Geometry of second order operators and odd symplectic structures

Geometry of second order operators and

Geometry of second order operators and odd symplectic structures Hovhannes Khudaverdian University of Manchester, Manchester, UK

INTEGRABLE SYSTEMS AND QUANTUM SYMMETRIES 24 – 28 August 2010 Yerevan


Contents Geometry of second order operators 2-nd order operators Connection on volume forms Divergence operator 2-nd order operator=Symmetric tensor+Connection Algebra of densities Second order self-adjoint operators on the algebra of densities and the canonical operator pencils Operators depending on a class of connections Second order operator on semidensitites and Batalin-Vilkovisky groupoid of connections ∆-operator on odd symplectic supermanifolds Invariant density on surfaces in odd symplectic sumpermanifold


References O.M. Khudaverdian, A.S. Schwarz, Yu.S. Tyupkin. Integral invariants for Supercanonical Transformations. Lett. Math. Phys., v. 5 (1981), p. 517-522.


References O.M. Khudaverdian, A.S. Schwarz, Yu.S. Tyupkin. Integral invariants for Supercanonical Transformations. Lett. Math. Phys., v. 5 (1981), p. 517-522. O.M. Khudaverdian, R.L. Mkrtchian. Integral Invariants of Buttin Bracket. Lett. Math. Phys. v. 18 (1989), p. 229-231 (Preprint EFI–918–69–86- Yerevan (1986)).


I.A.Batalin, G.A Vilkovisky. Gauge algebra and quantization. Phys.Lett., 102B (1981), 27-31. O.M. Khudaverdian, A.P. Nersessian. On Geometry of Batalin-Vilkovisky Formalism. Mod. Phys. Lett. A, v. 8 (1993), No. 25, p. 2377-2385. Batalin-Vilkovisky Formalism and Integration Theory on Manifolds. J. Math. Phys., v. 37 (1996), p. 3713-3724. A.S. Schwarz. Geometry of Batalin Vilkovsiky formalism. Comm.Math.Phys., 155, (1993), no.2, 249–260.


H.M. Khudaverdian. Geometry of Superspace with Even and Odd Brackets. J. Math. Phys. v. 32 (1991) p. 1934-1937 (Preprint of the Geneva University, UGVA–DPT 1989/05–613). Odd Invariant Semidensity and Divergence-like Operators on Odd Symplectic Superspace. Comm. Math. Phys., v. 198 (1998), p. 591-606. Semidensities on odd symplectic supermanifold., Comm. Math. Phys., v. 247 (2004), pp. 353-390 (Preprint of Max-Planck-Institut fur ¨ Mathematik, MPI-135 (1999), Bonn.)

H.M. Khudaverdian, T.Voronov On Odd Laplace operators. Lett. Math. Phys. 62 (2002), 127-142 On odd Laplace operators. II. In Amer. Math. Soc. Transl. (2), Vol. 212, 2004, pp.179–205 Differential forms and odd symplectic geometry. Amer. Math. Soc. Transl (2) Vol 224, 2008 pp.159—171

Geometry of second order operators and Geometry of second order operators 2-nd order operators

First order operator (on functions) ∂ ∂a ↔ ∂ xa

a

L = T (x)∂a + R(x) , 0

Change of coordinates x a = x a (x a ) 0

∂a =

0 xaa ∂a 0 ,

0 xaa

∂ xa = ∂ xa 0

!

L = T a (x)∂a + R(x) = T a (x)xaa ∂a 0 + R(x) | {z } Ta

0

L = T a (x)∂a + R(x) | {z } | {z } vector field scalar


Second order operator (on functions) 1 ∆ = S ab (x)∂a ∂b + T a (x)∂a + R(x) , 2 0

Change of coordinates x a = x a (x a ) 0 1 0 1 ∆ = S ab (x)∂a ∂b + · · · = xaa S ab xbb ∂a 0 ∂b 0 + . . . 2 2 | {z }

Sa

0b 0

S ab defines symmetric contravariant tensor of rank 2 on M.


Second order operator (on functions) 1 ∆ = S ab (x)∂a ∂b + T a (x)∂a + R(x) , 2 0

Change of coordinates x a = x a (x a ) 0 1 0 1 ∆ = S ab (x)∂a ∂b + · · · = xaa S ab xbb ∂a 0 ∂b 0 + . . . 2 2 | {z }

Sa

0b 0

S ab defines symmetric contravariant tensor of rank 2 on M. Quadratic polynomial H∆ = 12 S ab pa pb on T ∗ M is the principal symbol of the operator ∆ = 21 ∂a S ab (x)∂b + . . . . (Linear polynomial HL = T a pa on T ∗ M is the principal symbol of the operator L = T a ∂a + . . . .)


∆=

1 2

S ab (x)∂a ∂b +T a (x)∂a + R(x) | {z } top component

(1)

The top component symmetric tensor field S ab ∂a ⊗ ∂b on M. If S ≡ 0 then ∆ becomes first order operator T a ∂a is vector field. What about the geometrical meaning of the term T a ∂a if S 6= 0?


∆=

1 2

S ab (x)∂a ∂b +T a (x)∂a + R(x) | {z } top component

(1)

The top component symmetric tensor field S ab ∂a ⊗ ∂b on M. If S ≡ 0 then ∆ becomes first order operator T a ∂a is vector field. What about the geometrical meaning of the term T a ∂a if S 6= 0? To study this question consider the difference ∆+ − ∆ , where ∆+ is defined via a scalar product h∆f , gi = hf , ∆+ gi .


Scalar product and volume form

hf , giρ =

Z

f (x)g(x)ρ(x)Dx,

ρ(x)Dx is a volume form .

M

Dx 0 0 0 ∂ xa ρ(x)Dx = ρ(x(x )) Dx 0 Dx = ρ(x(x )) det 0 a Dx ∂x 0

h∆f , giρ = Z

= M

Z

∆ (f (x)) g(x)ρ(x)Dx

M

f (x)∆+ (g(x)) ρ(x)Dx = hf , ∆+ giρ .


By integrating by parts we have Z 1 ab a h∆f , giρ = S (x)∂a ∂b f + T (x)∂a f + R(x)f g(x)ρ(x)Dx = M 2 {z } | ∆f

1 ab 1 a f (x) ∂a ∂b S ρg − ∂a T ρg + Rg ρ(x)Dx = hf , ∆+ giρ 2ρ ρ M | {z }

Z

+

∆ g

∆+ − ∆ = ∂b S ab − 2T a + S ab ∂b log ρ ∂a + . . . | {z } vector field


By integrating by parts we have Z 1 ab a h∆f , giρ = S (x)∂a ∂b f + T (x)∂a f + R(x)f g(x)ρ(x)Dx = M 2 {z } | ∆f

1 ab 1 a f (x) ∂a ∂b S ρg − ∂a T ρg + Rg ρ(x)Dx = hf , ∆+ giρ 2ρ ρ M | {z }

Z

+

∆ g

∆+ − ∆ = ∂b S ab − 2T a + S ab ∂b log ρ ∂a + . . . | {z } vector field Claim: for an operator ∆ = S ab ∂a ∂b + T a ∂a + R, the expression γ a = ∂b S ab − 2T a , is an (upper) connection on volume forms.

Geometry of second order operators and Geometry of second order operators Connection on volume forms

Connection on volume forms Connection ∇ on volume forms defines covariant derivative ∇a (ρDx) = (∂a + γa )ρ(x)Dx,

γa Dx = ∇a (Dx) .


Connection on volume forms Connection ∇ on volume forms defines covariant derivative ∇a (ρDx) = (∂a + γa )ρ(x)Dx,

γa Dx = ∇a (Dx) . 0

Transformation of the symbol γa (x): x a = x a (x a ) ∂x ∂x a0 0 a0 γa Dx = xa ∇a 0 det = xa ∂a 0 log det + γa 0 Dx 0 Dx 0 ∂x ∂x 0

0

b γa = xaa γa 0 − xbb 0 xba .

The difference of two connections is a vector field: 0

γa − γa = vector field


Examples of connections Example (Connection defined by a chosen volume form) Let ρ(0) (x)Dx be a non-vanishing volume form. Define: (0)

∇a (ρ(x)Dx) = ∂a

ρDx ρ(0) Dx

! ρ(0) Dx = (∂a ρ(x)−∂a log ρ(0) (x))ρ(0) Dx,

(0)

γa = −∂a log ρ(0) (x) (0) ∇a ρ(0) (x)Dx ≡ 0 . It is the connection induced by a volume form


Example (Connection defined by a Riemannian structure) Let M be Riemannian manifold with metric gab dx a dx b . p (canonical volume form) ρ(g) (x)Dx = det gDx, 1 1 (g) γa = −∂a log ρ(g) (x) = − ∂a log det g = − g bc ∂a gbc = −Γbab , 2 2 a where Γbc are the Christoffel symbols of the Levi-Civita connection. Here connection on volume forms is the trace of Christoffel symbols.


Example (Connection defined by a Riemannian structure) Let M be Riemannian manifold with metric gab dx a dx b . p (canonical volume form) ρ(g) (x)Dx = det gDx, 1 1 (g) γa = −∂a log ρ(g) (x) = − ∂a log det g = − g bc ∂a gbc = −Γbab , 2 2 a where Γbc are the Christoffel symbols of the Levi-Civita connection. Here connection on volume forms is the trace of Christoffel symbols. Volume form connection γa = −∂a log ρ is a flat connection: its curvature vanishes fab = ∂a γb − ∂b γa = −∂a ∂b log ρ + ∂b ∂a log ρ = 0 .

Geometry of second order operators and Geometry of second order operators Divergence operator

Connection on volume forms and divergence If ∇ is a connection on volume forms ∇a ρDx = (∂a + γa )ρ(x)Dx then one can define a divergence operator on vector fields: For

X = X a ∂a , divγ X = ∂a X a − γa X a .

If the connection ∇ is induced by a volume form ρ(x)Dx (ρ)

γa

= −∂a log ρa ,

(ρ)

then

div γ (ρ) X = ∂a X a − γa X a = ∂a X a + X a ∂a log ρ(x) =

LX ρDx . ρDx

p If ρ(x)Dx = det gDx is the canonical volume form on a Riemannian manifold, then div X = (∂b + Γaab )X b .


Returning to operators For operator ∆ = 12 S ab ∂a ∂b + T a ∂a + R we consider adjoint operator ∆+ with respect to the scalar product induced by a volume form ρ(x)Dx. ∆+ − ∆ = ∂b S ab − 2T a + S ab ∂b log ρ ∂a + . . . {z } | vector field


Returning to operators For operator ∆ = 12 S ab ∂a ∂b + T a ∂a + R we consider adjoint operator ∆+ with respect to the scalar product induced by a volume form ρ(x)Dx. ∆+ − ∆ = ∂b S ab − 2T a + S ab ∂b log ρ ∂a + . . . {z } | vector field (ρ)

Consider the flat connection γa = −∂a log ρ induced by a volume form ρ(x)Dx. We obtain (ρ)

∂b S ab − 2T a = vector field − S ab ∂b log ρ = vector field + S ab γb


Returning to operators For operator ∆ = 12 S ab ∂a ∂b + T a ∂a + R we consider adjoint operator ∆+ with respect to the scalar product induced by a volume form ρ(x)Dx. ∆+ − ∆ = ∂b S ab − 2T a + S ab ∂b log ρ ∂a + . . . {z } | vector field (ρ)

Consider the flat connection γa = −∂a log ρ induced by a volume form ρ(x)Dx. We obtain (ρ)

∂b S ab − 2T a = vector field − S ab ∂b log ρ = vector field + S ab γb ∂b S ab − 2T a = vector field + γ a(ρ) = γ a upper connection

Geometry of second order operators and Geometry of second order operators 2-nd order operator=Symmetric tensor+Connection

Geometry of second order operator (on functions) ∂b S ab − 2T a = γ a upper connection on volume forms 1 1 1 ∂b S ba − γ a ∂a + R , ∆ = S ab ∂a ∂b f + T a ∂a + R = S ab ∂a ∂b + 2 2 2   1 1 ∆f = ∂a  |{z} S ab ∂b f  − γa ∂a f + |{z} R f, 2 2 |{z} tensor

connection

scalar


Geometry of second order operator (on functions) ∂b S ab − 2T a = γ a upper connection on volume forms 1 1 1 ∂b S ba − γ a ∂a + R , ∆ = S ab ∂a ∂b f + T a ∂a + R = S ab ∂a ∂b + 2 2 2   1 1 ∆f = ∂a  |{z} S ab ∂b f  − γa ∂a f + |{z} R f, 2 2 |{z} tensor

Upper connection

γa

connection

scalar

on volume forms defines contravariant derivative: ∇a (ρ(x)Dx) = S ab ∂b + γ a ρDx .

If γa is connection on volume form then γ a = S ab γb is upper connection.


Example: Laplace-Beltrami operator

Fix a volume form ρ(x)Dx and consider the induced flat connection γa = −∂a log ρ. Fix scalar R = 0. Then 1 1 ∆ = ∂a S ab ∂b − γ a ∂a + R = 2 2 1 ab 1 ab 1 1 ab ∂a S ∂b + S ∂b log ρ∂a = ∂a ρS ∂b . 2 2 2ρ p In the Riemannian case S ab = g ab and ρ(x) = det g.

Geometry of second order operators and Geometry of second order operators Algebra of densities

Algebra of densities Under a change of coordinates a density of weight σ is multiplied by the σ -th power of the Jacobian of the coordinate transformation: σ Dx σ 0 σ 0 0 0 ∂x σ det s(x)|Dx| = s(x(x )) |Dx |σ . 0 |Dx | = s(x(x )) 0 ∂x Dx


Algebra of densities Under a change of coordinates a density of weight σ is multiplied by the σ -th power of the Jacobian of the coordinate transformation: σ Dx σ 0 σ 0 0 0 ∂x σ det s(x)|Dx| = s(x(x )) |Dx |σ . 0 |Dx | = s(x(x )) 0 ∂x Dx Density of weight σ = 0 is a usual scalar function. Density of weight σ = 1 is a volume form. Wave function Ψ is a density of weight σ = 12 (semi-density).


Algebra of densities Under a change of coordinates a density of weight σ is multiplied by the σ -th power of the Jacobian of the coordinate transformation: σ Dx σ 0 σ 0 0 0 ∂x σ det s(x)|Dx| = s(x(x )) |Dx |σ . 0 |Dx | = s(x(x )) 0 ∂x Dx Density of weight σ = 0 is a usual scalar function. Density of weight σ = 1 is a volume form. Wave function Ψ is a density of weight σ = 12 (semi-density). Product of two densities: s1 (x)|Dx|σ1 · s2 (x)|Dx|σ2 = s 0 (x)|Dx|σ1 +σ2 .


Canonical scalar product of densities Definition σ1

σ2

hs1 (x)|Dx| , s2 (x)|Dx| i =

Z M

s1 (x)s2 (x)Dx,

hs1 (x)|Dx|σ1 , s2 (x)|Dx|σ2 i = 0

if σ1 + σ2 = 1 ,

if σ1 + σ2 6= 1


Canonical scalar product of densities Definition σ1

σ2

hs1 (x)|Dx| , s2 (x)|Dx| i =

Z M

s1 (x)s2 (x)Dx,

hs1 (x)|Dx|σ1 , s2 (x)|Dx|σ2 i = 0

if σ1 + σ2 = 1 ,

if σ1 + σ2 6= 1

Symbolic notation: s(x)|Dx|σ ↔ s(x)t σ . Density a(x, t) = ∑ ak t σk Z a(x, t)b(x, t) ha(x, t), b(x, t)i = Res Dx . t2 M


Differential operators on densities d Differential operators D = D(x, t, ∂∂x , dt ) act on densities σ σ σ k a(x, t) = ∑ ak (x)t , (t ↔ |Dx| ). Examples d d . t dt (a(x)t σ ) = σ a(x)t σ . Weight operator: σˆ = t dt Lie derivative: ∂ ∂Xa d LX = X a a + t ∂x ∂ x a dt ∂Xa σ a ∂ a(x) LX (a(x)|Dx| ) = X +σ a(x) |Dx|σ , . ∂ xa ∂ xa

Examples of adjoints ∂a+ = −∂a , t + = t,

d dt

+

d = − dt + 2t , σˆ + = 1 − σˆ .

Geometry of second order operators and Geometry of second order operators Second order self-adjoint operators on the algebra of densities and the canonical operator pencils

Second order operator on the density algebra Second order self-adjoint Contravariant tensor S ab , ←→ operator on algebra of upper connection γ a densities (H.Kh., T.Voronov 2003) ∆a(x, t) = ∆+ a(x, t) = 1 ∂a S ab ∂b + (2σˆ − 1)γ a ∂a + σˆ ∂a γ a + σˆ (σˆ − 1)θ a(x, t) . 2 Here θ = γ a Sab γ b = γ a γa (in the case if S ab is invertible).


Second order operator on the density algebra Second order self-adjoint Contravariant tensor S ab , ←→ operator on algebra of upper connection γ a densities (H.Kh., T.Voronov 2003) ∆a(x, t) = ∆+ a(x, t) = 1 ∂a S ab ∂b + (2σˆ − 1)γ a ∂a + σˆ ∂a γ a + σˆ (σˆ − 1)θ a(x, t) . 2 Here θ = γ a Sab γ b = γ a γa (in the case if S ab is invertible). In the general case θ is an object such that for an arbitrary connection γ θ − γ 0 a S ab γ 0 b − 2∂a (γ a − S ab γ 0 b ) is a scalar. It is a Brans-Dicke type ”scalar”.

0

a


Canonical pencil of operators Restricting the operator ∆ on densities of weight σ we arrive at the operator pencil ∆σ , ∆σ (a(x)|Dx|σ ) = 1 ∂a S ab ∂b + (2σ − 1)γ a ∂a + σ ∂a γ a + σ (σ − 1)θ a(x)|Dx|σ , 2 σ ∈ R.


Canonical pencil of operators Restricting the operator ∆ on densities of weight σ we arrive at the operator pencil ∆σ , ∆σ (a(x)|Dx|σ ) = 1 ∂a S ab ∂b + (2σ − 1)γ a ∂a + σ ∂a γ a + σ (σ − 1)θ a(x)|Dx|σ , 2 σ ∈ R.

Theorem (”Universality” property) Let L be an arbitrary second order operator acting on densities of the weight σ . If σ 6= 0, 12 , 1 then there exists a unique canonical pencil which passes through the operator L, L = ∆σ . (H.Kh., T.Voronov)

Geometry of second order operators and Operators depending on a class of connections Second order operator on semidensitites and Batalin-Vilkovisky groupoid of connections

Special case: operators on semidensities, σ = 21 .

Fix S ab . Choose an arbitrary connection γa . Consider the canonical pencil at σ = 12 . ∂ γa 1 p p γ γ a γa a ab ∆ 1 a(x) |Dx| = ∂a S ∂b a(x) + a(x) − a(x) |Dx| 2 2 2 4 How this operator changes if we change the connection γ? γ γ0 γ 1 1 ∆ 1 → ∆ 1 = ∆ 1 + ∂a X a − 2γa X a + Xa X a = 2 2 2 4 8 1 2 γ γ 1 1 1 2 a a ∆ 1 + ∂a X − γa X − X = ∆ 1 + div γ X − X . 2 2 4 8 4 2

γ → γ 0 = γ +X,

γ

γ0

2

2

∆1 = ∆1

⇔

1 div γ X − X2 = 0 . 2


Groupoid of connections Let A be an affine space of all connections on volume forms. X Arrow: γ −→γ 0 such that γ, γ 0 ∈ A and γ 0 = γ + X. X

Set S of admissible arrows: S = {γ −→γ 0 :

1 div γ X − X2 = 0} 2

−X

X

Inverse arrow: If γ −→γ 0 ∈ S then γ 0 −→γ ∈ S. (If div γ X − 12 X2 = 0 then −div γ+X X − 12 X2 = 0). X

Y

Multiplication of arrows: if γ1 −→γ2 , γ2 −→γ3 ∈ S then X+Y

γ1 −→γ3 ∈ S. 1 1 1 (if div γ1 X − X2 = div γ2 Y − Y2 = 0 then div γ1 (X + Y) − (X + Y)2 = 0 .) 2 2 2

We call this groupoid the Batalin-Vilkovisky groupoid. (H.Kh., T. Voronov.)


Conclusion γ

Operator ∆ 1 depends not on a connection but only on its 2

equivalence class, the groupoid orbit Oγ of a connection γ, Oγ = {γ 0 : γ

γ0

2

2

∆1 = ∆1

⇔

X

γ −→ γ 0 ∈ S}. 1 div γ X − X2 = 0 . 2


Conclusion γ

Operator ∆ 1 depends not on a connection but only on its 2

equivalence class, the groupoid orbit Oγ of a connection γ, Oγ = {γ 0 : γ

γ0

2

2

∆1 = ∆1

⇔

X

γ −→ γ 0 ∈ S}. 1 div γ X − X2 = 0 . 2

Where such operators naturally arise?

Geometry of second order operators and Operators depending on a class of connections ∆-operator on odd symplectic supermanifolds

Consider a supermanifold M with coordinates z A = { |{z} x a , |{z} θ α }. Let S AB be a (super)symmetric contravariant even odd tensor on M: S AB = S BA (−1)p(A)p(B) . It defines ∆ = S AB ∂A ∂B + . . . . Suppose S AB is invertible.

1-st case . S AB is an even tensor: p(S AB ) = p(A) + p(B). S AB = g AB defines an even Riemannian structure. There exists the canonical volume form and the canonical flat connection on volume forms: p ρ(z)|Dz| = Ber gAB , γA = −∂A log ρ(z) . Moreover there exists the unique Levi-Civita connection ΓABC and γA = −∂A log ρ(z)|Dz| = −(−1)B ΓBBA .


2-nd case . S AB is an odd tensor: p(S AB ) = 1 + p(A) + p(B). S AB = ΩAB defines an odd symplectic structure 1 : {z A , z B } = (−1)A ΩAB . There are no canonical volume form (no Liouville Theorem!) and no canonical flat connection on volume forms. There are many affine connections compatible with the symplectic structure. One cannot choose a unique ”Levi-Civita” connection ΓABC . One cannot choose a distinguished connection on volume forms. Can we choose a class of connections? 1 We

need to impose the additional condition (ΩAB πA πB , ΩAB πA πB ) = 0 where (, ) is a canonical Poisson bracket on the cotangent bundle T ∗ M, providing the Jacobi identity for the odd bracket {f , g} = (f , (ΩAB πA πB , g)).



Canonical class of connections Definition We say that γA is a Darboux flat connection if there exist Darboux coordinates such that γA ≡ 0 in these Darboux coordinates.

Theorem All Darboux flat connections belong to the same orbit of the Batalin-Vilkovisky groupoid. That means that for two Darboux flat connections γ1 , γ2 1 X γ1 −→γ2 ∈ S, i.e. div X − X2 = 0 , 2 (I.A.Batalin, G.A.Vilkovisky 2 —H.Kh.—H.Kh.,T.Voronov) 2 The

statement relies on the Batalin-Vilkovisky identity: r A 0 ΩAB ∂A ∂B Ber ∂ zA 0 = 0 for Darboux coordinates z A , z A .


Example. Canonical ∆-operator on semidensitites Let γ be an arbitrary Darboux flat connection and {z A } be arbitrary Darboux coordinates. Then p O ∆ 1 γ a(z) |Dz| = 2



∂ γA p γ A γA 1 A AB ∂A Ω ∂B a(z) + a(z) − a(z) |Dz| 2 2 4



∂ γA p γ A γA 1 A AB ∂A Ω ∂B a(z) + a(z) − a(z) |Dz| 2 2 4 p 1 = ΩBA ∂A ∂B a(z) |Dz|, 2 since ΩBA is a constant tensor in Darboux coordinates and A A according to Theorem above, ∂A2γ − γ 4γA = 0 for an arbitrary Darboux flat connection.

Geometry of second order operators and Operators depending on a class of connections Invariant density on surfaces in odd symplectic sumpermanifold

Analogue of mean curvature for an odd symplectic structure. Let M be an odd symplectic supermanifold equipped with a volume form ρ(z)|Dz|. Let C be a surface of codimension (1|1) in M and Ψ(z) be an odd vector field which is symplectoorthogonal to the surface M. Consider A(∇, Ψ) = Tr (Π (∇Ψ)) − div ρ Ψ , where Π is the projector on (1|1)-dimensional plane symplectoorthogonal to the surface C, and ∇ is an arbitrary affine connection on M. (H.Kh., O. Little)


Analogue of mean curvature for an odd symplectic structure. Let M be an odd symplectic supermanifold equipped with a volume form ρ(z)|Dz|. Let C be a surface of codimension (1|1) in M and Ψ(z) be an odd vector field which is symplectoorthogonal to the surface M. Consider A(∇, Ψ) = Tr (Π (∇Ψ)) − div ρ Ψ , where Π is the projector on (1|1)-dimensional plane symplectoorthogonal to the surface C, and ∇ is an arbitrary affine connection on M. (H.Kh., O. Little) In the even Riemannian case (surface of codimension (1|0)) one can take the canonical Levi-Civita connection ∇LC and the Riemannian volume form. Then A(∇LC , Ψ) = |Ψ| · mean curvature of the surface C .


In the odd symplectic case there is no preferred affine connection compatible with the symplectic structure. Consider the class of Darboux flat affine connections. (Connection is Darboux flat if there exist Darboux coordinates such that Christoffel symbols ΓA B C ≡ 0 in these Darboux coordinates)

Theorem The magnitude A(∇, Ψ) does not depend on a connection in the class of Darboux flat connections: A(∇, Ψ) = A(∇ 0 , Ψ) for two arbitrary Darboux flat connections ∇ and ∇ 0 .


In the odd symplectic case there is no preferred affine connection compatible with the symplectic structure. Consider the class of Darboux flat affine connections. (Connection is Darboux flat if there exist Darboux coordinates such that Christoffel symbols ΓA B C ≡ 0 in these Darboux coordinates)

Theorem The magnitude A(∇, Ψ) does not depend on a connection in the class of Darboux flat connections: A(∇, Ψ) = A(∇ 0 , Ψ) for two arbitrary Darboux flat connections ∇ and ∇ 0 . This construction reveals the geometrical meaning of odd invariant semidensity obtained in 1984 (H.Kh., R.Mkrtchyan).


References O.M. Khudaverdian, A.S. Schwarz, Yu.S. Tyupkin. Integral invariants for Supercanonical Transformations. Lett. Math. Phys., v. 5 (1981), p. 517-522.


References O.M. Khudaverdian, A.S. Schwarz, Yu.S. Tyupkin. Integral invariants for Supercanonical Transformations. Lett. Math. Phys., v. 5 (1981), p. 517-522. O.M. Khudaverdian, R.L. Mkrtchian. Integral Invariants of Buttin Bracket. Lett. Math. Phys. v. 18 (1989), p. 229-231 (Preprint EFI–918–69–86- Yerevan (1986)).


I.A.Batalin, G.A Vilkovisky. Gauge algebra and quantization. Phys.Lett., 102B (1981), 27-31. O.M. Khudaverdian, A.P. Nersessian. On Geometry of Batalin-Vilkovisky Formalism. Mod. Phys. Lett. A, v. 8 (1993), No. 25, p. 2377-2385. Batalin-Vilkovisky Formalism and Integration Theory on Manifolds. J. Math. Phys., v. 37 (1996), p. 3713-3724. A.S. Schwarz. Geometry of Batalin Vilkovsiky formalism. Comm.Math.Phys., 155, (1993), no.2, 249–260.


H.M. Khudaverdian. Geometry of Superspace with Even and Odd Brackets. J. Math. Phys. v. 32 (1991) p. 1934-1937 (Preprint of the Geneva University, UGVA–DPT 1989/05–613). Odd Invariant Semidensity and Divergence-like Operators on Odd Symplectic Superspace. Comm. Math. Phys., v. 198 (1998), p. 591-606. Semidensities on odd symplectic supermanifold., Comm. Math. Phys., v. 247 (2004), pp. 353-390 (Preprint of Max-Planck-Institut fur ¨ Mathematik, MPI-135 (1999), Bonn.)

H.M. Khudaverdian, T.Voronov On Odd Laplace operators. Lett. Math. Phys. 62 (2002), 127-142 On odd Laplace operators. II. In Amer. Math. Soc. Transl. (2), Vol. 212, 2004, pp.179–205 Differential forms and odd symplectic geometry. Amer. Math. Soc. Transl (2) Vol 224, 2008 pp.159—171