Fermionic screenings and line bundle twisted chiral de Rham complex - PDF document

Fermionic screenings and line bundle twisted chiral de Rham complex on toric manifolds. S. E. Parkhomenko Landau Institute for Theoretical Physics Chernogolovka, Russia. 1. Introduction Chiral de Rham (CDR) complex is a certain sheaf of vertex operator algebras which is a string generalization of the usual de Rham complex. It has been introduced for every smooth manifold by Malikov Shechtman and Vaintrob. The CDR on Calabi-Yau (CY) com- pact manifold is a most important case to consider in string theory. In this situation CDR is closely related to the space of states of the (open) string on CY manifold. 1.1 Chiral de Rham complex in local coordinates. On the affine space A with local coordinates a 1 , ..., a d it is given by the set of bosonic fields a µ [ n ] z − n , � a µ ( z ) = n ∈ Z a ∗ a ∗ µ [ n ] z − n − 1 , � µ ( z ) = (1) n ∈ Z

and fermionic fields α µ [ n ] z − n − 1 � α µ ( z ) = 2 , n ∈ Z µ [ n ] z − n − 1 α ∗ α ∗ � µ ( z ) = 2 , n ∈ Z µ = 1 , ..., d (2) with the commutation relations between the modes [ a ∗ µ [ n ] , a ν [ m ]] = δ µν δ n + m, 0 , α ∗ � � µ [ n ] , α ν [ m ] = δ µν δ n + m, 0 (3) The space of sections M A of the chiral de Rham complex over the affine space A is gen- erated from vacuum state | 0 > by the non- positive modes of the fields a µ ( z ), α ν ( z ) and by negative modes of a ∗ µ ( z ), α ∗ ν ( z ).

1.2. The change of coordinates. The important property is the behavior of the ( bcβγ )fields (1), (2) under the local change of coordinates on A (Malikov, Shechtman, Vaintrob). For each new set of coordinates b µ = g µ ( a 1 , ..., a d ) a µ = f µ ( b 1 , ..., b d ) (4) the isomorphic bcβγ system of fields is given by b µ ( z ) = g µ ( a 1 ( z ) , ..., a d ( z )) µ ( z ) = ∂f ν b ∗ ( a 1 ( z ) , ..., a d ( z )) a ∗ ν ( z ) + ∂b µ ∂ 2 f λ ∂g ν ( a 1 ( z ) , ..., a d ( z )) α ∗ λ ( z ) α ρ ( z ) ∂b µ ∂b ν ∂a ρ β µ ( z ) = ∂g µ ( a 1 ( z ) , ..., a d ( z )) α ν ( z ) ∂a ν µ ( z ) = ∂f ν β ∗ ( a 1 ( z ) , ..., a d ( z )) α ∗ ν ( z ) ∂b µ (5) (the normal ordering is implied).

1.3. The geometric interpretation of the fields: a µ ( z ) ↔ coordinate a µ on A ∂ a ∗ µ ( z ) ↔ ∂a µ α µ ( z ) ↔ da µ α ∗ µ ( z ) ↔ conjugated to da µ , (6)

1.4. N = 2 Virasoro superalgebra currents and Calabi-Yau condition. On M A the N = 2 Virasoro superalgebra acts by the currents α ∗ J [ n ] z − n − 1 � � J ( z ) = µ ( z ) α µ ( z ) = µ n ∈ Z µ ∂ z a µ + 1 ( a ∗ 2( ∂ z α ∗ µ α µ − α ∗ � T ( z ) = µ ∂ z α µ )) = µ L [ n ] z − n − 2 � n ∈ Z G + [ n ] z − n − 3 G + ( z ) = − α ∗ � � µ ( z ) ∂ z a µ ( z ) = 2 µ n ∈ Z G − [ n ] z − n − 3 G − ( z ) = α µ ( z ) a ∗ � � µ ( z ) = 2 µ n ∈ Z (7)

Commutation relations [ L [ n ] , L [ m ]] = ( n − m ) L [ n + m ] + d 4( n 3 − n ) δ n + m, 0 , G + [ n ] , G − [ m ] � � = 2 L [ n + m ] + ( n − m ) J [ n + m ] + d ( n 2 − 1 4) δ n + m, 0 , [ J [ n ] , J [ m ]] = ndδ n + m, 0 , = ( n � L [ n ] , G ± [ m ] � 2 − m ) G ± [ n + m ] , � J [ n ] , G ± [ m ] � = ± G ± [ n + m ] , [ L [ n ] , J [ m ]] = − mJ [ n + m ] (8)

One can see that G − [0] = α µ [ n ] a ∗ � � µ [ − n ] (9) µ n ∈ Z is the string generalization of the de Rham differential and on zero level with respect to L [0]-grading we have the usual de Rham com- plex α λ [0] ...α γ [0] a µ [0] ...a ν [0] | 0 > ∈ M A ↔ forms α µ [0] a ∗ � d DR = µ [0] ↔ de Rham differential µ (10) • N = 2 Virasoro superalgebra is defined globally when the manifold is Calabi-Yau (CY) This is most important for the string theory applications. If it is not, only zero modes G + [0], G − [0], J [0], L [0] are defined globally. For the case of toric manifold as well as CY hypersurface in toric manifold the explicit con- struction of chiral de Rham complex has been found recently by L.Borisov.

In this talk I would like to represent a gen- eralization of Borisov construction to include chiral de Rham complex twisted by line bun- dle defined on toric manifold. (In case of d-dimensional CY manifold it may describe the state of states of bound (d,d-2) bound system of D -branes.)

1.5. Approach. The main object of the construction is a set of fermionic screening currents S ∗ S ∗ i [ n ] z − n − 1 � i ( z ) = (11) n associated to the points of the polytope ∆ ∗ defining the toric manifold P ∆ ∗ . Zero modes of these currents are used to build up a dif- ferential S ∗ � D ∆ ∗ = i [0] (12) i whose cohomology calculated in some lattice vertex algebra gives the global sections of a chiral de Rham complex on P ∆ ∗ . The ap- proach can also be applied for the case when we have a pair of dual (reflexive) polytopes ∆ ∗ and ∆ defining CY hypersurface in toric manifold P ∆ ∗ . 1.6. Generalization. We consider toric manifold P ∆ ∗ and general- ize differential allowing also non-zero modes for screening currents: S ∗ D ∆ ∗ → ˜ � D ∆ ∗ = i [ N i ] (13) i

Proposition. The numbers N i of screening current modes from ˜ D ∆ ∗ define the support function of toric divisor of a line bundle on P ∆ ∗ . By this means, the chiral de Rham complex on P ∆ ∗ appears to be twisted by the line bun- dle.

2. Chiral de Rham complex on P d . P ∆ ∗ = P d (14) Let me review the construction of chiral de Rham complex and its cohomology for the case of projective space P d . 2.1. The fan of P d . Let Λ = Z e 1 ⊕ ... ⊕ Z e d (15) be the lattice in R d . The vertices of the poly- tope ∆ ∗ ⊂ Λ are given by the vectors e 1 , ..., e d and the vector e 0 e 0 = − e 1 − ... − e d (16) Then one considers the fan Σ ⊂ Λ (17) coding the toric dates of P d . The fan Σ is a union of finite number of cones.

In our case d-dimensional cones are C i = Span ( e 0 , ..., ˆ e i , ..., e d ) ⊂ Σ , i = 0 , 1 , 2 , ..., d, (18) where the vector e i is missing. The standard toric construction associates to the cone C i the affine space A i ≈ C d (19) The affine spaces A i , i = 0 , 1 , ..., d cover the toric manifold P d P d = ∪ A i (20)

The intersection of an arbitrary number of cones C i is also a cone in Σ: C i ∩ C j ... ∩ C k = C ij...k ⊂ Σ (21) For positive codimensions cones C ij , C ijk ,... the toric construction associates the spaces A ij = A i ∩ A j ≈ C d − 1 × C ∗ , A ijk = A i ∩ A j ∩ A k ≈ C d − 2 × ( C ∗ ) 2 ... (22) The collection of spaces A i , A ij ,..., A 012 ...d can be used to calculate ˇ C ech cohomology of anything on P d .

2.2. Chiral de Rham complex over A i in logarithmic coordinates. Let X µ [ n ] z − n � X µ ( z ) = x µ + X µ [0] ln( z ) − n n � =0 X ∗ µ [ n ] X ∗ µ ( z ) = x ∗ µ + X ∗ z − n � µ [0] ln( z ) − n n � =0 ψ µ [ n ] z − n − 1 � ψ µ ( z ) = 2 n ∈ Z µ [ n ] z − n − 1 ψ ∗ ψ ∗ � µ ( z ) = 2 , n ∈ Z µ = 1 , ..., d (23) be free bosonic and free fermionic fields � = δ µν , � x µ , X ∗ � x ∗ � ν [0] µ , X ν [0] = δ µν , � = nδ µν δ n + m, 0 , n, m � = 0 , � X µ [ n ] , X ∗ ν [ m ] { ψ µ [ n ] , ψ ∗ ν [ m ] } = δ µν δ n + m, 0 (24) For the lattice Γ = Λ ⊕ Λ ∗ (25)

where Λ ∗ is dual to Λ we introduce the direct sum of Fock modules Φ Γ = ⊕ ( p,p ∗ ) ∈ Γ F ( p,p ∗ ) (26) where F ( p,p ∗ ) is the Fock module generated from the vacuum | p, p ∗ > µ [0] | p, p ∗ > = p ∗ µ | p, p ∗ > X ∗ X µ [0] | p, p ∗ > = p µ | p, p ∗ > (27) by the negative modes X µ [ n ], X ∗ µ [ n ], ψ ∗ µ [ n ], n < 0 and by non-positive modes ψ µ [ n ], n ≤ 0.

2.3. Fermionic screenings. For every vector e i , i = 0 , 1 , ..., d generat- ing 1-dimensional cone from Σ, one defines the fermionic screening current and screening charge i ( z ) = e i · ψ ∗ exp( e i · X ∗ )( z ) S ∗ � dzS ∗ i ( z ) = S ∗ i [0] (28) We form the BRST operator for every maxi- mal dimension cone C i D ∗ i = S ∗ S ∗ i [0] + ... + S ∗ 0 [0] + ... + ˆ d [0] , i = 0 , ..., d (29) where S ∗ i [0] is missing. Then, one considers the space Φ C i ⊗ Λ ∗ = ⊕ ( p,p ∗ ) ∈ C i ⊗ Λ ∗ F ( p,p ∗ ) (30) Proposition (L.Borisov). The space of sections M i of the chiral de Rham complex over the affine space A i is given by the cohomology of Φ C i ⊗ Λ ∗ with re- spect to the operator D ∗ i .

M i is generated by the creation modes of the bcβγ system of fields a iµ ( z ) = exp [ w ∗ iµ · X ]( z ) iµ ( z ) = ( e µ · ∂X ∗ − a ∗ w ∗ iµ · ψe µ · ψ ∗ ) exp [ − w ∗ iµ · X ]( z ) α iµ ( z ) = w ∗ iµ · ψ exp [ w ∗ iµ · X ]( z ) , iµ ( z ) = e µ · ψ ∗ exp [ − w ∗ α ∗ iµ · X ]( z ) , µ = 0 , ... ˆ (31) i, ...d from the vacuum state | 0 > = | ( p = 0 , p ∗ = 0) > (32) where w ∗ iµ ( e ν ) = δ µν , µ, ν = 0 , ..., ˆ (33) i, ..., d