New Topological Field Theories from Dimensional Reduction of Nonlinear Gauge Theories

New Topological Field Theories from Dimensional Reduction of Nonlinear Gauge Theories Noriaki Ikeda Ritsumeikan University, Japan Collaboration with K.-I. Izawa and T. Tokunaga, N. I., Izawa, hep-th/0407243, N. I., Tokunaga, hep-th/0609098,

1. Introduction Purpose 1. Construct new gauge theories and apply to physics and mathematics A new gauge theory { new physics (Yang-Mills, gravity, ) new mathematics( ) Consistency of quantum field theories a Batalin-Vilkovisky Structure (S, S) = 0 (classical) (S q, S q ) 2i h S q = 0 (quantum) 1

Purpose 2. Unify (and classify) all geometries and algebras as (a kind of substructures of) (super) Poisson geometry Super Poisson geometry a Batalin-Vilkovisky Structure (S, S) = 0 (classical) geometry (S q, S q ) 2i h S q = 0 (quantum geometry) Many geometric structures are constructed as a BV structure. Lie, Poisson, complex, symplectic, Kähler, Calabi-Yau, Courant, Dirac, algebroid, generalized geometry, 2

We consider a topological field theory with a BF type (Schwarz type) kinetic term S = S 0 + S 1 = ΠT X BdA + S 1, as an interesting play ground. A nonlinear gauge theory is a most general topological field theory with a BF type (Schwarz type) kinetic term S = ΠT X BdA + S 1 in any dimension, which includes the Poisson sigma model, A-model, B-model,, and describes a topological membrane. A nonlinear gauge theory constructed from superfields with the nonnegative total degree has been analyzed. We know general S 1. N.I. 01 3

This Talk We construct toplogical field theories which include superfields with the negative total degree, consistent with dimensional reduction and deformation, and consider an Application to a generalized complex structure (and complex, symplectic geometry). Technique Deformation Theory Barnich, Henneaux 93, Barnich, Brandt, Henneaux, 95 Dimensional Reduction Kaluza, 21, Klein, 26 4

Plan of Talk Review of the AKSZ formalism and nonlinear gauge theories Dimensional reduction from 3 dimensions to 2 dimensions The Poisson sigma model with a two-form Application to a generalized complex structure 5

2. AKSZ Formalism of Nonlinear Gauge Theories X: a manifold in n dimensions (worldsheet or worldvolume) M: a manifold in d dimensions (target space) a map ϕ : X M The AKSZ Formalism of the Batalin-Vilkovisky formalism is formulated by three elements, Supermanifold (Superfield Φ) P-structure (Antibracket (, )) Q-structure (BV action S) Alexandrov, Kontsevich, Schwartz, Zaboronsky 97 6

Supermanifold (Superfield) X to supermanifold T X: a tangent bundle ΠT X: A supermanifold is a tangent bundle with reversed parity of the fiber. local coordinates {σ µ } on X, where µ = 1, 2,, n {θ µ } on T σ X, fermionic supercoordinate Def: form degree deg σ = 0 and deg θ = 1 We extend a smooth map ϕ : X M to a map ϕ : ΠT X M. 7

This procedure introduces ghosts and antifields systematically. function on ΠT X is called a superfield. A ϕ i = ϕ (0)i + θ µ 1ϕ ( 1)i µ 1 + 1 2! θµ 1θ µ 2ϕ ( 2)i µ 1 µ 2 + + 1 n! θµ 1 θ µ n ϕ ( n)i µ 1 µ n. M to supermanifold (a graded manifold) We introduce superfield pairs with the nonnegative total degrees and with the sum of the total degree n 1. {A p a p, B n p 1,ap }, p = 0, 1,, n 1 2 8

1. T M: a cotangent bundle T [n 1]M: A graded cotangent bundle is a cotangent bundle with the degree of the fiber n 1. local coordinates {ϕ i, B n 1,i } {ϕ i = A 0 i }: a map from ΠT X to M, where i = 1, 2,, d. {B n 1,i }: a superfield on ΠT X, which take a value on ϕ (T [n 1]M) p even = {B p,i } bosonic, p odd = {B p,i } fermionic Def: total degree: ϕ = 0 and B n 1 = n 1 Def: ghost number: ghf = F degf, where F is a superfield. 9

2. E: a vector bundle on M E p [p]: a vector bundle with the degree of the fiber p We consider E p [p] E p[n p 1]: p is an integer with 1 p n 2. local coordinates {A p a p, B n p 1,ap } A p a p : a total degree p superfield on ΠT X, which take a value on ϕ (E[p]) B n p 1,ap : a total degree n p 1 superfield on ΠT X, which take a value on ϕ (E [n p 1]) x : the floor function which gives the largest integer less than or equal to x R. 10

If n 2 p n 2, we can identify E[p] E [n p 1] with the dual bundle E [n p 1] (E ) [p], We consider graded bundle, T [n 1] n 1 2 p=1 E p [p], a {A p p, B n p 1,ap }, p = 0, 1,, n 1 2 in the local coordinate expression 11

n = 2 T [1]M. 12

P-structure is a graded Poisson structure on a graded (or super) manifold Ñ, whose bracket called the antibracket (, ) with the total degree p satisfies the following identities: (F, G) = ( 1) ( F p)( G p) (G, F ), (F, GH) = (F, G)H + ( 1) ( F p) G G(F, H), (F G, H) = F (G, H) + ( 1) G ( H p) (F, H)G, ( 1) ( F p)( H p) (F, (G, H)) + cyclic permutations = 0, 13

1. T [n 1]M has a natural P-structure induced from a natural Poisson (symplectic) structure on T M: (F, G) F ϕ i G F B n 1,i B n 1,i ϕ ig. in a Darboux coordinate. The total degree is n + 1. 2. An antibracket on E p [p] E p[n p 1] (F, G) F A p a p B n p 1,ap G ( 1) p(n p 1) F B n p 1,ap A p a p G in a Darboux coordinate. The total degree is n + 1. 14

Q-structure called a BV action S is a function on a target graded bundle, and a functional on ΠT X, which satisfies the classical master equation (S, S) = 0. We require ghs = 0. δf = (S, F ): a BRST transformation, satisfies δ 2 = 0. 15

3. BV Action of Nonlinear Gauge Theories Q-structure (BV Action) S = S 0 + S 1, where S 0 = n 1 2 p=0 ( 1) n p ΠT X B n p 1,ap da p a p, where d = θ µ µ, ΠT X ΠT X dn θd n σ. 16

The general solution S 1 = F (Φ), where F is an arbitrary ΠT X function of all superfields Φ with the total degree n, which satisfies (S 1, S 1 ) = 0, up to total derivatives and BRST exact terms. Higher order deformations vanish. Izawa, 00, N.I, 00 01 Bizdadea, Ciobirca, Cioroianu, Saliu, Sararu 00 01 02 03 We call a resulting field theory S = S 0 + S 1 a nonlinear gauge theory in n dimensions, a topological sigma model in n dimensions, or a topological (n 1)-brane. 17

Example.1. n = 2 Supermanifold T [1]M Antibracket (F, G) F ϕ i B 1,i G F B 1,i BV Action is the Poisson sigma model S 0 = ΠT X B 1i dϕ i, S 1 = ΠT X ϕ i G. 1 2 f ij (ϕ)b 1i B 1j, (S 1, S 1 ) = 0 derives f kl f ij + f il f jk + f jl f ki = 0 ϕ l ϕ l ϕ l π = f ij is a Poisson bivector field. ϕ l ϕ l 18

Example.2. n = 3 Supermanifold T [2]E[1] Antibracket (F, G) F ϕ i G F B 2,i BV Action The Courant sigma model B 2,i ϕ ig + F A 1 a S = S 0 + S 1, [ S 0 = B2i dϕ i + B 1a da ] a 1, ΠT X B 1,a G + F B 1,a A 1 a G 19

S 1 = ΠT X [f 1a i (ϕ)b 2i A 1 a + f ib 2 (ϕ)b 2i B 1b + 1 3! f 3abc(ϕ)A 1 a A 1 b A 1 c + 1 2 f 4ab c (ϕ)a 1 a A 1 b B 1c + 1 2 f 5a bc (ϕ)a 1 a B 1b B 1c + 1 3! f 6 abc (ϕ)b 1a B 1b B 1c ], (1) (S 1, S 1 ) = 0 six f s are structure functions of a Courant algebroid. Example.3. n =general has a structure to the n-algebroid. N.I. 02, Hofman, Park 02, Roytenberg 01, 06 20

4. Dimensional Reduction Dimensional reduction of X in n dimensions to Σ in m dimensions (n > m). Reduction of each superfield Φ is Φ(σ 1,, σ n ) = Φ Φ (σ 1,, σ m ) + n m p=1,µ q =n m+1,,n where deg θ = 1, θ = 1, ghθ = 0. θ µ1 θ µ p ϕ Φ p,µ1 µ p (σ 1,, σ m ), This procedure produces a new topological field theory on Σ, including a superfield with the negative total degree. 21

5. Dimensional Reduction of the Courant Sigma Model in 3 Dimensions to 2 Dimensions Reduction X to Σ, (σ 1, σ 2, σ 3 ) (σ 1, σ 2 ), Procedure 1. Reduction of Superfields ϕ i (σ µ, θ µ ) = ϕ i (σ 1, σ 2, θ 1, θ 2 ) + θ 3 ϕ 1 i (σ 1, σ 2, θ 1, θ 2 ), B 2i (σ µ, θ µ ) = B 2 i(σ 1, σ 2, θ 1, θ 2 ) + θ 3 β 1 i(σ 1, σ 2, θ 1, θ 2 ), A 1 a (σ µ, θ µ ) = Ã 1 a (σ 1, σ 2, θ 1, θ 2 ) + θ 3 α 0 a (σ 1, σ 2, θ 1, θ 2 ), B 1a (σ µ, θ µ ) = B 1 a(σ 1, σ 2, θ 1, θ 2 ) + θ 3 β0 a(σ 1, σ 2, θ 1, θ 2 ), (2) All these superfields do not depend on σ 3 and θ 3. ϕ i = α 0 a = 22

β 0 a = 0, ϕ 1 i = 1, Ã1 a = B 1 a = β 1 i = 1, B 2 i = 2. Procedure 2. Substitute to 3D Action do not derive the correct AKSZ action in 2 dimensions. (θ 3 ) 2 = 0 drops some terms in three dimensions but generally the extra terms appear in a reduced action. The existence of the negative total degree superfield ϕ 1 i complexifies the dimensional reduction in the AKSZ formalism. In order to derive the correct AKSZ action, first we should consider the dimensional reduction via the non-bv formalism. 23

3D AKSZ Action 3D non-bv Action 2D AKSZ Action 2D non-bv Action Procedure 1. Expansion of superfields by the ghost numbers where ϕ i = ϕ (0)i, etc. ϕ i = ϕ i + ϕ ( 1)i + ϕ ( 2)i + ϕ ( 3)i, B 2,i = B (2) 2,i + B(1) 2,i + B 2,i + B ( 1) 2,i, B 1a = B (1) 1,a + B 1,a + B ( 1) 1,a + B ( 2) 1,a, A 1 a = A (1)a 1 + A a 1 + A ( 1)a 1 + A ( 2)a 1, (3) 24

Procedure 2. Setting Φ (p) = 0 for p 0 in the action (antifield & ghost= 0) produces the 3D non-bv action. Procedure 3. Dimensional Reduction ϕ i (σ µ, θ µ ) = ϕ i (σ 1, σ 2, θ 1, θ 2 ), B 2i (σ µ, θ µ ) = B 2 i(σ 1, σ 2, θ 1, θ 2 ) + θ 3 β1 i(σ 1, σ 2, θ 1, θ 2 ), A a 1(σ µ, θ µ ) = Ã1a (σ 1, σ 2, θ 1, θ 2 ) + θ 3 α a 0 (σ 1, σ 2, θ 1, θ 2 ), B 1a (σ µ, θ µ ) = B 1 a(σ 1, σ 2, θ 1, θ 2 ) + θ 3 β0 a(σ 1, σ 2, θ 1, θ 2 ), (4) 3D action to 2D action S = S 0 + S 1 25

S 0 = S 1 = ( dθ 3 θ 3 dσ 3 β1 i d ϕ i + Ã1a d β0 a + β ) a 1 a d α 0 dθ 3 θ 3 dσ 3 ΠT Σ ΠT Σ f 1a i à 1a β1 i + f 2 ib B1 b β 1 i + 1 2 (f 3abc α 0 c + f 4ab c β0 c)ã1aã 1 b + ( f4abc α 0 b + f 5a cb β0 b)ã1a B1 c + 1 2 (f 5a bc α 0 a + f 6 abc β0 a) B 1 b B 1 c + (f 1bi α 0 b + f 2 ia β0 a) B 2 i, (5) Procedure 4. We need a new theory. 2D AKSZ Action 26

6. The Poisson sigma model with a 2-form S t = ΠT Σ B 1 Ad Φ A + 1 2 F AB ( Φ) B 1 A B 1 B + G A ( Φ) B 2 A, (6) where B 2 A is a total degree 2 superfield. The reduced action (5) is obtained by setting Φ A = ( ϕ i, β 0 a, α b 0 ), B 1 A = ( β 1 i, Ã1a, B1 b), B 2 A = ( B 2 i, 0, 0), i id 0 f 1c f 2 F AB = j f 1a f 3ace α e e 0 + f 4ac β0 e f 4aed α e de 0 + f 5a β0 e jb f 2 f 4bec α e ce bd 0 f 5b β0 e f 5e α e 0 + f6 bde β 0 e G A = (f 1bi α 0 b + f 2 ia β0 a, 0, 0), (7), 27

We take A = ( i, a, b ) indices on M, E ϕ and E ϕ. The action (5) is a particular case of the action (6) on E E. The action has the following gauge symmetry: δ Φ A = F AB c B, δ B 1 A = dc A + F BC Φ A B 1 B B 1 C GB Φ A t B, δ B 2 A = dt A + U A BC ( B 1 Bt C B 2 Cc B ) + 1 2 X A BCD B1 B B 1 Cc D, where c A is a gauge parameter with the total degree 1 and t A is a gauge parameter with the total degree 2. X A BCD is completely antisymmetric with respect to the indices BCD. 28

In fact, the action S is gauge invariant δs t = d(c A dφ A + G A t A ), ΠT Σ if and only if F, G, U and X satisfy the identities F D[A F BC] Φ D = GD X D ABC. F AB GC Φ A + U A BC G A = 0, Deformation of the Poisson bivector by the 3-forms G D X D ABC. cf. WZ Poisson sigma model Klimcik, Strobl 01 29

7. AKSZ action of the Poisson sigma model with a 2-form Fields Superfields B 1 A = B 1 (1) A + B 1 A + B 1 ( 1) A, Φ A = Φ A + Φ ( 1)A + Φ ( 2)A, B 2 A = B (2) 2 i + B (1) 2 i + B 2 i ϕ 1 A = ϕ 1 ( 1)i + ϕ 1 ( 2)i + ϕ 1 ( 3)i where ϕ 1 A = 1. 30

P-structure (Antibracket) (F, G) = F Φ A G F B 1 A B 1 A Φ AG +F ϕ G, F A 1 B 2 A B 2 A ϕ A G 1 BV Action S = S 0 + S 1 S 0 = ΠT Σ B 1 Ad Φ A B 2 Ad ϕ 1 A, 31

Procedure to obtain S 1 1. Consider most general (consistent) deformation of S 0 S in the sense of BBH. (Write down all the possible terms) 2. The BV action of S t is a particular case of the resulting action S. 3. If S ghs=0 = S 0 + S 1 ghs=0 = S t, it is the correct AKSZ action. Def: neg(f ): a negative total degree of a superfield F neg( ϕ 1 A ) = 1 and neg (other superfields) = 0 32

We expand S 1 by the negative total degrees: S 1 = p=0 S (p) 1 = dim M p=0 ΠT Σ L (p) 1 = p ΠT Σ ϕ 1 i1 ϕ 1 i p L (p) i 1 i p ( Φ, B 1, B 2 ) (S 0, S 1 ) = 0 = L (0) 1 = 1 2 f AB 1 ( Φ) B 1 A B 1 B + f A 2 ( Φ) B 2 A, L (1) 1 = 1 3! f 3A BCD ( Φ) ϕ 1A B 1 B B 1 C B 1 D +f 4A BC ( Φ) ϕ 1A B 1 B B 2 C, L (2) 1 =, (8) 33

and so on, where f i ( Φ) is a function of Φ A. (S 1, S 1 ) = p (S 1, S 1 ) (p) = 0, (9) determines the identities of f i (Φ) recursively, where (S 1, S 1 ) (p) is the negative total degree p part of (S 1, S 1 ). Batalin, Marnelius 01, N.I., Izawa 04 34

0-th order: (S 1, S 1 ) (0) = 0 = f D[A 1 f AB 1 f BC] 1 Φ D f D 2 f 3D ABC = 0, f C 2 Φ A + f 4A BC f A 2 = 0. f AB 1 = F AB, f A 2 = G A, f 3A BCD = X A BCD, f 4A BC = U A BC, S ghs=0 = S t. The action obtained by deformation is the correct aciton compatible with dimensional reduction. Substitute (7). 35

8. Application Generalized Complex Structure 3D topological sigma model with a generalized complex structure on T M T M (H = 0): N.I. 04 S = = ΠT X 1 2 0 + B 2, d(ϕ + 0) + 1 4 A 1 + B 1, d(a 1 + B 1 ) 0 + B 2, J (A 1 + B 1 ) 1 2 A 1 + B 1, A i J 1 ϕ i(a 1 + B 1 ) 1 2 B 2idϕ i + 1 2 B 1idA i 1 J i jb 2i A j 1 P ij B 2i B 1j ΠT X + 1 Q jk 2 ϕ i Ai 1A j 1 Ak 1 + 1 2 ( J k j ϕ i + J k ) i ϕ j A i 1A j 1 B 1k + 1 P jk 2 ϕ i Ai 1B 1j B 1k. 36

is derived by setting, f 1j i = J i j, f 2 ij = P ij, f 3ijk = Q jk ϕ i f 4ij k = J k j ϕ i + J k i ϕ j, f 5i jk = + Q ki ϕ j + Q ij ϕ k, P jk ϕ i, f 6 ijk = 0, (10) in the Courant sigma model, where J is a generalized complex structure: ( ) J P J = Q t. J (S, S) = 0 J is a generalized complex structure. 37

This action derives the Zucchini model (the Hitchin sigma model) as a boundary action if Σ = X: Zucchini 04 S Z = ΠT Σ B 1 id ϕ i + J i j B 1 id ϕ j + 1 2 P ij B 1 i B 1 j + 1 2 Q ijd ϕ i d ϕ j Dimensional reduction provides a new 2D action with GCS S 0 = S (0) 1 = ΠT Σ ΠT Σ + 1 2 1 2 ( β 1 id ϕ i B 2 id ϕ 1 i + B 1 id α 0 i + Ã 1 i d β 0 i) J i jã1j β1 i + P ij B β 1 i 1 j ) (( Q jk ϕ i + Q ij ϕ k + Q ki ϕ j α 0 k + ( J k j ϕ i ) + J k i ϕ j β 0 k ) 38

S (1) 1 = (( j J k j à 1ià 1 + ϕ i ( ) + 1 P jk i 2 ϕ i α 0 ΠT Σ ϕ 1 l J i j ϕ l ( 1 2 ϕ l J k j ϕ i ) ) J k i ϕ j α j jk P 0 β ϕ i 0 j à 1i B 1 k ( J i j α j 0 + P ij β0 j) B 2 i, B 1 j B 1 k j P B ij 2 iã1 + B B ϕ l 2 i 1 j 1 2 ) + J k i ϕ j 2 Q jk iã 1jà 1 k ϕ i ϕ lã1 à 1ià 1j B 1 k 1 2 P jk i B 2 B ϕ i ϕ lã1 1 j 1 k. S (2) 1 =, 39

9. Two Special Reductions to Complex Geometry and Symplectic Geometry P = Q = 0 (Complex geometry) (S, S) = 0 J = complex structure This condition is invariant under the redefinition (λ: constant): ϕ i = ϕ i, ϕ 1 i = λ ϕ 1i, B 2 i = λ B 2i, β1 i = 1 2 β 1i, i 1 Ã 1 = i 1, i α0 = λ α i 0, B 1 i = λ B 1i, β0 i = β 0i, (11) 40

S 0 = S (0) 1 = S (1) 1 = ΠT Σ ΠT Σ 1 4 1 4 ( β d ϕ i i 1i à 1 d β 0 i) + λ2 2 ( J i β j 1ià j J k j 1 + ( + λ 2 1 J k j 2 ϕ i α 0 [ l λ λ ϕ 1 2 ΠT Σ J i j ϕ i β 0 kã 1 ( B 2i d ϕ 1i + B 1 i d α 0i ), iã ) 1j jã i 1 B 1 k J i α j j 0 B 2 i j λ ϕ l B 2 iã 1 + 4 ) 2 J k j ϕ l ϕ iã 1, iã j 1 B 1 k ], S (p) 1 λ p. 41

If λ 0, S (p) 1 0 for p > 0. The 2D action is S J = 1 4 ΠT Σ β 1 id ϕ i à 1 i d β 0 i + J i j β 1 iã1j + J i k ϕ j β jã k 0 iã1 1. This action is nothing but the B-model action on T [1](T M) in AKSZ. (S J, S J ) = 0 J = complex structure. 42

J = 0 (Symplectic(Poisson) geometry) (S, S) = 0 Q = P 1 = symplectic structure The condition is invariant under the redefinition: (µ: constant.) ϕ i = ϕ i, i i ϕ 1 = µ ϕ 1, B 2 i = µ B 2i, β 1 i = 1 2 β 1i, i à 1 = µ à 1i, i α0 = α 0i, B 1 i = 1 2 B 1i, β0 i = µ β 0i, (12) 43

S 0 = S (0) 1 = S (1) 1 = ΠT Σ ΠT Σ 1 4 1 4 +µ 2 ( 1 2 ΠT Σ ( β d ϕ i 1i + B 1i d α 0i ) ( + µ2 2 B 2i d ϕ 1i à i 1 d β 0 i) ( ) P ij B β 1 i 1j + 1 P jk 2 ϕ i α i 0 B B 1 j 1k 1 P jk 2 ϕ i β i 0 jã 1 B 1 k ( Q jk ϕ i [ µ ϕ l 1 µ 2 + Q ij ϕ k + Q ki ϕ j P ij µ 2 P jk i 8 ϕ i ϕ lã 1 B B 1 j 1k ) α kã 0 iã 1 j 1 + P ij β B 0 j 2i B B ϕ l 2 i 1 j µ3 2 Q iã jk 2 ϕ i ϕ lã 1 jã 1 1k ], ), 44

S (p) 1 µ p. If µ 0, S (p) 1 0 for p > 0. After taking the limit µ 0 with preserving the symplectic structure, S (p) 1 for p > 0 reduces to zero, and the 2D action is S P = 1 4 ΠT Σ β 1 id ϕ i + B 1 id α i 0 + P ij B β 1 i 1 j + 1 P jk i 2 ϕ i α 0 B B 1 j 1 k. (13) (S P, S P ) = 0 P ij = Poisson This action on T [1](T M) is a different realization of a Poisson structure from A-model (the Poisson sigma model). 45

10. Summary and Outlook We have obtained a systematic method to construct a new topological field theory including superfields with the negative total degree by the AKSZ formalism, which is compatible with dimensional reduction. We have obtained one-parameter families of topological sigma models with a complex structure or a symplectic structure. Dimensional reduction via the AKSZ formalism (or the BV formalism) does not work. We need a general formalism. Quantization is direct and clear as a physical theory but 46

the calculation is complicated and the mathematical structure is unknown. A general theory in general dimensions is not constructed. 47