Suppose M is a holomorphic symplectic manifold of complex dimension 2 k for ex

Suppose m is a holomorphic symplectic manifold of

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Suppose M is a holomorphic symplectic manifold of complex dimension 2 k , for ex- ample a hyperk¨ahler manifold. It has a holomorphic non-degenerate (2 , 0)-form ω c = ω 1 + 2 . The real and imaginary parts ω 1 2 are themselves real symplectic forms on M . Let t negationslash = 0 be a real number then exp( 2 /t ) is the generalized Calabi-Yau manifold defined by the symplectic form ω 2 /t . Apply the B-field B = ω 1 /t and we obtain exp(( ω 1 + 2 ) /t ) 9
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Multiply by the constant t k and we have a family of generalized Calabi-Yau structures defined by ϕ t = t k exp(( ω 1 + 2 ) /t ) = t k + ... + 1 k ! ( ω 1 + 2 ) k . Thus as t 0, these B-field transforms of the symplectic structure ω 2 /t converge to the Calabi-Yau structure defined by the (2 k, 0) form ( ω 1 + 2 ) k /k !. In this way we can think of B as interpolating between two extreme types of generalized Calabi-Yau manifold. 4.3 Dimension 2 Let M be a closed oriented surface. We consider the possible generalized Calabi-Yau structures on it. First consider the odd type , defined by a form in Ω 1 C . All such forms are pure. We thus have a closed complex 1-form ϕ such that 0 negationslash = ( ϕ, ¯ ϕ ) = ϕ ¯ ϕ. This non-vanishing 1-form is a (1 , 0)-form for a complex structure, and since it is closed, is holomorphic. This is therefore an ordinary Calabi-Yau – an elliptic curve . Now consider the even type . Here ϕ = c + β , c Ω 0 C Ω 2 C . Since ϕ is closed, c is a constant. Again, ϕ is always pure, but we also have 0 negationslash = ( ϕ, ¯ ϕ ) = c ¯ β ¯ cβ. (5) In particular c negationslash = 0 and then from (5) β/c ¯ β/ ¯ c = 2 negationslash = 0 so that ω is a symplectic form and ϕ = c exp( B + ) where 2 B = β/c + ¯ β/ ¯ c. Thus the structure is the B-field transform of a symplectic manifold. 4.4 Dimension 4 On a 4-manifold M a generalized Calabi-Yau structure of odd type is defined by ϕ = β + γ 10
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where β is a complex closed 1-form and γ a complex closed 3-form. The form ϕ must define a complex pure spinor for T T * . Here we are looking at the spin representation S - of the complexification Spin (8 , C ) of Spin (4 , 4). In eight dimensions however, we have the special feature of triality – the vector representation and the two spin representations are related by an outer automorphism of Spin (8 , C ). For us this means in particular that the two spin spaces S ± have the same structure as the vector representation – an 8-dimensional space with a non-degenerate quadratic form. The pure spinors are then just the null vectors in this space. It follows that ϕ is pure if 0 = ( ϕ,ϕ ) = β γ. (6) We also have the condition 0 negationslash = ( ϕ, ¯ ϕ ) = β ¯ γ + ¯ β γ negationslash = 0 (7) which shows in particular that β is nowhere vanishing. Thus from (6), γ = β ν for some 2-form ν , well-defined modulo β . Using (7) again, β ¯ β ( ν ¯ ν ) negationslash = 0 (8) and from this we can see that locally , the structure on M is defined by a map f : M C (where df = β ) defining a fibration over an open set, a symplectic structure ν and a B-field ν on the fibres. A global example is the product of an odd and an
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  • Fall '13
  • Physics, φ, generalized calabi-yau

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