Holomorphic symplectic fermions Ingo Runkel Hamburg University - PowerPoint PPT Presentation

Holomorphic symplectic fermions Ingo Runkel Hamburg University joint with Alexei Davydov

Outline Investigate holomorphic extensions of symplectic fermions ◮ via embedding into a holomorphic VOA (existence) ◮ via study of commutative algebras (+ extra properties) in a braided tensor category (uniqueness – up to two assumptions)

Why? Symplectic fermions: ◮ first described in [Kausch ’95] , by now best studied example of a logarithmic CFT ◮ L 0 -action on representations may not be diagonalisable, thus have non-semisimple representation theory ◮ finite number of irreducible representations, projective covers have finite length, . . . Holomorphic VOAs: ◮ VOAs V , all of whose modules are isom. to direct sums of V ◮ have (almost) modular invariant character ◮ all examples I know are lattice VOAs and orbifolds thereof Can one find new examples by studying extensions of VOAs which have logarithmic modules?

Why? Symplectic fermions: ◮ first described in [Kausch ’95] , by now best studied example of a logarithmic CFT ◮ L 0 -action on representations may not be diagonalisable, thus have non-semisimple representation theory ◮ finite number of irreducible representations, projective covers have finite length, . . . Holomorphic VOAs: ◮ VOAs V , all of whose modules are isom. to direct sums of V ◮ have (almost) modular invariant character ◮ all examples I know are lattice VOAs and orbifolds thereof Can one find new examples by studying extensions of VOAs which have logarithmic modules? Answer for symplectic fermions (up to two assumptions) : No.

Free super-bosons Following [Frenkel, Lepowsky, Meurmann ’87], [Kac ’98] (see [IR ’12] for treatment of non-semisimple aspects): Fix ◮ a finite-dimensional super-vector space h ◮ a super-symmetric non-degenerate pairing ( − , − ) : h ⊗ h → C i.e. ( a , b ) = δ | a | , | b | ( − 1) | a | ( b , a ) Define the affine Lie super-algebras � h = h ⊗ C C [ t , t − 1 ] ⊕ C K 1 � 2 C [ t , t − 1 ] ⊕ C K h tw = h ⊗ C t with K even and central and super-bracket [ a m , b n ] = ( a , b ) m δ m + n , 0 K a , b ∈ h , m , n ∈ Z resp. m , n ∈ Z + 1 2 .

. . . free super-bosons – representations Consider � h and � h tw modules M ◮ where K acts as id ◮ which are bounded below : For each v ∈ M there is N > 0 s.t. m k v = 0 for all a i ∈ h , m 1 + · · · + m k ≥ N . a 1 m 1 · · · a k ◮ where the space of ground states M gnd = { v ∈ M | a m v = 0 for all a ∈ h , m > 0 } is finite-dimensional. Categories of representations of above type: ♭, 1 ( � ♭, 1 ( � Rep fd Rep fd h ) and h tw )

� � . . . free super-bosons – representations Rep fd ( h ) : category of finite-dimensional h -modules (not semisimple) Induction: ◮ N ∈ Rep fd ( h ) gives � h -module N := U ( � � h ) ⊗ U ( h ) ≥ 0 ⊕ C K N ( K acts as 1, a m with m > 0 acts as zero, a 0 acts as a ) ◮ super-vector space V gives � h tw -module N := U ( � � h tw ) ⊗ U ( h tw ) > 0 ⊕ C K V Theorem: The following functors are mutually inverse equivalences: ( − ) gnd � ( − ) gnd � ♭, 1 ( � ♭, 1 ( � Rep fd Rep fd ( h ) Rep fd s V ect fd h ) and h tw ) � � ( − ) ( − )

Symplectic fermions h -module � The vacuum � C 1 | 0 is a vertex operator super-algebra (VOSA) (central charge is super-dimension of h ) . For h purely odd, this is the symplectic fermion VOSA V ( d ) , where dim h = d . V ( d ) ev : parity-even subspace, a vertex operator algebra (VOA). Properties of V ( d ) ev : Abe ’05 ◮ central charge c = − d ◮ C 2 -cofinite ◮ has 4 irreducible representations S 1 S 2 S 3 S 4 − d − d 16 + 1 lowest L 0 -weight 0 1 16 2 character χ 1 ( τ ) χ 2 ( τ ) χ 3 ( τ ) χ 4 ( τ ) ◮ Rep V ( d ) ev is not semisimple

Modular invariance Question: Are there non-zero linear combinations of χ 1 , . . . , χ 4 with non-negative integral coefficients which are “almost” modular invariant? Almost modular invariant: f ( − 1 /τ ) = ξ f ( τ ) and f ( τ + 1) = ζ f ( τ ) for some ξ, ζ ∈ C ×

Modular invariance Question: Are there non-zero linear combinations of χ 1 , . . . , χ 4 with non-negative integral coefficients which are “almost” modular invariant? Almost modular invariant: f ( − 1 /τ ) = ξ f ( τ ) and f ( τ + 1) = ζ f ( τ ) for some ξ, ζ ∈ C × Answer: Davydov, IR ≈ ’15 Combinations as above exist iff d ∈ 16 Z > 0 . In this case, the only possibilities are Z ( τ ) = z 1 χ 1 ( τ ) + · · · + z 4 χ 4 ( τ ) with d d 2 − 1 , 2 2 − 1 , 1 , 0) Z > 0 . ( z 1 , z 2 , z 3 , z 4 ) ∈ (2 Z ( τ ) is modular invariant iff d ∈ 48 Z > 0 .

More questions Minimal almost modular invariant solution: 2 − 1 � � d Z min ( τ ) = 2 χ 1 ( τ ) + χ 2 ( τ ) + χ 3 ( τ ) ( d ∈ 16 Z > 0 ) Questions: Q1 Is Z min ( τ ) the character of a holomorphic extension of V ( d ) ev ? Q2 What are all holomorphic extensions of V ( d ) ev ?

More questions Minimal almost modular invariant solution: 2 − 1 � � d Z min ( τ ) = 2 χ 1 ( τ ) + χ 2 ( τ ) + χ 3 ( τ ) ( d ∈ 16 Z > 0 ) � � = 1 2 η ( τ ) − d d d d 2 + θ 3 ( τ ) 2 + θ 4 ( τ ) θ 2 ( τ ) 2 2 Questions: Q1 Is Z min ( τ ) the character of a holomorphic extension of V ( d ) ev ? Q2 What are all holomorphic extensions of V ( d ) ev ?

More questions Minimal almost modular invariant solution: 2 − 1 � � d Z min ( τ ) = 2 χ 1 ( τ ) + χ 2 ( τ ) + χ 3 ( τ ) ( d ∈ 16 Z > 0 ) � � = 1 2 η ( τ ) − d d d d 2 + θ 3 ( τ ) 2 + θ 4 ( τ ) θ 2 ( τ ) 2 2 Questions: Q1 Is Z min ( τ ) the character of a holomorphic extension of V ( d ) ev ? Q2 What are all holomorphic extensions of V ( d ) ev ? Summary of the answers: Q1 Yes, a possible extension is the lattice VOA for the even self-dual lattice D + d / 2 with modified stress tensor. Q2 Making two assumptions, the answer to Q1 provides all holomorphic extensions of V ( d ) ev .

Symplectic fermions as a sub-VOSA Lattice Z r with standard inner product gives VOSA V Z r . Sublattice D r = { m ∈ Z r | m 1 + · · · + m r ∈ 2 Z } is so (2 r ) root lattice and gives parity-even part: ( V Z r ) ev = V D r . We will need a non-standard stress tensor (aka. conformal vector or Virasoro element) for V Z r given by r � T FF = 1 ( H i − 1 H i − 1 − H i − 2 )Ω , 2 i =1 where H i m , i = 1 , . . . , r generate Heisenberg algebra Hei ( r ) . Central charge c FF = − 2 r . Appear e.g. as “free boson with background charge” or “Feigin-Fuchs free boson”. Detailed study in context of lattice VOAs in [Dong, Mason ’04].

. . . symplectic fermions as a sub-VOSA Theorem: Davydov, IR ≈ ’15 For every r ∈ Z > 0 , there is an embedding ι : V (2 r ) → V Z r of VOSAs which satisfies ι ( T SF ) = T FF . Sketch of proof: ◮ Pick a symplectic basis a 1 , . . . , a r , b 1 , . . . , b r of h , s.t. ( a i , b j ) = δ i , j . ◮ V (2 r ) generated by a i ( x ) , b i ( x ) , OPE a i ( x ) b j (0) = δ i , j x − 2 + reg Kausch ’95, Fuchs, Hwang, ◮ Free field construction: Semikhatov, Tipunin ’03 For m ∈ Z r write | m � for corresponding highest weight state in V Z r . Then ( e i : standard basis vectors of Z r ) f ∗ i := | e i � f i := − H i and − 1 |− e i � have OPE in f ∗ i ( x ) f j (0) = δ i , j x − 2 + reg . ◮ � h -module generated by Ω ∈ V Z r is isomorphic to V (2 r ) .

. . . symplectic fermions as a sub-VOSA For r ∈ 8 Z have the even self-dual lattice D + r = D r ∪ ( D r + [1]) with [1] = ( 1 2 , . . . , 1 2 ) . r . Since ( V Z r ) ev = V D r get: In particular, V D r ⊂ V D + Corollary: V (2 r ) ev is a sub-VOA of V D + r .

Recall questions: Q1 Is Z min ( τ ) the character of a holomorphic extension of V ( d ) ev ? Q2 What are all holomorphic extensions of V ( d ) ev ? Summary of the answers: Q1 Yes, a possible extension is the lattice VOA for the even self-dual lattice D + d / 2 with modified stress tensor. � Q2 Making two assumptions, the answer to Q1 provides all holomorphic extensions of V ( d ) ev . – next

The braided tensor category SF ( d ) Take super-vector space h to be purely odd. Had equivalences ♭, 1 ( � h ) ∼ ♭, 1 ( � h tw ) ∼ Rep fd = Rep fd ( h ) Rep fd = s V ect fd , Write SF ( d ) = SF 0 + SF 1 with SF 1 = s V ect fd . SF 0 = Rep fd ( h ) , Aim: 1. Use “vertex operators” and conformal blocks for � h ( tw ) to equip SF ( d ) with ◮ tensor product ◮ associator ◮ braiding 2. Find commutative algebras in SF ( d ) with certain extra properties

Vertex operators A slight generalisation of free boson vertex operators: IR ’12 Definition: Let A , B , C ∈ SF 0 . A vertex operator from A , B to C is a map V : R > 0 × ( A ⊗ � → � B ) − C ( � C is the algebraic completion) such that (i) even linear in A ⊗ � B , smooth in x (ii) L − 1 ◦ V ( x ) − V ( x ) ◦ ( id A ⊗ L − 1 ) = d dx V ( x ) � � x m a ⊗ id + id ⊗ a m (iii) for all a ∈ h , a m V ( x ) = V ( x ) + three more version when any two of A , B , C are in SF 1 . Vector space of all vertex operators from A , B to C : V C A , B Same definition works for super-vector spaces h which are not purely odd, i.e. for free super-bosons in general.