Odd Scalar Curvature in Field-Antifield Formalism

0.23 MB·English

by  Igor A. Batalin

Odd Scalar Curvature in Field-Antifield Formalism
Igor A. Batalinab and Klaus Beringac
aThe Niels Bohr Institute
Blegdamsvej 17
DK-2100 Copenhagen
Denmark
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bI.E. Tamm Theory Division
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P.N. Lebedev Physics Institute
n
a Russian Academy of Sciences
J 53 Leninisky Prospect

4
Moscow 119991
2
Russia

]
h
t cInstitute for Theoretical Physics & Astrophysics
-
p Masaryk University
e
Kotl´aˇrsk´a 2
h
[ CZ-611 37 Brno

Czech Republic
6
v
0 February 1, 2008
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Abstract
7
0
WeconsiderthepossibilityofaddingaGrassmann-oddfunctionν totheoddLaplacian. Requir-
:
v ingthetotal∆operatortobenilpotentleadstoadifferentialconditionforν,whichisintegrable. It
i
X turnsoutthattheoddfunctionν isnotanindependentgeometricobject,butisinsteadcompletely
specified by the antisymplectic structureE andthe density ρ. The main impactofintroducing the
r
a ν term is that it makes compatibility relations between E and ρ obsolete. We give a geometric
interpretation of ν as (minus 1/8 times) the odd scalar curvature of an arbitrary antisymplectic,
torsion-freeandρ-compatibleconnection. Weshowthatthetotal∆operatorisaρ-dressedversion
of Khudaverdian’s ∆E operator, which takes semidensities to semidensities. We also show that
the construction generalizes to the situation where ρ is replaced by a non-flat line bundle connec-
tion F. This generalization is implemented by breaking the nilpotency of ∆ with an arbitrary
Grassmann-even second-order operator source.
PACS number(s): 02.40.-k; 03.65.Ca; 04.60.Gw; 11.10.-z; 11.10.Ef; 11.15.Bt.
Keywords: BVField-AntifieldFormalism;OddLaplacian;AntisymplecticGeometry;Semi-
density; Antisymplectic Connection; Odd Scalar Curvature.
bE-mail: [email protected] cE-mail: [email protected]
1
1 Introduction
Conventionally [1, 2, 3, 4] the geometric arena for quantization of Lagrangian theories in the field-
antifield formalism [5,6,7]istaken tobean antisymplectic manifold (M;E) withameasuredensityρ.
Each point in themanifold M with local coordinates ΓA and Grassmann parity ε ε(ΓA) represents
A ≡
a field-antifield configuration ΓA= φα;φ , the antisymplectic structure E provides the antibracket
∗α
{ }
(, ), and the density ρ yields the path integral measure. However, up until recently, it has been
· ·
necessary to impose a compatibility condition [2, 8] between the two geometric structures E and ρ to
ensure nilpotency of the odd Laplacian
∆ (−1)εA∂→l ρEAB∂→l , ∂→l →∂l . (1.1)
ρ ≡ 2ρ A B A ≡ ∂ΓA
In this paper, we show that the compatibility condition between E and ρ can be omitted if one adds
an odd scalar function ν to the odd Laplacian ∆ ,
ρ
∆ = ∆ +ν (1.2)
ρ
such that the total ∆ operator is nilpotent
∆2 = 0 . (1.3)
Nilpotency is important for the field-antifield formalism in many ways, for instance in securing that
the physical partition function is independent of gauge-choice, see Appendix A. (More precisely,
Z
whatisreally vitalis thenilpotencyoftheunderlying∆ operator, cf.Sections 8-9.) Inphysicsterms,
E
the addition of the ν function to the odd Laplacian ∆ implies that the quantum master equation
ρ
∆eh¯iW = 0 (1.4)
is modified with a ν term at the two-loop order (h¯2):
O
1
(W,W) = i¯h∆ W +h¯2ν , (1.5)
2 ρ
and∆ isingeneralnolongeranilpotentoperator. Itturnsoutthatthezeroth-orderν termisuniquely
ρ
determinedfromthenilpotencyrequirement(1.3)apartfromanoddconstant. Oneparticularsolution
to the zeroth-order term, which we call ν , takes a special form [9]
ρ
ν(1) ν(2)
ν ν(0)+ , (1.6)
ρ ≡ ρ 8 − 24
where ν(0), ν(1) and ν(2) are defined as
ρ
1
ν(0) (∆ √ρ) , (1.7)
ρ ≡ √ρ 1
ν(1) ( 1)εA(∂→Bl ∂→Al EAB) , (1.8)
≡ −
ν(2) ( 1)εAεC(∂→Dl EAB)EBC(∂→Al ECD) (1.9)
≡ −
= ( 1)εB(∂→Al EBC)ECD(∂→Dl EBA) . (1.10)
− −
Here, ∆ in eq. (1.7) denotes the expression (1.1) for the odd Laplacian ∆ with ρ replaced by 1. In
1 ρ=1
particular, theoddscalar ν is a function of E and ρ, so thereis no call for new independentgeometric
ρ
2
structures on the manifold M. In Sections 2–6 we show that ∆ +ν is the only possible ∆ operator
ρ
within the set of all second-order differential operators. The now obsolete compatibility condition
[2, 8] between E and ρ can be recast as ν = odd constant, thereby making contact to the previous
ρ
approach [2], which uses the odd Laplacian ∆ only. The explicit formula (1.6) for ν is proven in
ρ ρ
Section 7 and Appendix B. The formula (1.6) first appeared in Ref. [9]. That paper was devoted to
Khudaverdian’s ∆ operator [10, 11, 12, 13], which takes semidensities to semidensities. This is no
E
coincidence: At the bare level of mathematical formulas the construction is intimately related to the
∆ operator, as shown in Sections 8-9. However the starting point is different. On one hand, Ref. [9]
E
studied the ∆ operator in its minimal and purestsetting, which is a manifold with an antisymplectic
E
structure E but without a density ρ. On the other hand, the starting point of the current paper is a
∆ operator that takes scalar functions to scalar functions, and this implies that a choice of ρ (or F,
cf. below) should be made. Later in Sections 10 and 11 we interpret the odd ν function as (minus
ρ
1/8 times) the odd scalar curvature R of an arbitrary antisymplectic, torsion-free and ρ-compatible
connection,
R
ν = . (1.11)
ρ − 8
One of the main priorities for the current article is to ensure that all arguments are handled in com-
pletely general coordinates without resorting to Darboux coordinates at any stage. This is important
to give a physical theory a natural, coordinate-independent, geometric status in the antisymplectic
phasespace. We shall also throughoutthepaper often addressthe question of generalizing thedensity
ρ to a non-flat line bundle connection F. It is well-known [2] that a density ρ gives rise to a flat line
bundle connection
F = (∂→l lnρ) . (1.12)
A A
In fact, several mathematical objects, for instance the odd Laplacian ∆ and the odd scalar ν ,
ρ ρ
can be formulated entirely using F instead of ρ. Surprisingly, many of these objects continue to be
well-defined for non-flat F’s as well, where the nilpotency (and the ordinary physical description) is
broken down. In Section 5 we shall therefore temporarily digress to contemplate a modification of the
nilpotency condition that addresses these mathematical observations. Finally, Section 12 contains our
conclusions.
General remark about notation. Wehavetwotypesofgrading: AGrassmanngradingε andanexterior
form degree p. The sign conventions are such that two exterior forms ξ and η, of Grassmann parity
ε , ε and exterior form degree p , p , respectively, commute in the following graded sense:
ξ η ξ η
η ξ = ( 1)εξεη+pξpηξ η (1.13)
∧ − ∧
inside the exterior algebra. We will often not write the exterior wedges “ ” explicitly.
∧
2 General Second-Order ∆ operator
We here introduce the setting and notation more carefully, and argue that the ∆ operator must be
equal to ∆ +ν upto an oddconstant. (Theundeterminedoddconstant comes from the fact that the
ρ ρ
square ∆2 = 1[∆,∆] does not change if ∆ is shifted by an odd constant.) Consider now an arbitrary
2
Grassmann-odd, second-order, differential operator ∆ that takes scalar functions to scalar functions.
In this paper, we shall only discuss the non-degenerate case, where the second-order term in ∆ is
of maximal rank, and hence provides for a non-degenerated antibracket (, ), cf. the Definition (2.6)
· ·
below. (Thenon-degeneracyassumptionismotivatedbythefactthatitissatisfiedforcurrentlyknown
applications. The degenerate case may be dealt with via for instance the antisymplectic conversion
3
mechanism [14, 15].) Due to the non-degeneracy assumption, it is always possible to organize ∆ as
∆ = ∆ +ν , (2.1)
F
where ν is a zeroth-order term and ∆ is an operator with terms of second and first order [2]
F
∆ (−1)εA(∂→l +F )EAB∂→l . (2.2)
F ≡ 2 A A B
Here,EAB=EAB(Γ),F =F (Γ)andν=ν(Γ)isa(2,0)-tensor,alinebundleconnection,andascalar,
A A
respectively. We shall sometimes use the slightly longer notation ∆ ∆ to acknowledge that it
F ≡ F,E
dependson two inputs: F and E. The line bundleconnection F transforms undergeneral coordinate
A
transformations ΓA ΓB as
′
→
→∂l →∂l ∂ΓB
F = ( ΓB)F +( lnJ) , J sdet ′ . (2.3)
A ∂ΓA ′ B′ ∂ΓA ≡ ∂ΓA
Thesetransformation properties guarantee thatthe expressions(2.1) and (2.2) remain invariant under
general coordinate transformations. The Grassmann-parities are
ε(EAB) = ε +ε +1 , ε(F ) = ε , ε(ν) = 1 . (2.4)
A B A A
One may, without loss of generality, assume that the (2,0)-tensor EAB has a Grassmann-graded
skewsymmetry
EAB = ( 1)(εA+1)(εB+1)EBA . (2.5)
− −
The antibracket (f,g) of two functions f = f(Γ) and g = g(Γ) is defined via a double commutator
∗
[16] with the ∆-operator, acting on the constant unit function 1,
(f,g) ( 1)εf[[∆→,f],g]1 ( 1)εf∆(fg) ( 1)εf(∆f)g f(∆g)+( 1)εgfg(∆1)
≡ − ≡ − − − − −
= (f∂←r)EAB(∂→l g) = ( 1)(εf+1)(εg+1)(g,f) , (2.6)
A B
− −
whereuseismadeoftheskewsymmetry(2.5)inthethirdequality. Bythenon-degeneracyassumption,
there exists an inverse matrix E such that
AB
EABE = δA = E EBA . (2.7)
BC C CB
Since the tensor EAB possesses a graded A B skewsymmetry (2.5), the inverse tensor E must be
AB
↔
skewsymmetric,
EAB = ( 1)εAεBEBA . (2.8)
− −
In other words, E is a two-form
AB
1
E = dΓAE dΓB . (2.9)
AB
2 ∧
The Grassmann parity is
ε(E ) = ε +ε +1 . (2.10)
AB A B
∗Here,andthroughoutthepaper,[A,B]and{A,B}denotethegradedcommutator[A,B]≡AB−(−1)εAεBBAand
thegraded anticommutator {A,B}≡AB+(−1)εAεBBA, respectively.
4
3 Nilpotency Conditions: Part I
The square ∆2 = 1[∆,∆] of an odd second-order operator (2.1) is generally a third-order differential
2
operator, which we, for simplicity, imagine has been normal ordered, i.e. with all derivatives standing
to the right. Nilpotency (1.3) of the ∆ operator leads to conditions on EAB, F and ν. Let us
A
therefore systematically, over the next four Sections 3–6, discuss order by order the consequences of
the nilpotency condition ∆2 = 0, starting with the highest (third) order terms, and going down until
we reach the zeroth order.
The third-order terms of ∆2 vanish if and only if the Jacobi identity
(ε +1)(ε +1)
( 1) f h (f,(g,h)) = 0 (3.1)
−
cyclX. f,g,h
for the antibracket (, ) holds. We shall always assume this from now on. Equivalently, the two-form
· ·
E is closed,
AB
dE = 0 . (3.2)
In terms of the matrices EAB and E , the Jacobi identity (3.1) and the closeness condition (3.2)
AB
read
( 1)(εA+1)(εC+1)EAD(∂→Dl EBC) = 0 , (3.3)
−
cycl.XA,B,C
( 1)εAεC(∂→Al EBC) = 0 , (3.4)
−
cycl.XA,B,C
respectively. By definition, a non-degenerate tensor E with Grassmann-parity (2.10), skewsymme-
AB
try (2.8), and closeness relation (3.4) is called an antisymplectic structure.
Granted the Jacobi identity (3.1), the second-order terms of ∆2 can be written on the form
1
AB∂→l ∂→l , (3.5)
4R B A
where AB with upper indices is a shorthand for
R
AD EAB BCECD( 1)εC , (3.6)
R ≡ R −
and with lower indices is the curvature tensor for the line bundle connection F :
AB A
R
AB [∂→Al +FA,∂→Bl +FB] = (∂→Al FB) ( 1)εAεB(A B) . (3.7)
R ≡ − − ↔
Remarkably, the two tensors and AB carry opposite symmetry:
AB
R R
AB = ( 1)εAεB BA , (3.8)
R − − R
AB = ( 1)εAεB BA . (3.9)
R − R
It follows that in the non-degenerate case, the second-order terms of ∆2 vanish if and only if the line
bundle connection F has vanishing curvature
A
= 0 . (3.10)
AB
R
5
The zero curvature condition (3.10) is an integrability condition for the local existence of a density ρ,
F = (∂→l lnρ) . (3.11)
A A
Under the F ρ identification (3.11) the ∆ operator (2.2) just becomes the ordinary odd Laplacian
↔ F
∆ from eq. (1.1),
ρ
∆ = ∆ . (3.12)
F ρ
Conventionally the field-antifield formalism requires the F ρ identification (3.11) to hold globally.
↔
Nevertheless, we shall present many of the constructions below using F rather than ρ, to be as general
as possible.
There exists a descriptive characterization: Granted the Jacobi identity (3.1), the second-order terms
of ∆2 vanish if and only if there is a Leibniz rule for the interplay of the so-called “one-bracket”
Φ1 ∆ (∆1)=∆ and the “two-bracket” (, )
∆≡ − F · ·
ε
∆ (f,g) = (∆ f,g) ( 1) f(f,∆ g) . (3.13)
F F F
− −
See Ref. [16, 17] for more details.
4 A Non-Zero F-Curvature?
In eq. (3.10) of the previous Section 3 we learned that the nilpotency condition (1.3) completely kills
the line bundle curvature . Nevertheless, several constructions continue to be well-defined for non-
R
zero . For instance, both the important scalars ν and R fall into this category, cf. eqs. (7.1) and
R F
(11.7) below. Another example, which turns out to be related to our discussion, is the Grassmann-
odd 2-cocycle of Khudaverdian and Voronov [8, 11, 18]. It is defined using two (possibly non-flat) line
bundle connections F(1) and F(2) as follows:
ν(F(1);F(2),E) 1div X (−1)εA(∂→l + F(1) +F(2))(EAB(F(1) F(2))) , (4.1)
≡ 4 F(12) (12) ≡ 4 A 2 B − B
where the divergence “div” is defined in eq. (10.13),
F(1) +F(2)
F(12) , (4.2)
≡ 2
and
XA EAB(F(1) F(2)) . (4.3)
(12) ≡ B − B
ItisclearfromDefinition(4.1)thatν(F(1);F(2),E)behavesasascalarundergeneralcoordinatetrans-
formations. This is because the average F(12) is again a line bundle connection, and X is a vector
(12)
field since the difference F(1) F(2) is a co-vector (=one-form), cf. eq. (2.3). That ν(F(1);F(2),E) is
B − B
a 2-cocycle
ν(F(1);F(2),E)+ν(F(2);F(3),E)+ν(F(3);F(1),E) = 0 (4.4)
follows easily by rewriting Definition (4.1) as
ν(F(1);F(2),E) = ν(0) ν(0) , (4.5)
F(1) − F(2)
(0)
where ν generalizes eq. (1.7):
F
ν(0) (−1)εA(∂→l + FA)(EABF ) . (4.6)
F ≡ 4 A 2 B
6
Note that Definitions (4.1) and (4.6) continue to make sense for non-flat F’s. We should stress that
(0)
ν itself is not a scalar, but we shall soon see that it can be replaced in eq. (4.5) by a scalar ν , cf.
F F
eq. (7.1) below. In other words, ν(F(1);F(2),E) is a 2-coboundary.
The F-curvature is also an interesting geometric object in its own right. It can be identified
AB
R
with a Ricci two-form of a tangent bundle connection , cf. eq. (11.4) in Section 11 below. The Ricci
∇
two-form
1
R = 2dΓARAB ∧dΓB(−1)εB (4.7)
is closed
d = 0 , (4.8)
R
due to the Bianchi identity
( 1)εAεC(∂→Al BC) = 0 , (4.9)
− R
cycl.XA,B,C
so the two-form (4.7) defines a cohomology class.
5 Breaking the Nilpotency
Due to the above mathematical reasons we shall digress in this Section 5 to contemplate how a non-
zero F-curvature could arise in antisymplectic geometry, although we should stress that it remains
unclear if it is useful in physics. Nevertheless, the strategy that we shall adapt here is to append a
general Grassmann-even (possibly degenerate) second-order operator source 1∆ to the right-hand
2
side of the nilpotency condition (1.3): R
1
∆2 = ∆ . (5.1)
2
R
A covariant and general way of realizing the second-order ∆ operator is to write
R
∆ ∆ +V +n , (5.2)
F,
R ≡ R R R
where
∆ (−1)εA(∂→l +F ) AB∂→l (5.3)
F,R ≡ 2 A A R B
is an Grassmann-even Laplacian based on F and AB. We have included a Grassmann-even vector
A
R
field
V VA∂→l (5.4)
A
R ≡ R
anda scalar function n togive asystematic treatment. Note that thevector field V is thedifference
of the subleading connRection terms inside ∆ and ∆ . We shall show below thaRt the n term is
F,
completely determined by consistency, whileRV in prinRciple can be any locally HamiltoniRan vector
field subjected to the following restriction: BotRh VA and n should be proportional to the -source
R
(or its derivatives) in order to restore nilpotency (1R.3) in thRe limit 0.
R→
The new condition (5.1) still imposes the Jacobi identity (3.1) for the antibracket (, ) at the third
· ·
order, since the modification is just of second order. (We mention, for later, that the Jacobi identity
aloneguarantees theexistenceofanilpotent∆ operatoranditsquantization scheme,cf.Sections8-9,
E
regardless of how the nilpotency (5.1) of ∆ is broken at lower orders.) The second-order terms in eq.
(5.1) implies that the F-curvature AB defined in eq. (3.7) should be identified with the principal
R
7
symbol AB appearing inside the ∆ operator (5.3), thereby justifying the notation. Note that the
R F,
Leibniz rule (3.13) is no longer valid. RTo see this, it is useful to define an even -bracket [19]
R
(f,g) [[∆→ ,f],g]1 ∆ (fg) (∆ f)g f(∆ g)+fg(∆ 1)
R ≡ R ≡ R − R − R R
= (f∂←r) AB(∂→l g) = ( 1)εfεg(g,f) . (5.5)
A B
R − R
It turns out that the -bracket (, ) measures the failure of the Leibniz rule:
R · ·
R
1
ε ε
(f,g) = ( 1) f∆ (f,g) ( 1) f(∆ f,g)+(f,∆ g) . (5.6)
2 R − F − − F F
Note that this -bracket (, ) does not satisfy a Jacobi identity. (In fact, we shall see that the
R · ·
closeness relation (4.8) for R will instead lead to a compatibility relation (5.8) below.) Since
AB
R
∆2 1∆ is a first-order operator, cf. eqs. (2.1) and (5.1), the commutator
F−2 F,
R
1 1
[∆ ,∆ ] = [∆ ,∆2 ∆ ] (5.7)
2 F,R F F F−2 F,R
becomes a second-order operator at most. (We shall improve this estimate in Lemma 5.1 below.) This
fact already implies that the two brackets (, ) and (, ) are compatible in the sense that
· · · ·
R
ε (ε +1) ε (ε +1)+ε
( 1) f h ((f,g),h) = ( 1) f h g((f,g) ,h) . (5.8)
cyclX. f,g,h − R cyclX. f,g,h − R
Phrased differently, one may define a one-parameter family of antisymplectic two-forms
1
E(θ) E +θ E+ θ = dΓAE (θ) dΓB , dE(θ) = 0 , (5.9)
AB
≡ R ≡ R 2 ∧
which depends on a Grassmann-odd parameter θ. In components it reads
E (θ) = E + θ , (5.10)
AB AB AB
R
EAB(θ) = EAB +( 1)εAθ AB = EAB + ABθ( 1)εB . (5.11)
− R R −
There exists locally an antisymplectic one-form potential
U(θ) U (θ)dΓA , U (θ) U +F θ ,
≡ A A ≡ A A
(5.12)
dU(θ) = E(θ) , ∂→Al UB(θ)−(−1)εAεB(A ↔ B) = EAB(θ) .
We will now improve the estimate from eq. (5.7):
Lemma 5.1 The commutator [∆ ,∆ ] is always a first-order operator at most.
F F,
R
Proof of Lemma 5.1: Note that the commutator [∆ ,∆ ] appears inside the square
F F,
R
(∆ (θ))2 = ∆2 +θ[∆ ,∆ ] = ∆2 +[∆ ,∆ ]θ (5.13)
F F F, F F F F,
R R
of the Grassmann-odd second-order operator
∆ (θ) ∆ +θ∆ ∆ +∆ θ = (−1)εA(∂→l +F )EAB(θ)∂→l . (5.14)
F ≡ F F,R ≡ F F,R 2 A A B
8
One knows from the general discussion in the previous Section 3 that the third-order terms in the
square (5.13) vanish because EAB(θ) satisfies the Jacobi identity (3.3). Moreover, the second-order
terms in the square (5.13) are of the form
(−1)εCEAB(θ) ECD(θ) ∂→l ∂→l = 1 AB∂→l ∂→l , (5.15)
4 RBC D A 4R B A
cf. eqs. (3.5) and (3.6). It is easy to see that the two θ-dependent terms inside the left-hand side of eq.
(5.15)cancelagainsteachother. Infact,eachofthetwotermsvanishseparatelyduetoskewsymmetry:
( 1)εC+εFEAB BCECD DFEFG = ACECD DG = ( 1)(εA+1)(εG+1)(A G) . (5.16)
− R R R R − ↔
Therefore, the θ-dependent part of the square (5.13) must be of first order at most.
(One may also give a proof of Lemma 5.1 based on Lemma B.1 in Appendix B.) Lemma 5.1 implies
(for instance via the technology of Ref. [16]) that
ε ε
∆ (f,g) (∆ f,g) (f,∆ g) = ( 1) f∆ (f,g) ( 1) f(∆ f,g)
F, F, F, F F
R − R − R −(f,∆ g) , R − − R (5.17)
F
− R
1 1 1
(∆2 ∆ )(f,g) = ((∆2 ∆ )f,g)+(f,(∆2 ∆ )g) . (5.18)
F−2 F,R F−2 F,R F−2 F,R
More generally, there exists a superformulation
∆(θ) ∆+θ∆ ∆+∆ θ = (−1)εA(∂→l +F (θ))EAB(θ)∂→l +ν(θ) , (5.19)
≡ R ≡ R 2 A A B
where
ν(θ) ν +θn ν +n θ , (5.20)
≡ R ≡ R
and
F (θ) F +2E VBθ F 2VBE θ . (5.21)
A A AB A BA
≡ R ≡ − R
The nilpotency condition
1 ∂ 2
∆(θ) = 0 (5.22)
(cid:18) − 2∂θ(cid:19)
precisely encodes the deformed condition (5.1) and its consistency relation
0 = [∆,[∆,∆]] = [∆,∆ ] = [∆ +ν,∆ +V +n ]
F F,
= [∆ ,∆ ]+[∆ ,VR]+[∆ ,n ] [∆R +RV ,νR] . (5.23)
F F, F F F,
R R R − R R
Note in the last line of eq. (5.23) that the first term [∆ ,∆ ] and the two last terms [∆ ,n ] and
F F, F
[∆ +V ,ν] are all of first order. Hence, the second term [R∆ ,V ] must be of first order aRs well.
F, F
ThisRin tuRrn implies that V should be a generating vector field forRan anticanonical transformation:
R
V (f,g) = (V (f),g)+(f,V (g)) . (5.24)
R R R
Sincetheantibracket is non-degenerated, itfollows that V mustbealocally Hamiltonian vector field,
which we, for simplicity, will assume is a globally HamiltoRnian vector field
V = 2(ν , ) , (5.25)
R − R ·
with some Fermionic globally defined Hamiltonian ν . The factor “ 2” in eq. (5.25) is chosen for
−
later convenience. The Hamiltonian ν in eq. (5.25) sRhould be considered as an additional geometric
R
9
input, which labels the different ways (5.1) of breaking the nilpotency of ∆. It is a priori only defined
in eq. (5.25) up to an odd constant. We fix this constant by requiring that
ν 0 for 0 . (5.26)
R → R →
Altogether, the Hamiltonian ν does not contribute to the curvature
R
∂→Al FB(θ) ( 1)εAεB(A B) = AB (5.27)
− − ↔ R
of the line bundle connection
F (θ) = F +4(∂→l ν )θ . (5.28)
A A A
R
Now let us continue the investigation of the deformed condition (5.1). The first-order terms of eq.
(5.1) cancel if and only if
1
∆2 ∆ = (ν ν , ) . (5.29)
F − 2 F,R − R ·
This is a differential equation for the function ν=ν(Γ), or, equivalently, for the difference ν ν . It
−
nowbecomes clear thattheν functionprovidesanauxiliary curvaturebackgroundfortheν funcRtion.
Since we assume that ν is gRiven, we will now focus on the difference ν ν rather than on ν itself.
−
TheFrobeniusintegrabiRlity condition foreq.(5.29) comesfromthefactthattRheoperator∆2 1∆
F − 2 F,
differentiates the antibracket, cf. eq. (5.18). This implies that the difference ν ν can be written aRs
−
a contour integral R
Γ 1
(ν ν )(Γ) = (ν ν )(Γ )+ ((∆2 ∆ )ΓA)E dΓB (5.30)
− R − R 0 ZΓ0 F−2 F,R AB(cid:12)(cid:12)Γ Γ′ ′
(cid:12) →
(cid:12)
that is independent of the curve (aside from the two endpoints). It only depends on E, F, and an
odd integration constant (ν ν )(Γ ). In particular, we conclude that the difference ν ν does not
0
− −
introduce any new geometric sRtructures. The first-order commutator from Lemma 5.1 cRan now be
expressed in terms of the difference ν ν as follows:
−
R
1 1
[∆ ,∆ ] = [∆ ,∆2 ∆ ] = ∆ (ν ν , ) (ν ν ,∆ ())
2 F,R F F F − 2 F,R F − R · − − R F ·
1
= (∆ (ν ν ), ) (ν ν , ) . (5.31)
F − R · − 2 − R · R
Here, eq. (5.29) is used in the second equality and the deformed Leibniz rule (5.6) is used in the third
(=last) equality.
Finally, the zeroth-order terms of eq. (5.1) cancel if and only if
n = 2(∆ ν) , (5.32)
F
R
sothisfixescompletelytheGrassmann-evenfunctionn . OnecanshowthatiftheHamiltonianvector
field VA vanishes in the flat limit 0, then the n Rfunction, defined via eq. (5.32), automatically
R→
does thRe same, cf. eq. (6.2) below. The nilpotency-Rbreaking operator ∆ will therefore vanish for
0, as it should. R
R→
6 Nilpotency Conditions: Part II
After this digression into non-zero curvature, let us now return to the nilpotent (and ordinary
R
physical) situation ∆2 = 0, where , VA and n are all zero. Not much changes for the condition
R
R R
10