Path Integral Approach to Fermionic Vacuum Energy in Non-parallel D1-Branes
aa r X i v : . [ h e p - t h ] N ov Path Integral Approach to Fermionic Vacuum Energy inNon-parallel D1-Branes
A. JahanDepartment of Physics, Amirkabir University of Technology, Tehran, [email protected]
Abstract
The fermionic one loop vacuum energy of the superstring theory in a system ofnon-parallel D1-branes is derived by applying the path integral formalism.
The present work is sequel to the previous one in which we applied the path integral tech-nique to derive the one loop vacuum energy (zero point energy) of a bosonic string in asystem of non-parallel D1-branes [1]. Here we shall derive the fermionic zero point energyby engaging the path integral formalism for a superstring in the same system of D1-branes.The path integral approach to the superstring theory is not a new subject. Indeed it standsas an alternative approach to unveil the physics of string theory through calculating thesuperstring S-matrix elements [2-9]. So, after a quick review of the path integral derivationof the fermionic partition function of the open superstrings, a similar approach is followedto derive the fermionic partition function in a system of angled D1-branes in section 3. Weshow that the result is in agreement with one derived earlier using the harmonic oscillatorrepresentation [10-12]. 1
Fermionic Partition Function: Path Integral Approach
From the fermionic part of the superstring action S F = i π Z d σ ¯Ψ µ ρ a ∂ a Ψ µ + S [ β, γ ] (1)the partition function can be achieved by evaluating the path integral Z F = Z D Ψ µ DβDγe iS [Ψ , ¯Ψ]+ iS [ β,γ ] (2)Here the action S [ β, γ ] stands for the superconformal ghosts action. We skip the explicitderivation of the contribution arising from the integration over the superconformal ghostfields β and γ as its net effect is to decrease the space-time dimensions by 2. After somealgebra and upon introducing Ψ µ = ( ψ µ , ˜ ψ µ ) T the action (1) (with Euclidean signature forworld-sheet and target space manifolds) takes a more simple form S [ ψ, ˜ ψ ] = i π Z d σ ( ψ µ ¯ ∂ψ µ + ˜ ψ µ ∂ ˜ ψ µ ) (3)where ∂ = ∂ τ + i∂ σ . Now, we introduce the notation ( ± , ± ) to distinguish the four fermionicspin structures in such a way that the upper (lower) sign denotes the periodic (anti-periodic)boundary conditions along the τ and σ directions, respectively [13, 14]. Therefore, the spinstructures ( ± , +) arise from the Ramond sector, which for the off-shell fluctuations implies ψ µ ( τ, σ ) = X m ∈ Z + , n ∈ Z ψ µmn √ u mn , ψ µ ( τ, σ ) = X m ∈ Z , n ∈ Z ψ µmn √ u mn (4)with the eigen-mode u mn = e iτω m e − inσ . For the spin structures ( ± , − ), arising from theNeveu-Schwrtz sector, we have ψ µ ( τ, σ ) = X m ∈ Z , n ∈ Z + ψ µmn √ u mn , ψ µ ( τ, σ ) = X m, n ∈ Z + ψ µmn √ u mn (5)In a similar way the fourier expansions of ˜ ψ µ associated with different spin structures canbe achieved via the substitution u mn ( τ, σ ) → u mn ( τ, − σ ) ≡ ˜ u mn ( τ, σ ) in equations (4) and(5). The eigen-modes fulfill the orthogonality relation h u mn u m ′ n ′ + e u mn e u m ′ n ′ i = 2 πsδ m + m ′ δ n + n ′ (6)2ere we have defined h Q i = Z s dτ Z π dσ Q (7)Hence, by taking into account the Grassmannian nature of coefficients, i.e. { ψ µmn , ψ νm ′ n ′ } = 0one obtains the partition function of the open superstring Z ψ = Z Dψ µ e − S [ ψ µ , ˜ ψ µ ] = Y mn λ d mn (8)with λ mn = − s ( ω m + in ). The above infinite product can be easily calculated with the aidof identities Y m ∈ Z ( mx + y ) = 2 sinh (cid:16) iπyx (cid:17) , Y m ∈ Z + ( mx + y ) = 2 cosh (cid:16) iπyx (cid:17) (9)and the zeta-function regularizations X m ∈ N m = 112 , X m ∈ N − m = 124 (10)Therefore we find the open superstring partition function as [9] ( q = e − s ) Z ψ = Y mn λ d mn = d q d Q n ∈ N (1 + q n ) d , m ∈ Z + q − d Q n ∈ N (1 + q n − ) d , m ∈ Z + q − d Q n ∈ N (1 − q n − ) d , m ∈ Z (11)We shall assign the symbols Z − + ψ , Z −− ψ and Z + − ψ to the terms of equation (11) from aboveto below, respectively. One must note that Z ++ ψ = 0 because of the well-known property ofthe Grassmann variables Z dψ µmn = 0 (12) We specify the position of first D1-brane by X i ( τ,
0) = 0 , i = 2 , ..., d (13)and the second one by X ( τ, π ) cos α = X ( τ, π ) sin α (14) X r ( τ, π ) = l r r = 3 , ..., d . We denote the deflection angle by α = πa and 0 ≤ a ≤
1. Then, theconditions satisfied by the ends of an open string at the boundaries, imposed by the classicalequations of motion, read [1, 10-12] ∂ σ X (0 , τ ) = X ( τ,
0) = 0 , (15) ∂ σ X ( τ, π, ) cos α = − ∂ σ X ( τ, π ) sin α Similarly, for the fermionic degrees of freedom we find¯ ǫρ ρ ψ (0 , τ ) = ¯ ǫρ ρ ψ (0 , τ ) = 0 (16)and ¯ ǫρ ρ ( ψ + tan αψ ) = ¯ ǫρ ρ ( ψ + tan αψ ) = 0 (17)at the other end σ = π . So, for the classical solutions one finds [10] ψ ˜ ψ = 12 i X n ψ n e in a ( τ − σ ) ± e in a ( τ + σ ) + complex conjugate (18)and ψ ˜ ψ = 12 X n ψ n e in a ( τ − σ ) ∓ e in a ( τ + σ ) + complex conjugate (19)The lower sign in expression (18) and (19) corresponds to the NS sector. Now let us considerthe fluctuations around the classical solutions in both sectors as ψ = 12 i X m,n ∈ Z ( ψ mn u amn − ¯ ψ mn ¯ u amn ) (20) ψ = 12 X m,n ∈ Z ( ψ mn u amn + ¯ ψ mn ¯ u amn ) (21)and ˜ ψ = 12 i X m,n ∈ Z ( ψ mn ˜ u amn − ¯ ψ mn ¯˜ u amn ) (22)˜ ψ = 12 X m,n ∈ Z ( ψ mn ˜ u amn + ¯ ψ mn ¯˜ u amn ) (23)where u amn = e iω m τ e − in a σ and n a = n + a . Thus one finds X A =1 h ψ A ¯ ∂ψ A i = 14 X mn X m ′ n ′ Ψt mn − l am ′ n ′ h u amn ¯ u am ′ n ′ i l am ′ n ′ h ¯ u amn u am ′ n ′ i Ψ m ′ n ′ (24)4here we have introduced Ψt mn = ( ψ mn , ¯ ψ mn ) and l amn = ( iω m − n a ). This expression whencombined with P A =1 h ˜ ψ A ∂ ˜ ψ A i yields the diagonalized action S = X mn Ψt mn − λ amn λ amn Ψ mn (25)where we have gained h ¯ u amn u am ′ n ′ + ¯˜ u amn ˜ u am ′ n ′ i = 2 πsδ n,n ′ δ m,m ′ (26)Thus integration over the fields ψ and ψ , or equivalently over Ψ, yields Z ψ ψ = Z D Ψ e − S [Ψ] = Y mn det − λ amn λ amn = Y mn λ amn (27)Therefore, on invoking the well-known formula ∞ X n =1 ( n + a ) = 124 − (cid:16) a + 12 (cid:17) (28)we obtain Z ψ ψ = q + a ( a − (1 + q a ) Q n ∈ N (1 + q n +2 a )(1 + q n − a ) , m ∈ Z + q − + a Q n ∈ N (1 + q n +2 a − )(1 + q n − a − ) , m ∈ Z + q − + a Q n ∈ N (1 − q n +2 a − )(1 − q n − a − ) , m ∈ Z (29)The remaining bosonic degrees of freedom either satisfy the Neumann or the Dirichletboundary condition. So, the corresponding fermionic degrees of freedom are characterizedby the condition ψ ( τ, π ) = ± η ˜ ψ ( τ, π ) (30)with η = 1 ( η = −
1) for the Neumann (Dirichlet) boundary condition. The minus signrefers to the Dirichlet boundary condition [15]. However, for a typical fermionic degree offreedom the partition function regardless to the boundary condition satisfied by its bosoniccounterpart is given by equation (11) (with d = 1). So, by taking into account the invarianceunder the modular transformation, we find the fermionic partition function Z F = Z ψ Z βγ in d = 10 dimensions as Z F = 12 h − Z + − ψ ψ ( Z + − ψ ) + Z −− ψ ψ ( Z −− ψ ) + Z − + ψ ψ ( Z − + ψ ) i (31)5here the factor comes from the GSO projection. A similar result for fermionic partitionfunction is derived earlier by applying the harmonic oscillator representation [10-12]. Theone loop vacuum energy of the system becomes A = ln Z = Z ∞ dss Z ( s ) (32)where the superstring partition function in non-parallel D1-brane setup will be Z = Z F Z B (33)= T q Y π (1 − q a ) − Y n ∈ N (1 − q n ) − (1 − q n +2 a ) − (1 − q n − a ) − × h − q a ) Y n =1 (1 + q n ) (1 + q n +2 a )(1 + q n − a )+ q − a Y n ∈ N (1 + q n − ) (1 + q n +2 a − )(1 + q n − a − )+ q − a Y n ∈ N (1 − q n − ) (1 − q n +2 a − )(1 − q n − a − ) i Here the partition function of bosonic part (in 26 dimensions) is [1, 10-12] Z B = T √ πs q Y π + a ( a − − − q a Y n ∈ N (1 − q n ) − (1 − q n +2 a ) − (1 − q n − a ) − (34)where Y stands for the distance between D-branes. The total interval of interaction time T arises from integration over the zero mode of X . References [1] A. Jahan, Mod. Phys. Lett.
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