Double Sigma Models and Double Field Theory Neil Copland, CQUeST, - PowerPoint PPT Presentation

Double Sigma Models and Double Field Theory Neil Copland, CQUeST, Sogang University. 14/3/2012, Edinburgh Mathematical Physics Group Wednesday, March 14, 12

Overview T-duality is an important property of strings that doesn’t exist for point particles: String theory on a circle of radius is equivalent to string R theory on a circle of radius . 1 /R The (quantised) momentum modes are exchanged with winding around the circle. Splitting the string co-ordinate then X = X L + X R ˜ the duality replaces it with . X = X L − X R For a string theory on a d -dimensional torus, the T-duality group is enlarged to O(d,d). There have been many attempts to make this symmetry manifest in the action, usually involving a doubling of co- ˜ ordinates to include those dual to winding, like , and this always X comes at a price. Here we seek to connect worldsheet and field theory pictures. Wednesday, March 14, 12

Plan ✤ The doubled formalism ✤ Chirality constraint and integration into action ✤ The background field method ✤ Double field theory and generalised Ricci tensor ✤ Agreement on a ‘fibred’ background ✤ A more general double sigma model Wednesday, March 14, 12

A duality-invariant picture ✤ Look for O(d,d) invariance and and new structures which emerge ✤ A more unified picture of and g, b φ ✤ Doubled geometry and differential geometry ✤ Geometric description of T-folds; string backgrounds where transition functions can be T-dualities - new compactifications ✤ String field theoretic motivation for double field theory, dependent ˜ X vertex operators - truly doubled theories Wednesday, March 14, 12

The Doubled Formalism ✤ A sigma model describing a torus fibration in which the fibre co- X A = ( X i , ˜ ordinates are doubled [Hull], . X i ) ✤ Various other earlier works on doubled sigma models [Tseytlin, Maharana, Schwarz, Sen, Duff,...] L = 1 4 H AB d X A ∧ ∗ d X B + L ( Y ) ✤ Minimal Lagrangian: ✤ Generalised metric and O(d,d) invariant metric: � h − 1 − h − 1 b ⇥ � ⇥ 0 1 1 H AB ( Y ) = , L AB = bh − 1 h − bh − 1 b 1 0 1 H − 1 = L − 1 H L − 1 Wednesday, March 14, 12

The Constraint Wednesday, March 14, 12

The Constraint ✤ We have doubled the number of co-ordinates, if we want to describe the same original string theory we need something else, a constraint which halves the degrees of freedom d X A = L AB H BC ∗ d X C Wednesday, March 14, 12

The Constraint ✤ We have doubled the number of co-ordinates, if we want to describe the same original string theory we need something else, a constraint which halves the degrees of freedom d X A = L AB H BC ∗ d X C ✤ Introducing a vielbein one can move to frame where � ⇥ � ⇥ 1 0 1 0 1 1 B ( y ) = B = H ¯ , L ¯ . A ¯ A ¯ 0 1 0 1 1 − 1 Wednesday, March 14, 12

The Constraint ✤ We have doubled the number of co-ordinates, if we want to describe the same original string theory we need something else, a constraint which halves the degrees of freedom d X A = L AB H BC ∗ d X C ✤ Introducing a vielbein one can move to frame where � ⇥ � ⇥ 1 0 1 0 1 1 B ( y ) = B = H ¯ , L ¯ . A ¯ A ¯ 0 1 0 1 1 − 1 ✤ In this frame the constraint is a chirality constraint. Half the co- ordinates are left-moving, and the other half right moving Wednesday, March 14, 12

The Constraint ✤ We have doubled the number of co-ordinates, if we want to describe the same original string theory we need something else, a constraint which halves the degrees of freedom d X A = L AB H BC ∗ d X C ✤ Introducing a vielbein one can move to frame where � ⇥ � ⇥ 1 0 1 0 1 1 B ( y ) = B = H ¯ , L ¯ . A ¯ A ¯ 0 1 0 1 1 − 1 ✤ In this frame the constraint is a chirality constraint. Half the co- ordinates are left-moving, and the other half right moving ✤ Explicitly in the simplest case (circle of radius R ) P = RX + R − 1 ˜ X, ∂ − P = 0 , Q = RX − R − 1 ˜ X, ∂ + Q = 0 . Wednesday, March 14, 12

Incorporating the constraint ✤ At the classical level the action + constraint give the ordinary string equations of motion. To check quantum equivalence we first incorporate the constraint into the action [Berman, NBC, Thompson]. ✤ We first go to the chiral frame: there we can impose the chirality constraint a la PST. ✤ Written in terms of the chiral P and Q the action has the form S d = 1 Z dP ∧ ∗ dP + 1 Z dQ ∧ ∗ dQ . 8 8 ✤ We also define vanishing one-forms P = dP − ∗ dP, Q = dQ + ∗ dQ . Wednesday, March 14, 12

The modified action ✤ We introduce two closed one-forms to the action ✓ ( P m u m ) 2 + ( Q m v m ) 2 ◆ S P ST = 1 Z dP ∧ ∗ dP + 1 Z dQ ∧ ∗ dQ − 1 Z d 2 σ 8 8 8 u 2 v 2 ✤ Simplest way to proceed is to fix them to be time like. The resulting action is loses manifest Lorentz invariance on the worldsheet S = 1 Z d 2 σ ( ∂ 1 P ∂ − P − ∂ 1 Q ∂ + Q ) . 4 = 1 Z h X ) 2 + 2 ∂ 0 X ∂ 1 ˜ i − ( R ∂ 1 X ) 2 − ( R − 1 ∂ 1 ˜ d 2 σ X . 2 ✤ In the more general case the action takes the following simple form on the fibre, with the base remaining the same L fib = − H AB ∂ 1 X A ∂ 1 X B + L AB ∂ 0 X A ∂ 1 X B ✤ The equation of motion integrates to give the constraint. Wednesday, March 14, 12

Background Field Method ✤ For the classical Weyl invariance of the string to extend to the quantum theory the beta functional must vanish. ✤ This can be calculated by expanding a quantum fluctuation around a X α = X α classical background [Honercamp;Alvarez-Gaume, cl + π α Freedman, Mukhi]. ✤ As does not transform covariantly, one does a more refined π α ξ α expansion to maintain the covariance of the action. is the tangent vector to the geodesic from to with length equal to that X α X α cl + π α cl of the geodesic. ✤ The fluctuation propagator can then be obtained and the fluctuations Wick contracted out. Wednesday, March 14, 12

Algorithmic Expansion ✤ Thanks to [Mukhi] we know a simple algorithmic method to background field expand, simply acting on the Lagrangian n times with the operator Z d 2 σξ α ( σ ) D σ α ✤ The action is given by Z d 2 σ ξ α ( σ ) D σ α ξ β ( σ 0 ) = 0 , Z d 2 σ ξ α ( σ ) D σ α ∂ µ X β ( σ 0 ) = D µ ξ β ( σ 0 ) , Z α D µ ξ β ( σ 0 ) = R β d 2 σ ξ α ( σ ) D σ αγδ ∂ µ X δ ξ α ξ γ ( σ 0 ) , Z d 2 σ ξ α ( σ ) D σ α T α 1 α 2 ... α n ( X ( σ 0 )) = D β T α 1 α 2 ... α n ξ β ( σ 0 ) , Wednesday, March 14, 12

Expansion and propagators ✤ At second order the result is 2 L (2) = − G αβ D 1 ξ α D 1 ξ β + L αβ D 0 ξ α D 1 ξ β + K αβ D 0 ξ α D 0 ξ β − R γαβδ ξ α ξ β ∂ 1 X γ ∂ 1 X δ + L αβ ; γ ξ γ ( D 0 ξ α ∂ 1 X β + ∂ 0 X α D 1 ξ β ) + 1 2 D α D β L γδ ξ α ξ β ∂ 0 X γ ∂ 1 X δ + 1 L γσ R σ αβδ + L δσ R σ ξ α ξ β ∂ 0 X γ ∂ 1 X δ � � αβγ 2 + 2 K αβ ; γ ξ γ D 0 ξ α ∂ 0 X β + 1 2 D α D β K γδ ξ α ξ β ∂ 0 X γ ∂ 0 X δ + K γσ R σ αβδ ξ α ξ β ∂ 0 X γ ∂ 0 X δ ✤ From the kinetic terms in the chiral frame the contractions can be determined to be h ξ α ( z ) ξ β ( z ) i = ∆ 0 G αβ + θ L αβ , ⇣ G α [ τ G ρ ] γ � L α [ τ L ρ ] γ ⌘ h ξ γ ∂ 1 ξ α ∂ 1 ξ ρ ξ τ i = � ∆ 0 , G α [ τ L ρ ] γ + L α [ τ G ρ ] γ ⌘ ⇣ h ξ γ ∂ 1 ξ α ∂ 0 ξ ρ ξ τ i = ∆ 0 + 2 θ L α [ τ L ρ ] γ , ⇣ G α [ τ G ρ ] γ + 3 L α [ τ L ρ ] γ ⌘ ⇣ G α [ τ L ρ ] γ + L α [ τ G ρ ] γ ⌘ h ξ γ ∂ 0 ξ α ∂ 0 ξ ρ ξ τ i = ∆ 0 + 2 θ . Wednesday, March 14, 12

Results ✤ After much manipulation we are left with the following divergent terms Z S W eyl = 1 − W GD ∂ 1 X G ∂ 1 X D + W gd ∂ µ Y g ∂ µ Y d ⇤ d 2 σ ⇥ ∆ 0 2 W GD = 1 2 ∂ 2 H GD − 1 GD − 1 ( ∂ a H ) H − 1 ( ∂ a H ) 2 Γ t ab g ab ∂ t H GD � � 2 R gd − 1 W gd = − ˆ 8 ∂ g H AB ∂ d H AB ✤ The terms proportional to vanish showing Lorentz invariance is θ maintained. ✤ After regularising and renormalising the beta functionals vanish if W does. W is not the Ricci tensor of . More work shows the vanishing H of W is the same as the and beta functional equations of the h b undoubled string. Wednesday, March 14, 12

Doubled beta functional Beta-functional BFE Ordinary Background field equation R AB = 0 String of string Wednesday, March 14, 12

Doubled beta functional Beta-functional BFE Ordinary Background field equation R AB = 0 String of string equiv Doubled Doubled version includes B W αβ = 0 Formalism Wednesday, March 14, 12

Doubled beta functional Beta-functional BFE Ordinary Background field equation R AB = 0 String of string equiv Doubled Doubled version includes B W αβ = 0 Formalism The doubled formalism calculation reproduces the string background field equations, including and dilaton, after a lot of work [BCT]. b The ordinary string background field equations can be obtained as equations of motion of an certain action: The string effective action. Wednesday, March 14, 12