Families of gauge conditions in BV formalism
Revised: May
Families of gauge conditions in BV formalism
S~ao Paulo
Brazil
Davis
U.S.A.
Andrei Mikhailov 0 3
Albert Schwarz 2
0 On leave from Institute for Theoretical and Experimental Physics , Moscow , Russia
1 sica Teorica, Universidade Estadual Paulista
2 Department of Mathematics, University of California , USA
3 Instituto de Fi
In BV formalism we can consider a Lagrangian submanifold as a gauge condition. Starting with the BV action functional we construct a closed form on the space of Lagrangian submanifolds. If the action functional is invariant with respect to some
BRST Quantization; Gauge Symmetry; Topological Strings

group H and
is an Hinvariant family of Lagrangian submanifold then under certain
conditions we construct a form on
that descends to a closed form on
=H. Integrating
the latter form over a cycle in
=H we obtain numbers that can have interesting physical
meaning. We show that one can get string amplitudes this way. Applying this construction
to topological quantum
eld theories one obtains topological invariants.
1 Introduction
4 Gauge symmetries
5 From BRST to BV
2 Families of equivalent action functionals
3 Families of Lagrangian submanifolds in BV phase space
6 Topological quantum
eld theories. Bosonic strings
7 String amplitudes
7.1
String amplitudes for critical string
8 Pure spinor superstring
A Some useful formulas
B De nition of using marked points
C Central extension of the group of canonical transformations
C.1 De nition of b C.2 How to build a form on LAG starting from b
family of physically equivalent action functionals. As was noticed in [1] this is wrong. The
consideration of a family of equivalent action functionals or family of gauge conditions
labeled by points of (super) manifold
leads to a construction of a closed di erential form
{ 1 {
(a closed pseudodi erential form if
is a supermanifold). If our action functionals
are invariant with respect to some group H then the form
is Hinvariant, but it does not
necessarily descend to
=H. Under some conditions we construct a closed Hequivariant
form
H and show that this equivariant form is homologous to a form descending to
=H.
This allows us to obtain interesting physical quantities integrating over cycles in
=H.
For example, we can start with topological quantum
eld theory on some manifold .
One can apply our results to the family of equivalent action functionals labeled by metrics
on
. We obtain topological invariants of
this way; it would be interesting to calculate
them and compare with known invariants.
This machinery can be applied to string amplitudes. The worldsheet of bosonic string
HJEP07(21)63
can be considered as twodimensional topological quantum
eld theory. Considering
as a
space of metrics and H as a group generated by di eomorphisms and Weyl transformations
we get formulas for string amplitudes; for appropriate choice of Lagrangian submanifolds
these formulas coincide with the standard ones. Similar constructions work for other types
of strings.
Some remarks about terminology and notations.
We are saying \manifold" instead
of \supermanifold", \group" instead of \supergroup", etc. We understand an element of
super Lie algebra as a linear combination P ATA where TA are even or odd generators of
Z2 graded Lie algebra and A are even or odd elements of some Grassmann algebra; hence
in our understanding an element of super Lie algebra is always an even object (see [
2, 3
]
for the de nitions of supermanifold, super Lie algebra, etc. that we are using).
We work in BVformalism assuming that the BV action functionals are de ned on
odd symplectic manifold M equipped with volume element (SPmanifold in terminology
of [4, 5]). In this situation the odd Laplacian
is de ned on the space of functions on M .
It was noticed in [6] that in the absence of the volume element the odd Laplacian is de ned
on semidensities; this allows the reformulation of BVformalism for any odd symplectic
manifold. In appendix C we show how to prove our main results in this more general
setting. Some basic formulas of BVformalism are listed in appendix A.
The space of (smooth) functions on a supermanifold M is denoted Fun(M ).This space
is Z2graded: Fun(M ) = Fun0(M ) + Fun1(M ). Functions on
T M (on the space of
tangent bundle with reversed parity of bers) are called pseudodi erential forms (PDF) on
M . (Di erential forms can be considered as polynomial functions on
T M .) Di stands
for the group of di eomorphisms, Vect for its Lie algebra (the algebra of vector elds),
Weyl for the group of Weyl transformations. As we have noticed an element of any super
Lie algebra (and hence a vector eld) is considered an even object.
We use the term \canonical transformation" for a transformation of (odd) symplectic
manifold preserving the symplectic form (another word for this notion is
\symplectomorphism"). On a simply connected manifold in nitesimal canonical transformations can be
characterized as Hamiltonian vector elds. Notice that in our terminology the Hamiltonian
on odd symplectic manifold is an odd function B; the rst order di erential operator
corresponding to the Hamiltonian vector eld with the Hamiltonian B is expressed in terms
of the odd Poisson bracket as an operator transforming a function G into fB; Gg; this
{ 2 {
operator is even (parity preserving). The condition
B = 0 means that the Hamiltonian
vector eld is volume preserving (= divergence free).
2
Families of equivalent action functionals
Let us consider a functional S de ned on an odd symplectic manifold M with volume
element and satisfying the quantum master equation
eSBV = 0. (Here
stands for
the odd Laplacian [23, BVformalism/OddLaplace.html].) Then the physical quantities
corresponding to the BV action functional SBV can be expressed as integrals R
L AeSBV
where L is a Lagrangian submanifold of M and the integral is taken with respect to
the volume element induced on this submanifold; A stands for quantum observable (i.e.
(AeSBV ) = 0 or equivalently
A + fA; SBVg = 0). These integrals depend only on the
homology class of the Lagrangian submanifold.
Let us consider now a family of physically equivalent BVaction functionals S ;
obeying fS ; S g = 0,
S = 0. We can consider S as a function on
M . We assume
that
is simply connected; then S being physically equivalent for di erent values of
is
equivalent to the existence of functions Ba such that:
for some Ba 2 Fun1(M ),
Ba = 0 (one can describe Ba as Hamiltonians of in nitesimal
volume preserving canonical transformations giving equivalence of functionals S for
innitesimally close
). The eq. (2.1) implies that
will assume a stronger condition:
Then the following PDF on is closed: Indeed using eqs. (A.13) and (A.14) we obtain
dB
fB; Bg = 0
where B = d aBa
( ; d ) =
exp (S + B)
Z
L
1
2
2
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
Z
L
d ( ; d ) =
fB; Sg +
fB; Bg eS+B =
eS+B = 0
Z
L
More generally, let us de ne:
Then:
hF i( ; d ) =
Z
L
the set of volume preserving canonical transformations transforming
xed point of .) A continuous (even di erentiable) section of
this bundle not necessarily exists globally, but always exists locally. It exists globally, in
particular, in the case when
is contractible. Di erentiating the section U
we obtain
and (2.2). (This is not quite correct: the operators B^ = d aB^a obey dB^
1=2[B^; B^] = 0,
but their Hamiltonians Ba speci ed via B^a = fBa; g are de ned only up to a dependent
constant and (2.2) is true only for an appropriate choice of these constants; see appendix C
. Their Hamiltonians Ba obey (2.1)
3
Families of Lagrangian submanifolds in BV phase space
We will show that one can construct some interesting quantities (including string
amplitudes) considering families of Lagrangian submanifolds instead of families of action
Let us x a connected family
of simply connected Lagrangian submanifolds. In other
words we assume that L depends on parameters
1; : : : ; k; : : : (these parameters can be
odd, but for simplicity we assume that they are even). Let G be the group of canonical
transformations of M (transformations preserving the odd symplectic structure), and g its
Lie algebra. Elements of g correspond to odd functions on M (Hamiltonians).
Tentative de nition of the closed form .
We want to de ne a closed pseudo
di erential form
on the space LAG of all simplyconnected Lagrangian submanifolds:
(3.1)
Roughly speaking, the value of
at a point v 2
T LAG is computed as follows. Notice
that v corresponds to a pair (L; ) where L 2 LAG and
2 Fun(L) is an odd function
on L describing the tangent vector.1 The variation of L can be described by in nitesimal
1As in Classical Mechanics, a function on Lagrangian manifold L speci es a tangent vector to LAG
(an in nitesimal deformation of L). In our case the symplectic form is odd, hence the correspondence is
parity reversing. These functions are called \in nitesimal gauge fermions". We have assumed that L is
transformation. (Notice that the canonical transformation is not unique, but the restriction
of its Hamiltonian to L is well de ned up to a constant summand.) In other words, for any
vector eld v inducing a tangent vector to LAG at L we have:
d
=
( v!)jL :
The function
depends on v 2
hence it can be considered as a oneform on LAG.
TL LAG (on odd tangent vector to LAG at L) linearly,
By de nition:
More generally, for every function F on M we de ne:
As a complication, the oneform is de ned only up to a constant:
(L; v) =
eSBV+
hF i(L; v) =
Therefore the de nition of by eq. (3.3) is strictly speaking ambiguous. We will prove that it is always possible to resolve this ambiguity in such a way, that the form is closed.
Moreover,
1; : : : ; n. This means that we have a family of Lagrangian submanifolds (L( )).
Let us nd a family of volume preserving canonical transformations g( ) such that:
not participate in any way in the de nition of ; we will use it just to compute d . Using
g( ) we can construct a family of physically equivalent action functionals S obeying
Z
L0
e
S =
eS :
Z
L
Here S is obtained from S by means of the transformation g . It is easy to check that the
form
introduced in present section coincides with the form constructed in the section 2
for the family S
and denoted by the same symbol; hence it is closed. (The second
summand in the de nition of
in section 2 is a Hamiltonian H of the in nitesimal canonical
transformation governing the variation of S . The Hamiltonian governing the variation of
L enters the de nition of in present section. These two Hamiltonians coincide up to a constant; resolving the ambiguity in the de nition of second Hamiltonian in appropriate way we can say that the Hamiltonians coincide.)
If we know the precise de nition of
we can give also a precise de nition of
hF i.
The formula (3.6) follows from (2.6). A more formal proof of the results of this section is given in appendix C. { 5 {
Gauge symmetries
Form is not necessarily base with respect to gauge symmetries.
We assume
that the action functional S, the observable A, the volume element on M , and the family
are invariant under a subgroup H
G (or Lie algebra h
g).2 We denote by h^ the set
of Hamiltonians of elements of h; then the hinvariance of S; A and volume element means
that for every h 2 h^ we have fS; hg = 0; fA; hg = 0 and
a weaker requirement:
h = 0. (It is enough to impose
h + fSBV; hg = 0;
(4.1)
HJEP07(21)63
see [7].) It follows from these assumptions that the form
is also Hinvariant (or
hinvariant). In general the form
is not horizontal, and therefore does not descend to
=H [23, omega/Descent To Double Coset.html]. However, in some important cases, in
particular in string theory, the form
does descend to
=H for appropriate choice of the
family of Lagrangian submanifolds.
We will now construct a modi ed form
which is base.
Under the assumptions of previous section, let us make the following additional
assumption [23, introduction/Proof of equivariance.html]. Suppose that there exists a map
: h^ ! Fun(M ) such that every Hamiltonian h 2 h^ satis es:
h = fSBV; (h)g +
(h) +
f (h); (h)g
1
2
(Notice that the Hamiltonian h is odd, but
(h) is even.) We will also require that
satis es the following \equivariance" property. For any two elements h 2 h^ and h~ 2 h^:
(4.2)
(4.3)
(4.4)
fh; (h~)g =
(fh; h~g)
form.)
the form
Let us suppose that the action of h on
comes from a free action of the corresponding
Lie group H (this Lie group is not necessarily connected). Then we can construct [23,
omega/Descent To Double Coset.html#(part. .Modi ed .P.D.F)] closed form
H , which
descends to
=H. (In other words this is a base form, i.e. Hinvariant and Hhorizontal
Technically, we use the formalism of equivariant cohomology [23,
equivariantcohomology/index.html]. The conditions we impose on the map
allow us to prove that
CH ( ; d ; h) =
eSBV+ + (h)
Z
L
represents a class of Hequivariant cohomology of
in the Cartan model [23,
equivariantcohomology/Equivariant Cohomology.html]. (We consider here
as a oneform on .)
Recall that in this model an equivariant cohomology class is represented by a di
erential form depending on an element of h and belonging to the kernel of Cartan di erential
d
h where h 2 h. (The dependence of h should agree with the action of the group H.)
2Notice, that H is not necessarily the full group of automorphisms. In string worldsheet theory, typically
H is the group of di eomorphisms.
{ 6 {
which is essentially the de nition of .
commutator hence fh; hg = 0; combining this with eq. (4.3) we get:
In the case when H is a conventional group the Poisson bracket corresponds to usual
(this also can be derived just from eqs. (4.1) and (4.2)).
From Cartan to base. If the action of H on
is free the Hequivariant cohomology is
isomorphic to the cohomology of
=H. An explicit formula for a base form belonging to
the same class of equivariant cohomology as
Ch can be written as follows ( [23,
equivariantcohomology/Direct Computation.html#(elem. .Def.Underline.Alpha)]). We need to choose
a connection
on
(the cohomology class of the resulting base form will not depend on the
choice of ). Then we have to replace
with the horizontal projection of , and substitute
the curvature f = d
2
1 2 for h (see [8] for a review):
base =
Z
L
exp SBV + (
( ) ) +
d
The proof of the fact that eq. (4.4) is equivariantly closed uses (3.6) and the relation [23,
omega/As Intertwiner.html]
where r 2 g and R stands for the corresponding Hamiltonian. This formula immediately
follows from:
the ber bundle
with our map
in eq. (4.8).
!
The second term
( ) is the horizontal projection of . The third term d
should be understood as follows. Consider the curvature d
2
1 2 of the connection in
=H; this is an Hequivariant hvalued 2form on . Composing it
we get a twoform with values in Fun(M ), which is denoted d
The considerations above are rigorous in
nitedimensional case, however, we will
use them in in nitedimensional case where they can be justi ed in the framework of
perturbation theory. Notice in the case when the dimension is in nite one should impose
some additional conditions. In particular, the quadratic part of the BV action functional
restricted to the Lagrangian submanifold should be nondegenerate. This condition
(nondegeneracy condition) is necessary to have well de ned perturbation theory. It is not
needed in
nitedimensional case when the integral has a de nition independent of the
perturbation theory and the integral of degenerate functional makes sense. The situation
with the completeness condition is similar: it is necessary only in in nitedimensional case.
The conditions of irreducibility and completeness are discussed in more details in [16].
The odd Laplacian is illde ned in the in nitedimensional case unless we are working in the framework of perturbation theory when we can apply the methods of [9] or [10]. { 7 {
However the equation
S = 0 does make sense; it just means that the nilpotent vector
eld Q corresponding to the
rst order di erential operator transforming a function f
into ff; Sg is volume preserving. (There exist standard ways to check that an operator
in in nitedimensional space is volume preserving; for example a method based on the
calculation of Seeley coe cients is explained in [11].) Replacing S by exp S~
we can write
the quantum master equation
e S~ = 0 as fS; Sg + ~ S = 0; in in nitedimensional case
we assume that both summands vanish: fS; Sg = 0 (classical classical master equation)
and
S = 0. Similarly, we assume that in (4.2)
= 0. In in nitedimensional case we
require that a quantum observable A satis es the equations
A = 0 and fA; Sg = 0.
Let us suppose that we have a functional S( ) with an odd symmetry QBRST (BRST
symmetry) that is nilpotent o shell (i.e. nilpotent without using the equations of motion).
Then we can construct an odd symplectic manifold adding anti elds
and solution to
the classical Master Equation given by the formula
SBV = S( ) + (QBRST i) i?
(5.1)
In the case when QBRST is volume preserving (divergencefree) SBV obeys also quantum
master equation
SBV = 0. This statement is rigorous in
nitedimensional situation; it
remains true also in the in nitedimensional case.
A special case of this construction comes from the \standard" BRST formalism. It
works for gauge theories as YangMills/QCD or ChernSimons, and also for the bosonic
string worldsheet theory [23, bosonicstring/index.html] and the RNS superstring [23,
HeteroticRNS/index.html].
One starts from the \classical action" Scl('), which is invariant with respect to group
H, hence with respect to its Lie algebra H with generators TA ("gauge symmetry"). Then
one introduces additional variables c
A (\the ghosts") with the quantum numbers of the
symmetry parameter, but opposite statistics.
The nilpotent symmetry Q is de ned by the following formulas [23,
BRSTformalism/Construction.html]:
QBRST'i = T AicA ; QBRSTcA =
12 fBAC c c
B C
(5.2)
where fBAC are structure constants of the Lie algebra H. To continue from BRST to
BV [23, BRSTformalism/BV from BRST.html], we de ne an odd symplectic manifold
adding to 'i; cA their anti elds 'i ; cA having opposite parity (geometrically this means
that we consider cotangent bundle with reversed parity of bers). Here 'i is the collective
notation for the \old elds". In such a situation, a solution of the classical Master Equation
(a special case of (5.1)) can be written in the form:
SBV = Scl(') +
Our goal will be to solve the eq. (4.2) for BV action functional (5.3). Notice that this
action functional is invariant with respect to the action of the group H and its Lie algebra
H; the hamiltonian of the element
=
ATA 2 H has the form h = T i A'i? + [ ; c]Ac? .
A A
There exists a solution mapping this Hamiltonian into
(h) =
Ac?A; it satis es the
conditions f (h); (h)g =
To check (4.2) it is su cient to notice that
f SBV; A ?
cA g = TA
i A'i? + [ ; c]Ac?
A
[ ; c]Ac? ; ~Ac?Ag = fBAC
A
B ~C c?A = [ ; ~]Ac? .
A
A solution of (4.2) should obey (4.3). To verify this condition we notice that fT Ai A'i? +
Physically meaningful quantities should be obtained from integrating eSBV over a
suitably chosen Lagrangian submanifold. However, a suitable choice of Lagrangian submanifold
is not completely trivial. The most straightforward guess would be to put all anti elds to
zero, i.e. '? = c
? = 0. But this is a wrong choice, because the restriction of SBV to
such a Lagrangian submanifold would be just Scl, a degenerate functional. The next guess
would be to consider some deformation of '? = c
? = 0, such that the restriction of SBV
be nonzero. But this also does not quite work, for the following reason: we want to keep
the ghost number symmetry. It turns out that in the class of Lagrangian submanifolds
respecting the ghost number symmetry, the one given by '? = c? = 0 is typically rigid. It
has no deformations, because there are no elds with negative ghost number.
In fact there are several solutions to this problem.
One solution is to consider a
family [23, BRSTformalism/Family of Lagrangian submanifolds.html] of Lagrangian
submanifolds. Each individual member of a family may be not ghostnumberinvariant, but we
require that the family as a whole be closed under the action of the ghost number
symmetry. Another solution is to add a eld/anti eld quartet, in other words nonminimal elds,
with some nonminimal elds carrying negative ghost number. (In
nitedimensional case
this gives the same answer [23, omega/As integration over single L.html] as considering
a family.) Yet another approach is to start with a Lagrangian submanifold [23,
BRSTformalism/Family of Lagrangian submanifolds.html] where some anti elds and some elds
are zero. Schematically:
'?1 = '2 = c1? = c2 = 0
(5.5)
In the particular case of bosonic string worldsheet theory in section 6, where H is the group
of di eomorphisms, we will use this third method.
Comment about anti elds. If
is a scalar eld, we will consider ? a density (i.e. a
volume form, or an area form in the twodimensional case). This is very natural:
(5.4)
The odd symplectic form is given by the integral of the density ( 1)
! = R ( 1)
^
?
^
?, i.e.
7!
A local in nitesimal eld rede nition
+ "V ( ) is generated by the odd
Hamiltonian R V ( ) ? (in order for this integral to make sense, ? should be a density).
In the same sense, we actually think of the \variational derivative"
as a density; it is
\generated by" ? in terms of odd Poisson bracket.
{ 9 {
Topological quantum
In BRST formalism a topological quantum
eld theory is de ned by a family of action
functionals depending on riemannian metric on some manifold X and satisfying the
condition that the variation of the action functional by in nitesimal variation of the metric
is BRST exact (topological quantum
eld theories of Witten type). In BV formalism we
should have solutions to the master equation fS; Sg = 0 depending on riemannian metric
and obeying dS = fb; Sg where d is the de Rham di erential on the space MET of all
metrics and b is a 1form on this space. (If V is a vector
eld on the space of metrics
we can write dS=dV = fb(V ); Sg.) Alternatively we can assume that the solution to the
master equation is xed, but the Lagrangian submanifold depends on the choice of metric.
We can construct an nform
n on MET integrating b(V1) : : : b(Vn)eS over some
Lagrangian submanifold L in the space of elds. Summing the forms
n we can get an
inhomogeneous closed form
that can be obtained by integrating eS+b over L. Under
certain conditions (see section 2) one can prove that this form is closed and descends to the
quotient space of MET with respect to the action of the group Di of di eomorphisms of
X. We obtain a closed form on the quotient MET=Di ; integrating this form over a cycle
we can get new invariants. In particular, applying these ideas to ChernSimons theory one
obtains invariants constructed by Kontsevich [12]; see [1] for detail. (Another construction
of these invariants was given in [13].)
In the rest of this section we will outline applications of these ideas to string
perturbation theory. The target of string theory can be regarded as twodimensional topological
quantum
eld theory; the above considerations can be applied to this TQFT. We will show
that string amplitudes are particular cases of new invariants we have mentioned. Instead
of formalism of families of equivalent action functionals we will use more exible formalism
of families of Lagrangian submanifolds.
Bosonic string. Master action in terms of world sheet metric.
The construction
outlined in section 5 works for both bosonic string [23, bosonicstring/index.html] and RNS
superstring [23, HeteroticRNS/index.html].
Let us consider bosonic string. For de niteness we are writing formulas for bosonic
string in at space. (To avoid anomalies we should assume that we work in the dimension
26.) We start with the action functional
Smat[g; x] =
2
1 Z pgg
(6.1)
We integrate here over a compact surface of genus h with metric g . We always assume
that h > 1. The subindex mat stands for \matter", although this action also involves
the dynamical metric g . This functional is invariant with respect to di eomorphisms
and Weyl transformations g0
= e g ; hence we can construct a BV action functional
introducing di eomorphism ghosts c, Weyl ghosts
and anti elds to g , xm and ghosts.3
3BV formalism was previously applied to bosonic string in [14].
HJEP07(21)63
SBV = Smat[g; x] +
+
Z
Following the general scheme outlined in section 5 we obtain:
We now choose the Lagrangian submanifold in the following way:
g
= g(0) ; x? = c? = ? = 0
where g(0) is a xed metric
The family (6.3) of Lagrangian submanifolds is closed under the action of di eomorphisms.
On Lagrangian submanifold (6.3) the action is quadratic and the form
is equal to:4
(g(0); g(0)) =
[Dx Dg? Dc D ] exp SBV +
[Dx Dg? Dc D ] exp Smat +
[Dx Db Dc ] exp Smat +
[Dx Dg? Dc D ]
d2z(L g )g?
[Dx Dg? Dc D ]
d2z
exp SBV +
g(0) g?
= 0
exp SBV +
Z
Z
g(0) g?
=
(Lcg) g
?
+ g g
?
+ ((c @ )xm)x?m + [c; c] c? + (Lc ) ?
1
2
(6.2)
(6.3)
(6.4)
(6.5)
(6.6)
(6.7)
(6.8)
(6.9)
HJEP07(21)63
We introduced the notation t = g g
? ; b
= traceless part of g? ; i.e. g?
= b
+
12 tg . In the transition to the last line we integrated over
and t.
Nondegeneracy. The exponential in (6.5) is nondegenerate.
(The restriction
of SBV to the Lagrangian submanifold of eq. (6.3) is nondegenerate [23,
BRSTformalism/Family of Lagrangian submanifolds.html#%28part. .Nondegeneracy%29]
modulo a
nitedimensional space of zero modes of b .
This nitedimensional degeneracy is removed by the second term in the exponential of (6.5).)
Symmetries. The form
is invariant with respect to di eomorphisms; moreover on the
family (6.3) it is a base form, because for any worldsheet vector eld :
To check that the last line is zero we notice that the derivative with respect to c under
the sign of twodimensional integral can be replaced be variational derivative under the
sign of in nitedimensional integral.
Let us study the behavior of this form with respect to Weyl transformations g0
=
e g . The 0th component
0 of inhomogeneous form
can be regarded as a partition
function of conformal eld theory. The variation of partition function by in nitesimal Weyl
4We denote the de Rham di erential on the in nitedimensional space of metrics by instead of d.
transformation is governed by trace anomaly Z=
= ( c1R2 + const)Z where c stands for
the central charge and R denotes the curvature of the worldsheet. In our case the central
charge vanishes (we are working in critical dimension d = 26; in general the central charge
is equal to d
26). We see that
0 does not change by Weyl transformations. The
kth component of the form
can be expressed in terms of correlation functions of the
same conformal theory. The behavior of correlation functions by Weyl transformations is
governed by conformal dimensions
i of elds i
([22], formula (13,50)). To check the Weyl invariance of
we notice that the dimension of
b
is 2 (it coincides with conformal dimension) and the dimension of g
is
is Weyl invariant. Moreover, it descends not
only to MET=Di , but also to MET=Di o Weyl; that can be identi ed with the moduli
space of complex structures on a compact surface of genus h. (A formal proof of the fact
that
is a base form for the Weyl group repeats the proof of similar statement for Di .)
We can get the partition function of bosonic string integrating the form over this moduli
space. (Notice that we are working with inhomogeneous forms, but the integration singles
out one component of this form.)
We
can
solve
eq.
(4.2)
using
the
general
considerations
[23,
omega/Case Of Standard BRST.html] of section 5.
Namely, we should take a map
sending a worldsheet vector eld
(z; z) plus in nitesimal Weyl transformation '(z; z)
to:
( ; ') =
Z
c? + ' ?
Then the functional fS; ( ; ')g can be considered as a Hamiltonian of in nitesimal
transformation of elds corresponding to the vector eld
and Weyl factor '. This means
eq. (6.11) de nes a solution of eq. (4.2) for the Lie algebra of the group Di o Weyl acting
on the space of elds. This allows us to construct an equivariant form
Z
gL
C
Lie(Di oWeyl)( ; ') =
exp SBV +
+
c? + ' ?
Z
(6.11)
(6.12)
We can then construct the corresponding base form [23, omega/Base Form.html] which
descends to =(Di o Weyl). On the standard family of Lagrangian submanifolds given by
eq. (6.3) c? = ? = 0. Therefore
C
Lie(Di oWeyl)( ; ') becomes essentially
of eq. (6.5).
Singular metrics.
Notice that in the action functional (6.1) we can allow slightly
singular metrics. We say that the worldsheet metric on a surface of genus h is slightly singular
if on some real curves one of the eigenvalues of the metric g
vanishes and another
eigenvalue remains positive. More precisely we suppose that g = det g
vanishes on a family
of closed real curves and in the neighborhood of one of these curves it takes the form
where
= 0 is the equation of the curve and
is a positive function.5 It is easy to check
2
5The simplest example of this picture is a cylinder with coordinates ( ; ) and metric ds2 = d 2 + 2d 2.
Here a <
< a; 0
that under these conditions the action functional (6.1) is nite if we make an additional
assumption that xm is constant on every closed curve where the metric is singular. The
formulas for BV action (6.2) and Lagrangian submanifold (6.3) can be applied to slightly
singular metrics. We obtain a family of Lagrangian manifolds labelled by these metrics.
Factorizing the topological space
of slightly singular metrics with respect to di
eomorphisms and Weyl transformations we obtain the space
=Di o Weyl. Points of this space
can be identi ed with complex curves having simplest singularities (nodes). (Every closed
curve where the metric is singular should be contracted to a point; the metric speci es a
complex structure in the complement to these points.) A part of this space that consists of
stable curves (curves having only
nite number of automorphisms) can be identi ed with
DeligneMumford compacti cation of the moduli space of algebraic curves of genus h. This
is a good topological space (an orbifold). The remaining part is a \bad" (nonseparable)
space, but it does not play any role (a heuristic explanation of this fact is the remark
that its contribution to the partition function is suppressed by the in nite volume of the
automorphism group). The form of eq. (6.5) descends to DeligneMumford space as a form
having some singularities. To obtain physical quantities we should integrate the form over
a cycle in DeligneMumford space; to obtain the partition function we should integrate over
the fundamental cycle. (Of course, this is only a formal calculationdue to the tachyon in
the spectrum of bosonic string the integral is divergent.)
Master equation in terms of complex structures.
A worldsheet complex structure can be speci ed by a eld of linear operators I acting on tangent spaces and obeying
I2 =
1. Another way to specify a complex structure is to x a complex vector eld e such
that the complex conjugate vector eld e together with e speci es a basis of complexi ed
tangent space. (To relate these descriptions we de ne e as the eigenvector of I having
eigenvalue i.) Notice that e is only de ned up to multiplication: e
ue, where u is a
Due to Weyl invariance one can express the functional (6.1) in terms of complex
struccomplex function on the worldsheet.
tures. We obtain the following functional:
Z
Smat[I; x] =
(6.13)
where the measure
on the worldsheet is speci ed by the condition the vectors e; e span
a parallelogram of measure 1 in tangent space. The functional is invariant with respect to
di eomorphisms. We can now follow the standard procedure by rst introducing the di
eomorphism ghosts [23, BRSTformalism/Construction.html] c (BRST formalism) and then
adding anti elds [23, BRSTformalism/BV from BRST.html]. The result is the Master
Action of the following form:
Z
SBV = Smat[I; x] +
(LcI) I
? + (Lcx)x? + [c; c] c?
(6.14)
1
2
In the expression for the action we integrate over a worldsheet. In the hloop contribution
the worldsheet is a surface of genus h.
Notice that one can introduce a notion of slightly degenerate complex structure
assuming that the vectors e and e can be linearly dependent on a family of closed curves on a
worldsheet. (In a neighborhood of such a curve we should have a relation e =
e + f + : : :
where tangent vectors e and f are linearly independent,
= 0 is the equation of the curve
and : : : are higher order terms with respect to .)
String amplitudes for critical string
To represent the string theory in BV form we have applied the general constructions of the
HJEP07(21)63
section 5 to the action functional Smat[g; x]. This functional depends on the metric g
on
the worldsheet (on a compact surface of genus h) and a map x( ) = xm( ) of this surface to
Rd. This functional is invariant with respect to di eomorphisms and Weyl transformations.
We applied the standard BRST construction in this setting and used (5.1) to get the BV
action. To describe string amplitudes we should add marked points (punctures) ( 1; : : : ; n)
on the worldsheet to this picture. Following [14] we will consider i as dynamical variables
on equal footing with the metric.
extra term c ( i) i? :
Using again the constructions of the section 5 we get the new BV action SB0V with an
SB0V = SBV + c ( i) i?
where SBV is de ned by (6.14). As was noticed in section 5 this functional obeys quantum
master equation in the case when the volume is Qinvariant; this remark forces us to use
the di eomorphism invariant measure pg( 1)d2 1
pg( n)d2 n on the space of marked
points.
Let us consider functionals Vi( i) (vertices) which are invariant under di eomorphisms.
The typical examples of such vertices are tachyoinic vertex eipx( ) and graviton vertex
klg
SB00V = SB0V + X iVi( i)
where i are in nitesimally small.
To de ne string amplitudes it is convenient to work with BVaction functional that is
obtained from (7.2) by means of \integrating out" Weyl ghosts.6 We obtain the new BV
action S~BV given by the formula
eS~BV = eSmat[g;x] +R (Lcg) g? +((c @ )xm)x?m+ 21 [c;c] c? c ( i) i?+P iVi( i)
(g?
g )
Denoting the traceless part of g?
by b
we can represent this action functional in the form
S~BV = S^BV + X iVi( i)
6If a solution A of the equation
A = 0 is de ned on direct product of two odd symplectic manifolds
Y0 and Y00 we can obtain a solution of similar equation on Y0 integrating over Lagrangian submanifolds
L 2 Y00. (See for example [10].) In our case we integrate over Lagrangian submanifold
= 0 of manifold
with coordinates ; .
(7.1)
(7.2)
(7.3)
where
taking
S^BV = Smat[g; x] +
Z
(Lcg) b
c ( i) i?
(7.4)
1
2
Now we can use the standard construction of the form
starting with the action functional
S~BV. However, we prefer to construct the form
starting with the functional S^BV and
including the factor V1 : : : Vn into de ning integral. (The form coming from the second
construction can be obtained from the rst one by means of di erentiation with respect to
parameters.) We consider a family of Lagrangian submanifolds parameterized by g(0) ; i(0)
HJEP07(21)63
g
= g(0) ; i = i(0) ; x? = c? = 0
The form , restricted to one of these Lagrangian submanifolds looks as follows:
(g(0) ; i(0) g(0) ; d i(0) ) =
=
Z
[Dx Db D
q
i Dc] g( 1(0))V1( 1(0))
q
g( n(0))Vn( n(0))
Z
exp Smat +
Using this formula we can get an expression of
in terms of correlation functions of
conformal eld theory. This allows us to analyze the behavior of
with respect to Weyl
transformations. It is easy to see that in our case of critical string this form is Weyl invariant
if conformal elds corresponding to vertices Vi have conformal dimension 2 (dimension
(1; 1) in the language of complex geometry). In this case the form descends to the moduli
space Mh;n of compact complex curves of genus h with n marked points and to its
DeligneMumford compacti cation Mh;n. Integrating over the fundamental cycle of Mh;n we obtain
the hloop contribution to string amplitudes. To check this we notice that after integration
over d ? (and omitting indices (0) for brevity) we get:
Z
[Dx Db Dc] j
g( j )( d j1 + c1( j ))( d j2 + c2( j ))Vj ( j )
Z
q
exp Smat +
This result is equivalent to the standard expression for the string amplitude [15]. To see
this we notice that j (d j1 +c1( j ))(d j2 +c2( j )) consist on 2n summands; one of them gives
the standard expression for string amplitudes with nonintegrated vertices, another gives
the standard expression with integrated vertices, and the rest correspond to the situation
when some vertices are integrated and some are nonintegrated. All these summands are
equal, hence we obtain the standard answer up to a factor 2n.
Another way to calculate the string amplitudes is to work with in nitesimal
deformations of BV action functional. Such deformations can be identi ed with (classical or
(7.5)
(7.6)
(7.7)
(7.8)
(7.9)
quantum) observables. In string theory they can be considered as integrated vertices.
Applying our approach to the deformation of BV action we obtain the standard expression of
string amplitudes in terms of integrated vertices (see [16] for detail).
An important method of calculation of scattering amplitudes in string theory is based
on the consideration of o shell string amplitudes.
This is the best method to calcu
late amplitudes when the mass gets quantum corrections.The o shell amplitudes should
be de ned in such a way that the particle masses correspond to their poles (in
momentum representation) and scattering amplitudes should be expressed in terms of residues in
these poles.
To de ne o shell string amplitudes for critical string one can consider surfaces with
marked points and local coordinate systems in the neighborhoods of these points [17], [18].
This is equivalent to consideration of surfaces with boundary. The BV formalism on
manifolds with boundary was analyzed in [10]. It should be possible to combine our approach [23,
boundary/index.html] with BVBFV formalism of [10]; these would lead to generalization
of de nitions given in [17, 18].
For noncritical strings very nice de nition of o shell amplitudes was suggested by A.
Polyakov [19]; it works well in our setting. Polyakov considers maps x( ) = xm( ) of a
surface with marked points 1; : : : ; k into Rd and includes the factor
i
Z
(xi
x( i))pg( i)d2 i
in the functional integral that de nes the partition function. Geometrically this means
that we integrate over all surfaces in Rd that contain the points x1; : : : ; xk 2 Rd(surfaces
with pinned points fxig in Polyakov's terminology). Doing the functional integral we
obtain a function G(x1; : : : ; xk) that can be interpreted as o shell amplitude in coordinate
representation. The o  shell amplitude in the momentum representation G(p1; : : : ; pk)
can be de ned as Fourier transform of G(x1; : : : ; xk) or directly as a functional integral for
partition function with insertion
j
Z
eipjx( j)q
g( j )d2 j
(7.10)
(7.11)
Polyakov considers o shell amplitudes only at tree level (genus zero surfaces), however
they can be considered also in multiloop case.
8
Pure spinor superstring
We hope that our ideas will lead to better understanding of pure spinor formalism in
superstring theory and to simpli ed expressions for amplitudes in this formalism.
The worldsheet sigmamodel of the pure spinor sigmamodel has di erent versions
which are quasiisomorphic to each other, as usual in the topological eld theory. There is
a \minimal version", which (in case of Type II theory7) describes matter elds (x; L; R)
and \ghost elds" L; R constrained to live on the pure spinor cone:
( L
m
L) = ( R
m
R) = 0
(8.1)
7For the heterotic string the rightmoving variables are those of the heterotic RNS formalism.
The at space sigmamodel [24] requires introduction of the momenta pL+ and pR conjugate
to L and R, and the fermionic part of the action is of the rst order in derivatives:
The action for pure spinors is, schematically:
Z
Z
where the \conjugate momenta" w+L; wR take values in the cotangent bundle of the pure
HJEP07(21)63
The model is invariant under a fermionic nilpotent symmetry Q. Importantly, it splits
(for Type II case) into the sum of left and right symmetries:
Q = QL + QR
such that the conserved currents corresponding to QL and QR are holomorphic and
antiholomorphic, respectively.
In the case of
at target space, it is easy to obtain the corresponding BV action
functional: for every eld
one should add its anti eld
and a term in the action having
the form (Q )
see (5.1).)
pure spinor string is:
However, the solution of eq. (4.2) requires di erent methods. As a rst step, let us
restrict ourselves to the left sector.9 The explicit form of eq. (4.2) for the left sector of the
. (This is a special case of general construction described in Sec 5;
(8.2)
(8.3)
(8.4)
(8.5)
(8.6)
1
2
fSBV ; a( )g +
fa( ); a( )g = Hh i
+ ( z@z L) ?L + (L w+)w?+
 this has to be solved for the unknown a( ); notice that Hh i is linear in , but a( ) does
not have to be linear in . (We have assumed that
a( ) = 0; otherwise we should add
an illde ned term
a( ).) One solution can be obtained as follows. Since the worldsheet
theory is conformal, a holomorphic vector eld + is symmetry; it is generated by +T++. It
was shown by Berkovits that the energymomentum tensor is BRSTtrivial: T++ = QLb++
(even o shell) where b++ is a composite bghost. This means that one should expect
that the worldsheet action can be included into topological conformal eld theory. A
rigorous proof of this statement is still unknown; the most convincing treatment of this
problem was given in [20].10 Notice that +b++ is a holomorphic current and therefore also
8One way of describing the pure spinor system is to cover the cone with patches. On each patch, both
L and wL take values in
at 11dimensional space.
9Applying this to the \full" sigmamodel, i.e. left plus right sector, is work in progress in collaboration
with R. Lipinski Jusinskas.
10That paper contains also the calculation of superstring amplitudes in the framework of BVformalism;
some ideas of this calculation can be used in our approach.
corresponds to some symmetry. We can identify a( ) =
h i, a BV Hamiltonian generating
the in nitesimal action of that symmetry. Then the second term in (8.5) vanishes and
this equation is satis ed. However we hope that there exist simpler solutions of eq. (8.5)
with nonvanishing second term; we leave this question for future work. We believe, that
applying the techniques described above one can not only justify the pure spinor formalism,
but also simplify the formulas (hopefully we can avoid using the complicated and not very
well de ned \composite bghost").
A
Some useful formulas
BV phase space [23, BVformalism/index.html] is an odd symplectic supermanifold M with
a nondegenerate closed odd 2form !. For any F 2 Fun(M ) we can de ne its Hamiltonian
vector eld. We will think of this vector eld as a rst order linear di erential operator,
acting on Fun(M ):
HJEP07(21)63
and denote this operator fF; g. (Here
nition:
AB(Z) is a matrix inverse to !AB(Z).) By de
where is the operator of contraction, satisfying [ V ; d] = LV . This implies:
In coordinates:
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
G 7! fF; Gg = F
AB(Z)
dF = ( )F +1
fF; g!
fF; Gg = fF; gdG = ( )G+1
fF; g fG; g!
! = dZAdZB!AB
!AB = ( 1)(A+1)(B+1)!BA
V = V A
F = divfF; g
Locally it is possible to choose the Darboux coordinates:
0
If the manifold M is equipped with a volume element (with a density) we can de ne
the odd Laplacian acing on functions by the formula
In Darboux coordinates
is:
where div stands for the divergence of vector eld with respect to the volume element.
The volume element should be chosen in such a way that
2 = 0. The relation between
odd Laplace operator [23, BVformalism/OddLaplace.html] and f ; g is:
(XY ) =( X)Y + ( )
X
Y + ( )X fX; Y g
e =
+
f ; g e
1
2
One can prove that
X given by this formula does not depend on the choice of Darboux
coordinates if X transforms as a semidensity (recall that semidensities transform as square
roots of densities= volume elements). Hence for any odd symplectic manifold one can de ne
on semidensities (volume element is not necessary), see [6].
B
De nition of
using marked points
Let LAG+ denote the space of Lagrangian submanifolds with marked points. A point of
LAG+ is a pair (L; a) where L 2 LAG and a 2 L. This de nes the double bration:
M
p
LAG+
! LAG
Given v 2
T(L;a)LAG+, we can consider two projections
v 2
TLLAG and p v 2
TaM . We will de ne
is a pseudodi erential form, i.e. a function of L; a; v. It will
depend on v only through
v. We can characterize
v as a section of
T M jL modulo
T L. We then de ne
as follows:
2 Fun(L)
d
=
(a) = 0
(
v!)jL
This de nition speci es
as a linear function of v, i.e. as a oneform on LAG+ In order
to make sense of
v! we must think of
v as a section of
T M ; the fact that it is only de ned up to tangent to T L does not matter because L is isotropic. Eq. (B.3) eliminates the ambiguity, and we can now safely de ne a function on
T LAG+ (a pseudodi erential form on LAG+) as in eq. (3.3):
More generally, for every function F on M we de ne:
(L; a; v) =
eSBV+
hF i(L; a; v) =
We will now prove the following formula:
(d
p !) hF i =
h F + fSBV ; F gi
(A.12)
(A.13)
(A.14)
As a straightforward generalization, we can consider a product of
with the
pullback under p of any di erential or pseudodi erential form
on M . It satis es: [23,
omega/Descent To LAGs.html#(part. .Upgrade to )]
d (p
hF i) = ( )j j+1p
h F + fSBV ; F gi + p (d + ! )
hF i
(B.7)
Notice the appearance of the nilpotent operator d + ! which was studied in [21].
Proof.
We take a family of Lagrangian submanifolds with marked points (L( ); a( ))
and represent it in the form
where g( ) are volume preserving canonical transformations (locally this is always possible).
It is su cient to analyze the restriction
hF i( ; d ) of the form (B.5) to this family.
As in section 3 using the canonical transformations g( ) we can construct a family
of action functionals S
These forms do not coincide with the forms
and corresponding forms that will be denoted by ~ and ~ hF i.
hF i( ; d ) constructed by means of family of
Lagrangian submanifolds with marked points, but they are closely related. As we noticed
in section 3 the second summand in the exponential in the formula de ning ~ hF i( ; d ) is
the Hamiltonian of the in nitesimal canonical transformation governing the variation of S .
The second summand in the formula de ning
hF i( ; d ) is the Hamiltonian H( ; d ) of
the in nitesimal canonical transformation11 governing the variation of L . They coincide
up to a constant summand. This constant can be calculated from (B.2). We obtain [23,
omega/Descent To LAGs.html#(part. .Upgrade to )]
hF i( ; d ) = C ~ hF i( ; d )
where C = e H( ;d )(g( )a0). (One can say that C is expressed in terms of the value of the
Hamiltonian of the in nitesimal canonical transformation at the marked point.)
We have calculated already the di erential of ~ hF i( ; d ). But we also have to evaluate
d of the prefactor C. Using eq. (A.3), appendix, and p ! = 12
! we get:
1
2
1
2
1
2
1
2
d e H( ;d )(ga0) =
= e H( ;d )(ga0) ( (d H( ; d ))(ga0)
fH( ; d ) ; H( ; d )g(ga0)) =
=
e H( ;d )(ga0)
fH( ; d ) ; H( ; d )g(ga0) =
= e H( ;d )(ga0)(( fH( ;d ) ; g)2!)(ga0) =
= e H( ;d )(ga0)(
{ 20 {
(B.10)
(B.11)
(B.12)
(B.13)
(B.14)
(B.15)
(B.16)
+ = (p e
)
Z
L
e
C+ = (p )
eS+ + (h)
2 Fun (( T M )
h)
d + !
fh; g + h
= 0
d
fh; g
C+ = 0
fh; g
= h
h(a)
Proof of
C+ being equivariantly closed.
We have to prove that:
where d is the de Rham di erential on LAG+. The action of d is given by eq. (B.7). The
is essentially as in eq. (4.6), but we have to remember to subtract the
action of fh; g
on
The vanishing of (d
computation.
compensating constant to make sure that
vanishes at the marked point; therefore:
fh; g) C+ when eqs. (4.2) and (B.20) are satis ed follows from direct
This concludes the proof. follows:
Given a \symplectic potential" satisfying d = ! we can construct a closed form as We will choose the following ansatz for the equivariantly closed analogue of :
The expression de ned in eq. (B.18) is a cocycle of the Cartan complex of equivariant
cohomology of LAG+ if in addition to (4.2) we have
Even though
lives in the same space as cochains of the Cartan complex, the di
erential de ned by eq. (B.20) is di erent. (The Cartan di erential would be d
fh; g.)
Comment. In particular, when we can choose an Hinvariant \symplectic potential"
such that d
= !, eq. (B.20) has a simple solution:
where
is of the same formal type as a Cartan cochain:
Central extension of the group of canonical transformations
In this section we will give a precise de nition of
using a wellde ned closed PDF b on
a central extension Gb of the group of canonical transformations.12 This group is in
nitedimensional, however, in this section we will keep the notation d for the de Rham di erential
on the group and on the space of Lagrangian submanifolds LAG.
12The existence of a central extension of the group of canonical transformations (symplectomorphisms) of
odd symplectic manifold M can be proven in the same way as for an even symplectic manifold. Namely, as in
the even case one constructs a bundle with connection over M , the ber of this bundle is a onedimensional
odd vector space. The group Gb can be de ned as a group of transformations of the total space of the bundle
that are compatible with the
bration (transform
bers into
bers), induce canonical transformation on
the base and are compatible with connection.
Z
L
= e
(B.17)
(B.18)
(B.19)
(B.20)
(B.21)
(B.22)
(B.23)
the fact that the Lie algebra of Gb is
equation:
Here following [6] we consider exp(SBV) as a semidensity, dgg 1 is the rightinvariant form
on Gb taking values in the Lie algebra (MaurerCartan form),and g stands for an element
of G corresponding to gb 2 Gb. In eq. (C.3) we consider dgbgb 1 as a function on M , using
bb
Fun(M ). This form satis es the MaurerCartan
d(dgbgb 1) + 2 fdgbgb 1 ; dgbgb 1g = 0
This b is closed as a PDF on Gb, i.e.:
b 2 Fun(LAG
T Gb)
b(L; gb; dgb) =
exp SBV + dgbgb 1
Z
gL
1
db = 0
The proof of eq. (C.5) is a straightforward computation [23, omega/De nition.html] very
similar to the computations in section 2.
(C.1)
(C.2)
(C.3)
(C.4)
(C.5)
(C.6)
(C.7)
(C.8)
De nition of b
Let us consider the Lie superalgebra
Fun(M ) with the commutator given by the odd
Poisson bracket. It is a central extension of the Lie superalgebra of Hamiltonian vector
elds which we denote g; therefore we denote it gb:
gb =
Fun(M )
We consider the central extension of the group of canonical transformations Gb, whose Lie
algebra is gb.
closed PDFs on Gb, which we will call b:
As a variation on our theme, we will now construct a map from LAG to the space of
We must stress that this b is wellde ned (does not contain any ambiguities).
C.2
How to build a form on LAG starting from b
Since G (and therefore Gb) acts on LAG, there is a natural projection:
b : LAG
T Gb !
T LAG
restriction of on L0 is a constant c. Using gbbgb 1 = b g 1 we get:
However, it is not true that b is constant along the bers of b. Indeed, for a
where St(L0) stands for the stable subgroup of L0 2 LAG in Gb one can check that the
2 Lie(St(L0)),
b(L0; gb; dgb + gbb) = k b(L0; gb; dgb)
where k is some number. Therefore b does not automatically provide a PDF on LAG.
would guarantee that k = 0.
g( ) 2 G;
a closed form
2
We could impose some additional restrictions, such as ghost number symmetry,13 which
Let us suppose now that a subset of LAG is represented in the form g( )L0 where
. Assume that we can nd a \lift" g( ) of g( ) to Gb. Then we can de ne
(L; dL) = b(L0; gb( ); d(gb( )))
(C.9)
This coincides with the \tentative" de nition of section 3, because the restriction of
dg^g^ 1 to gL0 gives . This is a general fact, true both in classical mechanics and in BV
formalism. In classical mechanics it is essentially the HamiltonJacobi equation, which
describes the evolution of a Lagrangian submanifold (speci ed by a generating function
cohomology class.
Acknowledgments
usually called S) under the Hamiltonian
on L plus a constant (which can depend on t).
Notice that by the variation of gb( ) the form
(L; dL) obviously remains in the same
We are grateful to Nathan Berkovits, Alberto Cattaneo, Alexei Kotov, Misha Movshev,
John Murray, Pavel Mnev, Sasha Polyakov and Kostas Skenderis for useful discussions.
The work of A.M. was partially supported by the FAPESP grant 2014/186349 \Dualidade
Gravitac; a~o/Teoria de Gauge", and in part by the RFBR grant 150199504 \String theory
and integrable systems".
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
Phys. 158 (1993) 373 [hepth/9210115] [INSPIRE].
BatalinVilkovisky Geometry, math/9909117.
[1] A. Schwarz, Topological quantum
eld theories, Proceedings of 13th International Congress in
Mathematical Physics ICMP, London U.K. (2000).
[2] A.S. Schwarz, Supergravity, complex geometry and Gstructures, Commun. Math. Phys. 87
[6] O.M. Khudaverdian, DeltaOperator on Semidensities and Integral Invariants in the
have ghost number
13We do not require that the semidensity be invariant under the ghost number symmetry; just that dgbgb 1
31 (1994) 299 [hepth/9310124] [INSPIRE].
eld theory, American Mathematical Society, Providence U.S.A. (2011).
manifolds with boundary, arXiv:1507.01221 [INSPIRE].
HJEP07(21)63
(1979) 1 [INSPIRE].
001 [hepth/0503038] [INSPIRE].
Press, Boca Raton U.S.A. (1987).
[3] A.S. Shvarts , On the de nition of superspace, Theor. Math. Phys. 60 ( 1984 ) 657 [Teor . Mat. [4] A.S. Schwarz , Geometry of BatalinVilkovisky quantization , Commun. Math. Phys . 155 [5] A.S. Schwarz , Semiclassical approximation in BatalinVilkovisky formalism , Commun. Math. [17] P.C. Nelson , Covariant insertion of general vertex operators , Phys. Rev. Lett . 62 ( 1989 ) 993 [18] A. Sen , O shell Amplitudes in Superstring Theory, Fortsch. Phys . 63 ( 2015 ) 149 [19] A.M. Polyakov , Contemporary Concepts in Physics . Vol. 3 : Gauge Fields and Strings, CRC [7] A.S. Schwarz , Symmetry transformations in BatalinVilkovisky formalism , Lett. Math. Phys. [8] E. Meinrenken , Equivariant cohomology and the cartan model , [10] A.S. Cattaneo , P. Mnev and N. Reshetikhin , Perturbative quantum gauge theories on [11] A.S. Schwarz , The Partition Function of a Degenerate Functional , Commun. Math. Phys. 67 [14] B. Craps and K. Skenderis , Comments on BRST quantization of strings , JHEP 05 ( 2005 ) [15] J. Polchinski , String Theory, Cambridge University Press, Cambridge U.K. ( 2005 ).