Table Of ContentGeneralized sigma model with dynamical antisymplectic
potential and non-Abelian de Rham’s differential
Igor A. Batalin(a,b)1 , Peter M. Lavrov(b,c)2
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1
0 (a) P.N. Lebedev Physics Institute,
2
Leninsky Prospect 53, 119991 Moscow, Russia
b
e
F (b)Tomsk State Pedagogical University,
7 Kievskaya St. 60, 634061 Tomsk, Russia
] (c)National Research Tomsk State University,
h
t
- Lenin Av. 36, 634050 Tomsk, Russia
p
e
h
[
4
v
2
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Abstract
5
0
0
For topological sigma models, we propose that their local Lagragian density is allowed
.
1
0 to depend non-linearly on the de Rham’s ”velocities” DZA. Then, by differentiating
7
the Lagrangian density with respect to the latter de Rham’s ”velocities”, we define a
1
: ”dynamical” anti-symplectic potential, in terms of which a ”dynamical” anti-symplectic
v
i metric is defined, as well. We define the local and the functional antibracket via the
X
r dynamical anti-symplectic metric. Finally, we show that the generalized action of the
a
sigma model satisfies the functional master equation, as required.
Keywords: sigma-model, de Rham’s differential, supermanifold, superfield, master equation
PACS numbers: 11.10.Ef, 11.15.Bt
1E-mail: [email protected]
2E-mail: [email protected]
1 Introduction
When formulating a topological sigma model, one proceeds usually with an anti-symplectic
configuration space ZA ε(ZA) =: ε whose anti-symplectic potential V (Z),ε(V ) =: ε +1,
A A A A
{ | }
is given originally [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. At the same time, it is assumed usually that
the kinetic part of the original local Lagrangian density has the form (DZA)V (Z), where
A
L −
D is the de Rham’s differential.
In the present paper, we generalize the local original Lagrangian density as to take the form
(Z,DZ) allowed to depend non-linearly on the de Rham’s ”velocities” DZA. Then, we define
L
a ”dynamical” anti-symplectic potential, V (Z,DZ) as the derivatives of the new Lagangian
A
density with respect to the mentioned de Rham’s ”velocities”. We define a local anti-symplectic
metric in its covariant components E (Z,DZ) as the standard vorticity of the ”dynamical”
AB
anti-symplectic potential V (Z,DZ), in terms of explicit ZA derivatives.
A
We define both the local and the functional antibracket by the standard formulae via the
”dynamical” anti-symplectic metric in its contravariant components EAB(Z,DZ). Finally, we
show that the new action Σ =: dµ (Z,DZ)satisfies the functional master equation, provided
L
the function S(Z,DZ) =: (Z,RDZ)+DZAV (Z,DZ) satisfies the local master equation.
A
L
2 Non-Abelian de Rham’s differential
Let Γ be an intrinsic configuration super-manifold,
Γ =: Xa,Ca ε(Xa) = 0, ε(Ca) = 1,a = 1,...,2m . (2.1)
{ | }
Let D be a non-Abelian de Rham’s differential, as defined by the conditions
1
ε(D) = 1, D2 = [D,D] = 0, D = D†, (2.2)
2 −
whose solution is sought for in the form,
∂ 1 ∂
D =: CaΛb(X) + CbCa d(X) , (2.3)
a ∂Xb 2 Uab ∂Cd
with Λb being invertible, and Ud being antisymmetric in its subscripts a,b [6]. The conditions
a ab
(2.2) imply
Λc∂ Λd (a b) = cΛd, (2.4)
a c b − ↔ Uab c
( Λe∂ d + e d)+cyclic perm.(a,b,c) = 0. (2.5)
− a eUbc UabUec
The Jacobi relation (2.5) provides for the integrability of the Maurer-Cartan equation (2.4). In
terms of the Boson integration measure,
dµ(Γ) =: ρ(X)[dX][dC], ρ =: det(Λ−1) = (det(Λ))−1, (2.6)
2
the anti-Hermiticity of the differential D implies
ρ−1∂ (ρΛb)+ b = 0. (2.7)
b a Uab
In turn, it follows from (2.7),
dµ(Γ)DF(Γ) = 0, F(Γ) = 0. (2.8)
Z (cid:12)∂Γ
(cid:12)
(cid:12)
3 Generalized sigma model
Let Γ′ be an antisymplectic phase space,
Γ′ =: ZA ε(ZA) = ε , A = 1,...,2N , (3.1)
A
{ | }
with N being an equal number of Boson and Fermion variables among ZA. Let us define the
local action of the generalized sigma model in the form,
Σ =: dµ(Γ) (Z(Γ),DZ(Γ)), (3.2)
Z L
with the measure dµ(Γ) being defined in (2.6). A functional derivative of the action (3.2) has
the form,
δΣ ∂
= ∂ +DV ( 1)εA, V =: L , (3.3)
δZA(Γ) AL A − A −∂(DZA)
in terms of explicit ZA-derivatives ∂ . In turn, we have
A
DV ( 1)εA = DZB∂ V ( 1)εA = ∂ V DZB( 1)εB(εA+1). (3.4)
A B A B A
− − − −
By inserting that into the relation (3.3), we get
δΣ
= E DZB( 1)εB +∂ S, (3.5)
δZA(Γ) AB − A
where
E =: ∂ V ∂ V ( 1)εAεB, (3.6)
AB A B B A
− −
S =: V DZA( 1)εA = (1 N ) , (3.7)
A DZ
L− − − L
∂
N =: DZA . (3.8)
DZ ∂(DZA)
3
4 Functional and local master equations
Let the metric (3.6) be invertible, and EAB be its inverse,
E EBC = δC. (4.1)
AB A
Let F(Z,DZ),G(Z,DZ) be two arbitrary local functions; their local antibracket is defined as
(F,G)(Z,DZ) =: F(Z,DZ)←∂− EAB(Z,DZ)−→∂ G(Z,DZ), (4.2)
A B
in terms of explicit ZA - derivatives ∂ .
A
In turn, let F[Z],G[Z] be two arbitrary functionals; their functional antibracket is defined
as
←−δ −→δ
(F,G)′[Z] =: F[Z] dµ(Γ) EAB(Z,DZ) G[Z]. (4.3)
Z δZA δZB
Due to the definition of the V , the second in (3.3), together with the definition (3.6) of the
A
E , each of the antibrackets, (4.2) and (4.3), satisfies its polarized Jacobi identity,3
AB
((ZA,ZB),ZC)( 1)(εA+1)(εC+1) +cyclic perm.(A,B,C) = 0, (4.4)
−
((ZA(Γ),ZB(Γ′))′,ZC(Γ′′))′( 1)(εA+1)(εC+1) +cyclic perm.(A,Γ;B,Γ′;C,Γ′′) = 0. (4.5)
−
Now, we are in a position to show that the functional master equation
1
(Σ,Σ)′ = 0, (4.6)
2
is satisfied as for the action Σ[Z] (4.2), provided that the local master equation
1
(S,S) = 0, (4.7)
2
is satisfied as for the function S(Z,DZ) (3.7). Indeed, by inserting the functional derivative
(3.5) into the left-hand side of the equation (4.6), we have
1 1
(Σ,Σ)′ = dµ(Γ) (S,S)+D = 0, (4.8)
2 Z (cid:18)2 L(cid:19)
due to the local master equation (4.7), together with the boundary condition,
= 0. (4.9)
∂Γ
L|
3Here in (4.5), Γ′ and Γ′′ mean (X′,C′) and (X′′,C′′) as to stand for (X,C) in (2.1). Not to be confused
with (3.1).
4
As a simple example of the Jacobi identity as for the functional antibracket (4.3), consider
only the first equality in the formula (4.8), before the use of the equation (4.7),
(Σ,Σ)′ = dµ(Γ)(S,S), (4.10)
Z
which is valid for any functional of the form (3.2), with S defined by (3.7). Then, we get the
non-polarized form of the functional Jacobi identity,
←∂− ←∂−
((Σ,Σ)′,Σ)′ = dµ(Γ)(S,S) + D DZA +(S,ZA) ( 1)εA = 0. (4.11)
Z (cid:16)∂ZA ∂(DZA) (cid:17)(cid:16) (cid:17) −
In fact, the second equality in eq.(4.11) holds due to the polarized functional Jacobi identity
(4.5) derived in Appendix A. The integrand in eq.(4.11) demonstrates explicitly the structure of
the original terms characteristic for the non-polarized functional Jacobi identity. Among other
formal properties involved, the simplest one is the nilpotency of the de Rham’s differential D,
the second in the conditions (2.2), as well as the appearance of the boundary terms of the form
”D(anything)”, and the local non-polarized Jacobi identity,
((S,S),S) = 0, (4.12)
together with its (DZ) - dual identity,
←∂−
(S,S) D(ZA,S) = 0. (4.13)
∂(DZA)
Finally, if we introduce the nilpotent functional odd Laplacian,
1 δ δ
∆′ =: (ρ′[Z])−1 dµ(Γ)( 1)εA ρ′[Z]EAB(Z,DZ) , (4.14)
2 Z − δZA δZB
with ρ′[Z] being a local functional measure,
lnρ′[Z] =: dµ(Γ) (Z,DZ), (4.15)
Z M
then one can consider formally the functional quantum master equation
i 1 ~
∆′exp W = 0 (W,W)′ + ∆′W = 0. (4.16)
(cid:26)~ (cid:27) ⇔ 2 i
It looks natural to seek for a solution to the quantum master action functional W in the form
of a power series expansion in i~,
W = Σ+i~W +.... (4.17)
1
Then, to the i~ order, it follows from (4.16),
(Σ,W )′ ∆′Σ = 0. (4.18)
1
−
Here in (4.18), the second term contains ill-defined functional factor, δ(Γ,Γ). It seems rather
difficulttomakesensetotheseexpressionsinaconsistentway. Thus, thestatusofthefunctional
quantum master equation (4.16) remains an open question. Perhaps, the most realistic idea is
that the mentioned ill-defined factor might be canceled from the functional path integral over
the field-antifield variables ZA(Γ).
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Acknowledgments
TheworkofI.A.BatalinissupportedinpartbytheRFBRgrants17-01-00429and17-02-00317.
The work of P. M. Lavrov is supported in part by the RFBR grant 15-02-03594.
Appendix A
In this Appendix, we derive in detail the polarized Jacobi identity (4.5) as for the functional
antibracket. First, we rewrite the relation (4.5) in the form4
←−δ
E (Z(Γ),DZ(Γ))δ(Γ,Γ′) ( 1)εAεC +cyclic perm.(A,Γ′;B,Γ;C,Γ′′) = 0. (A.1)
AB δZC(Γ′′) −
Now, the following relation holds
δ
V (Z(Γ),DZ(Γ)) (A,Γ′ B,Γ)( 1)εAεB =
δZA(Γ′) B − ↔ −
= δ(Γ,Γ′)E (Z(Γ),DZ(Γ))+ (A.2)
AB
∂
+( 1)εA(Dδ(Γ,Γ′)) V (Z(Γ),DZ(Γ)) (A B)( 1)(εA+1)(εB+1) .
− (cid:18)∂(DZA(Γ)) B − ↔ − (cid:19)
The second term in the right-hand side in (A.2) equals to zero due to the second in (3.3). Thus,
we get from (A.2),
←−δ ←−δ
V (Z(Γ),DZ(Γ)) ( 1)εAεB V (Z(Γ′),D′Z(Γ′)) =
B δZA(Γ′) − − A δZB(Γ)
= δ(Γ,Γ′)E (Z(Γ),DZ(Γ)). (A.3)
AB
By expressing the right-hand side from (A.3) and then inserting into the left-hand side in (A.1),
we observe that each term cancels its counterpart in the cyclic sum, so that the polarized Jacobi
identity (4.5) is confirmed.
Notice that the nilpotency of the functional odd Laplacian (4.14) implies the polarized
Jacobi identities (4.5) to hold, together with their consequence that the ∆′ is a differentiation
as for the functional polarized antibracket,
∆′(ZA(Γ),ZB(Γ′))′ = (∆′ZA(Γ),ZB(Γ′))′ (ZA(Γ),∆′ZB(Γ′))′( 1)εA, (A.4)
− −
and the property [12],
1
(∆′)2 +ad′ ∆′ ρ′ = 0, ∆′ =: ∆′ . (A.5)
1 √ρ′ 1 1 ρ′=1
(cid:16) p (cid:17) (cid:12)
(cid:12)
4See footnote on page 4.
6
Appendix B
In this Appendix B, we consider in short the local measure,
1 δlnρ′
∆′Σ = dµ(Γ) δ(Γ,Γ)∂ (ZA,S)( 1)εA + DZA +(S,ZA) . (B.1)
A
2 Z (cid:16) − δZA (cid:17)
(cid:0) (cid:1)
In analogy with (3.2), (3.5), we have from (4.15),
δlnρ′
= E˜ DZB( 1)εB +∂ S˜, (B.2)
δZA(Γ) AB − A
E˜ =: ∂ V˜ ∂ V˜ ( 1)εAεB, (B.3)
AB A B B A
− −
∂
V˜ =: M , (B.4)
A
−∂(DZA)
S˜ =: (1 N ) . (B.5)
DZ
− M
Due to the relation (B.2), the second term in (B.1) rewrites in the form
(S,S˜)+S←∂− ECAE˜ δC DZB( 1)εB +D 2V˜ DZB( 1)εB +S +S˜ , (B.6)
C AB − B − B −
(cid:0) (cid:1) (cid:0) (cid:1)
which is a natural counterpart to the formula (4.8).
Appendix C
Here, we consider in short the Legendre transformation from the de Rham’s ”velocities”
DZA to new antifield variables Z∗. Provided that the Hessian matrix
A
∂2 (Z,DZ)
L , (C.1)
∂(DZA)∂(DZB)
is invertible, one can introduce new antifield variables Z∗, by resolving the definition
A
∂ (Z,DZ)
Z∗ =: L , (C.2)
A − ∂(DZA)
with respect to the de Rham’s ”velocities” (DZA)(Z,Z∗). Then, we consider the function of
ZA,Z∗,
A
S(Z,Z∗) =: (Z,(DZ)(Z,Z∗))+(DZA)(Z,Z∗)Z∗, (C.3)
L A
in terms of which the general equation of motion for any dynamical quantity (Z,Z∗) reads
O
D +(S, ) = 0, (C.4)
ext
O O
where the only nonzero elementary extended antibrackets are
(ZA,Z∗) =: δA. (C.5)
B ext B
7
As for the function (C.3) by itself, we have
DS = 0, (S,S) = 0. (C.6)
ext
By choosing in the general equation (C.4) = ZA, and then = Z∗, we have
O O A
DZA = (S,ZA) , DZ∗ = (S,Z∗) . (C.7)
− ext A − A ext
These anticanonical equations are certainly generated by the usual action linear in the de
Rham’s ”velocities”,
Σ =: dµ(Γ) Z∗DZA( 1)εA +S(Z,Z∗) . (C.8)
Z A −
(cid:0) (cid:1)
Thus, provided that the Hessian matrix (C.1) is invertible, we arrive at the usual situation
although formulated within the phase space whose dimension is twice as more than the one of
the original antisymplectic phase space (3.1).
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