊗id +id ⊗m1⊗id +id ⊗id ⊗m1) = 0,(2.6)
where the signs arise by passing arguments past the degree 1 maps m1. In general, we insert Koszul
signs, for example:
(f⊗g)(x⊗y) = (−1)|g||x|(f(x)⊗g(y)) .(2.7)
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The complete A∞-algebra is then given by linear maps mi:a×i→aof degree 2 −isuch that
associativity holds up to coherent homotopy,
∑
i+j+k=n
(−1)i j+kmi+k+1(id⊗i⊗mj⊗id⊗k) = 0 (2.8)
for all n∈
N
+.
The strong homotopy version of a Lie algebra is defined analogously. Here, we have a
Z
-
graded vector space g=⊕i∈
Z
gitogether with the higher products µi:g×i→gwhich are totally
antisymmetric linear maps of degree 2 −isuch that for each n∈
N
+, the homotopy Jacobi identity
holds,
0=∑
i+j=n
(−1)jµj+1◦(µi⊗id⊗j)◦∑
σ∈Sh(i;n)
σ∧,(2.9)
where the unshuffle operator σ∧creates all graded permutations of narguments `1⊗`2⊗ ·· · ⊗ `n
preserving the relative order of (`σ(1),...,`σ(j))and (`σ(j+1),. . ., `σ(n))and inserting the appropri-
ate sign for graded antisymmetry. Explicitly, the homotopy Jacobi identity reads as
∑
i+j=n
∑
σ∈Sh(j;i)
(−1)jχ(σ;`1,...,`n)µj+1(µi(`σ(1),...,vσ(i)), `σ(i+1),.. . ,`σ(n)) = 0,(2.10)
where χ(σ;`1,...`i)is the sign necessary to obtain the permutation σfrom graded antisymmetry:
(`1∧ · · · ∧ `n) = χ(σ;`1,...,`n)(`σ(1)∧ · · · ∧ `σ(n)).(2.11)
Just as any matrix algebra comes with a natural Lie bracket given by the commutator, any
A∞-algebra comes with the higher products of an L∞-algebra,
µi(`1,...,`n) = mi◦∑
σ∈Sn
σ∧!(`1,. . ., `n) = ∑
σ∈Sn
χ(σ;`1,...`i)mi(`σ(1), ... , `σ(n)),(2.12)
which is reasonably obvious comparing (2.8) and (2.9). If the A∞-algebra is just a matrix algebra,
the corresponding L∞-algebra is indeed just the usual matrix Lie algebra induced by the commuta-
tor.
We note that the definition of both A∞- and L∞-algebras readily extends from graded vector
spaces to graded modules over rings over
R
.
2.2 Codifferential graded coalgebras and differential graded algebras
The higher products miand µiintroduced above for A∞- and L∞-algebras are always of de-
gree 2 −i. If we shift the underlying graded vector spaces aand gby −1, switching to a[1]and
g[1], then the degree of all higher products changes to 1. We use the usual notation,
a[k] = M
i∈
Z
(a[k])iwith a[k]i=ak+i.(2.13)
Once this is done, we can continue the higher products miand µito coderivations on the tensor
algebras4
O•a[1] =
R
⊕a[1]⊕a[1]⊗a[1]⊕a[1]⊗a[1]⊗a[1]⊕.. . ,
K•g[1] =
R
⊕g[1]⊕g[1]g[1]⊕g[1]g[1]g[1]⊕.. . , (2.14)
4Strictly speaking, one either needs to remove the summand
R
from the tensor algebras or admit curved homotopy
algebras, containing a higher product m0or µ0. For simplicity, we ignore this point in our discussion.
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respectively. Recall that a coderivation D satisfies the co-Leibniz rule
∆◦D= (D⊗id)◦∆+ (id ⊗D)◦∆,(2.15)
where the relevant coproducts ∆⊗and ∆on N•a[1]and J•g[1]are the shuffle coproduct and its
symmetrised form:
∆⊗(a1⊗ · · · ⊗ an) =
n
∑
k=0
(a1⊗ · · · ⊗ ak)O(ak+1⊗ ·· · ⊗ ak),
∆(`1 · · ·  `n) =
n
∑
k=0
∑
σ∈Sh(k;n)
ε(σ;`1,. . ., `n)(`σ(1) · · ·  `σ(k))O(`σ(j+1) · · ·  `σ(n))
(2.16)
for ai∈a[1]and `i∈g[1], where ε(σ;`1,...`i)is the sign necessary to obtain the permutation σ
from graded symmetry:
(`1 · · ·  `n) = ε(σ;`1,. .., `n)(`σ(1) · ··  `σ(n)).(2.17)
For an A∞-algebra, for example, the continuation of m2to a coderivation m2on N•a[1]reads
as
m2(r) = 0,m2(a1) = 0,m2(a1⊗a2) = m2(a1,a2),
m2(a1⊗a2⊗a3) = m2(a1,a2)⊗a3+ (−1)|a1|a1⊗m2(a2,a3), .. . (2.18)
for all r∈
R
,a1,2,3∈a[1]. Note that we slightly abused notation, using the same symbol for m2
and the map it induces on a[1]. Also, Koszul signs arise as above. We now combine the higher
products into single coderivations of degree 1:
D⊗=m1+m2+m3+.. . and D=µ1+µ2+µ3+. .. . (2.19)
In both cases, the identity for associativity up to coherent homotopy (2.8) as well as the homotopy
Jacobi identity (2.9) amount to D⊗and Dbeing differentials:
D2
⊗=0 and D2
=0.(2.20)
We thus saw that A∞-algebras correspond to codifferential graded (cofree5) coalgebras or codgcoas
for short. In the case of L∞-algebras, the codgcoas are also cocommutative.
In many cases, in particular when the homogeneously graded subspaces of gand aare finite
dimensional, one can dualise this picture. The result is a differential graded algebra or dga for
short. Explicitly, we consider the algebras
O•a[1]∗=
R
⊕a[1]∗⊕a[1]∗⊗a[1]∗⊕a[1]∗⊗a[1]∗⊗a[1]∗⊕.. . ,
K•g[1]∗=
R
⊕g[1]∗⊕g[1]∗g[1]∗⊕g[1]∗g[1]∗g[1]∗⊕.. . , (2.21)
and the codifferentials Dand D⊗dualise to differentials
Q:=D∗
and Q⊗:=D∗
⊗(2.22)
with
Q2
=0 and Q2
⊗=0.(2.23)
To physicists, these differentials are familiar, e.g. from the BRST and BV complexes.
5which implies that they arise as the tensor algebra on some graded vector space
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2.3 Examples
To illustrate the above constructions, let us briefly consider the example of an ordinary matrix
algebra a=a0. The codifferential D⊗on N•a[1]is simply given by the matrix product, continued
to a coderivation. For example,
D⊗(a1⊗a2⊗a3⊗a4):=a1a2⊗a3⊗a4−a1⊗a2a3⊗a4+a1⊗a2⊗a3a4,(2.24)
where ai∈a[1]. Clearly, D2
⊗=0 amounts to associativity, for example
D⊗(D⊗(a1⊗a2⊗a3)) :=D⊗(a1a2⊗a3−a1⊗a2a3)
= (a1a2)a3−a1(a2a3).(2.25)
On J•g[1]for g=a, the symmetrisation in shifted degrees leads to the coproduct
D(a1a2a3):= (a1a2−a2a1)a3−(a1a3−a3a1)a2+ (a2a3−a3a2)a1,(2.26)
where we used the fact that the ai∈a[1]are all of degree −1. Here, D2
=0 leads to the Jacobi
identity:
D(D(a1a2a3)) :=D([a1,a2]a3−[a1,a3]a2+ [a2,a3]a1)
= [[a1,a2],a3]−[[a1,a3],a2] + [[a2,a3],a1].(2.27)
To simplify the dualisation, let us assume that ais finite-dimensional and introduce a basis τa,
a=1,...,dim(a)on a. We then have structure constants
τaτb=:sc
abτcand [τa,τb] =:fc
abτc.(2.28)
Together with the coordinate functions ξa:a[1]→
R
dual to the shifted basis τaon a[1], we have
the differentials
Q⊗ξa=−sa
bcξb⊗ξcand Qξa=−1
2fa
bcξbξc,(2.29)
which both satisfy the Leibniz rule,
Q⊗(p1⊗p2) = (Q⊗p1)⊗p2+ (−1)|p1|p1⊗(Q⊗p2),
Q(p1p2) = (Qp1)p2+ (−1)|p1|p1(Qp2)(2.30)
for any elements p1,2of N•a[1]or J•a[1], respectively. The fact that Q⊗and Qare differen-
tials, and thus square to zero, corresponds again to associativity and the Jabobi identity for the
commutator, respectively.
Genuine A∞- and L∞-algebras arise in a number of contexts in physics. We postpone discussing
any further examples to section 3, where homotopy algebras arise in the form of BRST and BV
complexes of classical field theories.
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2.4 Cyclic structures
To write down action principles for usual gauge theories requires an invariant metric on the
gauge Lie algebra, i.e. the data of a quadratic Lie algebra. There is a straightforward generalisation
to a differential graded Lie algebra g, where an inner product is a bilinear, graded symmetric, and
non-degenerate map h−,−i :g×g→
R
satisfying
hd`1,`2i+ (−1)|a1|h`1,d`2i=0,
h[`1,`2], `3i+ (−1)|`1| |`2|h`2,[`1, `3]i=0.(2.31)
These relations have a straightforward generalisation to L∞-algebras:
hµi(`1,. . ., `i−1, `i),`i+1i+ (−1)|`i|(i+|`1|+···+|`i−1|)h`i,µi(`1,. . ., `i−1, `i+1)i,(2.32)
which is equivalent to
h`1,µi(`2,...,`i+1)i= (−1)i+i(|`1|+|`i+1|)+|`i+1|∑i
j=1|`j|h`i+1,µi(`1,...,`i)i(2.33)
for all `i∈g. Such inner products are called cyclic structures and an L∞-algebra endowed with a
cyclic structure is simply called cyclic.
This definition does not rely on symmetrisation, and it thus also applies to A∞-algebras. That is,
acyclic A∞-algebra is an A∞-algebra (a,mi)with a bilinear, graded symmetric, and non-degenerate
map h−,−i :a×a→
R
satisfying
ha1,mi(a2,...,ai+1)i= (−1)i+i(|a1|+|ai+1|)+|ai+1|∑i
j=1|aj|hai+1,mi(a1,...,ai)i(2.34)
for all ai∈a. Note that a strict A∞-algebras is nothing but a differential (noncommutative) Frobenius
algebra.
3. Homotopy algebras from classical field theories
In this section, we show how the action of a gauge group and the equations of motion both lead
to differentials that nicely combine into the BV differential. The resulting BV complex is dual to an
L∞-algebra, and thus field theories correspond to strong homotopy Lie algebras. This assignment
is “functorial” in the sense that embeddings, projections and equivalences of field theories map to
corresponding morphisms of L∞-algebras. Moreover, any L∞-algebra comes with a natural “field
theory,” called homotopy Maurer–Cartan theory, and the homotopy Maurer–Cartan action of the
L∞-algebra of a field theory is just the field theory’s original action.
3.1 Homotopy Maurer–Cartan theory
Recall that a Maurer–Cartan element aof a differential graded Lie algebra (g,d,[−,−]) is an
element of degree 1 satisfying the Maurer–Cartan equation
da+1
2[a,a] = 0.(3.1)
If gis endowed with an invariant inner product h−,−i compatible with the differential, this equa-
tion is the equation of motion of the Maurer–Cartan action
SMC =1
2ha,dai+1
3! ha,[a,a]i,(3.2)
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where h−,−i is an inner product on gsatisfying (2.31).
The evident generalisation of this action to homotopy Maurer–Cartan theory reads as
ShMC[a]:=∑
i≥1
1
(i+1)!ha,µi(a,...,a)i,(3.3)
where we call a∈g1the gauge potential. Its curvature is defined as
f:=µ1(a) + 1
2µ2(a,a) + ·· · =∑
i≥1
1
i!µi(a,...,a),(3.4)
and f=0 are the stationary points of (3.3), defining homotopy Maurer–Cartan elements a. The
homotopy Jacobi identities induce a Bianchi identity,
∑
i≥0
(−1)i
i!µi+1(f,a,...,a) = 0,(3.5)
and the action (3.3) is invariant under the infinitesimal gauge transformations
δc0a=∑
i≥0
(−1)i
i!µi+1(c0,a,...,a),(3.6)
where c0∈g0is the gauge parameter. There are also higher gauge transformations; for details
see [37]. Similarly, there are higher Bianchi identities.
We conclude that any L∞-algebra gcomes with the full kinematical data of a gauge theory,
together with equations of motion f=0. If gis cyclic, the equations of motion arise from an action
principle. Clearly, the theory is essentially trivial, if there are no gauge potentials, which is the case
for g1=∗. We note the following roles of the homogeneously graded subspaces giof g:
◦the gauge potential or the general field takes values in g1;
◦the curvature takes values in g2;
◦the “left-hand side” of the Bianchi identity takes values in g3;
◦gauge symmetries are parametrised by g0;
◦higher gauge symmetries are parametrised by gi,i<0, and the “left-hand sides” of higher
Bianchi identities take values in gi,i>3.
A homotopy Maurer–Cartan action also exists for A∞-algebras a, where we similarly have a
gauge potential a∈a1with curvature
f:=m1(a) + m2(a,a) + ·· · =∑
i≥1
mi(a,...,a),(3.7)
satisfying the Bianchi identity
∑
i≥0
i−1
∑
j=0
(−1)i+jmi+1(a,...,a
| {z }
j
,f,a,...,a
| {z }
i−j
) = 0.(3.8)
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Flatness f=0 arises as the equation of motion of the action
ShMC[a]:=∑
i≥1
1
i+1ha,mi(a,...,a)i,(3.9)
which is invariant under infinitesimal gauge transformations
δc0a=∑
i≥0
i−1
∑
j=0
(−1)i+jmi+1(a,...,a
| {z }
j
,c0,a,...,a
| {z }
i−j
),(3.10)
where c0∈a0again parametrises the gauge transformations. Note that these formulas yield those
for an L∞-algebra after graded antisymmetrisation (2.12).
3.2 The BV formalism
The BV formalism is commonly used in the quantisation of field theories with “open” gauge
symmetries, i.e. gauge symmetries that only close on-shell. This is the case e.g. in ordinary higher
gauge theories. Usually, constructing the BV complex is a two-step process. In a first step, one
introduces ghosts, which are fermionic fields parametrising infinitesimal gauge transformations.
This leads to the BRST differential and the BRST complex, which encodes the action of gauge
transformations (as well as higher gauge transformations encoded in ghosts for ghosts) on the vari-
ous fields. The BV complex is then obtained by introducing an antifield for each field, ghost, ghost
for ghosts, etc., together with a graded Poisson bracket {−,−}BV called antibracket. One then
extends the classical action to the (classical) BV action SBV =Sclassical +.. . by terms containing
the antifields such that classical master equation
{SBV,SBV }=0 (3.11)
holds. This recipe appears quite ad-hoc, but regarded from a more mathematical perspective, the
situation becomes much clearer. The aim of the classical part of the BV formalism is to gain
a good description of classical observables. These are functions on field space modulo gauge
transformations and restricted to fields satisfying the equations of motion.
The first step is to quotient by gauge symmetries. We know that quotient spaces can be prob-
lematic to deal with, so it is better to consider “derived quotients”. That is, we encode the whole
gauge structure in an action groupoid, i.e. a category with invertible morphisms, where the objects
are the field configurations and the morphisms are gauge transformations. Differentiating an action
groupoid yields an action Lie algebroid (a special kind of L∞-algebroid), and if the field space is re-
garded as a vector space, we obtain an L∞-algebra. In the special case of an ordinary gauge theory,
this L∞-algebra is of the form gBRST =g0⊕g1, where g1are the fields and g0the infinitesimal gauge
transformations. The dual description of this L∞-algebra in terms of differential graded algebras is
nothing but the usual BRST complex, and this is known as a Chevalley–Eilenberg resolution of the
functions on field space modulo gauge transformations. Here, gauge-trivial observables are func-
tions on field space which are QBRST-exact and gauge invariant observables are functions which
are QBRST-closed.
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To restrict to fields satisfying the equations of motion, we perform a Koszul–Tate resolution.
That is, we enlarge the L∞-algebra gBRST to its cotangent space gBV =T∗[−1]gBRST with the du-
als of the fields, ghosts and higher ghosts called antifields (and antifields of ghosts, etc.). As a
cotangent space, gBV comes with the natural symplectic form
ωBV =dΦA∧dΦ+
A,(3.12)
where ΦAis a coordinate function on gBRST. This implies that Ais a multi-index running over
fields, ghosts, and their antighosts as well as all their labels such as momenta, tensor and gauge
labels. This symplectic form induces the antibracket, a Poisson bracket {−,−}BV of degree 1.6
Moreover, we extend QBRST to a Hamiltonian vector field QBV ={SBV,−}BV , where SBV is the
minimal extension of the classical action Scl such that QBV restricts to QBRST for Φ+
A=0 and such
that SBV satisfies the classical master equation
{SBV,SBV }=0.(3.13)
It follows that QBV encodes the equation of motion obtained by varying the classical action with
respect to the field φ:
QBVφ+={SBV ,φ+}=δSBV
δφ .(3.14)
In the absence of ghosts, the right hand side is the equation of motion and functions on field space
differing by QBV-exact terms are on-shell equivalent. If ghosts are added to the picture, then such
functions are on-shell equivalent after taking into account gauge symmetries. Observables are thus
encoded in the QBV-cohomology.
Altogether, the BV formalism assigns to any Lagrangian field theory an L∞-algebra given by
the dga encoded in the BV complex. In practice, however, it will be much more convenient to
derive the L∞-algebra directly: either by a direct construction, or by matching the action and gauge
transformations to a homotopy Maurer–Cartan theory. We shall discuss this in the following.
3.3 Higher Chern–Simons theories
Homotopy Maurer–Cartan theory is clearly a vast generalisation of Chern–Simons theory,
and one can derive higher Chern–Simons theories directly by constructing suitable L∞-algebras,
cf. [40].
We first observe that the tensor product of a differential graded algebra and an L∞-algebra
comes with a natural L∞-algebra structure. For concreteness’ sake, let us focus on the case where
the differential graded algebra is the de Rham complex (Ω•(M),d)on some manifold M. Then the
L∞-algebra Ω•(M,g) = Ω•(M)⊗ghas higher products
ˆ
µ1:=d⊗id +id ⊗µ1,
ˆ
µi:=id ⊗µifor i≥2.(3.15)
Note that the total degree of elements in Ω•(M,g)is the sum of the individual degrees.
6Poisson algebras with Poisson brackets of degree 1 are also known as Gerstenhaber algebras.
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If Mis compact and gis a cyclic L∞-algebra, then Ω•(M,g)is also cyclic with
h−,−iΩ•(M,g):=ZM
h−,−ig.(3.16)
The homotopy Maurer–Cartan action for Ω•(M,g)now provides a higher Chern–Simons the-
ory on M. For the gauge potentials to be connections on higher non-abelian gerbes, one should
restrict gto be trivial in positive degrees: gi=∗for i>0. This amounts to restricting gto a
categorified Lie algebra. Moreover, if the manifold Mis of dimension n+3, then gshould be
non-trivial in degrees kfor −n≤k≤0.
As an example, consider a four-dimensional manifold Mand an L∞-algebra of the form g=
g−1⊕g0. The gauge potential decomposes as
a=A+B∈Ω1(M,g0)⊕Ω2(M,g−1),(3.17)
and its curvature reads as
f=ˆ
µ1(a) + 1
2ˆ
µ2(a,a) + 1
3! ˆ
µ3(a,a,a) = F+H∈Ω2(M,g0)⊕Ω3(M,g−1),
F=dA+1
2µ2(A,A) + µ1(B),
H=dB+µ2(A,B)−1
3! µ3(A,A,A),
(3.18)
where the higher products µ3here do not “see” the form part of a gauge potential. The homotopy
Maurer–Cartan action is then higher Chern–Simons theory7on a four-dimensional manifold M,
ShCS =ZMnhB,dA+1
2µ2(A,A) + 1
2µ1(B)i+1
4! hµ3(A,A,A),Aio.(3.19)
Our construction here can be regarded as a concrete example of the AKSZ-formalism [41],
see [37] for a brief summary. This formalism directly yields a BV complex from two differential
graded algebras representing a source and target space.
3.4 Examples
It turns out that we can usually circumvent the BV procedure and obtain rather directly an A∞-
algebra, by matching the action of the field theory under consideration to the homotopy Maurer–
Cartan action. Also, it is often more convenient to work with A∞-algebras which become the
L∞-algebra that would be obtained from the BV formalism after graded antisymmetrisation (2.12).
As a first example, consider scalar field theory with cubic and quartic interactions on four-
dimensional Minkowski space
R
1,3with metric η,
Sscalar :=−Zd4xn1
2ϕϕ+κ
3! ϕ3+λ
4! ϕ4o,(3.20)
where :=ηµν ∂µ∂νand κ,λ∈
R
. The graded vector space underlying the relevant A∞-algebra
is [36]
a=a1⊕a2:=C∞(
R
1,3)⊕C∞(
R
1,3)(3.21a)
7This is the theory for a topologically trivial underlying non-abelian gerbe. Just as in the case of ordinary Chern–
Simons theory, one can generalise this action and glue together this local description to a global picture.
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and the non-trivial higher products are maps mi:a×i
1→a2,
m1(ϕ1):=−ϕ1,m2(ϕ1,ϕ2):=−κϕ1ϕ2,m3(ϕ1,ϕ2,ϕ3):=−λϕ1ϕ2ϕ3,(3.21b)
for ϕi∈a1. The cyclic structure is
hϕ,ϕ+i:=Zd4xϕ(x)ϕ+(x)(3.21c)
for ϕ∈a1and ϕ+∈a2. One readily checks that the homotopy Maurer–Cartan action (3.9) for this
A∞-algebra reproduces (3.20).
A more interesting example is Yang–Mills theory on
R
1,3. We start from the usual action
SYM :=Ztrn1
2F∧?Fowith F=dA+1
2[A,A](3.22)
for a gauge potential one form Ataking values in a gauge matrix Lie algebra h. Here, we know that
we should also take into account gauge transformations, and the graded vector space underlying
the A∞-algebra is thus of the form
a=a0⊕a1⊕a2⊕a3:=Ω0(M,g)⊕Ω1(M,g)⊕Ω1(M,g)⊕Ω0(M,g),(3.23a)
where we will have ghosts ci∈a0, fields Ai∈a1, antifields A+
i∈a2and antifields to the ghosts
c+
i∈a3. We define the evident inner product
hc1+A1+A+
1+c+
1,c2+A2+A+
2+c+
2i
:=Z
R
1,3trnc†
1∧?c+
2+c†
2∧?c+
1−A†
1∧?A+
2−A†
2∧?A+
1o(3.23b)
on a. The differential m1now encodes linearised gauge transformations and equations of motions,
completed cyclically:
m1(c1) = −dc1∈a1,m1(A1) = −d†dA1∈a2,m1(A+
1) = −d†A+
1∈a3.(3.23c)
The non-abelian terms are incorporated by the higher products,
m2(c1+A1+A+
1+c+
1,c2+A2+A+
2+c+
2)
=κc1c2
|{z}
∈a0
+κ(c1A2+A1c2)
| {z }
∈a1
+κ(−c1A+
2+A+
1c2)
| {z }
∈a2
+
+κ(d†(A1∧A2) + ?(A1∧?dA2)−?(?(dA1∧A2))
| {z }
∈a2
+κ(c1c+
2−c+
2c1) + κ(−?(A1∧?A+
2) + ?(A+
1∧?A2))
| {z }
∈a3
,
m3(A1,A2,A3) = κ2(?(A1∧?(A2∧A3)) −?(?(A1∧A2)∧A3)∈a2.
(3.23d)
These follow from non-abelian gauge transformations, field equations and completion via the cyclic
structure (3.23b). Again, the homotopy Maurer–Cartan action for areproduces the Yang–Mills
action (3.22).
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4. Equivalence of field theories
The correspondence between Lagrangian field theories and L∞-algebras is functorial, and in
particular, physical equivalence of classical field theories amounts to equivalence between L∞-
algebras. The appropriate mathematical notion for this is a quasi-isomorphism.
4.1 Quasi-isomorphisms
Both A∞- and L∞-algebras are differential complexes, and one may be tempted to define mor-
phisms between two such homotopy algebras as chain maps between the underlying complexes,
which respect the higher products. These morphisms are called strict, but they are too restrictive
for most purposes.
A more appropriate notion is found in the differential graded algebra and codifferential graded
coalgebra descriptions, where there is an obvious notion of morphism. This notion contains the
above mentioned strict morphisms. Translated back to the homotopy algebras, morphisms
aφ
−→ ˜
aand gψ
−−→ ˜
g(4.1)
are encoded in maps
φ= (φi:a×i→a),|φi|=1−iand ψ= (ψi:g∧i→g),|ψi|=1−i(4.2)
for i∈
N
+. These link the higher products in the two homotopy algebras. For example, in the case
of L∞-algebras, we have the following relation:
∑
j+k=i
∑
σ∈Sh(j;i)
(−1)kχ(σ;`1,. . ., `i)ψk+1(µj(`σ(1),...,`σ(j)),`σ(j+1),...,`σ(i))
=
i
∑
j=1
1
j!∑
k1+···+kj=i
∑
σ∈Sh(k1,...,kj−1;i)
χ(σ;`1,. . ., `i)ζ(σ;`1,...,`i)×
ט
µjψk1`σ(1),...,`σ(k1),...,ψkj`σ(k1+···+kj−1+1),...,`σ(i),
(4.3a)
where χ(σ;`1,. . ., `i)is again the Koszul sign (2.11) and
ζ(σ;`1,. . ., `i):= (−1)∑1≤m<n≤jkmkn+∑j−1
m=1km(j−m)+∑j
m=2(1−km)∑k1+···+km−1
k=1|`σ(k)|.(4.3b)
We then have two notions of isomorphism. First, an isomorphism of A∞- or L∞-algebra is a
morphism of A∞- or L∞-algebras for which the chain map φ1or ψ1is an isomorphism. Because
φ1or ψ1are always chain maps and thus descend to the cohomologies of the differentials m1
and µ1, we can extend the notion of quasi-isomorphism from ordinary chain complexes: a quasi-
isomorphism is a morphism of A∞- or L∞-algebra for which φ1or ψ1induces an isomorphism on
the cohomology of the respective differential:
aφ
−→ ˜
a,φ1:H•
m1(a)∼
=
−−→ H•
˜m1(˜
a)and gψ
−−→ ˜
g,ψ1:H•
µ1(g)∼
=
−−→ H•
˜
µ1(˜
g).(4.4)
It turns out that in essentially all situations, quasi-isomorphic A∞- or L∞-algebras should be
regarded as equivalent. The L∞-algebras of two field theories are quasi-isomorphic if they have the
same observables. That means in particular that they can be related by field redefinition, factoring
out symmetries, integrating out fields, etc.
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4.2 Structural theorems and field theories
There are a number of structural theorems that help us work with L∞-algebras and that also
have concrete meaning for the discussion of field theories. First, let us define the following special
classes of A∞- and L∞-algebras:
◦astrict A∞- or L∞-algebra is one in which the higher products mior µivanish for i>2;
◦askeletal A∞- or L∞-algebra is one in which the differential m1or µ1vanishes;
◦alinearly contractible A∞- or L∞-algebra is one in which the only non-trivial higher product
is the differential m1or µ1and in which the corresponding cohomologies vanish.
Also, it is clear that we can construct the direct sums of homotopy algebras. Then we have the
following structural theorems:
◦the decomposition theorem [23] states that any A∞- or L∞-algebra is isomorphic to the sum
of a skeletal and a linearly contractible A∞- or L∞-algebra;
◦this directly implies the minimal model theorem [22,23], which states that any A∞- or L∞-
algebra is quasi-isomorphic to a skeletal one, which is called a minimal model as it embeds
into all quasi-isomorphic L∞-algebras;
◦the strictification theorem [42,43] states that any A∞- or L∞-algebra is quasi-isomorphic to a
strict one.
These theorems are now useful for the homological algebraic perspective on field theories.
First, the strictification theorem implies that any field theory is equivalent to a field theory with
only cubic interaction terms. That is easily seen for scalar field theories, where one can incorporate
auxiliary fields with algebraic equations to render the action cubic, cf. [33]. It is also well-known
that Yang–Mills theory allows for a first-order formulation with cubic interaction vertices, cf. the
discussion in [37]. The abstract perspective makes this clear for any field theory.
Second, the minimal model theorem tells us that for any field theory, there is an equivalent field
theory with the same observables but without kinematical term (and thus without propagator). This
equivalent field theory therefore has to encode the tree level scattering amplitudes of the original
field theory. It is thus important to compute the minimal model of an L∞-algebra, which is best
done with the homological perturbation lemma [27,26,28].
4.3 Minimal models from the homological perturbation lemma
Let us focus on the case of an A∞-algebra a; the discussion for an L∞-algebra is fully analogous,
albeit slightly complicated by the graded symmetrisation operations that need to be included ev-
erywhere. The homological perturbation lemma (HPL) starts from the differential complex (a,m1)
underlying aand the diagram
(a,m1) (a◦,0),
h
p
e(4.5)
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where a◦is the cohomology H•
m1(a),pis the projection, eis an embedding of a◦into (a,m1), which
requires a choice (i.e. gauge fixing in the case of a gauge theory), and his a contracting homotopy:
id =m1◦h+h◦m1+e◦p,p◦e=id .(4.6)
One can always redefine the maps e,pand hsuch that the following relations are satisfied:
p◦h=h◦e=h◦h=0,p◦m1=m1◦e=0,(4.7)
which simplifies the situation a bit more. The higher products miare then regarded as a perturbation
of the differential m1and the HPL gives the perturbations to the other maps so that a perturbed form
of diagram (4.5) is recovered.
To simplify the discussion and to link it up to Feynman diagrams, we switch to the codifferen-
tial coalgebra picture. That is, we extend both pand eto coalgebra morphisms P0and E0between
⊗•aand ⊗•a◦,
P0|Nka:=p⊗kand E0|Nka◦:=e⊗k.(4.8a)
The contracting homotopy his extended to a map H0:N•a→N•aas follows:
H0|Nka:=∑
i+j=k−1
id⊗i⊗h⊗(e◦p)⊗j,(4.8b)
which is sometimes called the “tensor trick.” We then arrive at a diagram
(⊗•a,D0) (⊗•a◦,0),
H0
P0
E0
(4.9)
where D0is the continuation of m1to a codifferential on ⊗•aand equations (4.7) induce the rela-
tions
id =D0◦H0+H0◦D0+E0◦P0,
P0◦E0=id ,P0◦H0=H0◦E0=H0◦H0=0,P0◦D0=D0◦E0=0.(4.10)
We now perturb D0to D=D0+Dint, regarding Dint as small. The HPL then states that the
corresponding deformation of (4.9) is given by the maps
P=P0◦(1+Dint ◦H0)−1,H=H0◦(1+Dint ◦H0)−1,
E= (1+H0◦Dint)−1◦E0,D◦=P◦Dint ◦E0,(4.11)
where D◦is now the codifferential on ⊗•a◦, encoding the higher products on the minimal model
which we are after. By construction, Eand Pare compatible with the codifferentials,
P◦D=D◦◦Pand D◦E=E◦D◦.(4.12)
Later we use the fact that D◦is given by the morphism E, which is constructed recursively:
D◦=P0◦Dint ◦E,E=E0−H0◦Dint ◦E.(4.13)
This recursion relation is responsible for recursion relations in computations of scattering ampli-
tudes.
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5. Scattering amplitudes
The idea of perturbative quantum field theory is quite literally implemented in the homological
perturbation lemma. Given an A∞-algebra afor a Lagrangian field theory, the space a1is the
set of fields while the space a◦
1is the set of free on-shell fields. The recursion relation (4.13)
encodes precisely the tree level Feynman diagram expansion. A slight extension of the homological
perturbation lemma incorporates loops and yet again, useful recursion relations are obtained.
5.1 Tree level
We start with the example of scalar field theory, using the A∞-algebra adefined in (3.21). We
would like to compute the tree level four-point amplitude. This is given by the higher products in
the minimal model of the L∞-algebra corresponding to aor to the symmetrised expression
A(ϕ1,ϕ2,ϕ3,ϕ4) = ∑
σ∈S4/
Z
4
hϕσ(4),m◦
3(ϕσ(1),ϕσ(2),ϕσ(3))i
=∑
σ∈S3
hϕ4,m◦
3(ϕσ(1),ϕσ(2),ϕσ(3))i.(5.1)
The higher product m◦
3appears now in the codifferential D◦, and m◦
3is the restriction of D◦to the
domain ⊗3a, with subsequent projection onto a. Let us introduce simplifying notation. For any
operator O:⊗•a1→ ⊗•a2, we denote the restriction and projection onto jinputs and ioutputs by
Oi,j:=pr⊗ia2◦O⊗ja1.(5.2)
In particular, m3=D◦1,3. The recursion relation (4.13) translates to the explicit recursion
D◦1,3=P1,1
0◦(D1,2
int ◦E2,3+D1,3
int ◦E3,3),
Ei,j=δi jEi,i
0−H0◦
i+2
∑
k=2
Di,k
int ◦Ek,j,(5.3)
so that
D◦1,3=P1,1
0◦D1,2
int ◦H0◦D2,3
int +D1,3
int ◦E3,3
0.(5.4)
Explicitly, we compute
m◦
3(ϕ1,ϕ2,ϕ3) = D◦1,3(ϕ1⊗ϕ2⊗ϕ3)
=P1,1
0◦D1,2
int ◦H0◦D2,3
int +D1,3
int ◦E3,3
0(ϕ1⊗ϕ2⊗ϕ3)
=P1,1
0◦D1,2
int ◦H0◦D2,3
int +D1,3
int (e(ϕ1)⊗e(ϕ2)⊗e(ϕ3))
=pm2(h(m2(e(ϕ1),e(ϕ2))),e(ϕ3)) + m2(e(ϕ1),ϕ,h(m2(e(ϕ2),e(ϕ3))))+
+m3(e(ϕ1),e(ϕ2),e(ϕ3)).
(5.5)
Here, eembeds the asymptotic on-shell fields ϕiinto the full set of interacting fields. The rela-
tion (4.6) shows that his the inverse to the kinematical operator m1on field space with the free
fields removed, i.e. the propagator. The maps m2and m3correspond to cubic and quartic inter-
action vertices. We can thus translate the expression (5.5) into Feynman diagrams for a “current”
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(three incoming fields, ϕ1,ϕ2, and ϕ3, and one outgoing antifield, ϕ+
4) and plugging this into
formula (5.1), we recover the expected four-point tree level Feynman diagrams:
ϕ1ϕ2
ϕ4ϕ3
κ κ
ϕ1ϕ2
ϕ4ϕ3
κ
κ
ϕ1ϕ2
ϕ4ϕ3
κ
κ
ϕ1ϕ2
ϕ4ϕ3
λ(5.6)
A more precise analysis requires us to restrict the function space C∞(
R
1,3)we used in the
graded vector space underlying a, for example to on-shell states with boundary values of compact
support and Schwarz functions on
R
1,3. This discussion can be found in [33], but we suppress
these details here and in the following, as they are mostly technical.
The same procedure can be applied to Yang–Mills theory. Here, the A∞-algebra (3.23) fac-
torises into the tensor product of a kinematical A∞-algebra and a colour Lie algebra [36]. Moreover,
the recursion relation (4.13) for Einduces a recursion relation for currents, which is known as the
Berends–Giele recursion relation [1], which was used in the same paper to prove the Parke–Taylor
formula conjectured in [44].
5.2 Loop level
Let us return to the homological perturbation lemma. We note that the BV formalism gives us
a guideline as to how to incorporate the quantum effects and to go to full loop level. Namely, the
BV differential QBV should be replaced by a linear combination of the BV differential with the BV
Laplacian ∆, with a transition from classical to quantum master equation:
QBV :={SBV,−} ,{SBV,SBV}=0−→ ¯
h∆+{SBV,−} ,2¯
h∆SBV +{SBV,SBV }=0.(5.7)
In some cases, the quantum master equation requires corrections to the classical SBV in powers of
¯
h. For (unregularised) scalar field theory and Yang–Mills theory, however, the classical BV action
also satisfies the quantum master equation.
We have to translate the replacement (5.7) to the dual, codifferential coalgebra picture that
we have been using at tree level. Recall that the BV Laplacian is a product of two functional
derivatives,
∆=∂2
∂ φ A∂ φ +
A
,(5.8)
one with respect to a field φAand one with respect to the corresponding antifield φ+
Awith identical
labels A(momentum, polarisation, colour factor, etc.). Dually, ∆∗should insert pairs of fields
and antifields everywhere into the tensor product. In the case of a scalar field theory with fields
ϕ1,2∈a1, for example, we expect
∆∗(ϕ1⊗ϕ2) = Zd4k
(2π)4nψ(k)⊗ψ+(k)⊗ϕ1⊗ϕ2+ψ(k)⊗ϕ1⊗ψ+(k)⊗ϕ2+·· ·
+ψ+(k)⊗ψ(k)⊗ϕ1⊗ϕ2+ψ+(k)⊗ϕ1⊗ψ(k)⊗ϕ2+·· · o,
(5.9)
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where ψ(k)is a (momentum space) basis of the field space a1and ψ+(k)of the antifield space a2.
The transition from classical to quantum master equation (5.7) corresponds then to a replace-
ment
Dint →Dint −i¯
h∆∗,(5.10)
and we can still treat Dint −i¯
h∆∗as a perturbation and thus apply the homological perturbation
lemma. The recursion relation (4.13) then takes the form
D◦=P0◦(Dint −i¯
h∆∗)◦E,E=E0−H0◦(Dint −i¯
h∆∗)◦E.(5.11)
Note that the map Dint −i¯
h∆∗is no longer a coderivation as it fails to satisfy the co-Leibniz
rule (2.15). As a result, the maps D◦and Eare no longer coalgebra morphisms, in general, and
the structure encoded by D◦is not a classical A∞-algebra, but a quantum A∞-algebra. In a quan-
tum homotopy algebra, the homotopy relations are corrected by orders in ¯
h, cf. [45] for details on
quantum L∞-algebras. It is this quantum minimal model that describes the full Feynman diagram
expansion.
Just as at tree level, we also have a finite recursion relation at loop level [36]. Using again the
notation (5.2), the morphism Ei,j, restricted to `loops and vinteraction vertices satisfies
Ei,j
`,v=δ0
`δ0
vδi jEii
0−H0◦
i+2
∑
k=2
Di,k
int ◦Ek,j
`,v−1+i¯
hH0◦∆∗◦Ei−2,j
`−1,v,(5.12)
where Ei,j
`,v=0 for ` < 0 or v<0. This formula can directly be inserted into a computer algebra
programme to generate the perturbative expansion to any desired order.
As an application, let us briefly examine the 1-loop correction to the 2-point function in scalar
field theory with A∞-algebra (3.21). That is, we compute D◦1,1up to one loop. Note that (5.11)
and (5.12) simplify to
D◦1,1=P1,1
0◦(D1,3
int ◦E3,1
1,0+D1,2
int ◦E3,1
1,1)
=P1,1
0◦(D1,3
int ◦i¯
h∆∗◦E1,1
0+D1,2
int ◦H0◦D2,3◦i¯
h∆∗◦E1,1
0).(5.13)
We now apply this expression to a field ϕ∈a◦
1. First, we note that
(∆∗◦E1,1
0)(ϕ) = Zd4k
(2π)4nψ(k)⊗ψ+(k)⊗e(ϕ) + ψ(k)⊗e(ϕ)⊗ψ+(k)+
+e(ϕ)⊗ψ(k)⊗ψ+(k) + ψ+(k)⊗ψ(k)⊗e(ϕ)+
+ψ+(k)⊗e(ϕ)⊗ψ(k) + e(ϕ)⊗ψ+(k)⊗ψ(k)o,
(5.14)
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Perturbative Quantum Field Theory and Homotopy Algebras Christian Saemann
and the full formula (5.13) is best depicted as the following diagrams:
p
m3
h
∆∗
e
ϕ
p
m3
h
∆∗
e
ϕ
p
m3
h
∆∗
e
ϕ
+·· ·
p
m2
h
m2
h
∆∗
e
ϕ
p
m2
h
m2
h
∆∗
e
ϕ
p
m2
h
m2
h
∆∗
e
ϕ
+·· ·
Here, ϕdenotes the incoming on-shell field, the map eembeds it into the set of the full interacting
fields, dashed lines denote antifields, which are produced by the higher products m2,3:a×2,3
1→
a2and the dual BV Laplacian ∆∗, and pis the projection of an antifield back onto the on-shell
antifields. The ellipses in the first line represent another three diagrams where the order in the
field/antifield pair produced by ∆∗is swapped. In the second line, the ellipsis refer to another
nine diagrams, three more with the order of field/antifield pairs produced by ∆∗as given, and a
second copy of these altogether six diagrams with the order interchanged. We note that strings of
operations as the summands in (5.13) encode several Feynman diagrams.
The 2-point amplitude at one loop is given by
A1 loop
2(ϕ1,ϕ2) = hϕ1,D◦1,1(ϕ2)i.(5.15)
and the above diagrams turn indeed into the expected Feynman diagrams with the expected sym-
metry factors.
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Perturbative Quantum Field Theory and Homotopy Algebras Christian Saemann
5.3 Applications
In this last section, let us briefly point out some applications of the homological algebraic
perspective on perturbative quantum field theory. So far, we merely saw that any quantum field
theory can be recast in an equivalent quantum field theory with cubic interaction vertices and that
the Berends–Giele tree level recursion relations for Yang–Mills currents generalise to loop level
and to an arbitrary Lagrangian field theory.
One of the key reasons to consider A∞-algebras over L∞-algebras is that they allow for an easy
analysis of the colour structure of loop amplitudes in Yang–Mills theory. Recall that a Yang–Mills
field can either be regarded as a one-form taking values in the vector space underlying the gauge
Lie algebra or, in the case of a matrix gauge algebra, as a matrix-valued one-form. In the second
case, Feynman diagrams “thicken” to ribbon graphs and each line is replaced by a pair of lines,
one for each matrix index. The Lie bracket is replaced by the associative matrix product, and this
is best encoded in an A∞-algebra. Following the contractions of indices in loop diagrams, which is
particularly easy in our formalism, it becomes clear that Feynman diagrams split into planar and
non-planar diagrams and that planar diagrams dominate non-planar ones by at least one factor N,
where Nis the rank of the gauge group. A more precise formula is readily derived [36] in our
picture. Moreover, the one-loop amplitude of Yang–Mills theory is fully determined by its planar
part. This has been known for some time [46], but the proof of this relation is simplified in our
formalism [36].
In general, our formalism is well-suited to prove combinatorial identities. As a new example,
let us give a short proof that the tree level Berends–Giele currents in Yang–Mills theory vanish
when summed over all shuffles; a traditional proof is found already in [47], see also [48] or [49,
Appendix A]. We start from the colour-stripped form of the Yang–Mills A∞-algebra (3.23), cf. [36],
which has in particular higher products8
m2(A1,A2) = κ(d†(A1∧A2) + ?(A1∧?dA2)−?(?(dA1∧A2)) ,
m3(A1,A2,A3) = κ2(?(A1∧?(A2∧A3)) −?(?(A1∧A2)∧A3),(5.16)
where here the Aiare plain differential forms. Consider now the colour-stripped current
E◦1,n(A1,...,An),(5.17)
and partition the input fields Aiinto two pairwise disjoint, non-empty subsets Φ1= (A1,...,Am)
and Φ2= (Am+1,...,An). Then, at tree level,
∑
σ∈Sh(m;n)
E◦1,n(Aσ−1(1)⊗ · · · ⊗ Aσ−1(n)) = 0,(5.18)
where the sum runs over all shuffles, i.e. permutations preserving the relative order of the Aiin
Φ1and Φ2. The map E1,ncan be depicted as a sum of trees with binary and ternary nodes, whose
leaves are the sequence of input fields Aσ−1(1)⊗ · ·· ⊗ Aσ−1(n). Each such tree will contain maximal
subtrees where all the leaves are either from Φ1or Φ2, which we call pure subtrees of type Φ1or
Φ2. Two or three of these pure subtrees are then joined together by nodes m2or m3, originating
8denoted by mito distinguish from the full higher products mi
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Perturbative Quantum Field Theory and Homotopy Algebras Christian Saemann
from actions of Dint. There are now eight distinct such ordered joins, depicted below. If one of
a diagrams appears in a shuffle sum, then so do the others in the same row. Moreover, due to the
symmetry properties of m2and m3, these sums vanish:
m2
Φ1Φ2
+m2
Φ2Φ1
=0 (5.19a)
m3
Φ1˜
Φ1Φ2
+m3
Φ1Φ2˜
Φ1
+m3
Φ2Φ1˜
Φ1
=0 (5.19b)
m3
Φ1Φ2˜
Φ2
+m3
Φ2˜
Φ2Φ1
+m3
Φ2Φ1˜
Φ2
=0 (5.19c)
Here Φ1,˜
Φ1and Φ2,˜
Φ2depict pure trees of type Φ1and Φ2, respectively. We thus conclude (5.18),
which, as pointed out in [49], implies the Kleiss–Kuijf relations [5,6].
5.4 Outlook
An obvious problem to attack form the homological algebraic perspective is certainly the
derivation of the BCJ double copy formulas [50,51] which state that tree level amplitudes in grav-
ity can be obtained from two copies of colour-stripped Yang–Mills tree level amplitudes. Since the
origin of this relation lies in the open/closed duality of strings and because the underlying string
field theories are best described using homotopy algebras, our formalism seems to be the natural
framework to attack this problem.
Finally, we saw that physical equivalence of classical field theories translates to quasi-isomor-
phisms of their corresponding homotopy algebras. It would be very interesting to analyse the
corresponding statement for the minimal models of quantum homotopy algebras. In this context,
the renormalisation group flow should be a quasi-isomorphism of quantum homotopy algebras.
References
[1] F. A. Berends and W. T. Giele, Recursive calculations for processes with n gluons, Nucl. Phys. B 306
(1988) 759.
[2] B. Zwiebach, Closed string field theory: Quantum action and the B–V master equation, Nucl. Phys. B
390 (1993) 33 [hep-th/9206084].
[3] I. A. Batalin and G. A. Vilkovisky, Gauge algebra and quantization, Phys. Lett. B 102 (1981) 27.
[4] A. Schwarz, Geometry of Batalin–Vilkovisky quantization, Commun. Math. Phys. 155 (1993)
249 [hep-th/9205088].
21

PoS(CORFU2019)199
Perturbative Quantum Field Theory and Homotopy Algebras Christian Saemann
[5] R. Kleiss and H. Kuijf, Multi-gluon cross-sections and five jet production at hadron colliders, Nucl.
Phys. B 312 (1989) 616.
[6] V. Del Duca, L. Dixon, and F. Maltoni, New color decompositions for gauge amplitudes at tree and
loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563].
[7] J. D. Stasheff, On the homotopy associativity of H-spaces, I., Trans. Amer. Math. Soc. 108 (1963) 275.
[8] J. D. Stasheff, On the homotopy associativity of H-spaces, II., Trans. Amer. Math. Soc. 108 (1963)
293.
[9] J. Stasheff, Deformation theory and the Batalin–Vilkovisky master equation, “Deformation theory and
symplectic geometry.” Proceedings, Meeting, Ascona, Switzerland, June 16–22, 1996
[q-alg/9702012].
[10] J. Stasheff, The (secret?) homological algebra of the Batalin–Vilkovisky approach, Contemp. Math.
219 (1998) 195 [hep-th/9712157].
[11] M. Movshev and A. Schwarz, On maximally supersymmetric Yang–Mills theories, Nucl. Phys. B 681
(2004) 324 [hep-th/0311132].
[12] M. Movshev and A. Schwarz, Algebraic structure of Yang–Mills theory, Prog. Math. 244 (2006)
473 [hep-th/0404183].
[13] A. R. Gover, P. Somberg, and V. Souˇ
cek, Yang–Mills detour complexes and conformal geometry,
Commun. Math. Phys. 278 (2008) 307 [math.DG/0606401].
[14] A. M. Zeitlin, Homotopy Lie superalgebra in Yang–Mills theory, JHEP 0709 (2007)
068 [0708.1773 [hep-th]].
[15] A. M. Zeitlin, Formal Maurer–Cartan structures: From CFT to classical field equations, JHEP 0712
(2007) 098 [0708.0955 [hep-th]].
[16] A. M. Zeitlin, Batalin–Vilkovisky Yang–Mills theory as a homotopy Chern–Simons theory via string
field theory, Int. J. Mod. Phys. A 24 (2009) 1309 [0709.1411 [hep-th]].
[17] A. M. Zeitlin, String field theory-inspired algebraic structures in gauge theories, J. Math. Phys. 50
(2009) 063501 [0711.3843 [hep-th]].
[18] A. M. Zeitlin, Conformal field theory and algebraic structure of gauge theory, 0812.1840
[hep-th].
[19] C. I. Lazaroiu, D. McNamee, C. Saemann, and A. Zejak, Strong homotopy Lie algebras, generalized
Nahm equations and multiple M2-branes, 0901.3905 [hep-th].
[20] O. Hohm and B. Zwiebach, L∞algebras and field theory, Fortsch. Phys. 65 (2017)
1700014 [1701.08824 [hep-th]].
[21] K. J. Costello, Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4, Pure
Appl. Math. Quart. 9(2013) 73 [1111.4234 [math.QA]].
[22] T. Kadeishvili, Algebraic structure in the homology of an A∞-algebra, Soobshch. Akad. Nauk. Gruz.
SSR 108 (1982) 249.
[23] H. Kajiura, Noncommutative homotopy algebras associated with open strings, Rev. Math. Phys. 19
(2007) 1 [math.QA/0306332].
[24] M. Kontsevich and Y. Soibelman, Homological mirror symmetry and torus fibrations,
math.SG/0011041.
22

PoS(CORFU2019)199
Perturbative Quantum Field Theory and Homotopy Algebras Christian Saemann
[25] O. Gwilliam and T. Johnson-Freyd, How to derive Feynman diagrams for finite-dimensional integrals
directly from the BV formalism, 1202.1554 [math-ph].
[26] V. K. A. M. Gugenheim, L. A. Lambe, and J. D. Stasheff, Perturbation theory in differential
homological algebra. II, Illinois J. Math. 35 (1991) 357.
[27] V. K. A. M. Gugenheim and L. A. Lambe, Perturbation theory in differential homological algebra I,
Illinois J. Maths. 33 (1989) 566.
[28] M. Crainic, On the perturbation lemma, and deformations, math.AT/0403266.
[29] T. Johnson-Freyd, Homological perturbation theory for nonperturbative integrals, Lett. Math. Phys.
105 (2015) 1605 [1206.5319 [math-ph]].
[30] M. Doubek, B. Jurˇ
co, and J. Pulmann, Quantum L∞algebras and the homological perturbation
lemma, Commun. Math. Phys. 367 (2019) 215 [1712.02696 [math-ph]].
[31] A. S. Arvanitakis, The L∞-algebra of the S-matrix, JHEP 1907 (2019) 115 [1903.05643
[hep-th]].
[32] A. Nützi and M. Reiterer, Scattering amplitudes in YM and GR as minimal model brackets and their
recursive characterization, 1812.06454 [math-ph].
[33] T. Macrelli, C. Saemann, and M. Wolf, Scattering amplitude recursion relations in BV quantisable
theories, Phys. Rev. D 100 (2019) 045017 [1903.05713 [hep-th]].
[34] M. Reiterer, A homotopy BV algebra for Yang–Mills and color-kinematics, 1912.03110
[math-ph].
[35] A. Nützi and M. Reiterer, Scattering amplitude annihilators, 1905.02224 [math-ph].
[36] B. Jurco, T. Macrelli, C. Saemann, and M. Wolf, Loop Amplitudes and Quantum Homotopy Algebras,
1912.06695 [hep-th].
[37] B. Jurˇ
co, L. Raspollini, C. Saemann, and M. Wolf, L∞-algebras of classical field theories and the
Batalin–Vilkovisky formalism, Fortsch. Phys. 67 (2019) 1900025 [1809.09899 [hep-th]].
[38] B. Jurˇ
co, T. Macrelli, L. Raspollini, C. Saemann, and M. Wolf, L∞-algebras, the BV formalism, and
classical fields, in: “Higher Structures in M-Theory,” proceedings of the LMS/EPSRC Durham
Symposium, 12-18 August 2018 [1903.02887 [hep-th]].
[39] B. Jurˇ
co, U. Schreiber, C. Saemann, and M. Wolf, Higher Structures in M-Theory, in: “Higher
Structures in M-Theory,” proceedings of the LMS/EPSRC Durham Symposium, 12-18 August 2018
[1903.02807 [hep-th]].
[40] C. Saemann and M. Wolf, Supersymmetric Yang–Mills theory as higher Chern–Simons theory, JHEP
1707 (2017) 111 [1702.04160 [hep-th]].
[41] M. Alexandrov, M. Kontsevich, A. Schwarz, and O. Zaboronsky, The geometry of the master equation
and topological quantum field theory, Int. J. Mod. Phys. A 12 (1997) 1405 [hep-th/9502010].
[42] I. Kriz and P. May, Operads, algebras, modules and motives, SMF, Paris, 1995, available online.
[43] C. Berger and I. Moerdijk, Resolution of coloured operads and rectification of homotopy algebras,
Contemp. Math. 431 (2007) 31 [math.AT/0512576].
[44] S. J. Parke and T. R. Taylor, An amplitude for n gluon scattering, Phys. Rev. Lett. 56 (1986) 2459.
[45] M. Markl, Loop homotopy algebras in closed string field theory, Commun. Math. Phys. 221 (2001)
367 [hep-th/9711045].
23

PoS(CORFU2019)199
Perturbative Quantum Field Theory and Homotopy Algebras Christian Saemann
[46] Z. Bern, L. Dixon, D. C. Dunbar, and D. A. Kosower, One-loop n-point gauge theory amplitudes,
unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226].
[47] F. A. Berends and W. T. Giele, Multiple soft gluon radiation in parton processes, Nucl. Phys. B 313
(1989) 595.
[48] B. Kol and R. Shir, Color structures and permutations, JHEP 1411 (2014) 20 [1403.6837
[hep-th]].
[49] S. Lee, C. R. Mafra, and O. Schlotterer, Non-linear gauge transformations in D =10 SYM theory and
the BCJ duality, JHEP 1603 (2016) 090 [1510.08843 [hep-th]].
[50] Z. Bern, J. J. M. Carrasco, and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev.
D78 (2008) 085011 [0805.3993 [hep-ph]].
[51] Z. Bern, J. J. M. Carrasco, and H. Johansson, Perturbative quantum gravity as a double copy of gauge
theory, Phys. Rev. Lett. 105 (2010) 061602 [1004.0476 [hep-th]].
24