Gauge-Stringy Instantons

Transcript

1 Gauge-Stringy Instantons Parsa Hossein Ghorbani Institute for Research in Fundamental Sciences (IPM) School of Particles and Accelerators INFN Sezione di Napoli July 2013 Parsa Hossein Ghorbani Gauge-Stringy Instantons 1/42

2 Brane Configuration Parsa Hossein Ghorbani Gauge-Stringy Instantons 2/42

3 Supersymmetry gauge theories in string theory can be realized by coinciding D-branes: The N = 4 D = 4 U(N) gauge theories lives on a stack of N D3-branes. Instanton in gauge theories are realized by Dp/D(p-4) brane configurations. Parsa Hossein Ghorbani Gauge-Stringy Instantons 3/42

4 Gauge instantons with charge k in N = 4 U(N) in 4D are realized by N D3-branes and k D( 1)-branes. x µ D3 D(-1) 3/3 strings D3-1/3 strings : Neumann boundary condition : Dirichlet boundary condition -1/-1 strings Neveu-Schwarz Ramond Spectrum: 3/3  µ, ˆΦ i ˆΛαA, ˆΛ αa (-1)/(-1) â µ, ˆχ i ˆM αa, ˆλ αa 3/(-1) & (-1)/3 ŵ α, ˆ w α ˆµ A, ˆ µ A Parsa Hossein Ghorbani Gauge-Stringy Instantons 4/42

5 Gauge instantons with charge k in N = 4 U(N) in 4D are realized by N D3-branes and k D( 1)-branes. D3 x µ D3 D(-1) 3/3 strings : Neumann boundary condition : Dirichlet boundary condition Neveu-Schwarz Ramond Spectrum: 3/3  µ, ˆΦ i ˆΛαA, ˆΛ αa (-1)/(-1) â µ, ˆχ i ˆM αa, ˆλ αa 3/(-1) & (-1)/3 ŵ α, ˆ w α ˆµ A, ˆ µ A Parsa Hossein Ghorbani Gauge-Stringy Instantons 5/42

6 Gauge instantons with charge k in N = 4 U(N) in 4D are realized by N D3-branes and k D( 1)-branes. D3 x µ D3 D(-1) : Neumann boundary condition : Dirichlet boundary condition -1/-1 strings Neveu-Schwarz Ramond Spectrum: 3/3  µ, ˆΦ i ˆΛαA, ˆΛ αa (-1)/(-1) â µ, ˆχ i ˆM αa, ˆλ αa 3/(-1) & (-1)/3 ŵ α, ˆ w α ˆµ A, ˆ µ A Parsa Hossein Ghorbani Gauge-Stringy Instantons 6/42

7 Gauge instantons with charge k in N = 4 U(N) in 4D are realized by N D3-branes and k D( 1)-branes. D3 x µ D3 D(-1) -1/3 strings : Neumann boundary condition : Dirichlet boundary condition Neveu-Schwarz Ramond Spectrum: 3/3  µ, ˆΦ i ˆΛαA, ˆΛ αa (-1)/(-1) â µ, ˆχ i ˆM αa, ˆλ αa 3/(-1) & (-1)/3 ŵ α, ˆ w α ˆµ A, ˆ µ A Parsa Hossein Ghorbani Gauge-Stringy Instantons 7/42

8 We are interested in studying instantons in N = 2 U(N) gauge theories in 4D. One way to reduce the number of of supersymmetries is to add orbifolds in the background. An example of orbifold group that we consider in our model: Z 3 = {1, ξ, ξ 1 } ξ = e 2πi 3 The orbifold group acts only on the first two complex coordinates in the internal space. The manifold in the internal space: C 2 /Z 3 C Parsa Hossein Ghorbani Gauge-Stringy Instantons 8/42

9 At the singularity of orbifold the SUSY breaks down: N = 4 U(N) N = 2 U(N 1 ) U(N 2 ) U(N 3 ) In the presence of the orbifold N D3-branes split into N 1,N 2 and N 3 fractional D3-branes. Three gauge theories on fractional branes can be demonstrated by a quiver diagram: U(N 1 ) N 1 N 2 U(N 2 ) N 3 U(N 3 ) Parsa Hossein Ghorbani Gauge-Stringy Instantons 9/42

10 To our model we include another background: An O3-plan along the D3-branes world-volume. Orientifold projection imposes extra symmetry on the world-sheet. Adding O3-plane do not change the number of supersymmetries; It reduces the number of moduli degrees of freedom. The orientifold projection also reduces the unitary groups: U(N), U(k) USp(N), O(k) Parsa Hossein Ghorbani Gauge-Stringy Instantons 10/42

11 Gauge instanton configuration: The gauge branes and D-instantons are in the same representation of the orbifold group, i.e. they occupy the same node of quiver diagram. U(N 2 ) U(N 2 ) U(k 2 ) U(k 2 ) Parsa Hossein Ghorbani Gauge-Stringy Instantons 11/42

12 Stringy instanton configuration: The gauge branes and D-instantons are in two different representations of the orbifold group,.i.e they occupy different nodes of quiver diagram. O(k 1 ) U(N 2 ) U(N 2 ) Parsa Hossein Ghorbani Gauge-Stringy Instantons 12/42

13 Gauge-Stringy Model: O(k 1 ) U(N 2 ) U(N 2 ) U(k 2 ) U(k 2 ) Parsa Hossein Ghorbani Gauge-Stringy Instantons 13/42

14 Gauge-Stringy Spectrum Parsa Hossein Ghorbani Gauge-Stringy Instantons 14/42

15 The NS sector of -1/-1 string states are called the Neutral Bosonic Moduli: φ M a µ χ p a µ χ i M = 0,.., 9; µ = 0,.., 3; i = 1, 2, 3 Under orbifold transformation: a µ = γ(g) a µ γ(g) 1 χ i = ξ i γ(g) χ i γ(g) 1 g(γ) is a representation of the orbifold group acting on Chan-Paton matrices: 1l ks 0 0 γ(g) = 0 ξ 1l kg ξ 1 1l kg Parsa Hossein Ghorbani Gauge-Stringy Instantons 15/42

16 Under orientifold transformation: a µ = γ + (Ω) a T µ γ + (Ω) 1 χ i = γ + (Ω) χ T i γ + (Ω) 1 γ + (Ω) is a symmetric representation of orientifold group acting on Chan-Paton matrices: 1l ks 0 0 γ + (Ω) = 0 0 1l kg 0 1l kg 0 1l ks and 1l kg are respectively k s k s and k g k g unit matrices. Parsa Hossein Ghorbani Gauge-Stringy Instantons 16/42

17 Neutral Bosonic Chan-Patons satisfying orbifold and orienfifold conditions: a µ (s) 0 0 χ (s) 0 0 a µ = 0 a µ (g) 0 χ 3 = 0 χ (g) a µ T 0 0 χ T (g) (g) χ 1 = a µ (s) = aµ T (s) χ (s) = χ T (s) 0 χ 1 (gs) χ χ 1 (gs) (g) χ 2 = χ 2 T (gs) 0 0 T χ 2 (g) 0 χ 1 (gs) χ 1 (g) = χ1 T (g) χ 2 (g) = χ2 T (g) Parsa Hossein Ghorbani Gauge-Stringy Instantons 17/42

18 The R sector of 1/ 1 string states are the Neutral Fermionic Moduli: Λ A λ αa M αa A = 1,.., 16 α, α = 1, 2 A = 1,.., 4 Through GSO projection we have chosen only the anti-chiral Ramond spinor i.e. Λ A. The index α (α) in λ αa (M αa ) is anti-chiral (chiral) in the Lorentz space. The lower (upper) index A is chiral (anti-chiral) in 6d internal space. Parsa Hossein Ghorbani Gauge-Stringy Instantons 18/42

19 The Chan-Paton structure of Neutral Fermionic Moduli: M αȧ (s) 0 0 M αȧ = 0 M(g) αȧ 0 M(s) αȧ = M αȧ T (s) 0 0 M(g) αȧ T λ αȧ = λ (s) αȧ λ (g) αȧ T λ (g) αȧ λ (s) αȧ = λ (s) αȧ T The entries in the Chan-Paton matrices are of either stringy or gauge type. Parsa Hossein Ghorbani Gauge-Stringy Instantons 19/42

20 The off-diagonal Chan-Patons of Neutral Fermionic Moduli: 0 M α M α3 (gs) 0 = 0 0 M(g) α M(gs)T α 0 0 M(g) α = M (g) α T 0 0 λ (gs) α λ α3 = T λ (gs) α 0 0 T λ (g) α = λ (g) α 0 λ (g) α 0 M α4 = λ α4 = 0 0 M α (gs) 0 0 M α (gs) T λ (gs) α 0 M α (g) 0 0 λ (gs) α λ (g) α T 0 0 M α (g) = M α (g) T λ (g) α = λ T (g) α Parsa Hossein Ghorbani Gauge-Stringy Instantons 20/42

21 The charged moduli stem from 1/3 string states which enjoy the mixed boundary conditions. The string endpoint on D3-branes transform under anti-symmetric representation γ (Ω) of orienfifold group and string endpoint on D-instanton transform under the symmetric representation γ + (Ω): γ (Ω) = ɛ N1 N l N N 0 1l N N 0 1l ks 0 0 γ + (Ω) = 0 0 1l kg 0 1l kg 0 Parsa Hossein Ghorbani Gauge-Stringy Instantons 21/42

22 Cherged bosonic moduli w α and w α are Weyl spinors in Lorentz space and scalars in internal space. Orbifold and orienfifold conditions on w α and w α are: w α = γ (g) w α γ (g) 1 w α = γ + (Ω) w Ṫ α γ (Ω) 1 Chan-Paton matrices takes the form: w α = 0 w (g) α 0 w α = 0 0 w (g) α w (g) T α w T (g) α Parsa Hossein Ghorbani Gauge-Stringy Instantons 22/42

23 The R sector of 3/ 1 string states µ A and µ A are charged fermionic moduli: µ A = R(g) A Bγ(g)µ B γ(g) 1 µ A = R(Ω) A Bγ + (Ω)(µ B ) T γ (Ω) µ a = 0 µ a (g) 0 µ a = 0 µ a T 0 0 µ a (g) 0 (g) 0 0 µ a T (g) µ T (s) 0 µ 3 = 0 0 µ (g) µ 3 = 0 0 µ T (g) µ (s) µ 4 = µ (s) µ (g) 0 µ 4 = 0 0 µ T (s) µ T (g) 0 Parsa Hossein Ghorbani Gauge-Stringy Instantons 23/42

24 The quartic terms in the moduli action can be rewritten in quadratic form by introducing auxiliary moduli : D c η c µν [a µ, a ν ] + ζ c mn [χ m, χ n ]. where D c = η c µν ζ c mn = 0 D(s) c D(g) c D(g) c T with D c (s) = Dc (s)t. Parsa Hossein Ghorbani Gauge-Stringy Instantons 24/42

25 where C αȧ ( σ µ ) α α [ a µ, χ αa] χȧb ( σ m )ȧb χ m χ a ḃ (σm ) a ḃ χ m therefore: C α3 = 0 C α (gs) C α (g) C α (gs)t 0 0 C α4 = 0 0 C α (gs) 0 0 C α (gs) T 0 C α (g) 0 C(g) α = Cα T (g) C α (g) = C α (g) T Parsa Hossein Ghorbani Gauge-Stringy Instantons 25/42

26 h a w α χ αa h 1 = 0 0 h (g) h1 = h (s) h 2 = h (s) 0 0 h 2 = 0 h (g) 0 0 h T (s) h (g) T h T (s) h (g) T 0 Parsa Hossein Ghorbani Gauge-Stringy Instantons 26/42

27 Gauge-Stringy Q-Exact Action Parsa Hossein Ghorbani Gauge-Stringy Instantons 27/42

28 The moduli action is known from ADHM construction. It can also be rederived from disk amplitudes couplings in type IIB: S = S cubic + S quartic + S charged g 2 0 S quartic = 1 2 D2 c + 1 ( η 2 Dc c [ µ ν ] µν a, a + ζ c [ m n ]) mn χ, χ 1 [ aµ, χ ] [ a µ, χ ] 4 1 [a µ, χ αa] [ ] 1 aµ, χ a α 2 4 [χ, χ] [χ, χ] 1 [ χ, χ αa] [ χ, χ a α ], 4 g 2 [ 0 S cubic = 4( σµ ) αβ M βa ], a µ λ α [ a + 4( σµ ) αβ M βȧ ], a µ λ α ȧ i [ 2 λ αa χ, λ αa] i [ 2 λ αȧ χ, λ αȧ] [ iλ αȧ χȧb, λ α ] b i 2 M αa [χ, M αa] i [ ] 2 M αȧ [ χ, M αȧ ] im αa χ a ḃ, M ḃ α g 2 0 S charged = 2i ( µ a w α + w α µ a) λ α a + 2i ( µȧw α + w α µȧ) λ α ȧ id c w α ( τ c) β α w β χȧb w α w α χ bȧ + 2χ w α w α χ ) +i µ a µ aχ + i µȧµȧ χ + i ( µ a µḃ µḃµ a χ a ḃ All moduli in the above action are 3 3 block Chan-Paton matrices. Parsa Hossein Ghorbani Gauge-Stringy Instantons 28/42

29 The prepotential of N = 2 SYM is obtained through the logarithm of the total partition function Z: Z = q k Z k k=1 q = µ γ(ks,kg) e 2πiτ k-instanton partition function is given by integrating out over all neutral and charged moduli. Z k = N k dx 4 dθ 4 d ˆM k e S(M k,φ) [dm k ] = µ γ Superspace coordinates θ αa trm αa and x µ tra µ are the center of the instanton. The moduli action does not depend on the center of instanton. The moduli integration except for leading instanton numbers is too difficult to perform. Parsa Hossein Ghorbani Gauge-Stringy Instantons 29/42

30 The action of the instanton moduli space enjoys an important holomorphicity property. The holomorphicity becomes evident by a topological twist: SU(2) R SU(2) I SU(2) = diag(su(2) R SU(2) I ) This identification reorganize the 4 supercharges Q αa into a singlet and a triplet: Q = 1 2 ɛ α β α β Q Q c = i 2 (τ α β c) α βq The moduli having an index of right-handed Lorentz subgroup or the internal subgroup are decomposed. λ αa λ α β 1 2 ɛ α β η + i 2 (τ c ) α βλ c M αa M α β 1 2 M µ(σ µ ) α β Parsa Hossein Ghorbani Gauge-Stringy Instantons 30/42

31 The action turns out to be Q-exact; S = Q Ξ: Ξ = i 4 M µ [ ] 1 χ, a [ ] µ + 2 A ηc µν λc α a µ, a ( ν w τ c) α β w β ( c λ + µ α w α + w α µ α) χ λc D c + i 4 [χ, χ] η 1 2 ( µ a ha + h a µ a ) ( w α µ a + µ a w α) χ a α +4 ( σ µ) α [ ] αa 1 χa α α, a µ M + 2 M αa C αa i [ 2 λ αa χ αa ], χ + 1 ζ c m n mn ( σ ) β [ σ α 2 λc χ a α ], χ βa Qχ = 0, Qa µ = M µ Qλ c = D c Qw α = µ α Q w α = µ α Qχ αa = λ αa Q χ = η Qχ a α = λ a α QM αa = C αa Qµ a = h a Q µ a = h a Qη = i [χ, χ] QM µ = i [ χ, a µ] QD c = i [ χ, λ c] Qµ α = iw α χ Q µ α = iχ w α Qλ αa = i [χ, χ αa] Qλ a α = i [χ, χ a α ] QC αa = i [ χ, M αa] Qh a = iµ a χ Q h a = iχ µ a Parsa Hossein Ghorbani Gauge-Stringy Instantons 31/42

32 The multi-instanton calculus becomes possible by localization of integral on the instanton moduli space through the introduction of an Ω-background. In IIB the Ω-background is provided by a R-R 3-form flux F LMN. The flux invariant under orbifold and orientifold projection: F µν F µνz and F µν F µν z, z and z along the third complex coordinate. Interaction with bosonic moduli: 1 tr {F µν a ν [ χ, a µ ] + i Fa µ [χ, a ν ] i F µν a µ F νρ a ρ} g 2 0 Interaction with fermionic moduli: 1 { tr 1 2 ɛ cdeλ c λ d f e f c λ c η + if c D c χ + F µν M µ M ν} g 2 0 Parsa Hossein Ghorbani Gauge-Stringy Instantons 32/42

33 The BRST transformation of moduli involve only holomorphic graviphoton field strength. Q Ω χ = 0, Q Ω χ = η Q Ω a µ = M µ Q Ω λ c = D c Q Ω η = i [χ, χ] Q Ω M µ = i [ χ, a µ] i F µν a ν Q Ω D c = i [ χ, λ c] + ɛ cde λ d f e Q Ω w α = µ α Q Ω µ α = iw α χ + iφw α 1 µν F µν ( σ ) w 2 α β β Q Ω w α = µ α Q Ω µ α = iχ w α i w α φ 1 µν F µν ( σ ) w 2 α β β Q Ω χ αa = λ αa Q Ω λ αa [ = i χ, χ αa] 1 µν F µν ( σ ) χ 2 α β βa Q Ω χ a α = λ a α Q Ω λ a α = i [χ, χ a α ] 1 2 µν F µν ( σ ) β α χ a β Q Ω M αa = C αa Q Ω C αa = i [ χ, M αa] 1 F µν ( ) α σ µν β 2 M βa Q Ω µ a = h a Q Ω µ a = h a Q Ω h a = iµ a χ iφµ a Q Ω ha = iχ µ a + i µ a φ The gauge fermion Ξ depends only on anti-holomorphic graviphoton field strength: Ξ = Ξ + Ξ F Ξ F = if cλ c χ + f µν a µ M ν + f µν ( σ µν ) α β w α µ β + f µν ( σ µν ) α β χ αa λ a β Parsa Hossein Ghorbani Gauge-Stringy Instantons 33/42

34 The dimension of the moduli space is defined as the sum over all canonical dimensions of the moduli. In number of degrees of freedom of the various Chan-Paton matrices and the moduli dimensions are given by the table: BRST pairs moduli gauge stringy gauge-stringy [L] (a µ, M µ ) k 2 g ( χ, η) k 2 g (λ c, D c ) k 2 g 1 2 ks (ks + 1) (L, L 1 2 ) 1 2 ks (ks 1) (L 1, L 3 2 ) 1 2 ks (ks 1) 3 (L 2, L 2 ) (µ a, h a ), ( µ a, h a) k gn k sn (L 1 2, L 0 ) ( w α, µ α), ( w α, µ α ) k gn (L, L 1 2 ) (M αa, C αa ) 1 2 k g ( kg + 1 ) k gk s (L 1 2, L 0 ) (χ αa, λ αa ) 1 ( 2 k g kg 1 ) k gk s (L 1, L 3 2 ) The dimension of the moduli space becomes then: [dm] = µ b 1(k g k s) Parsa Hossein Ghorbani Gauge-Stringy Instantons 34/42

35 Gauge-Stringy Partition Function Parsa Hossein Ghorbani Gauge-Stringy Instantons 35/42

36 In a certain localization limit the functional integral becomes Gaussian and easy to perform. Z (gs) k = dχd χ I (g) I (s) I (gs) where I (g) = P (g)( χ)r (g) ( χ)c (g) ( χ) Q (g) ( χ)l (g) ( χ)w (g) ( χ) I (s) = P (s)(χ)r (s) (χ) Q (s) (χ) I (gs) = C (gs)( χ, χ) L (gs) ( χ, χ) The functions above are the determinants of the BRST charge in different representations. χ χ s and χ χ g are unpaired Chan-Paton matrices. Parsa Hossein Ghorbani Gauge-Stringy Instantons 36/42

37 The integration still get simplified more if one goes to the Cartan basis. In Cartan basis the partition function integration takes the form: Z (gs) k = r[u(k)] dχ i d χ I ( d χ I 2πi ) I=1 r[so(k)] i=1 ( dχ i 2πi ) ( χ I) (χ i ) I (g) (χ I ) I (s) (χ i ) I (gs) (χ i, χ I ) where I (g) (χ I ) = P (g)( χ I )R (g) ( χ I )C (g) ( χ I ) Q (g) ( χ I )L (g) ( χ I )W (g) ( χ I ) I (s) (χ i ) = P (s)(χ i )R (s) (χ i ) Q (s) (χ i ) I (gs) (χ i, χ I ) = C (gs)( χ I, χ i ) L (gs) ( χ I, χ i ) Parsa Hossein Ghorbani Gauge-Stringy Instantons 37/42

38 Switching off any of the gauge or stringy moduli we reach respectively to stringy or gauge partition function. The partition function we obtain in our string theory calculations is exactly the same as the one extracted from ADHM calculations: Z (g) k = ɛ k (E 1 E 2 ) k k d χ I I=1 N 2 l=1 A=1 (2 χ I + E A )( χ I + φ l ) ( χ I + φ l ɛ)( χ I + φ l + ɛ) ( χ I χ J ) 2 [( χ I χ J ) 2 ɛ 2 ]( χ I + χ J + E A ) [( χ I χ J ) 2 E 2 A ][( χ I + χ J ) 2 ɛ 2 ] The stringy partition function is also coincides with the one we studied in a separate paper. Parsa Hossein Ghorbani Gauge-Stringy Instantons 38/42

39 The prepotential is the logarithm of the total partition function: F (n.p.) (Φ) = ɛ log Z tot φ Φ,EA 0 The prepotential itself come from the contribution of all instanton numbers: F (n.p.) = F k q k φ Φ,EA 0 k=1 Expanding the logarithm of the total partition function one arrives at F (gs) 1 = ɛz (gs) 1 F (gs) 2 = ɛz (gs) 2 F (gs) 2 1 /2ɛ Parsa Hossein Ghorbani Gauge-Stringy Instantons 39/42

40 Performing the integration even for small instanton numbers is very difficult. We have done the integration of the partition function for only k = 1 and k = 2: Z (gs) 1 = ( 1)N N 1 E1 2 2 Z (gs) 2 = ( 1)N N 2 E 4 ( 1 8trΦ 4 4E 2 3 1trΦ 2 + 5/16E1) 4 The prepotential corrections due to 1- and 2-instanton in U(N) gauge theory becomes: F (gs) 1 = ( 1)N N 1 E1 4 2 F (gs) 2 = ( 1)N N 2 E 6 ( 1 8trΦ 4 4E 2 3 1trΦ 2 + 5/16E1 4 ) N E6 1 Parsa Hossein Ghorbani Gauge-Stringy Instantons 40/42

41 Conclusion One may be interested in studying the gauge-instanton functions in other classical gauge theories. It is nice to see if the dimensionless moduli measure can exist in other gauge-stringy configurations. Obtaining higher instanton number calculation is still an open problem in gauge-stringy calculation. Parsa Hossein Ghorbani Gauge-Stringy Instantons 41/42

42 Thank You! Parsa Hossein Ghorbani Gauge-Stringy Instantons 42/42