Towards Field Theory of D-Branes Tamiaki Yoneya (University of - PowerPoint PPT Presentation

Towards Field Theory of D-Branes Tamiaki Yoneya (University of Tokyo, Komaba) ⋄ Motivations: D-brane field theory ? ⋄ Free-fermion representation of single-charge 1/2-BPS operators: Puzzles and resolution ⋄ A Toy Theory: D3-brane field theory restricted to 1/2-BPS sector: ⋄ D-brane exclusion principle and its realization ⋄ D-brane fields and their bilinears ⋄ Holographic interpretaion and ‘Superstar’ entropy ⋄ Discussion Based on T. Y. , hep-th/0510114 (JHEP12-028)

1 I. Why D-brane field theory? Basic motivation for second-quantized field theories for D-branes ⋄ Two main approaches to D-brane dynamics : • Open strings: Effective super Yang-Mills theories or open-string field theories of D-branes are ⋄ configuration-space (first-quantized) formulations of D-branes • Closed strings: D-branes ∼ soliton (or ‘lump’) solutions (‘VSFT’ also belongs to this category) ⋄ difficult to treat fluctutations with respect to creation and annihilation of D-branes Desirable to develop a truly second-quantized formulation of D-brane dynamics ∼ field theory of D-branes ⇓ Is it possible to treat the whole set { N=0, 1, 2, 3, . . . } of U(N) super Yang-Mills theory by some ‘Fock space’-like representation?

2 quantum-statistical symmetry (permutation of particles) in particle quantum mechanics ( x 1 , x 2 ) ↔ ( x 2 , x 1 ) ⇓ U ( N ) gauge symmetry (or Chan-Paton symmetry) of Yang-Mills theory X ij ↔ ( UXU − 1 ) ij continuous quantum-statistics ?! (For diagonal X ij = x i δ ij , gauge symmetry reduces to permutation symmetry) It is not easy to imagine workable (as physics) Fock space with continuous statistical symmetry. Mathematical notions such as K-theory, ... are useful for various topological characterizations of D-branes, but are not for discussing real quantum mechanical dynamics of D-branes.

3 Old c = 1 matrix model suggests a toy model for this situation : • Gauge invariant Hilbert space of one (hermitian) N × N matrix X ij is equivalent to the Hilbert Z [ dX ] Z � � Y space of N fermions N � ˜ Ψ 1 | X �� X | ˜ � � [ dU ] � Ψ 1 | X �� X | Ψ 2 � = Ψ 2 � Y dx i i =1 | {z } � X | ˜ Ψ � ≡ ( x i − x j ) ×� X | Ψ � : completely antisymmetric i<j Vandermonde determinant Second quantization of this N fermion system gives the Fock space of the c = 1 matrix model. Recent development on 1/2-BPS states in AdS/CFT suggests that this viewpoint might be a useful starting point to a possible second-quantized field theory for D-branes.

4 II. Free fermion representation of single-charge 1/2-BPS operators in AdS/CFT h i ⋄ Generic 1/2 BPS operators (on the Yang-Mills side) in AdS 5 /SYM 4 � � O ( k 1 ,k 2 ,...,kn ) ( x ) ≡ O k 1 ( x ) O k 2 ( x ) · · · O kn ( x ) (0 ,r, 0) O k ( x ) ≡ Tr φ { i 1 ( x ) φ i 2 ( x ) · · · φ ik } ( x ) φ i ( i = 1 , 2 , . . . , 6) ( ∼ transverse coordinates of D3-branes) (For n = 1 , KK modes of h αβ , a αβγδ on the sugra side) r = k 1 + k 2 + · · · k n � � � � � � ⋄ Pick up 2 ( i = 5 , 6 ) directions, O J Z ( x ) k 1 Z ( x ) k 2 Z ( x ) kn ( k 1 ,k 2 ,...,kn ) ( x ) ≡ Tr · · · Tr Tr , 1 J = r = angular momentum in 5 − 6 plane Z = √ ( φ 5 + iφ 6 ) , 2 For large J ∼ N ( n ≫ 1) , these correspond to the excitation of ‘Giant Gravitons’ large spherical D3-branes with dipole-like RR-fields in the bulk of AdS 5 × S 5

5 Suppose we compute two-point functions J ( ℓ 1 ,ℓ 2 ,...,ℓm ) ( x ) O J �O ( k 1 ,k 2 ,...,kn ) ( y ) � • Nonrenormalization property of 2 (and 3-point) functions of 1/2 BPS operators allows us to use the free-field limit of the SYM 4 . Z � dZ ( τ ) � • The free-field limit of SYM 4 is further replaced by the (complex) 1 dimensional model with dZ ( τ ) S 1 d = dτ Tr + Z ( τ ) Z ( τ ) dτ dτ ⇓ J ( ℓ 1 ,ℓ 2 ,...,ℓm ) ( τ 1 ) O J ( k 1 ,k 2 ,...,kn ) ( τ 2 ) � = f ( N ) e − J ( τ 1 − τ 2) �O if we make the following indentification e τ 1 − τ 2 = | x − y | 2 ( τ ∼ radial time ) to be regarded as an effective theory for spherical D3-branes travelling only along the 5-6 plane.

6 • By using the similar technique as for the hermitian 1-matrix model, this model can equivalently be X X described by the Fock space of free fermions on a 1 (complex) dimensional base space. b n p n ( z )e −| z | 2 n p n ( z ∗ )e −| z | 2 ψ ( z, z ∗ ) † = ψ ( z, z ∗ ) = b † r , n =0 n =0 2 n πn ! z n , p n ( z ) = { b n , b † b n | 0 � = 0 = � 0 | b † m } = δ nm , n . ∂ ∂z ∗ ) ψ ( z, z ∗ ) = 0 ( z + with the ‘lowest Landau level’ (LLL) condition : s Z � � X • Matrix traces for the 1/2 BPS operators = fermion bilinears ∞ 1 ( n + q )! Z n dzdz ∗ ψ † ( z, z ∗ ) z n ψ ( z, z ∗ ) = b † ↔ Tr n + q b q 2 n/ 2 q ! q =0 ⋄ sugra fields ∼ particle-hope pairs near the fermi sea (‘ripplons’) ⋄ giant gravitons ∼ higher (and deeper) excitations of particles and/or holes Z � � X • Exact two-point functions are reproduced for arbitrary N with Hamiltonian ∞ z ∂ dzdz ∗ ψ ( z, z ∗ ) † ∂z + zz ∗ ψ ( z, z ∗ ) = nb † H = n b n n =0

7 • On the bulk sugra side, LLM (Lin-Lunin-Maldacena) showed (singularity free) backgrounds satisfying the energy condition ∆ = J with symmetry | {z } | {z } × SO (4) × U (1) SO (4) S 3 in AdS 5 S 3 × R in S 5 � boundary condition defining droplets on a two-dimensional plane embedded in 10D. � classical approximation (fermi liquid) to the Hilbert space of the fermion fields ψ ( z, z ∗ ) holes deep in the fermi sea ∼ giants extended in S 5 – particles excited high above the fermi sea ∼ giants extended in AdS 5 –

8 In particular, the Ground state = circular fermi sea | N � ≡ b † N − 1 b † N − 2 · · · b † 0 | 0 � corresponds to the unique AdS 5 × S 5 geometry . ground state droplet z-plane 1 density of state ground state Let us now return to our problem.

9 Is this fermion field what we are seeking for? : Not quite! Puzzles • The ground state ( J = 0 ) should be independent of the choice of the direction of angular momentum. But, apparently, it looks as if the ground state has a nonzero angular momentum. Rotation in 5-6 plane z → e iθ z b † n → e inθ b † ∼ n ⇓ | N � = b † N − 1 b † N − 2 · · · b † 0 | 0 � → e iJ 0 θ | N � J 0 = N ( N − 1) / 2 Also the vacuum | 0 � depends on the choice of the U(1) plane. We could absorb this phase by assuming that the vacuum is transformed as | 0 � → e − iJ 0 θ | 0 � , but then the vacuum itself would be dependent of the number N of D3-branes. • More seriously, if we choose different directions for the angular momentum plane, one and the same ground state ( AdS 5 × S 5 itself) is represented by different Hilbert spaces with different excitation modes.

10 Note that the states with different SO(6) indices represent physically independent degrees of freedom of D3-branes, and also that the ground state corresponding to AdS 5 × S 5 geometry with given N must corresponds to a uniquely fixed state in the Fock space of D-branes. ⇓ • Desirable to develop an extended formalism in which all SO(6) directions are treated on an equal footing, and the ground state is manifestly singlet under SO(6). • It would be a first step towards a quantum field theory of D3-branes (and of other cases). ⋄ According to the usual logic, however, the fermion picture seems to depend crucially on the reduction to the special U(1) plane. - Matrix models with two or more matrices (spatial dim. ≥ 1 ) have never been reduced to fermion theories.

11 III. Extended fermion field theory of general ( multi-charge ) 1/2-BPS operators � � � � Consider general 1/2-BPS operators O I ( k 1 ,k 2 ,...,kn ) ( x ) ≡ w I φ i 1 · · · φ ik 1 · · · Tr φ ir − kn +1 · · · φ ir i 1 ··· ir Tr r = k 1 + k 2 + · · · + k n = ∆ { w I i 1 ··· ir } = basis for totally symmetric traceless tensors Due to the free-field contractions and traceless condition, two-point (three-point and also general extremal ) functions always take the factorized form M ( τ ) . | {z } �O I 1 ( k 1 ,k 2 ,...,kn ) ( τ 1 ) O I 2 � w I 1 w I 2 � G ( { k.ℓ } , N ) e − r ( τ 1 − τ 2) ( ℓ 1 ,ℓ 2 ,...,ℓn ) ( τ 2 ) = × invariant product G ( { k, ℓ } , N )e − r ( τ 1 − τ 2) = correlator of hermitian 1-matrix free-field theory | {z } | {z } = � : O r : : O r ( k 1 ,...,kn ) ( τ 1 ) ( ℓ 1 ,...,ℓn ) ( τ 2 ) : � M no contraction no contraction

12 � � � � � � The single hermitian matrix field is completely inert against SO(6). Z O r M k 1 M k 2 M kn ( k 1 ,...,kn ) ( τ 1 ) M ≡ Tr · · · Tr Tr S M = 1 M 2 + M 2 ) dτ Tr( ˙ 2 Lessons: • Emergence of free-fermion picture for 1/2 BPS operators is essentially owing to this factorization, not to the choice of a single U (1) plane. • The hermitian matrix degrees of freedom are actually invariant under SO(6). However, to manage the normal ordering prescription, we can go to the coherent-state representation, by introducing complex field Z instead of the real field M . The 1-matrix hermitian matrix model can then be equivalently treated as the complex 1-matrix model restricted to the lowest Landau level (LLL). ket states ↔ holomorphic wave functions – bra states ↔ anti-holomorphic wave functions – Namely, the origin of LLL condition is nothing other than the normal ordering prescription!