Vertex Operator Super Algebras on a Riemann Surface Alexander Zuevsky National University of Ireland, Galway In collaboration with: Geoffrey Mason, University of California Santa Cruz and Michael Tuite, National University of Ireland, Galway Alexander Zuevsky VOSAs on a Riemann Surface
Introduction 1 Sewing tori to form a genus two Riemann surface 2 The genus two partition function for a VOA 3 The Heisenberg VOA 4 The rank two fermionic Vertex Operator Super Algebra 5 Higher genus considerations Alexander Zuevsky VOSAs on a Riemann Surface
1. Sewing Tori to Form a Genus Two Riemann Surface Consider two oriented tori Σ a = C / Λ τ a with a = 1 , 2 for Λ τ a = 2 πi ( Z ⊕ τ a Z ) for τ a ∈ H 1 , the complex upper half plane. For z a ∈ Σ a the closed disk | z a | ≤ r a is contained in Σ a provided r a < 1 2 D ( τ a ) where D ( τ a ) = λ ∈ Λ τa ,λ � =0 | λ | = minimal lattice distance . min Introduce a sewing parameter ǫ ∈ C and excise the disks | z 1 | ≤ | ǫ | /r 2 and | z 2 | ≤ | ǫ | /r 1 where | ǫ | ≤ r 1 r 2 < 1 4 D ( τ 1 ) D ( τ 2 ) . Alexander Zuevsky VOSAs on a Riemann Surface
Identify annular regions | ǫ | /r 2 ≤ | z 1 | ≤ r 1 and | ǫ | /r 1 ≤ | z 2 | ≤ r 2 via the sewing relation z 1 z 2 = ǫ. z 1 = 0 z 2 = 0 ✬✩ ✬✩ ✎☞ ✎☞ ❅ r 1 ✍✌ ✍✌ ❅ ☛ ❯ Σ 1 Σ 2 r 2 ✫✪ ✗ ✫✪ � ✗ � | ǫ | /r 2 | ǫ | /r 1 Gives a genus two Riemann surface Σ (2) parameterized by the domain D ǫ = { ( τ 1 , τ 2 , ǫ ) ∈ H 1 × H 1 × C | | ǫ | < 1 4 D ( τ 1 ) D ( τ 2 ) } . Alexander Zuevsky VOSAs on a Riemann Surface
Structures on Σ (2) Constructed from Genus One Data Yamada (1980) describes how to compute the period matrix and other structures on a genus g Riemann surface in terms of lower genus data. For standard homology basis a i , b j with i = 1 , . . . , g on a genus g Riemann surface consider the normalized differential of the second kind which is a symmetric meromorphic form with dxdy ω ( x, y ) ∼ for local coordinates x ∼ y, ( x − y ) 2 � where a i ω ( x, · ) = 0 . A normalized basis of holomorphic 1-forms ν i and the period matrix Ω ij are given by � ν i ( x ) = ω ( x, · ) , b i � 1 Ω ij = ν i . 2 πi b i Alexander Zuevsky VOSAs on a Riemann Surface
ω (2) on the Sewn Surface Σ (2) ω (2) can be determined from ω (1) on each torus in Yamada’s sewing scheme [Yamada, Mason-Tuite]. For a torus Σ (1) = C / Λ τ the differential is ω (1) ( x, y ) = P 2 ( x − y, τ ) dx dy, P 2 ( z, τ ) = ℘ ( z, τ ) + E 2 ( τ ) , for Weierstrass function � ℘ ( z, τ ) = 1 ( k − 1) E k ( τ ) z k − 2 , z 2 + k ≥ 4 and Eisenstein series for k ≥ 2 � � ′ � � 1 1 E k ( τ ) = . (2 πi ) k ( mτ + n ) k m n E k vanishes for odd k and is a weight k modular form for k ≥ 4 . E 2 is a quasi-modular form. Alexander Zuevsky VOSAs on a Riemann Surface
Expanding � 1 C ( k, l ) x k − 1 y l − 1 , P 2 ( x − y, τ ) = ( x − y ) 2 + k,l ≥ 1 where ( k + l − 1)! C ( k, l ) = C ( k, l, τ ) = ( − 1) k +1 ( k − 1)!( l − 1)! E k + l ( τ ) , we compute ω (2) ( x, y ) in the sewing scheme in terms of the following genus one data A a ( k, l, τ a , ǫ ) = ǫ ( k + l ) / 2 C ( k, l, τ a ) = √ kl √ 3 ǫ 2 E 4 ( τ a ) ǫE 2 ( τ a ) 0 0 · · · √ − 3 ǫ 2 E 4 ( τ a ) 2 ǫ 3 E 6 ( τ a ) 0 0 − 5 · · · √ 3 ǫ 2 E 4 ( τ a ) 10 ǫ 3 E 6 ( τ a ) 0 0 · · · √ 2 ǫ 3 E 6 ( τ a ) − 35 ǫ 4 E 8 ( τ a ) 0 − 5 0 · · · . . . . ... . . . . . . . . Alexander Zuevsky VOSAs on a Riemann Surface
A Determinant and the Period Matrix Consider the infinite matrix I − A 1 A 2 where I is the infinite identity matrix and define det( I − A 1 A 2 ) by log det( I − A 1 A 2 ) = Tr log( I − A 1 A 2 ) � 1 n Tr(( A 1 A 2 ) n ) , = − n ≥ 1 as a formal power series in ǫ . Theorem (Mason-Tuite) (a) The infinite matrix � ( I − A 1 A 2 ) − 1 = ( A 1 A 2 ) n , n ≥ 0 is convergent for ( τ 1 , τ 2 , ǫ ) ∈ D ǫ . (b) det( I − A 1 A 2 ) is non-vanishing and holomorphic on D ǫ . Alexander Zuevsky VOSAs on a Riemann Surface
Furthermore we may obtain an explicit formula for the genus two period matrix Ω = Ω (2) on Σ (2) Theorem (Mason-Tuite) Ω = Ω( τ 1 , τ 2 , ǫ ) is holomorphic on D ǫ and is given by 2 πiτ 1 + ǫ ( A 2 ( I − A 1 A 2 ) − 1 )(1 , 1) , 2 πi Ω 11 = 2 πiτ 2 + ǫ ( A 1 ( I − A 2 A 1 ) − 1 )(1 , 1) , 2 πi Ω 22 = − ǫ ( I − A 1 A 2 ) − 1 (1 , 1) . 2 πi Ω 12 = Here (1 , 1) refers to the (1 , 1) -entry of a matrix. Alexander Zuevsky VOSAs on a Riemann Surface
The Szeg¨ o Kernel The Szeg¨ o Kernel is defined by � � �� x � � θ � α ϑ y ν 1 1 2 dy ∼ dx β 2 S ( x, y | Ω) = � � for x ∼ y, φ x − y α ϑ (0) E ( x, y ) β � � α with ϑ (0) � = 0 for Riemann theta series with real β characteristics α = ( α i ) , β = ( β i ) for i = 1 , . . . , g � α � � ϑ ( z | Ω) = exp ( iπ ( n + α ) . Ω . ( n + α ) + ( n + α ) . ( z + 2 πiβ )) , β n ∈ Z g θ j = − e − 2 πiβ j , φ j = − e 2 πiα j , j = 1 , . . . , g, and E ( x, y ) is the genus g prime form. Alexander Zuevsky VOSAs on a Riemann Surface
Genus One Szego Kernel On the torus Σ (1) the Szeg¨ o kernel for ( θ, φ ) � = (1 , 1) is � θ � � θ � 1 1 S (1) 2 dy 2 , ( x, y | τ ) = P 1 ( x − y, τ ) dx φ φ where � α � � θ � ϑ ( z, τ ) β ∂ z ϑ 1 (0 , τ ) P 1 ( z, τ ) = � α � ϑ 1 ( z, τ ) , φ ϑ (0 , τ ) β � � 1 for ϑ 1 ( z, τ ) = ϑ ( z, τ ) . 2 1 2 Alexander Zuevsky VOSAs on a Riemann Surface
Twisted Eisenstein Series We define ‘twisted’ modular weight k Eisenstein series [DLM, Mason-Tuite-Z] � θ � � θ � � 1 ( τ ) z k − 1 , P 1 ( z, τ ) = z − E k φ φ k ≥ 1 � � � θ � ′ � � θ m φ n 1 E k ( τ ) = . φ (2 πi ) k ( mτ + n ) k m n It is also useful to note that � θ � θ � � � 1 ( k, l ) x k − 1 y l − 1 , P 1 ( x − y, τ ) = x − y + C φ φ k,l ≥ 1 � θ � θ � � ( k, l, τ ) = ( − 1) l � k + l − 2 � where C E k + l − 1 ( τ ) . k − 1 φ φ Alexander Zuevsky VOSAs on a Riemann Surface
Modular Properties Define the standard left action of the modular group for � a � b γ = ∈ Γ = SL (2 , Z ) on ( z, τ ) ∈ C × H with c d � � cτ + d, aτ + b z γ. ( z, τ ) = ( γ.z, γ.τ ) = . cτ + d We also define a left action of Γ on ( θ, φ ) � θ � � θ a φ b � γ. = . θ c φ d φ Then we obtain: Theorem (Mason-Tuite-Z) For ( θ, φ ) � = (1 , 1) we have � � θ �� � θ � ( γ.z, γ.τ ) = ( cτ + d ) k P k P k γ. ( z, τ ) . φ φ Alexander Zuevsky VOSAs on a Riemann Surface
Modular Properties Theorem (Mason-Tuite-Z) � θ � For ( θ, φ ) � = (1 , 1) , E k is a modular form of weight k where φ � � θ �� � θ � ( γ.τ ) = ( cτ + d ) k E k E k γ. ( τ ) . φ φ Alexander Zuevsky VOSAs on a Riemann Surface
o Kernel on Σ (2) and another Determinant The Szeg¨ We may compute S (2) � � θ ( x, y ) for θ = ( θ 1 , θ 2 ) in the sewing φ scheme in terms of the genus one data � θ a � � θ a � 1 2 ( k + l − 1) C F a ( k, l ) = F a ( k, l, τ a , ǫ ) = ǫ ( k, l, τ a ) . φ a φ a S (2) is described in terms of the infinite matrix I − Q for � � θ 1 0 ξ F 1 √ φ 1 � � , Q = ξ = − 1 . θ 2 − ξ F 2 0 φ 2 Theorem (Tuite-Z) (a) The infinite matrix ( I − Q ) − 1 = � n ≥ 0 Q n is convergent for ( τ 1 , τ 2 , ǫ ) ∈ D ǫ , (b) det( I − Q ) is non-vanishing and holomorphic on D ǫ . Alexander Zuevsky VOSAs on a Riemann Surface
2. Vertex Operator Super Algebras A Vertex Operator Superalgebra (VOSA) is a quadruple 1 = � ( V, Y, 1 , ω ) : V = V ¯ 0 ⊕ V ¯ n ≥ 0 V n is a superspace, Y is a linear map Y : V → (End V )[[ z, z − 1 ]] : so that for any vector (state) a ∈ V , � a ( n ) z − n − 1 , Y ( a, z ) = a ( k ) 1 = δ k, − 1 a, k ≥ − 1 , n ∈ Z a ( n ) V α ⊂ V α + p ( a ) , with locality property for all a , b ∈ V ( x − y ) N [ Y ( a, x ) , Y ( b, y )] = 0; 1 ∈ V ¯ 0 , 0 is the vacuum vector, Y ( 1 , z ) = Id V , and ω ∈ V ¯ 0 , 2 the conformal vector, � L ( n ) z − n − 2 , Y ( ω, z ) = n ∈ Z where L ( n ) forms a Virasoro algebra for central charge c [ L ( m ) , L ( n )] = ( m − n ) L ( m + n ) + c 12( m 3 − m ) δ m, − n . Alexander Zuevsky VOSAs on a Riemann Surface
L ( − 1) satisfies the translation property Y ( L ( − 1) a, z ) = d dz Y ( a, z ) . L (0) describes the grading with L (0) a = wt ( a ) a , and V n = { a ∈ V | wt ( a ) = n } . We quote also the standard commutator property of VOSAs � m � � Y ( a ( j ) b, z ) z m − j . [ a ( m ) , Y ( b, z )] = j j ≥ 0 Note also the associativity property for a , b ∈ V , Y ( a, x ) Y ( b, y ) = Y ( Y ( a, x − y ) b, y ) , Alexander Zuevsky VOSAs on a Riemann Surface
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