A-infinity structure in string theory and the Yang-Mills equation

0.22 MB·English

by  Dmitry Grigoryev

A
-structure in string theory and
∞
the Yang-Mills equation
Dmitry Grigoryev∗and Pavel Khromov†
Abstract
We consider local operators of CFT inserted at the boundary of the worldsheet and an
infiniteset of mapsthatact on aspaceof thelocal operators. Thesemaps havenaturalCFT
interpretation and form A -algebra. In terms of these operators we define the homotopical
∞
Maurer-Cartan equation, find its symmetries and explore its properties. Further we recover
the Yang-Mills equation from the homotopical Maurer-Cartan equation, identify the first α′-
1 correction to it and propose method for calculation all corrections.
1
0
1 Introduction
2

n
String theory on a nontrivial background is described in terms of essentially nonlinear sigma-
a
J models. For a string to consistently propagate in the background the corresponding sigma-model

7 must be conformal invariant. Condition of conformal invariance turns out to be an equations
2 of motion to a fields on the target space of the theory. Thus there is a well-known statement
[1,9]: equationofmotionforfieldsonatargetspaceofasigma-modelisabeta functionvanishing
]
h conditions for the corresponding two dimensional sigma-model. In the 80th using sigma model
t approach classical equations of motion were reproduced and α′-corrections the equations were
-
p calculated [2]-[10].
e In this paper we check the Losev’s conjecture. It says that particular representations of the
h
homotopical Maurer-Cartan (hMC) equation reproduce classical equations of motion (Yang-Mills
[
or Einstein, for example) together with the string corrections. By the hMC equation we mean

1 a generalization of common Maurer-Cartan equation to A -algebras. These algebras were firstly
∞
v
introducedbyStasheffin[11]. ReviewonissuesconcerningA -structuremaybefoundin[12]and
8 ∞
referencestherein. Higherstructuresfind applicationsin closedstringfieldtheory[13],topological
8
3 theories[14,15]andinstudyofgaugetheoriesandgravityviathehMCequationthatwassuggested
5 in [16] and developed in [17, 18, 19].
. To checkthe conjecturewe considerlocaloperatorsinsertedatthe boundaryof the worldsheet
1
0 and a set of maps that acts on the space of the local operators. The first map is the BRST
1 differential, the second is a binary map that is just an operator product expansion (depending on
1 the splitting betweenthe operators). It turns outthatthese twomaps maybe completedwith the
:
v infinite set of maps, also depending on a single parameter, such that they form an A∞-algebra,
i thus depending on the parameter too.
X
In this setup using a single proposal we reproduce the Yang-Mills equation from the hMC
r
a equation and identify the first α′-correction that is linear in α′. This approach of course can be
applied in a similar way to a calculation of α′-corrections of degree more that one, but in this
paper we have restricted ourselves to a discussion of the main idea and the simplest example.
The paper has the following structure: in section2 definition of maps that formA -algebrais
∞
given. Then we study properties of the hMC equation written in terms of this maps. In section 3
using a single proposal we reproduce the Yang-Mills equation and calculate the first α′-correction
toitfromthe hMC equation. Importantbutquite technicalissuesmaybe foundinthe Appendix.
Some speculations concerning suggested approach are collected in the Conclusion.
∗SteklovMathematicalInstitute, Moscow,Russia. [email protected]
†SteklovMathematicalInstitute, Moscow,Russia. [email protected]
1
2 The homotopical Maurer-Cartan equation
Todiscussthepropertiesoftheequationwefirstneedtodefinemapsintermsofwhichtheequation
will be written.
All our discussion will be associated with the following CFT
1
S = d2z η ∂Xµ∂Xν + d2z b∂¯c+c.c. (2.1)
2πα′ Z µν Z
Σ Σ
which governs a maps from the open string worldsheet Σ to the Minkowski space of dimension 26
with the standard flat metric on it. For the fields of the theory we have standard OPEs.
2.1 The maps
Letϕbealocaloperatorinsertedatthe boundaryofthe worldsheet. Ingeneralwewillbe dealing
with Lie algebra valued local operators, so they don’t commute. The open string worldsheet may
be brought to the upper half plane (UHP) of the complex plane by a proper conformal mapping.
In this representation the local operators are inserted at the real line.
Let Ωn be the space of local operators with ghost number n. Then we have the following
complex:
... −−−Q−→ Ω0 −−−Q−→ Ω1 −−−Q−→ Ω2 −−−Q−→ ... (2.2)
with a differential which is the BRST operator.
We consider a space of all operators H = ⊕n∈ZΩn and introduce A∞-algebra of maps M1,
M ,...that act on this space:
2
M :H⊗k →H. (2.3)
k
Herewefollowthefollowingroutetodefinemaps. First,weintroducegeometricdefinitionofmaps
M that depend on a single parameter. In terms of these maps we can define the hMC equation.
k
As long as the definition of the maps M involves notion of complicated Riemann surfaces, these
k
mapsareinconvenientfordoingpracticalcalculations. Sowemapthesemapstoafamiliarregion,
namely the UHP, and define m = f ◦M , where f is a proper conformal mapping to the UHP,
k k
whichisdifferentfordifferentmapsM . Nextweprovideanevidencethatinsomecases–andthe
k
calculation we do to reproduce the Yang-Mills belongs to this case – we even don’t need to know
the exact mapping, but all we need to know is a little piece of information about the mapping
(that lead to appearance of splittings) and some extra assumption (that lead to hierarchy).
We define the first map M in the following way:
1
M (ϕ)=Q(ϕ), (2.4)
1
where a local operator ϕ is inserted at the point −1 on a half disc (see fig. 2.1).
Then we define
m =f ◦M , (2.5)
1 1
which is an action of the BRST operator on the image of ϕ on the UHP. For a primary local
operator we can write
f ◦M (ϕ(−1))=CF ·M (ϕ(f(−1))), (2.6)
1 1
where CF denotes a conformal factor that arises due to transformational properties of operators
we are acting on.
ϕ(−1)
R jBRST
Figure 2.1: To the definition of M .
1
2
ϕ1
S χ ϕout χ
ϕ1(x) ϕ2(y)
ϕ2
ǫ
τ
Figure2.3: Tothedefinition
Figure 2.2: To the definition of M . Left correlator is calculated
2
of m .
on a surface glued from three unit half-discs with local operator 2
insertions and a strip of width 2 and length τ. There is a conical
singularityatthepointS withtotalangle3π(eachindividualangle
is π). Right correlator is calculated on a unit disc.
We define the second map M in the following way:
2
M (ϕ (w ),ϕ (w ))=e−τL0ϕ (w )ϕ (w ), (2.7)
2 1 1 2 2 1 1 2 2
whereL is a 0-thmode ofstress-energytensorofthe CFT andthe exponentactsonalloperators
0
that stand to the right.
As long as e−τL0 represents an insertion of a strip with a length τ we may think about result
of application ϕ = M (ϕ ,ϕ ) of M to operators ϕ and ϕ as a correlator calculated on a
out 2 1 2 2 1 2
Riemann surface glued from two half-discs and the τ-strip (see fig. 2.2) where equality holds for
any operator χ.
Let us note that at τ = 0 the map M coincides with the Witten’s product [20] of string
2
states corresponding to local operators ϕ and ϕ in the open string field theory, which is non-
1 2
commutative and associative. We study all maps with τ 6= 0 so M is non-commutative and
2
non-associative.
We define
m =f ◦M (2.8)
2 2
with f being a conformal mapping to the UHP. For a primary local operators we can write
f ◦M ϕ (z ),ϕ (z ) =CF ·m ϕ (f(z )),ϕ (f(z )) , (2.9)
2 1 1 2 2 2 1 1 2 2
(cid:0) (cid:1) (cid:0) (cid:1)
where again CF denotes a conformal factor that arises due to transformational properties of
operatorswe areacting on. But we actually aren’tinterested in the imagesof z andz under the
1 2
mapping f. So let us write f(z )= x and f(z )=x+ǫ. We call parameter ǫ splitting. Presence
1 2
of τ-strip in (2.7) results in behavior ǫ ∼ e−τ → 0 of the splitting between the local operators in
(2.9) in a limit τ →∞ (see fig. 2.3). Thus we can write 1
mǫ(ϕ ,ϕ )(x)=CF · ϕ (x)ϕ (x+ǫ) (2.10)
2 1 2 1 2
h ix
with the notation [O(x ,...,x )] meaning that we expand the expression O(x ,...,x ) at the
1 k x1 1 k
point x . So applying this map to operatorsϕ and ϕ we simply perform OPEof ϕ (x)ϕ (x+ǫ)
1 1 2 1 2
at the point x.
LetusfindwhatcorrespondstoacompositionM ◦M ontheUHP.Todothisweswitchtoan
2 2
equivalentdefinitionwithRiemannsurfacesandexaminepropertiesofϕ =M (M (ϕ ,ϕ ),ϕ ).
out 2 2 1 2 3
The composition can be calculated as follows: ϕ =M (ψ,ϕ ), where ψ =M (ϕ ,ϕ ) is a local
out 2 3 2 1 2
operator. In the correlator hM (ψ,ϕ ),χi used for defining ϕ we can replace ψ with a part
2 3 out
of a surface which gives the same correlator (see fig. 2.4). Since M is non-associative there are
2
two ways of composing M ◦M , namely M (M (ϕ ,ϕ ),ϕ ) and M (ϕ ,M (ϕ ,ϕ )). They are
2 2 2 2 1 2 3 2 1 2 2 3
depicted on fig. 2.5. For a primary local operators we can write
f ◦M (M (ϕ ,ϕ ),ϕ )=CF ·mǫ2(mǫ1(f ◦ϕ ,f ◦ϕ ),f ◦ϕ ),
2 2 1 2 3 2 2 1 2 3 (2.11)
f ◦M (ϕ ,M (ϕ ,ϕ ))=CF ·mǫ2(f ◦ϕ ,mǫ1(f ◦ϕ ,f ◦ϕ )),
2 1 2 2 3 2 1 2 2 3
where we manifestly indicate the splittings ǫ and ǫ between the images of insertion points of
1 2
1In general one should keep regular terms in splittings because they might cancel out divergences arising from
contractions. But one can convince himself that this happens at a higher (more than 3) orders in the coupling t
(seesection3),sothissituationiscrucialonlywhencalculatingα′-corrections.
3
χ
ϕ1 χ
ϕ1 S3 ϕ1 S3
ϕ2 S4 τ S4 τ
S1 S2 S2
τ χ ϕ2 ϕ3 τ
τ
ϕ3 S1
τ ϕ2 ϕ3
Figure 2.5: Two ways of composing M ◦M .
Figure 2.4: Gluing together two surfaces to 2 2
EachangleatpointsS andS isπ(totalangle
evaluate one way of a composition M ◦M . 1 2
2 2
is 3π). Each angle at points S and S is π/2.
3 4
the operators that in the limit τ → ∞ behave as ǫ ∼ e−2τ+o(τ) and ǫ ∼ e−τ+o(τ). Thus the
1 2
maps M being identical before the conformalmapping seem to become different maps m on the
2 2
UHP gaining what we will call a hierarchy of the splittings: the ”inner” m has ǫ as a splitting
2 1
betweentheir arguments,while the ”outer”m has splitting ǫ (see fig. 2.6). Of course,it doesn’t
2 2
mean that we have different maps on the UHP (they are the same, although are distorted by
the conformal mapping), but it implies that while doing calculation on the UHP we have to be
careful when encounter a composition of the maps, because composition results in the hierarchy
on the UHP.
ϕ1(x) ϕ2(y) ϕ3(z) ϕ1(x) ϕ2(y) ϕ3(z)
ǫ1 ǫ1
ǫ2 ǫ2
Figure 2.6: Compositions mǫ2(mǫ1(ϕ ,ϕ ),ϕ ) and mǫ2(ϕ ,mǫ1(ϕ ,ϕ )) correspondingly.
2 2 1 2 3 2 1 2 2 3
Define M this way 2
3
M (ϕ (w ),ϕ (w ),ϕ (w ))=
3 1 1 2 2 3 3
τ
−ta1ta2ta3(−1)|ϕ1| dse−τL0ϕa1(w )e(s−τ)L0{b ,ϕa2(w )}ϕa3(w )
hZ0 1 1 −1 2 2 3 3 (2.12)
τ
− dse−τL0ϕa3(w )e(s−τ)L0{b ,ϕa2(w )}ϕa1(w ) ,
Z0 3 3 −1 2 2 1 1 i
where |ϕ | is ghost number of ϕ and ta is a generator of corresponding Lie algebra so a local
1 1
operator can be written as ϕ=ϕata with ϕa being a scalar. Exponents act on all operators that
stand to the right. Fig. 2.7 is a geometrical(visual) representationof the definition. Thus we can
see that M is a homotopy between two ways of composing M ◦M . This is a strict statement
3 2 2
because, as we will see, M satisfy the third relation (2.18) of A -algebra.
3 ∞
IngeneralviamappingM totheUHPwegetsomethingcomplicatedbecausedoingconformal
3
mapping we have to take into account transformation properties of the operators we are acting
on while the fields may have non-zero conformal dimension and even be non-primary. But for a
primary local operators of conformal dimension 0 we can write a simple expression
x+ǫ2−ǫ1
mǫ1,ǫ2(ϕ ,ϕ ,ϕ )(x)=(−1)|ϕ1| ϕ (x) dy {b ,ϕ (y)}ϕ (x+ǫ ) (2.13)
3 1 2 3 (cid:20) 1 Z −1 2 3 2 (cid:21)
x+ǫ1 x
with the splittings
ǫ ∼e−2τ+o(τ), ǫ ∼e−τ+o(τ) when τ →∞. (2.14)
1 2
2Fromhereonby{a,b}wemeansupercommutator.
4
Aslongaswedon’thaveanexplicitconformalmappingwedon’tknowdependenceofthesplittings
on τ exactly. In actual calculations we need more precise relations between the splittings. Thus
we propose
ǫ /ǫ →0, ǫ2/ǫ →0 when τ →∞. (2.15)
1 2 2 1
So it is actually a prescription based on the mapping.
We see that in the case of primary local operatorswith conformaldimension 0 the mapping to
the UHP results only in the hierarchy of the splittings and we don’t need to know the mapping
exactly.
The first three maps M satisfy first three quadratic relations of A -algebra:
k ∞
M2 =0, (2.16)
1
M M (A,B)=M (M (A),B)+(−1)|A|M (A,M (B)), (2.17)
1 2 2 1 2 1
M (A,M (B,C))−M (M (A,B),C)=
2 2 2 2
=M M (A,B,C)
1 3
+M (M (A),B,C)+(−1)|A|M (A,M (B),C)+(−1)|A|+|B|M (A,B,M (C)). (2.18)
3 1 3 1 3 1
Indeed, this relations may be trivially checked using definitions (2.7) and (2.12).
Conformal mapping to the UHP does not spoil this property, but again one should be careful
when doing compositions. Thus writing similar conditions on the UHP we manifestly indicate the
splittings:
m2 =0, (2.19)
1
m mǫ(A,B)=mǫ(m (A),B)+(−1)|A|mǫ(A,m (B)), (2.20)
1 2 2 1 2 1
mǫ2(A,mǫ1(B,C))−mǫ2(mǫ1(A,B),C)=
2 2 2 2
=m mǫ1,ǫ2(A,B,C)
1 3
+mǫ1,ǫ2(m (A),B,C)+(−1)|A|mǫ1,ǫ2(A,m (B),C)+(−1)|A|+|B|mǫ1,ǫ2(A,B,m (C)). (2.21)
3 1 3 1 3 1
See Appendix A for manifest verification of this relations.
We know that the map M is a homotopy between two possible compositions ofM ◦M . The
3 2 2
map M is an integral over an edge of polytope called associahedra [11, 12]. Following this route
3
we can define each higher M as an integral over corresponding (k−2)-dimensional face of the
k
associahedra. Having in mind this homotopical definition we propose that the rest of conditions
of A -algebra are satisfied.
∞
χ
χ
ϕ1
τ
τ ϕ1
ds
τ −
s
R b
R0 (cid:16) ϕ2 ϕ3 (cid:17)
s R b
ϕ2 ϕ3
Figure 2.7: To the definition of m .
3
We still have m = f ◦M , but let us work out this definition on the UHP. In order to define
k k
m we take all ways of composing k−1 instances of m into a map H⊗k →H. Each composition
k 2
is an OPE of k operators inserted at the real line. Insertion points of the operators are governed
by hierarchy of splittings ǫ ,...,ǫ and structure of concrete composition. The coordinates of
1 k−1
insertionpointsdefineapointinRk. Soforeachwayofcomposingk−1instancesofm wehavea
2
pointinRk. Thereis (k−2)-dimensionalface D ofassociahedrawiththese points being vertices.
k
5
Define map m on a primary local operators of conformal dimension 0 as the following integral
k
mǫ1,...,ǫk−1(ϕ ,...,ϕ )(x )=
k 1 k 1
=(−1)(k−2)|ϕ1|...(−1)|ϕk−2| ϕ (x ) dx ...dx {b ,ϕ (x )}...{b ,ϕ (x )}ϕ (x )
1 1 2 k−1 −1 2 2 −1 k−1 k−1 k k
h ZDk ix1
(2.22)
From the definition it follows that the boundary of D is union of (k−3)-dimensional faces, each
k
of them being integration domain for an instance of m . Using decent relations we can obtain
k−1
quadratic relations of A -structure from the structure of associahedra. Schematic derivation is
∞
presented below (we suppress all signs and splittings for readability):
Qm (ϕ ,...,ϕ )−m (Qϕ ,...,ϕ )−...−m (ϕ ,...,Qϕ )
k 1 k k 1 k k 1 k
= ϕ (x )[L ,ϕ (x )]...{b ,ϕ (x )}ϕ (x )+...+
1 1 0 2 2 −1 k−1 k−1 k k
Z
Dk
ϕ (x ){b ,ϕ (x )}...[L ,ϕ (x )]ϕ (x )
1 1 −1 2 2 0 k−1 k−1 k k
Z
Dk
= ϕ (x )∂ϕ (x )...{b ,ϕ (x )}ϕ (x )+...+
1 1 2 2 −1 k−1 k−1 k k
Z
Dk
ϕ (x ){b ,ϕ (x )}...∂ϕ (x )ϕ (x )
1 1 −1 2 2 k−1 k−1 k k
Z
Dk
k−2
= ±m (ϕ ,...,ϕ ,m (ϕ ,...,ϕ ))±...±m (m (ϕ ,...,ϕ ),ϕ ,...,ϕ ))
n+1 1 n k−n n+1 k n+1 k−n 1 k−n k−n+1 k
nX=1(cid:2) (cid:3)
(2.23)
Example of m for k =3 was given above. Later we will present example for k =4 where D is a
k 4
region defined by relations (3.36) with the hierarchy of splittings (3.37).
In conclusion of the sectionlet us recallthat we have introduced the maps M that depend on
k
aparameterτ. ThesemapsformA -algebrathusdependingontheparametertoo. Laterinorder
∞
to reproduce the Yang-Mills equation we will take a limit τ → ∞. Under conformal mapping we
have obtained the maps m that depends on a set of a parameters (which depend on τ) – point
k
splittings. In the limit τ → ∞ the splittings go to zero with their own pace that is governed by
whatwecalledthehierarchyofthesplittings. Thehierarchyappearsviatheconformalmappingof
themapsM totheUHP.Werepeatournotethatinactualcalculationsweuseproposedhierarchy
k
of splittings based on actual hierarchy, but it is not proved that these hierarchies are equal.
2.2 The equation
Having defined maps let us consider the homotopical Maurer-Cartan equation. By the hMC
equation we mean a generalization of common Maurer-Cartan equation to A -algebras. In this
∞
section we study properties of the hMC equation written in terms of M , but all this properties
k
are also true for the equation written in terms of m , because they are related by the conformal
k
mapping. The equation associated with the maps M is
k
M (V)+M (V,V)−M (V,V,V)+...=0 (2.24)
1 2 3
with V being a ghost number 1 local operator.
Map M has ghost number 2−k, so by applying it to V we get a local operator M (V,...,V)
k k
with ghost number 2.
As long as the maps M form A -algebra the symmetry of the equation is
k ∞
δV =M (H)+M (V,H)−M (H,V)
1 2 2
(2.25)
−M (H,V,V)+M (V,H,V)−M (V,V,H)+...,
3 3 3
where parameter of the symmetry H is ghost number 0 local operator and ellipsis stands for the
subsequentmapswherewehavesubstitutedoneH insteadofV inadifferentplacesinM keeping
k
track of the sign.
Thus we consider a local operator from Ω1 as a variable. Later in order to obtain the Yang-
Mills equation we specialize to the local operators defined by means of a field A which we will
µ
6
calla parametrization. The equationonthis operatoris in Ω2. The parameterof symmetryof the
equation is in Ω0.
In attempt to solve the hMC equation with a local operator V in a particular parametriza-
tion A one may encounter an obstruction. Obstruction is a cohomology of the space Ω2 of ghost
µ
number 2 local operators. Thus for the local operator to be a solution of the hMC equation the
parametrizationmust satisfy an equation– an obstructionvanishing condition. This equationhas
a symmetry that is inherited from the symmetry δV of the hMC equation.
In order to obtain the Yang-Mills equation from the hMC equation we have to reproduce
structure of the Yang-Mills equation from the obstruction vanishing condition and the symmetry
δA oftheYang-MillsequationfromthesymmetryδV ofthehMCequation. Asaresult,weobtain
µ
(possibly) nonlinear dependence A =A (A) of a gauge potential A on the parametrization A
µ µ µ µ
that may lead to identification of α′-correctionto the Yang-Mills equation.
In general all the maps M depend on the parameter τ. We match the hMC and the Yang-
k
Mills equations in a limit τ → ∞. Thus we are only left with expressions that don’t depend on
the parameter and with singular ones. But we don’t expect appearance of the parameter in the
obstruction vanishing condition. So we expect singular in the parameter terms to be exact.
AslongaswewillbedoingallcalculationontheUHPitissensibletoreformulatethisprocedure
in terms of the splittings. Namely, we match the hMC and the Yang-Mills equations in a limit
when all the splittings go to zero, although each with their own pace governed by a hierarchy of
splittings. Conditions (2.15) provides us with the first example of relations in the hierarchy. Thus
weareonlyleftwithexpressionsthatdon’tdependonthe parametersandwithsingularoneswith
latter expected to be exact.
TorealizethisapproachletusstartwithexaminationofastructureoftheYang-Millsequation.
The equation can be rewritten as follows
Max(A)+YM2(A)+YM3(A)=0 (2.26)
with Max,YM2 and YM3 being linear, quadratic and cubic in gauge potential A parts of the
µ
Yang-Mills equation correspondingly
Max(A )=2A −∂ ∂ A , (2.27)
µ µ µ ν ν
YM2(A )=∂ [A ,A ]+[A ,∂ A −∂ A ], (2.28)
µ ν ν µ ν ν µ µ ν
YM3(A )=[A ,[A ,A ]]. (2.29)
µ ν ν µ
The symmetry of (2.26) is
δA =∂ ε+[A ,ε] (2.30)
µ µ µ
withε being aninfinitesimalparameterofthe symmetry. Seekingsolutioninaperturbationseries
in a coupling t
A =tA(1)+t2A(2)+..., (2.31)
µ µ µ
ε=tε(1)+t2ε(2)+... (2.32)
we get the following equations
t1 : Max(A(1))=0, (2.33)
µ
t2 : Max(A(2))+YM2(A(1))=0, (2.34)
µ µ
t3 : Max(A(3))+YM3(A(1))
µ µ
+2[A(1),∂ A(2)]+2[A(2),∂ A(1)]−[A(1),∂ A(2)]−[A(2),∂ A(1)]
ν ν µ ν ν µ ν µ ν ν µ ν
−[A(1),∂ A(2)]−[A(2),∂ A(1)]=0. (2.35)
µ ν ν µ ν ν
and so on. Their symmetries are
δA(1) =∂ ε(1), (2.36)
µ µ
δA(2) =∂ ε(2)+[A(1),ε(1)], (2.37)
µ µ µ
δA(3) =∂ ε(3)+[A(1),ε(2)]+[A(2),ε(1)] (2.38)
µ µ µ µ
and so on.
7
Next let us find a similar structures in the hMC equation and its symmetry. To do this we
again take an attempt to find a solution to the hMC equation (2.24) as a perturbation series in
the coupling t
V =tV +t2V +..., (2.39)
1 2
where each V is also a series in α′. This way we obtain the following equations
k
t1 : M (V )=0, (2.40)
1 1
t2 : M (V )+M (V ,V )=0, (2.41)
1 2 2 1 1
t3 : M (V )+M (V ,V )+M (V ,V )−M (V ,V ,V )=0 (2.42)
1 3 2 1 2 2 2 1 3 1 1 1
and so on. Corresponding equations on the UHP are
t1 : m (V )=0, (2.43)
1 1
t2 : m (V )+mǫ(V ,V )=0, (2.44)
1 2 2 1 1
t3 : m (V )+mǫ2(V ,V )+mǫ2(V ,V )−mǫ1,ǫ2(V ,V ,V )=0 (2.45)
1 3 2 1 2 2 2 1 3 1 1 1
andsoon. We willbe solvingthese particularequationsinsection3. As longasthe localoperator
becomes a series so do the parametrization A = tkA(k) and the symmetry parameter
µ k≥1 µ
H = tkH(k). P
k≥1
LePt us examine some properties that do not depend on a particular choice of parametrization.
The first equation is
M (V )=0. (2.46)
1 1
It has a symmetry that is simply shift by an exact element
δV =M (H(1)). (2.47)
1 1
The second equation is
M (V )+M (V ,V )=0. (2.48)
1 2 2 1 1
It has more complicated symmetry containing apart from an exact element m -commutator
2
δV =M (H(1)), (2.49)
1 1
δV =M (H(2))+M (V ,H(1))−M (H(1),V ). (2.50)
2 1 2 1 2 1
Similar reasoning could be done for a subsequent equations, but we better switch to a concrete
example.
3 Emergence of the Yang-Mills equation
In this section we present derivation of the Yang-Mills equation from the hMC equation (2.43)-
(2.45). To do this we start with the following parametrization of the local operator in the first
order in the coupling t
α′
V =cA(1)∂Xµ− ∂c∂ A(1), (3.1)
1 µ 4 µ µ
where A(1) =A(1)(X) is a functions on the target space. For V we have
µ µ 1
α′
Q(V )= c∂c 2A(1)−∂ ∂ A(1) ∂Xµ, (3.2)
1 4 µ µ ν ν
h i
so in this parametrization equation (2.43) takes form
α′
c∂c 2A(1)−∂ ∂ A(1) ∂Xµ =0. (3.3)
4 µ µ ν ν
h i
(1)
WeseethatV solves(2.43)whenparametrizationA satisfyMaxwellequation. Aswediscussed
1 µ
earlier(2.47)thisequationhasasymmetrywhichisashiftbyanexactelementV →V +Q(H(1)).
1 1
The form of the exact element is
α′
Q(H(1))=c∂ H(1)∂Xµ− ∂c2H(1). (3.4)
µ
4
8
This shift naturally generates Maxwell gauge transformation for (3.1)
δA(1) =∂ H(1). (3.5)
µ µ
Thus the symmetry of the hMC equation at the first order in coupling t implies the symmetry of
the Maxwell (or linear part of the Yang-Mills) equation.
SoatthefirstorderintwehavethefollowingidentificationsofgaugepotentialAandparameter
of gauge transformation ε with parametrizations of local operators:
A(1) =A(1), (3.6)
µ µ
ε(1) =H(1) (3.7)
together with the Maxwell equation
Max(A(1))=0 (3.8)
µ
and its gauge symmetry
δA(1) =∂ ε(1). (3.9)
µ µ
Due to a formofthe parametrization,structureofthe equationandthe symmetry there is no way
for α′-correction to appear at this stage.
Situation changes when we go to the second order in the coupling t. In order to find an
obstructionwe calculate mǫ(V ,V )in (2.44) upto terms O(α′3). The obstructionmay be written
2 1 1
in the form
α′ α′
Max A(2)− A(1)∂ A(1) +YM2(A(1))
4 h (cid:18) µ 2 α µ α (cid:19) µ
+α′ [∂ A(1),∂ F(1)]+[Max(A(1)),∂ A(1)]
α β µ αβ α µ α
(cid:16) (cid:17)
α′
− 2[∂ A(1),∂ ∂ A(1)]−[∂ A(1),∂ ∂ A(1)]−[∂ A(1),∂ ∂ A(1)] logǫ2+O(α′2) ∂Xµ =0.
2 α β α β µ α β α µ β α µ α β β
h i i
(3.10)
with F(1) =∂ A(1)−∂ A(1) being a prototype of a linear part of field strength.
αβ α β β α
The second line of (3.10) coincides with α′∂ [F(1),F(1)] that is a ∂3A2-part of a variation of
α µβ βα
Tr(F3)fromthenon-abeliangeneralizationofBorn-Infeldaction[21,22]moduloMaxwellequation
on A(1). Namely our result differs from the variation of Tr(F3) by [∂ A(1),Max(A(1))]. Thus
µ β µ β
instead of the second line in (3.10) we will write α′∂ [F(1),F(1)].
α µβ βα
As we can see this obstruction does depend on the splitting via logǫ. But structure with logǫ
is similar to the YM2-part (2.28). This fact allow us to do a redefinition below that will help us
to deal with logǫ-terms.
For V we have the following expression
2
α′ α′ c ∂c
V =cA(2)∂Xµ− ∂c∂ A(2)+ + A(1)A(1)+O(α′2) (3.11)
2 µ 4 µ µ 2 (cid:18)ǫ 2 (cid:19) µ µ
Its symmetry can be found this way. A variation of (2.44) is
Q(δV )+mǫ(δV ,V )+mǫ(V ,δV )=0. (3.12)
2 2 1 1 2 1 1
As long as δV =QH(1) and due to (2.43) we can write
1
Q δV −mǫ(V ,H(1))+mǫ(H(1),V ) =0 (3.13)
2 2 1 2 1
(cid:16) (cid:17)
so
δV =QH(2)+mǫ(V ,H(1))−mǫ(H(1),V ). (3.14)
2 2 1 2 1
Doing some math we arrive to
α′
δV =c ∂ H(2)+[A(1),H(1)] ∂Xµ− ∂c∂ ∂ H(2)+[A(1),H(1)]
2 µ µ 4 µ µ µ
h i h i
α′ c ∂c α′
+ + {A(1),∂ H(1)}+δ cA(1)∂ A(1) ∂Xµ (3.15)
2 (cid:18)ǫ 2 (cid:19) µ µ Max(cid:18) 2 α µ α (cid:19)
α′
− c[∂ A(1),∂ H(1)]∂Xµlogǫ2+O(α′2),
2 α µ α
9
which besides a variation of (3.11) contains some other terms and δ A(1) =∂ H(1).
Max µ µ
The following redefinition
α′
A˜(1) = 1− logǫ22 A(1), (3.16)
µ (cid:18) 4 (cid:19) µ
where2=ηµν∂ ∂ isd’Alambertianonthe targetspacehelps us to dealwith logǫ. The equation
µ ν
becomes
α′ α′
1+ logǫ22 Max A˜(2)− A˜(1)∂ A˜(1) +YM2(A˜(1))+α′∂ [F˜(1),F˜(1)] +O(α′2)=0.
(cid:20) 4 (cid:21)(cid:20) (cid:18) µ 2 α µ α (cid:19) µ α µβ βα (cid:21)
(3.17)
As long as 1+ α′ logǫ22 is invertible (at least perturbatively) we may omit it.
4
Doing more contractions in mǫ(V ,V ) will only have the effect of the redefinition (see App.
2 1 1
B). The full redefinition is
α′
A˜(1) =exp − logǫ22 A(1), (3.18)
µ (cid:18) 4 (cid:19) µ
which is similar to redefinition made in [22] while deriving the Yang-Mills action from the boson-
ic open string field theory action. Also, the Yang-Mills equation was obtained form the open
superstring field theory in [23].
After this redefinition the symmetry becomes
α′
δV =c ∂ H(2)+[A˜(1),H(1)] ∂Xµ− ∂c∂ ∂ H(2)+[A˜(1),H(1)]
2 µ µ 4 µ µ µ
h i h i (3.19)
α′ c ∂c α′
+ + {A˜(1),∂ H(1)}+δ cA˜(1)∂ A˜(1) ∂Xµ+O(α′2).
2 (cid:18)ǫ 2 (cid:19) µ µ Max(cid:18) 2 α µ α (cid:19)
The form of the equation (3.17) and its symmetry (3.19) forces us to do the following identifi-
cations at the second order in the coupling t
α′
A(2) =A˜(2)− A˜(1)∂ A˜(1), (3.20)
µ µ 2 α µ α
ε(2) =H(2). (3.21)
In this terms quadratic part of the Yang-Mills equation is
Max(A(2))+YM2(A(1))+α′∂ [F(1),F(1)]+O(α′2)=0, (3.22)
µ µ α µβ βα
where
F(1) =∂ A(1)−∂ A(1) (3.23)
αβ α β β α
is a linear part of field strength and the gauge symmetry is
δA(1) =∂ ε(1),
µ µ
(3.24)
δA(2) =∂ ε(2)+[A(1),ε(1)].
µ µ µ
So we see that using approach based on solving the hMC equation we have reproduced quadratic
part of the Yang-Mills equation together with α′-correction that is linear in α′ which arises from
the Tr(F3) term in the non-abelian generalizationof Born-Infeld action.
Thus we are able to propose a general method for calculating α′-corrections to the Yang-Mills
equationfromthehMC equation. Oneshouldcalculatearesultofeachmapuptodesiredorderin
α′ orin otherwordsdoa propernumberofcontractions. Ingeneralone shouldkeepregularterms
in the splittings which may cancelout singularterms, howeverin our calculationsthey have never
contributed to the equation. Then exact terms may be absorbed by V and non-exact terms form
k
an obstruction that after proper redefinition of the parametrization provide us with the equation
and desired α′-corrections to it.
Letus getacubic termofthe Yang-Mills equationformthe hMC equation. In orderto do this
we have to find an obstruction to the equation
Q(V )+mǫ2(V ,V )+mǫ2(V ,V )−mǫ1,ǫ2(V ,V ,V )=0 (3.25)
3 2 1 2 2 2 1 3 1 1 1
10