Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Speculations on Arithmetical Physics

In a lecture written for the 25th Arbeitstagung in

Bonn, Y Manin presented intriguing connections

between arithmetic geometry (especially Arakelov

geometry) and physics. The theme is also discussed

in Manin (1989). These considerations are based on a

philosophical viewpoint according to which funda-

mental physics might, like adeles, have Archimedean

(real or complex) as well as non-Archimedean

(p-adic) manifestations. Since adelic objects are

more fundamental and often simpler than their

Archimedean components, one can hope to use this

point of view in order to carry over some computa-

tion of physical relevance to the non-Archimedean

side where one can employ number-theoretic methods.

Adelic phy sics? Some of the results mentioned in

the previous sections seem to lend themselves well to

this adelic interpretation. The quantum statistical

mechanics of Q-lattices relies fundamentally on

adeles and it admits generalizations to systems

associated to other algebraic varieties (Shimura

varieties) that have an adelic descript ion and adelic

groups of symmetries. The result on the Polyakov

measure also has an adelic flavor, as it uses

essentially the Archimedean component of the

Faltings height function. The latter is in fact a

product of contributions from all the Archimedean

and non-Archimedean places of the field of defini-

tion of algebraic points in the moduli space, so that

one can expect that there would be an adelic

Polyakov measure, of which one normally sees the

Archimedean side only. The Freund–Witten adelic

product formula for the Veneziano string amplitude

fits in the same context, with p-adic amplitudes

ð; Þ¼

jxj

1

j1  xj

1

and B

(, )

1

(, ) (cf. Varadarajan

(2004)).

Adelic physics and motives A similar adelic philo-

sophy was taken up by other authors, who proposed

ways of introducing non-Archimedean and adelic

geometries in quantum physics. A recent survey is

given in Varadarajan (2004). For instance, Volovich

(1995) proposed spacetime model s based on

cohomological realizations of motives, with e´tale

topology ‘‘interpolating’’ between a proposed non-

Archimedean geometry at the Planck scale an d

Euclidean geometry at the macroscopic scale. In

this viewpoint, motivic L-functions appear as parti-

tion funct ions and actions of motivic Galois groups

govern the dynamics.

See also: Hopf Algebra Structure of Renormalizable

Quantum Field Theory; Mirror Symmetry: A Geometric

Survey; Quantum Ergodicity and Mixing of

Eigenfunctions; Random Matrix Theory in Physics;

Regularization for Dynamical Zeta Functions.

Further Reading

Beilinson A and Manin Yu (1986) The Mumford form and the

Polyakov measure in string theory. Communications in

Mathematical Physics 107(3): 359–376.

Bost JB and Connes A (1995) Hecke algebras, type III factors and

phase transitions with spontaneous symmetry breaking in

number theory. Selecta Mathematica (N.S.) 1(3): 411–457.

Broadhurst DJ and Kreimer D (1997) Association of multiple zeta

values with positive knots via Feynman diagrams up to 9

loops. Physics Letters 393(3–4): 403–412.

Cartier P (2002) Fonctions polylogarithmes, nombres polyzetas et

groupes pro-unipotents, Se´minaire Bourbaki, Vol. 2000/2001.

Aste´risque No. 282 (2002), Exp. No. 885, viii, 137–173.

Connes A, Marcolli M, and Ramachandran N (2005) KMS states

and complex multiplication. Preprint (to appear in Selecta

Mathematica), arXiv math.OA/0501424.

Degiovanni P (1994) Moore and Seiberg equations, topological

field theories and Galois theory. In: Leila S (ed.) The

Grothendieck Theory of Dessins D’enfants. Cambridge: Cam-

bridge University Press.

Julia BL, Moussa P, and Vanhove P (eds.) (2005) Frontiers in Number

Theory, Physics and Geometry, vols. I, II, Papers from the Meeting

held in Les Houches, March 9–21, 2003. Berlin: Springer.

Kreimer D (2000) Knots and Fe ynman Diagrams. Cambridge

Lecture Notes in Physics, vol. 13. Cambridge: Cambridge

University Press.

Lang S (1988) Introduction to Arakelov Theory. Berlin: Springer.

Manin Yu (1989) Reflections on arithmetical physics. In: Dita P

and Georgescu V (eds.) Conformal Invariance and String

Theory, pp. 293–303. Boston: Academic Press.

Manin Yu (1991) Three-dimensional hyperbolic geometry as

1-adic Arakelov geometry. Inventiones Mathematicae

104(2): 223–243.

Manin Yu and Marcolli M (2001) Holography principle and

arithmetic of algebraic curves. Advanced Theoretical and

Mathematical Physics 5(3): 617–650.

Moch S, Uwer P, and Weinzierl S (2002) Nested sums, expansion

of transcendental functions, and multiscale multiloop inte-

grals. Journal of Mathematical Physics 43(6): 3363–3386.

Moore G (1998) Arithmetic and attractors, hep-th/9807087.

Nahm W (2005) Conformal field theory and torsion elements of

the bloch group. In: Julia BL, Moussa P, and Vanhove P (eds.)

Frontiers in Number Theory, Physics and Geometry, vol. I.

Berlin: Springer.

Schneps L (ed.) (1994) The Grothendieck Theory of Dessins

d’enfants. Cambridge: Cambridge University Press.

Serre JP (1992) Motifs, in Journe´es Arithme´tiques, 1989 (Luminy,

1989). Aste´risque No. 198–200, (1991), 11, 333–349.

Stevenhagen P (2001) Hilbert’s 12th problem, complex multi-

plication and Shimura reciprocity. In: Class Field Theory – Its

Centenary and Prospect (Tokyo, 1998), 161–176, Adv. Stud.

Pure Math., 30, Math. Soc. Japan, Tokyo, 2001.

Varadarajan VS (2004) Arithmetic quantum physics: why, what

and whether. Proceedings of the Steklov Institute of Mathe-

matics 245: 258–265.

Volovich IV (1995) From p-adic strings to e´tale strings. Proceed-

ings of the Steklov Institute of Mathematics 203(3): 37–42.

Number Theory in Physics 607

Operads

J Stasheff, Lansdale, PA, USA

Introduction

An operad is an abstraction of a family of composable

functions of n variables for various n, useful for the

‘‘bookkeeping’’ and applications of such families.

Operads are particularly important and useful in

categories with a good notion of ‘‘homotopy,’’ where

they play a key role in organizing hierarchies of higher

homotopies, reflecting their original use as a tool in

homotopy theory, especially for studying (iterated)

loop spaces. For several years now, operads have

become increasingly important in mathematical

physics, especially in string field theory, where they

organize the terms of higher order in perturbed

actions, and in deformation quantization.

The major focus of this article will be on operads as

they are relevant to mathematical physics, but will also

include some background material from homotopy

theory, where they originated. A borderland where

homotopy theory and cohomological physics overlap is

the world of differential graded vector spaces, including

those of differential forms, ghosts, anti-ghosts, etc.,

sometimes lumped together as BRST theory. Here, as

elsewhere in contemporary mathematical physics, the

flow has been in both directions – sometimes physicists

have discovered or reinvented known mathematics but

finding new applications, at other times physics has

suggested new concepts for mathematicians to develop

further. In the case of operads, they have provided

general structure for varieties of algebras, some of

which are novel types contributed by physicists.

For a reasonably up-to-date introduction and

survey, consider Markl et al. (2002), although there

have been many developments since then. Two

particularly important original works are Boardman

and Vogt (1973) and May (1972).

Definitions and Examples

The term ‘‘operad’’ is due to May, building on work

of Stasheff and of Boardman–Vogt. The most

fundamental example of an operad is the endo-

morphism operad End

:= {Map(X

, X)}

n1

where,

for a set or topological space X, {Map(X

, X)}

means the set or space of functions or continuous

functions from the n-fold product of X with itself to

X, together with the operations



: MapðX

; XÞMapðX

; XÞ!MapðX

nþm1

; XÞ

given, for 1  i  n,by

ðf 

gÞðx

; ...; x

mþn1

¼ f ðx

; ...; x

i1

; gðx

; ...; x

iþm1

Þ; x

iþm

; ...Þ

In the endomorphism operad End

, there are

easily discovered relations involving iterated 

operations and the symmetric group 

actions on

the X

s. For example,

ðf 

gÞ

h ¼ f 

ðg 

jiþ1

hÞ

for i  j  i þ m  1

if g is a function of m variables, since only the name

of the position for the insertion is changed.

An operad (O, 

) consists of a collection

{O(n)}

n1

of objects and maps 

: O(n) O(m) !

O(n þ m  1) for m, n  1andi  n satisfying the

relations manifest in the example End

May’s original definition corre sponds to simulta-

neous insertions into all possible positions of inputs

into f 2 Map(X

, X). In most examples, the struc-

tures are ‘‘manifest’’ without appeal to the technical

definitions.

It helps to see graphic examples of operads,

particularly ones relevant for physics. Two kinds

that are particularly important are the tree operads

and the little disks (or cubes) operad s.

Let T (n) be the set of planar trees with one root

and n leaves labeled (arbitrarily) 1 through n. The

collection T = {T (n)}

n1

of sets of trees forms an

operad by grafting the root of g to the leaf of f

labeled i,asinFigure 1, where the leaves are

assumed labeled in order from left to right. Figure 1

can be interpreted as portraying the 

result of

inserting a 3-linear operation into a 5-linear one.

The little n-disks operad D

= {D

(j)}

j1

where

(j) consists of an ordered collection of jn-disks

embedded in the standard n-dimensional unit disk

with disjoint interiors, the embedding being of

the form az þ b with 0 < a 2 R. The operations are

given as indicated in Figure 2.

Just as group theory without representations is

rather sterile, so are operads best appreciated by

their representations known as (varieties of) alge-

bras, especially algebras with higher homotopies.

An algebra A over an operad P ‘‘is’’ a map of

operad P !End

. This is just a compact way

of saying that an algebra A has a coherent system

of maps P (n)  A

!A. Much of this article will

speak in terms of such algebras with the correspond-

ing operad being understood.

Operads in Homotopy Theory

A major motivation for the development of operads

was the desire to have a homotopy-invariant char-

acterization of based loop spaces and iterated loop

spaces. Precisely such coherent systems of higher

homotopies provided the answers. For based loop

spaces, the operad in question K= {K

}

n1

consists of

the polytopes known as ‘‘associahedra.’’ The usual

product of based loops is only homotopy associative.

If we fix a specific associating homotopy and

consider the five ways of parenthesizing the product

of four loops, there results a pentagon whose edges

correspond to a path of loops (Figure 3).

From the leftmost vertex to the rightmost, consider

the two paths of loops across the top or around the

bottom. By further adjustment of parameters, the

pentagon can be filled in by a family of such paths.

The associahedron K

can be described as a

convex polytope with one vertex for each way of

associating n ordered variables, that is, ways of

inserting parentheses in a meaningful way in a word

of n letters. The edges correspond to a single

application of an associating homotopy. More

generally, the cellular structure of the associahedra

is well described by planar rooted trees, the vertices

corresponding to binary trees and so forth (see

Figure 4).

For K

,seeFigure 5 or a rotatable image available at

http://igd.univ-lyon1.fr/chapoton/stasheff.html. The

facets are all products of two associahedra of lower

dimension and specific imbeddings can be given to play

theroleofthe

operations as in an operad.

An A

-space is a space Y which admits a coherent

family of maps

: K

 Y

! Y

so that they make Y an algebra over the operad

(without 

-actions) K= {K

}

n1

The main result by Stasheff is: A connected space

Y (of the homotopy type of a CW-complex) has the

homotopy type of a based loop space X for some

X if and only if Y is an A

-space.

Homotopy characterizat ion of iterated loop

spaces 

for some space X

required the full

power of the theory of operads with the symmetries.

Figure 1 Grafting with the leaves numbered from left to right.

Figure 2 The little 2-disks operad.

(ab)(cd )

((ab)c)d

(a(bc))da((bc)d

)

a(b(cd

))

Figure 3 The associahedron K

Figure 4 K

with vertices labeled by trees.

Figure 5 The associahedron K

610 Operads

An early motivation for the invention of a theory

of operads was the consideration of infinite loop

spaces, that is, a sequence of spaces X

such that

each X

is homotopy equivalent to X

nþ1

Although introduced originally in the category of

topological spaces, operads were available almost

immediately for differential graded (dg) vector

spaces, also known as chain complexes. In physics,

the differential is often called a BRST operator, a

term that should be reserved for a special kind of dg

algebra, see below.

Operads in Algebra

The 

notation first appeared in Gerstenhaber’s study

of the algebraic structure of the Hochschild cohomol-

ogy of an associative algebra, about the same time as

the construction of the associahedra where the opera-

tions were given in a less convenient notation. Recall

the Hochschild cohomology of an associative algebra

A is the homology of the complex Hom(A

n

, A)with

the coboundary given as follows (all signs below are

indicated as , any of the standard references will

specify conventions and signs): for f 2 Hom(A

n

, A)

and g 2 Hom(A

m

, A)let

f  g ¼ 

 f 

g ½1

Gerstenhaber then defines his bracket as [f , g] = f  g

g  f . With hindsight, he realized that the

Hochschild coboundary can be written as

h ¼½m; h½2

where m : A  A !A is the multiplication. More-

over, the associativity of m is equivalent to

½m; m¼0 ½3

-Algebras

In the setting of graded vector spaces V = 

r2Z

there are two conventions for defining A

-algebras,

which differ by a shift in grading. We adopt the

physics convention so that A here is the suspension

of that considered in the original papers. The

cellular chains of the associahedra form the A

operad, providing the following definition.

Definition 1 A

-algebra (Strong homotopy asso-

ciative algebra). Let A be a Z-graded vector space

A = 

r2Z

and suppose that there exists a collec-

tion of degree 1 multilinear maps

m :¼fm

: A

k

!Ag

k1

(A, m ) is called an A

-algebra when the multilinear

maps m

satisfy the following relation s:

pþq¼nþ1

i¼1

m



¼ 0 ½4

with an appropriate set of signs for n  1.

A weak A

-algebra consists of a collection of

degree 1 multilinear maps

m :¼fm

: A

k

! Ag

k0

satisfying the above relations, but for n  0 and in

particular with k, l  0.

Remark 1 The ‘‘weak’’ version is fairly new,

inspired by physics, where m

: C !A, regarded as

an element m

(1) 2 A, is related to what physicists

refer to as a ‘‘background.’’ The augmented relati on

then implies that m

(1) is a cycle, but m

need no

longer be 0, rather

¼m

ðm

 1Þ m

ð1  m

Þ½5

Just as associativity was captured by the equation

[m, m] = 0, so the defining relations of the definition

of an A

-algebra are captured by

½m ; m ¼0 ½6

Decades later it was realized that consideri ng

A =A

n

as a coalgebra with

ða

a

Þ¼

pþq

ða

a

ða

pþ1

a

we then have an isomorphism

HomðA

n

; AÞ’CoderðT

AÞ

Here Coder is the space of all coderivations of T

The Gerstenhaber bracket is indeed the ‘‘intrinsic’’

commutator bracket of coderivations via the above

isomorphism. As such, it satisf ies a graded version of

the Jacobi identity; after a shift in grading from the

original one of Hochschild, the Hochschild cochain

complex forms a dg Lie algebra.

-Algebras

Since an ordinary Lie algebra g is regarded as

ungraded, the defining bracket is regarded as skew-

symmetric. For dg Lie algebras and L

-algebras, we

need graded symmetry, which refers to symmetry with

signs determined by the grading. The basic operation is

 : x  y 7!ð1Þ

jxjjyj

y  x ½7

Also we adopt the convention that tensor products

of graded functions or operators have the signs built

Operads 611

in; for example, (f  g)(x  y) = (1)

jgjjxj

f (x)  g(y).

By decomposing each permutation as a product of

transpositions, there is then defined the sign of a

permutation of n graded elements, for example, for

any c

2 V,1  i  n, and any  2 S

, the permuta-

tion of n graded elements is defined by

ðc

; ...; c

Þ¼ð1Þ

ðÞ

ðc

ð1Þ

; ...; c

ðnÞ

Þ½8

The sign (1)

()

is often referred to as the Koszul

sign of the permutation.

Definition 2 (Graded symmetry). A graded sym-

metric multilinear map of a graded vector space V to

itself is a linear map f : V

n

!V such that for any

2 V,1  i  n, and any  2 S

(the permutation

group of n elements), the relation

f ðc

; ...; c

Þ¼ð1Þ

ðÞ

f ðc

ð1Þ

; ...; c

ðnÞ

Þ½9

holds.

Definition 3 By a (k, l)-uns huffle of c

, ..., c

with

n = k þ l is meant a permutation  such that for i <

j  k, we have (i) <(j) and similarly for k < i <

j  k þ l. We denote the subset of (k, l)-uns huffles in

kþl

by S

k, l

and by S

kþl = n

, the union of the subsets

k, l

with k þ l = n. Similarly, a (k

, ..., k

)-unshuffle

means a permutation  2 S

with n = k

þþk

such that the order is preserved within each block of

length k

, ..., k

. The subset of S

consisting of all

such unshuffles we denote by S

,..., k

Definition 4 L

-algebra (Strong homotopy Lie

algebra). Let L be a graded vector space and suppose

that a collection of degree 1 graded symmetric linear

maps l := {l

: L

k

!L}

k1

is given. (L, l)iscalledan

-algebra iff the maps satisfy the following relations:

2S

kþl¼n

ð1Þ

ðÞ

1þl

ðl

ðc

ð1Þ

; ...; c

ðkÞ

Þ;

½c

ðkþ1Þ

; ...; c

ðnÞ

Þ¼0 ½10

for n  1.

A weak L

-algebra consists of a collection of

degree 1 graded symmetric linear maps

l:= {l

: L

k

!L}

l0

satisfying the above relations,

but for n  0 and with k, l  0.

Remark 2 The alternate definition in which the

summation is over all permutations, rather than just

unshuffles, requires the inclusion of app ropriate

coefficients involving factorials.

Just as an A

-algebra can be described as a

coderivation of T

A, similarly an L

-algebra L can

be described as a coderivation on S

L, the symmetric

subcoalgebra of T

The operad of Lie algebras was defined rather

late, although it was earlier implicit in the work of

Fred Cohen. It is defined as the homology

n1

(Config(R

, n)) for n  1, where Config(R

, n)

denotes the configuration space of ordered n-tuples

of distinct points in R

. Equivalently, the configura-

tions can be thought of as the centers of the little

2-disks. The open disks being contractible to their

centers, this is a suboperad of the full homology



Just as a Lie algebra is obtained from an

associative algebra using the commutator as bracket

and, inversely, a Lie algebra gives rise to its

universal enveloping associative algebra, an

-algebra can be obtained from an A

-algebra

by n-variabl e analogs of commutators and there

is a universal enveloping A

-algebra of a given

-algebra.

Open–Closed Homotopy Algebras

Open–closed string field theory suggests interaction

between an L

-algebra H

and an A

-algebra H

including a strong homotopy representation of H

on H

by strong homotopy derivations. Here is the

formal definition:

Definition 5 Let H= H

H

be a graded vector

space and (H

, l) be a weak L

-algebra. Consider a

collection of multilinear maps

n :¼fn

k;l

: ðH

k

ðH

l

k;l0

each of which is graded symmetric on (H

)

l

.We

denote the collection also by n . We call (H, n , l)a

(partial) open-closed homotopy algebra (OCHA)

when n satisfies the following relations (up to some

factorial coefficients):

0 ¼

k;l0

mk

p¼0

2S

n

mþ1k;nl

ðo

; ...; o

;

k;l

ðo

pþ1

; ...; o

pþk

; c

ð1Þ

; ...; c

ðlÞ

Þ;

pþkþ1

; ...; o

; c

ðlþ1Þ

; ...; c

ðnÞ

2S

l¼1

n

m;nþ1l

ðo

; ...; o

;

ðc

ð1Þ

; ...; c

ðlÞ

Þ; c

ðlþ1Þ

; ...; c

ðnÞ

Þ½11

Other Algebras of Interest

The Hochschild complex also has a graded product

(without invoking the shift) known as the cup

product. Except for the signs and the grading, the

bracket and the product satisfy the Leibniz rule of a

Poisson algebra on the cohomology; the result is

612 Operads

axiomatized as a ‘‘Gerstenhaber algebra.’’ However,

on the cochain complex, the Lie bracket and the

associative product are compatible only up to

homotopy.

This naturally raises the issue of an operad for

strong homotopy Gerstenhaber algebras. The operad

G for Gerstenhaber algebras is the homology of the

little disks operad, H



). But now we have

choices: in addition to relaxing the Leibniz rule up

to homotopy, the bracket could be relaxed to be

part of an L

-algebra and/or the product could be

relaxed to be part of an A

-algebra. The choice

which is now known as the G

-operad is defined in

terms of a procedure which works for what are

known as quadratic operads, indicating they have

generators in O(2) and relations in O(3): the

corresponding O

has ‘‘dual’’ relations. For exam-

ple, this gives the classical Koszul duality between

Lie and commutat ive associative algebras. The G

operad can also be described as the ‘‘minimal

model’’ of G in the sense of Markl.

Another alternative is to consider just the ‘‘brace’’

operations, originally introduced by Kadeishvili and

later independently by Getzler, but described in the

Hochschild complex setting by Gerstenhaber–

Voronov. Together with the cup product, these

determine an operad denoted HG which acts on the

Hochschild complex; there is an operad map from

to HG, hence G

also acts on the Hochschild

complex. Finally, Tamarkin showed that G

quasi-isomorphic to the dg operad of singular chains

on the little disks operad, thus providing one of

several pr oofs of what had been a conjecture by

Deligne.

Algebras with invariant inner products < , >

are of considerable importance in mathematics and

especially in mathematical physics; invariance means

<a, bc > = < ab, c> or <a,[b, c] > = < [a, b], c> in,

respectively, the associative or the Lie case (with

appropriate signs in the graded case). Using the

inner product, n-ary operations A

n

!A can be

converted to operations A

nþ1

!C of which we can

require cyclic symmetry. To handle such algebras,

there is a notion of ‘‘cyclic operad.’’ In terms of

trees, the transition is to take a rooted tree and then

regard the root edge as just another leaf. This point

of view corresponds to an essential symmetry for

particle interactions.

Operads in Mathematical Physics

One reason for the explosive development of operad

theory in the 1990s was the introducti on of operadic

structures in field theories, for example, conformal

field theories (CFTs) and string field theories (SFTs).

These operadic structures were directly related to

the moduli spaces of Riemann surfaces with punc-

tures or boundaries (or other decorations) in these

physical theories.

Two special ‘‘higher-homotopy algebras’’ have

been emphasized because they are particularly

important in mathematical physics: A

for open-

string field theory and L

for closed-string field

theory and for deformation quantization. Open–

closed string field theory combines A

-algebra and

-algebra in a particular way known as an OCHA.

The operad for L

-algebras is given a very nice

and physically relevant geometric interpretation in

terms of a real compactification of the moduli space

of Riemann spheres with punctures, while for

OCHAs, there is a real compactification of the

moduli space of Riemann disks with punctures on

the boundary or in the interior (bulk). Thus , this

operad can be regarded as obtained from a moduli

space of configurations of points (punctures) in the

disk by compactifying the moduli spaces by adding

boundary strata where two (or more) points

(punctures) collide. Points on the boundary strata

can be visualized as ‘‘bubble trees’’ of disks and/or

spheres, see Figure 6. Alternatively, the little disks

operad can be regarded as being obtained by

‘‘decorating’’ the points with little disks, while for

OCHAs there is also a basic half-disk decorated

with little disks in the bulk and little half-disks for

the boundary points. The corresponding colored

operad is Voronov’s ‘‘Swiss-cheese operad.’’

‘‘Colored’’ refers to the fact that disks can be

inserted into half-disks but not vice versa. Compare

trees with two ‘‘colors’’ of edges with grafts

restricted to ones which match colors.

On-Shell versus Off-Shell

In cohomological physics, the ‘‘on-shell’’ states or

observables are usually given by the cohomology

with respect to an internal differential, which in

physics is called the BRST differential or BRST operator,

though originally this meant the Chevalley–Eilenberg

differential associated to the action of the Lie algebra of

Figure 6 Bubble tree for circle configurations.

Operads 613

gauge symmetries of a physical theory. The generators

of the Chevalley–Eilenberg cochain complex are known

as ‘ ‘ghosts’’. On-shell subspaces of algebras which are

not closed under the product of the larger ‘ ‘off-shell’’

algebra are called ‘ ‘open’’ algebras by physicists. Quite

generally, this situation gives rise to an algebra over an

appropriate operad. A special case involves a differential

graded algebra A and a linear imbedding H(A)  A.

The (co)homology is in turn a graded algebra (with 0 as

differential), but inherits a higher-homotopy structure

so that cohomology and original algebra are equivalent.

In the associative case, the inheritance is a result

of Kadeishvili:

Let (A, d) be a differential graded associative or

-algebra, then the homology H(A) inherits the

structure of an A

-algebra.

Even if the original algebra A is strictly associa-

tive, the inherited A

-structure generally has non-

trivial operations m

Analogous results hold for L

-algebras and

others. It is the L

-version that is relevant for

closed-string field theory (CSFT). Zwiebach showed

the quantum theory of covariant closed strings has

an action defined in terms of an infinite chain of

string field products. The genus-0 (tree level) string

field algebra is an L

-algebra inherited from the off-

shell state space modeled by the Batalin–Vilkovisky

(BV) construction. The higher-order brackets pro-

vide higher-order correlation or n-point functions

which play a crucial role in the extended Lagrangian

of the theory.

Batalin–Fradkin–Vilkovisky and Batalin–Vilkovisky

Constructions

The constructions of Batalin–Fradkin–Vilkovisky

(BFV) for constrained Hamiltonian systems and of

Batalin–Vilkovisky (BV) for Lagran gians with sym-

metries are important examples of L

-structures

derived from ‘‘open’’ algebra settings, though the

-structures were recognized quite a while after

the constructions.

The BFV setting is that of a symplectic manifold

W with a family of constraints, that is, a family of

functions 



2 C

(W). The constraints are called

‘‘first class’’ if the ideal they generate is closed under

the Poisson bracket. The vector space spanned by

the constraints will in general be an open algebra;

the structure of the bracket is give n by structure

functions, rather than structure constants. The zero

locus of all the constraints forms the constr aint

surface V. In the first-class case, the constraints are

in involution and determine a foliation F of V. If the

space of leaves V=F is a manifold, it would be

considered the true physical space and the physical

observables would be functions in C

(V=F). BFV

construct a differential graded Poisson algebra such

that the cohomology in degree 0 agrees with

(V=F) when that makes sense and, in the regular

case, the rest of the cohomology is that of the

differential forms along the leaves of the foliation.

The BFV differential is a deformation of the

Chevalley–Eilenberg/BRST differential and can be

constructed most efficiently by the same techniques

used in proving Kadeishivili’s inheritance theorem.

Crucially, it is an inner derivation with respect to

the Poisson bracket. After the fact, an L

-structure

can be observed in the extended algebra.

For a Lagrangian with symmetries, BV develop a

similar construction, the main difference being that

there is no Poisson bracket initially, but one is

constructed by adjoining ‘‘anti-fields’’ as conjugate to

the fields but of ghost degree 1 and the differential of

an anti-field being the Euler–Lagrange expression for

the corresponding field. Then, as in the Hamiltonian

case, ghosts and anti-ghosts, etc. are adjoined and the

construction proceeds in a parallel fashion.

Deformation Quantization

Once algebras over an operad P are considered, it is

natural to consider also morphisms of such algebras

over a fixed P .

From a homotopy point of view, the appropriate

maps need not respect the operad structure strictly

but only up to higher homotopy; indeed, there is a

related operad to define such maps. For L

algebras, such L

-maps play a key role in deforma-

tion quantization. That refers to deformation of the

commutative multiplication of a Poisson algebra in

the direction of the Poisson bracket; that is, to first

order, the deformation is given by the bracket.

More generally, for any associative algebra A with

multiplication m, one considers formal deformations

a ? b ¼ mða; bÞþtm

ða; bÞþt

ða; b Þþ ½12

where each m

2 Hom( A  A, A). The associativity

of ? provides a sequence of con straints on the m

.In

particular, m

must be a Hochschild cocycle and the

obstruction to the existence of m

is a class in the

Hochschild cohomology of degree 3. In fact, the

primary obstruction is represented by [m

, m

]. If it

is cohomologous to zero, that fact identifies candi-

dates for m

, that is,

½m

; m

¼2½m; m

½13

or, using the notation d = [m,],

 1=2½m

; m

¼0 ½14

614 Operads

once known as the integrability equation but now,

more frequently, as a Maurer–Cartan equation. For

a Poisson algebra, the Poisson bracket is a Hochs-

child cocycle but in general a full deformation need

not exist. However, for the algebra A of smooth

functions on a Poisson (e.g., symplectic) manifold

M, Kontsevich showed that such a full formal

deformation does exist.

The guiding philosophy is that deformations are

controlled by a dg Lie or L

-algebra L,uniqueupto

-homotopy equivalence. Therefore, the obstruc-

tions can be computed in any of the equivalent dg Lie

algebras. Moreover, the structure of the obstructions

is known sufficiently so that if there is an equivalent

dg Lie algebra with d in fact zero, then all the

obstructions to deformation quantization vanish. The

key to Kontsevich’s proof was the construction of an

-map, inducing an isomorphism in cohomology,

from the Lie algebra of polyvector fields on R

with

the Schouten bracket and d = 0totheLiealgebraof

multidifferential operators on A = C

)regardedas

a subalgebra of the Hochschild cochain complex for A

with the Gerstenhaber bracket.

BV Algebras

In addition to their construction of a differential

graded Gerstenhaber algebra (a differential graded

commutative algebra with a compatible Poisson

bracket of degree 1), BV introduced a new mathe-

matical structure, adding a second-order differential

operator  relating the commutative product and

the bracket. The operator  is a derivation of the

bracket and of square zero. Moreove r,

½a; b¼ðabÞðaÞb  aðbÞ½15

so that the failure of  to be a derivation of the

product is given by the bracket.

The definition of a BV algebra is then a Gerstenhaber

algebra with such an operator, though alternative

definitions exist in which  and the product are

primary and the bracket is defined by the above

equation. From the operadic/higher-homotopy point

of view, one can then go on to consider BV

algebras.

Recall that A

-algebras and L

-algebras (among

others) can be characterized by an ‘‘inner’’ coderiva-

tion d = [m, ] of square zero on an appropriate

‘‘standard’’ construction. In the context of BV

algebras, where the bracket is more commonly

written as {,}, the classical action is an element S

such that {S

, S

} = 0 or, equivalently, d = {S

, } is of

square zero. The quan tum analog S is a perturbation

of S

and satisfies instead

fS; Sg¼S ½16

This was originally called the ‘‘master equation,’’

but now is increasingly referred to as a ‘‘Maurer–

Cartan’’ equation.

Insertion Operads

There is another class of operads illustrated by trees

(and more generally graphs) with a very different

sort of ‘‘composition,’’ namely insertion of one

graph into another. The most directly relevant to

physics is the kind of insertion used by Connes and

Kreimer in their Hopf algebra constructed for

renormalization of Feynman diagrams. For example,

consider all finite graphs with exactly two external

edges and internal numbered edges. Given tw o

graphs 

, 

, define 





by cutting edge i of



and identifying the dangling edges with the two

external edges of 

For planar trees, yet another insertion operad is

obtained by Chapoton, isolating a part of a structure

due to Kontsevich, in which a small neighborhood

of a vertex of the second planar tree is removed and

the dangling edges are attached to a vertex of the

first tree by entering through the angles between the

edges at that vertex (Figure 7).

Inside the HG-operad is the operad Brace for an

abstract brace algeb ra (forgetting the cup product),

first described as such by Chapoton using the

insertion operations of Kontsevich and Soibelman.

-Categories

Also of importance for applications to mathematical

physics is the notion of an A

-category, first made

explicit by Fukaya and now playin g a major role in

string D-brane theory and homological mirror

symmetry. The D-branes are the objects of the A

category and the open strings with boundaries on

two (possibly equal) D-branes B

, B

are the

morphisms from B

to B

. The operations m

are

defined only on tuples (a

, ..., a

) of ‘‘composable’’

morphisms (e.g., strings).

PROPs

While an operad is an abstraction of a family of

composable functions of n variables for various n,a

PROP is an abstraction of a family of functions in

Figure 7 Angles determined by edges with leaves extended to

the semicircle.

Operads 615

Hom(A

p

, A

q

)forallp and q. Now the relevant

images are graphs with p input legs and q output

legs with composition being defined by grafting

output legs of one graph to i nputs of another.

Feynman diagrams are the obvious example in

physics or, in conformal field theory, tubular

neighborhoods of such graphs, which is to say,

Riemann surfaces with boundary circles: p as

inputs and q as outputs.

See also: Algebraic Approach to Quantum Field Theory;

Batalin–Vilkovisky Quantization; Constrained Systems;

Deformations of the Poisson Bracket on a Symplectic

Manifold; Deformation Quantization; Deformation Theory;

Hopf Algebra Structure of Renormalizable Quantum Field

Theory; String Field Theory.