Intoduction to Supergeometry Anton Galaev Masaryk University (Brno, - PowerPoint PPT Presentation

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Bilinear forms on a vector superspace. Let g : V ⊗ V → R be a bilinear form on the superspace V . g is symmetric if g ( y , x ) = ( − 1) | x || y | g ( x , y ); g is skew-symmetric if g ( y , x ) = − ( − 1) | x || y | g ( x , y ); g is even if g ( V ¯ 0 , V ¯ 1 ) = g ( V ¯ 1 , V ¯ 0 ) = 0; g is odd if g ( V ¯ 0 , V ¯ 0 ) = g ( V ¯ 1 , V ¯ 1 ) = 0. Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Let g be an even non-degenerate symmetric on R n | m = R n ⊕ Π( R m ), i.e. g ( R n , Π( R 2 k )) = g (Π( R 2 k ) , R n ) = 0, the restriction of g to R n is non-degenerate and symmetric (with some signature ( p , q ) , p + q = n ), the restriction of g to Π( R m ) is non-degenerate and skew-symmetric, i.e. m = 2 k . The orthosymplectic Lie superalgebra i | g ( ξ x , y )+( − 1) | x | ¯ i g ( x , ξ y ) = 0 } . osp ( p , q | 2 k ) ¯ i = { ξ ∈ gl ( n | 2 k , R ) ¯ Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Let e.g. the restriction of g to R n be positive definite   1 n 0 0     g =    . 0 0 1 k  0 − 1 k 0 Then,  �    �    �  A B 1 B 2      �    � A t = − A , C t 2 = C 2 , C t osp ( n | 2 k , R ) =  B t  � 3 = C 3 . C 1 C 2 2    �     �    − B t − C t � C 3 1 1 osp ( p , q | 2 k ) = ( so ( p , q ) ⊕ sp (2 k , R )) ⊕ R p , q ⊗ R 2 k Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Consider an odd non-degenerate supersymmetric form g on R n | n = R n ⊕ Π( R n ), i.e. g ( R n , R n ) = g (Π( R n ) , Π( R n )) = 0, and g ( x 0 , x 1 ) = g ( x 1 , x 0 ) for all x 0 ∈ R n , x 1 ∈ Π( R n ).    0 1 n There exists a basis of R n ⊕ Π( R n ) such that g =  . 1 n 0 The periplectic Lie superalgebra :    �  �   �  A B  � B = − B t , C = C t pe ( n , R ) = �   − A t � C pe ( n , R ) = gl ( n , R ) ⊕ ( S 2 R n ⊕ Λ 2 ( R n ) ∗ ) spe ( n , R ) = pe ( n , R ) ∩ sl ( n | n , R ) is simple. Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Consider an odd non-degenerate skew-symmetric form g on R n ⊕ Π( R n ). There exists a basis of R n ⊕ Π( R n ) such that   0 1 n   . g = − 1 n 0  �    �   �  A B  � pe sk ( n , R ) = B = B t , C = − C t  . �  − A t � C pe sk ( n , R ) ≃ pe ( n , R ) . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Let J be an odd complex structure on R n | n = R n ⊕ Π( R n ), i.e. J is an odd isomorphism of R n ⊕ Π( R n ) with J 2 = − id . The queer Lie superalgebra q ( n , R ) is the subalgebra of gl ( n | n , R ) commuting with J .   0 1 n There exists a basis of R n ⊕ Π( R n ) such that J =   . − 1 n 0 Then,        �  �      A B  A B � �   q ( n , R ) =  , sq ( n , R ) = trB = 0 �    � B A B A psq ( n , R ) = sq ( n , R ) / R E 2 n is simple. Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Examples of exceptional simple Lie superalgebras: 1 = C 7 ⊗ C 2 ; g = G (3) , 0 = G (2) ⊕ sl (2 , C ) , g ¯ g ¯ 1 = C 8 ⊗ C 2 . 0 = spin (7) ⊕ sl (2 , C ) , g = F (4) , g ¯ g ¯ Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Let V be a purely odd vector space, i.e. V = V ¯ 1 . By definition, S 2 V ∗ = { b : V ⊗ V → R | b ( x , y ) = ( − 1) | x || y | b ( y , x ) } , but | x | = | y | = ¯ 1, if x , y � = 0. This shows that b ( x , y ) = − b ( y , x ), S 2 V ∗ = Λ 2 Π V ∗ , S 2 V = Λ 2 Π V . Similarly, Λ 2 V = S 2 Π V . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras The odd vector superspace R 0 | m as the first example of a supermanifold 0 Consider R n . This is both a vector space and a smooth manifolds. The algebra of smooth functions on R n contains the dense subset of polynomial functions: S ∗ ( R n ) ∗ = ⊕ ∞ k =0 S k ( R n ) ∗ ⊂ C ∞ ( R n ) . Consider the odd vector space R 0 | m = Π R m . Then S ∗ (Π R m ) ∗ = ⊕ ∞ k =0 S k (Π R m ) ∗ = ⊕ ∞ k =0 Λ k ( R m ) ∗ = Λ ∗ ( R m ) ∗ = Λ( m ) . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras By this reason, C ∞ ( R 0 | m ) = Λ( m ) . Any f ∈ C ∞ ( R 0 | m ) has the form m � � f α 1 ··· α r ξ α 1 · · · ξ α r , f 0 , f α 1 ··· α r ∈ R . f = f 0 + r =1 1 ≤ α 1 < ··· <α r ≤ m The functions ξ α should play the role of coordinate functions on the ”manifold” R 0 | m . But ξ α ξ β + ξ β ξ α = 0 , ( ξ α ) 2 = 0 , i.e. these coordinate functions can not take real values (except 0). Since the coordinate functions should parametrise the points, we get only one point 0 in our ”manifold”. Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras By definition, R 0 | m is a supermanifold of superdimension 0 | m ; it is a pair R 0 | m = ( { 0 } , Λ( m )) , where 0 is the only point of R 0 | m and Λ( m ) is the algebra of superfunctions on R 0 | m . Define the value at the point 0 of the superfunction f ∈ C ∞ ( R 0 | m ) of the form � m � f α 1 ··· α r ξ α 1 · · · ξ α r , f 0 , f α 1 ··· α r ∈ R f = f 0 + r =1 1 ≤ α 1 < ··· <α r ≤ m by f (0) := f 0 ∈ R . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Consider the tangent space T 0 R 0 | m = { A : C ∞ ( R 0 | m ) → R | A ( fg ) = ( Af ) g (0)+( − 1) | A || f | f (0)( Ag ) } . Exercise. The odd vectors ( ∂ α ) 0 acting by ( ∂ α ) 0 f = ( ∂ α f ) 0 form a basis of T 0 R 0 | m , i.e. T 0 R 0 | m = R 0 | m . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Vector fields on R 0 | m : T R 0 | m = { A : C ∞ ( R 0 | m ) → C ∞ ( R 0 | m ) | A ( fg ) = ( Af ) g +( − 1) | A || f | f ( Ag ) } . ∂ξ α = ∂ α assuming ∂ α ξ β = δ β ∂ Define the odd vectorfields α . Exercise. T R 0 | m = Λ( m ) ⊗ R span R { ∂ 1 , ..., ∂ m } = Λ( m ) ⊗ R R 0 | m . Define the Lie superbrackets by [ A , B ] = A ◦ B − ( − 1) | A || B | B ◦ A . The Lie superalgebra T R 0 | m with this brackets is denoted by vect (0 | m , R ). It is a finite-dimensional Lie superalgebra. For m ≥ 2 it is simple . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras For X = X α ∂ α ∈ vect (0 | m , R ) define its divergence � ( − 1) | X α | ∂ α X α . div X = α Define the special (divergence-free) vectorial Lie superalgebra svect (0 | m ) = { X ∈ vect (0 | m , R ) | div X = 0 } . It is simple for m ≥ 3. Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Let m = 2 k . Consider the 2-form ω = � k α =1 d ξ α ◦ d ξ α + k . Assume 0, d ξ α ◦ d ξ β = d ξ β d ξ α . | d ξ α | = ¯ Define the Lie superalgebra of Hamiltonian vector fields ˜ h (0 | 2 k , R ) = { X ∈ vect (0 | 2 k , R ) | L X ω = 0 } . The Lie superalgebra h (0 | 2 k , R ) = [˜ h (0 | 2 k , R ) , ˜ h (0 | 2 k , R )] is simple. Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Classification of finite dim. simple complex Lie superalgebras: • classical type, i.e. the g ¯ 0 -module g ¯ 1 is completely reducible sl ( n | m , C ), psl ( n | n , C ), osp ( n | 2 m , C ), pe ( n , C ) , G (3), F (4),... • Cartan type vect (0 | n , C ), svect (0 | n , C ), h (0 | 2 k , C )... V. G. Kac, Lie superalgebras. Adv. Math., 26 (1977), 8–96. L. Frappat, A. Sciarrino, P. Sorba, Dictionary on Lie Superalgebras , arXiv:hep-th/9607161 Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Peculiarities: • zero Killing form e.g. on psl ( n | n , C ), pe ( n , C ); • in general no total reducibility of simple LSA; • semisimple LSA are of the from � g i ⊗ Λ( n i ); • there exist non-trivial irreducible representation of solvable LSA Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras The state of a quantum mechanical system is represented by a unit vector (defined up to a phase, i.e. a complex number of length 1) in a complex Hilbert space H . Let H describe the state of a single particle. Then the states of two identical particles v and v ′ is described by the tensor product H ⊗ H . Since the particles are identical, the states v ′ ⊗ v v ⊗ v ′ and must be the same. Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras But the state is defined up to a phase, consequently v ′ ⊗ v = λ v ⊗ v ′ . Applying this twice, we get λ 2 = 1, i.e. λ = ± 1. If λ = 1, then the particle is called boson . Two identical bosons are described by a vector in S 2 H . If λ = − 1, then the particle is called fermion . Two identical fermions are described by a vector in Λ 2 H . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras To unify the bosons and fermions consider the Hilbert superspace 0 ⊕ H ¯ H = H ¯ 1 , Where H ¯ 0 describes a boson and Π H ¯ 1 describes a fermion. Then � � S 2 H = S 2 H ¯ S 2 H ¯ H ¯ 0 ⊗ H ¯ 1 . 0 1 But S 2 H ¯ 1 = Λ 2 Π H ¯ 1 . Thus the summands of S 2 H describe two bosons, or a boson and a fermion, or two fermions. The sign rule of superalgebra encodes the statistics of a particle! Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Let A be a supercommutative superalgebra and M be a real vector superspace. M is a left A -supermodule if there exists a morphism · : A ⊗ R M → M , ( a , x ) �→ a · x , | a · x | = | a | + | x | . M can be also considered as a right A -supermodule if we put x · a = ( − 1) | x || a | a · x . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Let M and N be A -supermodules. A homogeneous map ϕ : M → N is called A -linear if ϕ ( ax ) = ( − 1) | ϕ || a | a ϕ ( x ) . Equivalently, ϕ ( xa ) = ϕ ( x ) a . Denote by Hom A ( M , N ) the vector superspace of all A -linear maps from M to N , and set End A ( M ) = Hom A ( M , M ). Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras We say that M over A is free of rank n | m if there exists a basis e 1 , ..., e n + m of M over A such that e 1 , ..., e n ∈ M ¯ 0 and e n +1 , ..., e n + m ∈ M ¯ 1 . This means that for any x ∈ M there exist x 1 , ..., x n + m ∈ A such that n + m � x a e a . x = a =1 Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Let M and N be free A -supermodules of ranks m | n and r | s . For an A -linear map ϕ : M → N define ϕ b a ∈ A , a = 1 , ..., n + m , b = 1 , ..., r + s such that r + s � f b ϕ b ϕ ( e a ) = a . b =1 We get an r + s × n + m matrix with elements from A . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Let x = � n + m a =1 e a x a ∈ M , y = ϕ ( x ) = � r + s b =1 f b y b ∈ N then � n + m � n + m n + m r + s � � � � ϕ ( e a ) x a = e a x a f b ϕ b a x a . ϕ ( x ) = ϕ = a =1 a =1 a =1 b =1 We get that n + m � y b = ϕ b a x a . a =1 In the matrix form       y 1 ϕ 1 ϕ 1 x 1 · · · n + m 1        .   . .   .  . . . .  ·    =     .  . . . .    ϕ r + s ϕ r + s y r + s x n + m · · · 1 n + m Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Since we have the decompositions M = M ¯ 0 ⊕ M ¯ 1 and N = N ¯ 0 ⊕ N ¯ 1 , the map ϕ can be divided into 4 parts. According to that we may write     ϕ 1 ϕ 1 · · · n + m 1    ϕ ¯ ϕ ¯  . .  0¯ 0¯ 0 1  = . . ϕ =    . . .  ϕ ¯ ϕ ¯ 1¯ 1¯ 0 1 ϕ r + s ϕ r + s · · · 1 n + m ϕ is even if and only if the entries of the matrices ϕ ¯ 0 and ϕ ¯ 1 are 0¯ 1¯ even and the entries of the matrices ϕ ¯ 0 and ϕ ¯ 1 are odd. 1¯ 0¯ Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras The dual space: M ∗ = Hom A ( M , A ). For ϕ ∈ Hom A ( M , N ) define ϕ ∗ ∈ Hom A ( N ∗ , M ∗ ), ϕ ∗ ( ξ ) = ( − 1) | ϕ || ξ | ξ ◦ ϕ. Then the matrix of ϕ ∗ w.r.t. the dual bases f ∗ b and e ∗ a has the form (exercise)   ϕ t ( − 1) | ϕ | +1 ϕ t ¯ 0¯ ¯ 1¯  0 0  . ( − 1) | ϕ | ϕ t ϕ t ¯ 0¯ ¯ 1¯ 1 1 Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Let L be an r + s × n + m matrix with elements form A    L ¯ L ¯ 0¯ 0¯ 0 1  L = L ¯ L ¯ 1¯ 1¯ 0 1 (i.e. it can be the matrix of a homomorphism from M to N ) We say that L is even if the entries of the matrices L ¯ 0 and L ¯ 1 are 0¯ 1¯ even and the entries of the matrices L ¯ 0 and L ¯ 1 are odd. 1¯ 0¯ Define the supertransposed matrix   ( − 1) | L | +1 L t L t L st = 0¯ ¯ ¯ 1¯  0 0  . ( − 1) | L | L t L t 0¯ ¯ ¯ 1¯ 1 1 Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras Consider set Mat A ( n | m ) of all squire matrices of order n + m with elements from A . It becomes an A -supermodule with respect to the multiplication   aL ¯ aL ¯ 0¯ 0¯ 0 1   . aL = ( − 1) | a | aL ¯ ( − 1) | a | aL ¯ 1¯ 1¯ 0 1 Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras    L ¯ L ¯ 0¯ 0¯ 0 1  define the supertrace For a homogenious L = L ¯ L ¯ 1¯ 1¯ 0 1 0 − ( − 1) | L | tr L ¯ str L = tr L ¯ 1 . 0¯ 1¯ Proposition. str ([ K , L ]) = 0 . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras The group GL A ( n | m ) = { L ∈ Mat A ( n | m ) | | L | = ¯ 0 , L is invertible } is called general linear supergroup of rank n | m over A . Example. GL R ( n | m ) = GL ( n , R ) × GL ( m , R ) . Theorem. Let L ∈ Mat A ( n | m ). Then L ∈ GL A ( n | m ) if and only if L ¯ 0 and L ¯ 1 are invertible. 0¯ 1¯ Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras    B 00 B 01  be a usual real matrix. Suppose that B 11 is Let B = B 10 B 11 invertible, then     B 01 B − 1  B 00 − B 01 B − 1  1 11 B 10 0 11   , B = 0 1 B 10 B 11 consequently, det B = det( B 00 − B 01 B − 1 11 B 10 ) · det B 11 . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Lie superalgebras Lie superalgebras of vector fields on R 0 | m Superdomains Supermanifolds About quantum particles and supersymmetry Supersymmetries Modules over supercommutative superalgebras For L ∈ GL A ( n | m ) define its superdeterminant or Berezian 1 L − 1 0 ) · det L − 1 BerL = det( L ¯ 0 − L ¯ 1 L ¯ 1 ∈ A ¯ 0 . 0¯ 0¯ 1¯ ¯ 1¯ 1¯ ¯ Theorem. Ber ( KL ) = Ber ( K ) · Ber ( L ) . Ber ( E n + m + ǫ L ) = 1 + ǫ str L , ǫ 2 = 0. Ber exp L = e str L . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries A superdomain of dimension n | m U = ( U , C ∞ ( U )) , C ∞ ( U ) = C ∞ ( U ) ⊗ Λ( m ) . U ⊂ R n , Let ξ 1 , ..., ξ m be generators of Λ( m ), then any f ∈ C ∞ ( U ) can be written as m � � f α 1 ...α r ξ α 1 · · · ξ α r , f = ˜ ˜ f , f α 1 ...α r ∈ C ∞ ( U ) . f + α 1 < ··· <α r r =1 f ( x ) := ˜ x ∈ U ⇒ f ( x ) ∈ R . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries A morphism of superdomains: ϕ : U = ( U , C ∞ ( U )) → V = ( V , C ∞ ( V )) is a pair ϕ ∗ : C ∞ ( V ) → C ∞ ( U ) ϕ, ϕ ∗ ) , ϕ : U → V , ϕ = ( ˜ ˜ such that ( ϕ ∗ f )( x ) = f ( ˜ ϕ ( x )) . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries If ψ = ( ˜ ψ, ψ ∗ ) : V → W is another morphism, then the decomposition is defined as ϕ, ϕ ∗ ◦ ψ ∗ ) : U → W . ψ ◦ ϕ = ( ˜ ψ ◦ ˜ ϕ : U → V is called a diffeomorphism if it admits an inverse morphism. Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Example. The inclusion i = (˜ ˜ i ∗ ( f ) = ˜ i , i ∗ ) : U → U , i ( x ) = x , f . The projection p , p ∗ ) : U → U , p ∗ ( f ) = f . p = (˜ ˜ p ( x ) = x , Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Proposition. For any morphism of superalgebras ϕ ∗ : C ∞ ( V ) → C ∞ ( U ) there exists a unique continuous map ϕ, ϕ ∗ ) is a morphism from U to V . ϕ : U → V such that ϕ = ( ˜ ˜ Proof. The composition C ∞ ( V ) → C ∞ ( V ) → C ∞ ( U ) → C ∞ ( U ) defines map ϕ : U → V , which is compatible with ϕ ∗ . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Corollary. For any morphism s : C ∞ ( U ) → R there exists a unique point x ∈ U such that s ( f ) = f ( x ). Proof. Since R = C ∞ ( pt ), ϕ ∗ = s defines ϕ : pt → U . Let x = ˜ ϕ ( pt ) ∈ U . Since ϕ ∗ ( f ) ∈ R , ϕ ∗ ( f ) = ϕ ∗ ( f )( pt ) = f ( ˜ ϕ ( pt )) = f ( x ) . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Systems of coordinates. Consider a superdomain U = ( U , C ∞ ( U ) = C ∞ ( U ) ⊗ Λ( m )) . Let x 1 , ..., x n be coordinates on U ; ξ 1 , ..., ξ m odd generators of Λ( m ). The superfunctions x 1 , ..., x n , ξ 1 , ..., ξ m are called coordinates on U . x n + α = ξ α . Denotation ( x i , ξ α ), or ( x a ) , Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Vector fields on U . T U = ( T U ) ¯ 0 ⊕ ( T U ) ¯ 1 ,  �  �   | X | = ¯ � i , X is R -linear  X : C ∞ ( U ) → C ∞ ( U ) � ( T U ) ¯ i = �  X ( fg ) = X ( f ) g + ( − 1) | f || X | fX ( g ) � Define the vector fields ∂ x i and ∂ ξ α assuming ∂ x i ( f ξ α 1 · · · ξ α r ) = ∂ f ∂ x i ξ α 1 · · · ξ α r , r � ∂ ξ α ( f ξ α 1 · · · ξ α r ) = ( − 1) s − 1 δ αα s f ξ α 1 · · · � ξ α s · · · ξ α r . s =1 Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Proposition. The C ∞ ( U )-module T U is free of rank n | m . T U = C ∞ ( U ) ⊗ R span R { ∂ x 1 , ..., ∂ ξ m } . Proof. Let X ∈ T U . We claim that X = ( Xx a ) ∂ a . Consider X ′ = X − ( Xx a ) ∂ a , X ′ ( fg ) = X ′ ( f ) g + ( − 1) | f || X ′ | fX ′ ( g ). For f ∈ C ∞ ( U ) let X ′ ( f ) = � X ′ α 1 ,...,α r ( f ) ξ α 1 · · · ξ α r , then X ′ α 1 ,...,α r : C ∞ ( U ) → C ∞ ( U ) , X ′ α 1 ,...,α r ( fg ) = X ′ α 1 ,...,α r ( f ) g + fX ′ X ′ α 1 ,...,α r ( x i ) = 0 , α 1 ,...,α r ( g ) , X ′ X ′ ( f ) = 0. = ⇒ α 1 ,...,α r = 0 , X ′ = 0. Moreover, X ′ ( ξ α ) = 0 = ⇒ Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Lemma. Let ϕ : U → V be a morphism, then � ∂ f � � ∂ϕ ∗ ( y b ) ∂ ∂ x a ( ϕ ∗ f ) = ϕ ∗ , ∂ x a ∂ y b b f ∈ C ∞ ( V ) . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Theorem. If ϕ : U → V is a morphism and y 1 , ..., y r , η 1 , ..., η s are coordinates on V , then the functions ϕ ∗ ( y 1 ) , ..., ϕ ∗ ( y r ) , ϕ ∗ ( η 1 ) , ..., ϕ ∗ ( η s ) uniquely define ϕ . Proof. Note: if g = � g α 1 ,...,α p ξ α 1 · · · ξ α p ∈ C ∞ ( U ), then g α 1 ,...,α p = ( ∂ ξ α p · · · ∂ ξ α 1 g ) ∼ . First let f = f ( y 1 , ..., y r ) ∈ C ∞ ( V ), then we may find ϕ ∗ ( f ) using the previous formula and the lemma: ϕ ∗ � � ϕ ∗ � � e.g. ∂ ξ α ϕ ∗ ( f ) = � = � ∂ϕ ∗ ( y b ) ∂ϕ ∗ ( y i ) ∂ f ∂ f . b ∂ξ α ∂ y b b ∂ξ α ∂ y i In general, if f = � f β 1 ,...,β p θ β 1 · · · θ β p ∈ C ∞ ( V ), then ϕ ∗ ( f ) = � ϕ ∗ ( f β 1 ,...,β p ) ϕ ∗ ( θ β 1 ) · · · ϕ ∗ ( θ β p ) . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries This gives the so-called symbolic way of calculation : if U and V are superdomains with coordinates ( x , ξ ) = ( x i , ξ α ) and ( y , θ ) = ( y k , θ β ), a morphism ϕ : U → V can be written symbolically ϕ : ( x , ξ ) �→ ( y , θ ) , y = y ( x , ξ ) , θ = θ ( x , ξ ) , where in fact y k = ϕ ∗ ( y k ) = y k ( x i , ξ α ) , θ β = ϕ ∗ ( θ β ) = θ β ( x i , ξ α ) . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries We may write ϕ ∗ ( f )( x i , ξ α ) = f ( y j ( x i , ξ α ) , θ ( x i , ξ α )) and find this function using the above proof. Example. Let U = V = R 1 | 2 with the coordinates x , ξ 1 , ξ 2 and ϕ is given by ϕ ∗ ( x ) = x + ξ 1 ξ 2 , ϕ ∗ ( ξ 1 ) = ξ 1 , ϕ ∗ ( ξ 2 ) = ξ 2 . Let f = f ( x ), then f ( x + ξ 1 ξ 2 ) = ( ϕ ∗ f )( x , ξ 1 , ξ 2 ) , ( ϕ ∗ f )( x , ξ 1 , ξ 2 ) = ( ϕ ∗ f ) ∼ ( x ) + ( ϕ ∗ f ) 12 ( x ) ξ 1 ξ 2 , ( ϕ ∗ f ) ∼ ( x ) = f ( x ) , Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries ( ϕ ∗ f ) 12 = ( ∂ ξ 2 ∂ ξ 1 ϕ ∗ ( f )) ∼ = ( ∂ ξ 2 ( ∂ ξ 1 ( ϕ ∗ ( x )) ϕ ∗ ( ∂ x f ))) ∼ = ( ∂ ξ 2 ( ξ 2 ϕ ∗ ( ∂ x f ))) ∼ = ( ϕ ∗ ( ∂ x f )) ∼ − ( ξ 2 ∂ ξ 2 ϕ ∗ ( ∂ x f )) ∼ = ∂ x f . Thus, f ( x + ξ 1 ξ 2 ) = f ( x ) + ∂ x f ( x ) ξ 1 ξ 2 Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries We see that if f ∈ C ∞ ( V r | s ), then we may consider the expression f ( g 1 , ..., g r , h 1 , ..., h s ) , where g 1 , ..., g r and h 1 , ..., h r are respectively even and odd functions on some U . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Let x 1 , ..., x n , ξ 1 , ..., ξ m be coordinates on U . If ϕ : U → V is a diffeomorphism and y 1 , ..., y n , η 1 , ..., η m are coordinates on V as above, then the functions ϕ ∗ ( y 1 ) , ..., ϕ ∗ ( y n ) , ϕ ∗ ( η 1 ) , ..., ϕ ∗ ( η m ) are also called coordinates on U . In that case ϕ ∗ ( y 1 ) , ..., ϕ ∗ ( y n ) are not necessary coordinates on U . By the above considerations, the expression f ( y j , θ β ) = f ( x i ( y j , θ β ) , ξ α ( y j , θ β )) makes sense. Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Examples of morphisms. 1. ϕ : R n → R k | m : since ( θ β ) 2 = 0 , ( ϕ ∗ ( θ β )) 2 = 0, but ϕ ∗ ( θ β ) ∈ C ∞ ( R n ) = ⇒ ϕ ∗ ( θ β ) = 0, ϕ : R n → R k . thus ϕ is given by ˜ Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries 2. ϕ : R 0 | 2 → M n | 0 : f ∈ C ∞ M = ϕ ∗ ( f ) = a ( f ) + b ( f ) ξ 1 ξ 2 , ⇒ a ( f ) , b ( f ) ∈ R . ⇒ a ( fg ) + b ( fg ) ξ 1 ξ 2 = ϕ ∗ ( fg ) = ϕ ∗ ( f ) ϕ ∗ ( g ) = a ( f ) b ( f ) + ( a ( g ) b ( f ) + a ( f ) b ( g )) ξ 1 ξ 2 , ⇒ a ( fg ) = a ( f ) a ( g ) , = b ( fg ) = a ( g ) b ( f ) + a ( f ) b ( g ) ⇒ a : C ∞ M → R is a homomorphism = ⇒ ∃ x ∈ M , a ( f ) = f ( x ), = finally, b ( fg ) = b ( g ) f ( x ) + f ( x ) b ( g ), i.e. b ∈ T x M . Thus, ϕ is defined by a point x ∈ m and a tangent vector b ∈ T x M , ϕ ∗ ( f ) = f ( x ) + b ( f ) ξ 1 ξ 2 . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Example. Let E → U be a vector bundle over U , U = ( U , Γ( U , Λ E )) . If ξ 1 , ..., ξ m are generators of Γ( U , Λ E ), then x 1 , ..., x n , ξ 1 , ..., ξ m are coordinates on U . Any automorphism ϕ of the bundle Λ E → U preserving the parity defines the automorphism of U : ϕ ∗ ( x i ) = ϕ 0 ( x 1 , ..., x n ) , � � α 1 ...α 2 r +1 ( x 1 , ..., x n ) ξ α 1 · · · ξ α 2 r +1 . ϕ ∗ ( ξ α ) = ϕ α r ≥ 0 α 1 < ··· <α 2 r +1 Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Any morphism of U has the coordinate form � � α 1 ...α 2 r ( x 1 , ..., x n ) ξ α 1 · · · ξ α 2 r , ϕ ∗ ( x i ) = ϕ 0 ( x 1 , ..., x n )+ ϕ α r ≥ 1 α 1 < ··· <α 2 r � � α 1 ...α 2 r +1 ( x 1 , ..., x n ) ξ α 1 · · · ξ α 2 r +1 . ϕ ∗ ( ξ α ) = ϕ α r ≥ 0 α 1 < ··· <α 2 r +1 Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Let ϕ : U → V be a morphism and X ∈ T U . We get the map X ◦ ϕ ∗ : C ∞ ( V ) → C ∞ ( U ). ϕ ∗ � � � ∂ ∂ x a ◦ ϕ ∗ � f = � ∂ϕ ∗ ( y b ) ∂ f , f ∈ C ∞ ( V ). Lemma. ∂ x a b ∂ y b In the matrix form:       ∂ ( ϕ ∗ f ) ∂ ( ϕ ∗ y j ) ∂ ( ϕ ∗ η β )  ϕ ∗ ∂ f  ∂ x i  =  ∂ x i ∂ x i  · ∂ y j  . ∂ ( ϕ ∗ f ) ∂ ( ϕ ∗ y j ) ∂ ( ϕ ∗ η β ) ϕ ∗ ∂ f ∂ξ α ∂ξ α ∂ξ α ∂η β Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Define the Jacoby matrix of ϕ :   st ∂ ( ϕ ∗ y j ) ∂ ( ϕ ∗ η β ) ∂ x i ∂ x i   J ( ϕ ) = = ∂ ( ϕ ∗ y j ) ∂ ( ϕ ∗ η β ) ∂ξ α ∂ξ α   ∂ ( ϕ ∗ y 1 ) ∂ ( ϕ ∗ y 1 ) − ∂ ( ϕ ∗ y 1 ) − ∂ ( ϕ ∗ y 1 ) · · · · · ·   ∂ x 1 ∂ x n ∂ξ 1 ∂ξ m   . . . .  . . . .  . . . .     ∂ ( ϕ ∗ y r ) ∂ ( ϕ ∗ y r ) − ∂ ( ϕ ∗ y r ) − ∂ ( ϕ ∗ y r )   · · · · · ·   ∂ x 1 ∂ x n ∂ξ 1 ∂ξ m   .  ∂ ( ϕ ∗ η 1 ) ∂ ( ϕ ∗ η 1 ) ∂ ( ϕ ∗ η 1 ) ∂ ( ϕ ∗ η 1 )  · · · · · ·   ∂ x 1 ∂ x n ∂ξ 1 ∂ξ m   . . . .   . . . . . . . .     ∂ ( ϕ ∗ η s ) ∂ ( ϕ ∗ η s ) ∂ ( ϕ ∗ η s ) ∂ ( ϕ ∗ η s ) · · · · · · ∂ x 1 ∂ x n ∂ξ 1 ∂ξ m Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Lemma. If ϕ : U → V and ψ : V → W are morphisms, then J ( ψ ◦ ϕ ) = ϕ ∗ ( J ( ψ )) · J ( ϕ ) . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Berezin integral. Let x 1 , ..., x n , ξ 1 , ..., ξ m be coordinates on U such that x 1 , ..., x n � are coordinates on U ; let f ∈ C ∞ ( U ). to define U f assume the following: � � d ξ α = 0 , ξ α d ξ α = 1 , ξ α d ξ β = − d ξ β · ξ α , ξ α dx i = dx i · ξ α . Using that, we get � � dx 1 · · · dx n d ξ 1 · · · d ξ m f = ( − 1) m ( m − 1) dx 1 · · · dx n f 1 ··· m . 2 U U Note that � � dx 1 · · · dx n d ξ 1 · · · d ξ m f = dx 1 · · · dx n ∂ ξ 1 · · · ∂ ξ m f . U U Anton Galaev Intoduction to Supergeometry

Linear superalgebra Superdomains Supermanifolds Supersymmetries Theorem. Let ϕ : U → V be a diffeomorphism of superdomains. Let f ∈ C ∞ ( V ) have a compact support. Then � � ϕ ∗ f · Ber ( J ( ϕ )) . f = V U Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries Sheaves. Let M be a topological space. A sheaf F of algebras (vector spaces, groups,...) on M is an assignment U �→ F ( U ) to each open subset U ⊂ M of an algebra (vector space, group) F ( U ) such that the following conditions are satisfied. If V ⊂ U , then there exists a homomorphism map ρ U , V : F ( U ) → F ( V ) , f �→ ρ U , V ( f ) such that 1) ρ U , U = id ; 2) ρ W , V = ρ U , V ◦ ρ W , U , V ⊂ U ⊂ W 3) if ( U i ) is a covering of U , f i ∈ F ( U i ), ρ U i , U i ∩ U j ( f i ) = ρ U j , U i ∩ U j ( f j ) , then there exists a unique f ∈ F ( U ) such that ρ U , U i f = f i . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries A morphism ϕ : F → T of two sheaves on M is a collection of maps ϕ ( U ) : F ( U ) → T ( U ) , U ⊂ M is open such that r U , V ◦ ϕ ( U ) = ϕ ( V ) ◦ ρ U , V , V ⊂ U . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries Example. M is a smooth manifold, and C ∞ M is the sheaf of smooth functions on M : C ∞ M ( U ) are smooth functions on the subset U ⊂ M . Note that a smooth manifolds may be defined as a pair ( M , C ∞ M ), where M is a Hausdorf topological space, and C ∞ M is a sheaf of commutative algebras on M locally isomorphic to the sheaf of smooth functions on an open subset of R n . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries Example. E → M is a vector bundle over a smooth manifold M , U �→ Γ( U , E ) is the sheaf of smooth sections of E . Note that this sheaf allows to reconstruct E . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries Definition of a supermanifold: A supermanifold of dimension n | m is a pair M = ( M , O M ), where M is a Hausdorf topological space, and O M is a sheaf of commutative superalgebras on M locally isomorphic to the sheaf of superfunctions on an open subset of R n | m . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries A morphism of two supermanifolds ϕ : M → N is a pair ϕ, ϕ ∗ ), where ˜ ϕ = ( ˜ ϕ : M → N is a continuous map and a morphism of sheaves ϕ ∗ : O N → ϕ ∗ O M , here ϕ ∗ O M is the induced sheaf on N : ϕ ∗ O M ( U ) = O M ( ϕ − 1 ( U )), U ⊂ N . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries Consider M and define the sheaf C ∞ M : C ∞ M ( U ) = O M ( U ) / ( O M ( U ) ¯ 1 ) . Then C ∞ M defines the structure of a smooth manifold on M . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries The inclusion i = (˜ i , i ∗ ) : M → M , ˜ i ∗ ( f ) = ˜ i ( x ) = x , f , where f ∈ O M ( U ) �→ ˜ f ∈ C ∞ M ( U ) = O M ( U ) / ( O M ( U ) ¯ 1 ) . If there exists a splitting O M ( U ) = C ∞ M ( U ) ⊕ ( O M ( U ) ¯ 1 ), then there is an inclusion C ∞ M ( U ) ⊂ O M ( U ), and one considers the projection p , p ∗ ) : M → M , p ∗ ( f ) = f , p = (˜ p ( x ) = x , ˜ . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries Example. Let E → M be a vector bundle over M , define O M ( U ) = Γ( U , Λ E ) , U ⊂ M . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries Definition of a supermanifold using local charts A coordinate chart on a topological space M is a pair ( U , c ), where U ⊂ R n | m is a superdomain, and c : U → M is a homeomorphism on c ( U ). Two charts ( U 1 , c 1 ) and ( U 2 , c 2 ) are compatible, if there exists a diffeomorphism γ 12 : ( U 12 , C ∞ U 1 | U 12 ) → ( U 21 , C ∞ U 2 | U 21 ) , γ 12 = c − 1 ˜ ◦ c 1 | U 12 2 U 12 = c − 1 U 21 = c − 1 1 ( c 1 ( U 1 ) ∩ c 2 ( U 2 )) , 2 ( c 1 ( U 1 ) ∩ c 2 ( U 2 )) Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries An atlas on a topological space M is a set of compatible charts (( U α , c α ) , γ αβ ) such that ∪ α c α ( U α ) = M , γ βα = γ − 1 αβ , γ αβ γ βδ γ δα = id . A supermanifold M is a pair: a topological space M and an atlas (( U α , c α ) , γ αβ ). Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries Product of supermanifolds If U and V are superdomains with the coordinates x 1 , ..., x n , ξ 1 , ...ξ m , y 1 , ..., y r , θ 1 , ..., θ s , then U × V is a superdomain with the base U × V and coordinates x 1 , ..., x m , y 1 , ..., y r , ξ 1 , ..., ξ m , θ 1 , ..., θ s . If M = ( M , ( U α , c α ) , γ αβ ) and N = ( N , ( V µ , c µ ) , γ µν ) are supermanifolds, then the product M × N is defined by ( M × N , ( U α × V µ , c α × c µ ) , γ αβ × γ µν ) . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries Theorem of Batchelor (1979). Let M = ( M , O M ) be a supermanifold. Then there exists a vector bundle E → M such that M ≃ ( M , Γ( · , Λ E )). Moreover, there is the following one-to-one correspondence:        vector bundles of rank m         Supermanifolds            over n -dim. smooth ← → of dim. n | m mod.         manifolds mod.           isomorphisms of supermf.     isom. of vector bundles. Morphisms of supermanifolds are in general not induced by morphisms of vector bundles! Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries The tangent sheaf: T M = ( T M ) ¯ 0 ⊕ ( T M ) ¯ 1 , ( T M ) ¯ i ( U ) =  �  �   | X | = ¯ � i , X is R -linear �  X : O M ( U ) → O M ( U ) �  X ( fg ) = X ( f ) g + ( − 1) | f || X | fX ( g ) � The vector fields ∂ i = ∂ x i , ∂ α = ∂ ξ α form a local basis of T M ( U ) ⇒ T M is a locally free sheaf of supermodules over O M Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries x ∈ M , the tangent space: T x M = { X : O M , x → R | X ( fg ) = X ( f ) g ( x )+( − 1) | f || X | f ( x ) X ( g ) } . The vectors ( ∂ x 1 ) x , ..., ( ∂ ξ m ) x span T x M (( ∂ x a ) x f = ( ∂ x a f )( x )). Note: ( T x M ) ¯ 0 = T x M . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries A Lie supergroup is a supermanifold G = ( G , O G ) together with e : R 0 | 0 → G three morphisms µ : G × G → G , i : G → G , G × G G × G id × µ ( id , e ) ✲ ✲ µ µ ✲ id ✲ G × R 0 | 0 = G G × G × G G G ✲ µ ✲ µ ✲ µ × id ( e , id ) ✲ ✲ G × G G × G G × G d ✲ i µ × i ✲ G e ✲ G ✲ µ ✲ i d × ✲ i G × G Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries Action of a Lie supergroup G on a supermanifold M : is a morphism a : G × M → M such that G × M id G × a ✲ a ✲ G × G × M G ✲ a µ × id M ✲ G × M Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries The Lie superalgebra of a Lie supergroup. A vectorfield X ∈ T G ( G ) is called left-invariant if (1 ⊗ X ) ◦ µ ∗ = µ ∗ ◦ X : O G ( G ) → O G×G ( G × G ) . The Lie superalgebra g of the Lie supergroup G is the Lie superalgebra of left-invariant vector fields on G . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries Proposition. The vector superspace g can be identified with the tangent space T e G . The isomorphism is given by X e ∈ T e G �→ X = (1 ⊗ X e ) ◦ µ ∗ ∈ g . Note: g ¯ 0 is the Lie algebra of G . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries Super Harish-Chandra pairs. The Lie supergroup G defines canonically the pair ( G , g ), g ¯ 0 = Lie ( G ); there exists Ad : G → gl ( g ), Ad | G × g ¯ d Ad | g ¯ 1 = [ · , · ] g ¯ 0 = Ad G , 1 . 0 × g ¯ 0 × g ¯ Conversely, any such pair ( G , g ) defines a Lie supergroup G . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries Example. An action a : G × M → M can be given by an action of G on M and by a morphism g → ( T M ( M )) 0 such that the differential of the action of G coincides with the representation of g ¯ 0 . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries Example. A representation of G on a vector superspace V consists of a representation of G on V and of a morphism g → gl ( V ) such that the differential of the representation of G coincides with the representation of g ¯ 0 . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries Example. GL ( n | m , R ) = ( GL ( n , R ) × GL ( m , R ) , gl ( n | m , R )) , OSp ( n | 2 m , R ) = ( O ( n ) × Sp (2 m , R ) , osp ( n | 2 m , R )) . Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries Functor of points. Let M be a fixed supermanifold, and S is another supermanifold. An S -point of M is a morphism S → M . The set of S -points of M : M ( S ) = Hom ( S , M ) . Any morphism ψ : S 1 → S 2 defines the morphism M ( ψ ) : M ( S 2 ) → M ( S 1 ) , ϕ �→ ψ ◦ ϕ. The map S �→ M ( S ) is a contravariant functor from the category of supermanifolds to the category of sets. Anton Galaev Intoduction to Supergeometry

Linear superalgebra Supermanifolds Superdomains Lie supergroups Supermanifolds Functor of points Supersymmetries A morphism of supermanifolds ϕ : M → N induces the map ϕ S : M ( S ) → N ( S ) , ψ �→ ϕ ◦ ψ. Yoneda’s Lemma. For given maps { f S : M ( S ) → N ( S ) } S that are functorial in S , there exists a unique morphism ϕ : M → N such that ϕ S = f S . α : T → S f S ✲ N ( S ) M ( S ) M ( α ) N ( α ) ❄ f T ✲ N ( T ) ❄ M ( T ) Anton Galaev Intoduction to Supergeometry