Instantons in (deformed) gauge theories from strings

Instantons in (deformed) gauge theories from RNS open strings 1,2 1 Dipartimento di Fisica Teorica Università di Torino 2 Istituto Nazionale di Fisica Nucleare Sezione di Torino S.I.S.S.A, Trieste - April 28, 2004

This talk is mostly based on... M. Billo, M. Frau, I. Pesando, F. Fucito, A. Lerda and A. Liccardo, Classical gauge instantons from open strings, JHEP 0302 (2003) 045 [arxiv:hep-th/0211250]. M. Billo, M. Frau, I. Pesando and A. Lerda, N = 1/2 gauge theory and its instanton moduli space from open strings in R-R background, arxiv:hep-th/0402160.

Outline Introduction The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution

Introduction

Field theory from strings String theory as a tool to study field theories. A single string scattering amplitude reproduces, for α 0, a sum of Feynman diagrams: α 0 + +... Moreover, String theory S-matrix elements Field theory eff. actions

String amplitudes A N-point string amplitude A N is schematically given by A N = Vφ1 V φn Σ V φi is the vertex for the emission of the field φ i : Σ V φi φ i V φi Σ is a Riemann surface of a given topology... Σ is the v.e.v. in C.F.T. on Σ.

Gauge theories and D-branes In the contemporary string perspective, we can in particular study gauge theories by considering the lightest d.o.f. of open strings suspended between D-branes in a well-suited limit α 0 with gauge quantities fixed.

Many useful outcomes perturbative amplitudes (many gluons,...) via string techniques; construction of realistic extensions of Standard model (D-brane worlds) AdS/CFT and its extensions to non-conformal cases; hints about non-perturbative aspects (Matrix models á la Dijkgraaf-Vafa, certain cases of gauge/gravity duality,...); description of gauge instantons moduli space by means of D3/D(-1) systems.

Non-perturbative aspects: instantons We will focus mostly on the stringy description of instantons. [Witten, 1995, Douglas, 1995, Dorey et al, 1999],... Our goal is to show how the stringy description of instantons via D3/D( 1) systems is more than a convenient book-keeping for the description of instanton moduli space á la ADHM. The D(-1) s represent indeed the sources responsible for the emission of the non-trivial gauge field profile in the instanton solution.

Deformations by closed string backgrounds Open strings interact with closed strings. We can turn on a closed string background and still look at the massless open string d.o.f.. In this way, deformations of the gauge theory are naturally suggested by their string realization. Such deformations are characterized by new geometry in (super)space-time; new mathematical structures; new types of interactions and couplings.

Non-(anti)commutative theories The most famous example is that of (gauge) field theories in the background of the B µν field of the NS-NS sector of closed string. One gets non-commutative field theories, i.e. theories defined on a non commutative space-time Another case, recently attracting attention, is that of gauge (and matter) fields in the background of a graviphoton field strength C µν from the Ramond-Ramond sector of closed strings. These turn out to be defined on a non-anticommutative superspace [Ooguri Vafa, 2003, de Boer et al, 2003, Seiberg, 2003],...

The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile

Usual string perturbation The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile The lowest-order world-sheets Σ in the string perturbative expansion are spheres for closed strings, disks for open strings. Closed or open vertices have vanishing tadpoles on them: Vφclosed sphere = 0, Vφ open disk = 0. No tadpoles these surfaces can describe only the trivial vacua around which ordinary perturbation theory is performed, but are inadequate to describe non-pertubative backgrounds!

The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile Closed string tadpoles and D-brane solutions The microscopic realization of supergravity p-brane solutions as Dp-branes (Polchinski) changes drastically the situation! φ closed φ closed V φclosed disk p 0 source Dp-brane (The F.T. of) this diagram gives directly the leading long-distance behaviour of the Dp-brane SUGRA solution.

Open string tadpoles and instantons The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile This approach can be be generalized to the non-perturbative sector of open strings, in particular to instantons of gauge theories. The world-sheets corresponding to istantonic backgrounds are mixed disks, with boundary partly on a D-instanton. φ open φ open V φopen mixed disk 0 mixed disk source In this case, this diagram should give the leading long-distance behaviour of the instanton solution.

The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile Instantons & and their moduli (flashing review) Consider the k = 1 instanton of SU(2) theory A c µ(x) = 2 ηc µν(x x 0 ) ν (x x 0 ) 2 + ρ 2 S 3 winding # 1 map R 4 S 3 = SU(2) η c µν are the self-dual t Hooft symbols, and F µν is self-dual.

Instantons:Singular gauge The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile With a singular gauge transf. so-called singular gauge: (F µν still self-dual despite the η c µν) Return A c µ(x) = 2ρ 2 η c µν 2ρ 2 η c µν (x x 0 ) ν (x x 0 ) 2 [ (x x 0 ) 2 + ρ 2 ] (x x 0 ) ν (x x 0 ) 4 ( ρ 2 ) 1 (x x 0 ) 2 +... S x0 3 R 4 winding # -1 map S 3 = SU(2)

Instantons: parameters The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile Parameters (moduli) of k = 1 sol. in SU(2) theory: moduli meaning # x µ 0 center 4 ρ size 1 θ orientation ( ) 3 ( ) from large gauge transf.s A U(θ)AU (θ)

Instantons: parameters The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile For an SU(N) theory, embed the SU(2) instanton in SU(N): ( ) 0N 2 N 2 0 A µ = U 0 A SU(2) U µ Thus, there are 4N 5 moduli parametrizing total # of paramenters: 4N. SU(N) SU(N 2) U(1) For instanton # k in SU(N): total # of moduli: 4Nk, described by ADHM construction: moduli space as a HiperKähler quotient. Atiyah,Drinfeld, Hitchin, Manin

Instanton charge and D-instantons The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile The world-volume action of N Dp-branes with a U(N) gauge field F is [ D. B. I. + C p+1 + 1 2 C p 3 Tr ( F F ) ] +... D p A gauge instanton ( i.e. Tr ( F F ) 0 ) a localized charge for the RR field C p 3 a localized D(p 3)-brane inside the Dp-branes. Instanton-charge k sol.s of 3+1 dims. SU(N) gauge theories k D-instantons inside N D3-branes [Witten, 1995, Douglas, 1995, Dorey et al, 1999],...

Stringy description of gauge instantons The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile 1 2 3 4 5 6 7 8 9 10 D3 D( 1) N D3 branes u = 1,..., N D3/D3, C-P: uv D(-1)/D(-1), C-P: ij i = 1,..., k k D( 1) branes D(-1)/D3, C-P: iu

Disk amplitudes and effective actions The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile Usual disks: D3 Disk amplitudes α 0 field theory limit D(-1) Effective actions Mixed disks: D3/D3 D(-1)/D(-1) and mixed D3 D(-1) N = 4 SYM action ADHM measure

Plan Introduction The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile We will now discuss a bit more in detail the stringy description of instantons, focusing on the case of pure SU(N), N = 1 SYM. Though for simplicity we well discuss mostly its bosonic part, this is the supersymmetic theory we will later deform to N = 1/2.

The C 3 /(Z 2 Z 2 ) orbifold The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile Type IIB string theory on target space R6 R 4 Z 2 Z 2 Decompose x M (x µ, x a ), (µ = 1,... 4, a = 5,..., 10). Z 2 Z 2 SO(6) is generated by g 1 : a rotation by π in the 7-8 and by π in the 9-10 plane; g 1 : a rotation by π in the 5-6 and by π in the 9-10 plane. The origin is a fixed point the orbifold is a singular, non-compact, Calabi-Yau space.

Residual supersymmetry The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile Of the 8 spinor weights of SO(6), λ = (± 1 2, ± 1 2, ± 1 2 ), only λ (+) = (+ 1 2, +1 2, +1 2 ), λ ( ) = ( 1 2, 1 2, 1 2 ) are invariant ones w.r.t. the generators g 1,2. They are the orbifold realization of the 2(= 8/4) Killing spinors of the CY. We remain with 8(= 32/4) real susies in the bulk. Only two spin fields survive the orbifold projection: S (±) = e ± i 2 (ϕ 1+ϕ 2 +ϕ 3 ), where ϕ 1,2,3 bosonize the SO(6) current algebra.

Fractional D3-branes The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile Fixed point orbifold Fractional D3 s along x 1,..., x 4 Place N fractional D3 branes, localized at the orbifold fixed point. The branes preserve 4 = 8/2 real supercharges. The Chan-Patons of open strings attached to fractional branes transform in an irrep of Z 2 Z 2. The fractional branes must sit at the orbifold fixed point.

The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile Fractional D3 branes and pure N = 1 gauge theory Spectrum of massless open strings attached to N fractional D3 s of a given type corresponds to N = 1 pure U(N) gauge theory. Schematically, { ψ µ A NS: µ ψ a no scalars! { S R: α S (+) Λ α S α S ( ) Λ α The standard action is retrieved from disk amplitudes in the α 0 limit: S = 1 ( 1 ) gym 2 d 4 x Tr 2 F µν 2 2 Λ α D/ αβ Λ β.

Auxiliary fields The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile The action can be obtained from cubic diagram only introducing the (anti-selfdual) auxiliary field H µν H c η c µν: S = 1 gym 2 { ( µ d 4 ) x Tr A ν ν A µ µ A ν [ + 2i µ A ν A µ, A ν] 2 Λ α D/ αβ Λ β + H c H c + H c η µν c [ A µ, A ν]}, Integrating out H c gives H µν [A µ, A ν ] and the usual action

Auxiliary fields in the open string set-up The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile The auxiliary field H µν is associated to the (non-brst invariant) vertex H A A V H (y; p) = (2πα ) H µν(p) 2 We have then, for instance, 1 2 V H V A V A = 1 ψ ν ψ µ (y) e i 2πα p X(y). g 2 YM + other ordering last term in the previous action. ( ) Tr H µν (p 1 )A µ (p 2 )A ν (p 3 )

Moduli spectrum in the N = 1 case The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile D(-1)/D(-1) strings With k D(-1) s, all vertices have Chan-Paton factors in the adjoint of U(k). D(-1) D(-1) Neveu-Schwarz sector The vertices surviving the orbifold projection are V a (y) = (2πα ) 1 2 g0 a µ ψ µ (y) e φ(y).

Moduli spectrum in the N = 1 case The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile D(-1)/D(-1) strings With k D(-1) s, all vertices have Chan-Paton factors in the adjoint of U(k). D(-1) D(-1) Ramond sector The vertices surviving the orbifold projection are V M (y) = (2πα ) 3 4 g 0 2 M α Sα (y)s ( ) (y) e 1 2 φ(y), V λ (y) = (2πα ) 3 4 λ α S α (y)s (+) (y) e 1 2 φ(y). M α has dimensions of (length) 1 2, λ α of (length) 3 2.

Moduli spectrum in the N = 1 case The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile D(-1)/D3 strings All vertices have Chan-Patons in the bifundamental of U(k) U(N). D(-1) D3 Neveu-Schwarz sector The vertices surviving the orbifold projection are V w (y) = (2πα ) 1 2 V w (y) = (2πα ) 1 2 g 0 2 w α (y) S α (y) e φ(y), g 0 2 w α (y) S α (y) e φ(y), The (anti-)twist fields, switch the b.c. s on the X µ string fields.

Moduli spectrum in the N = 1 case The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile D(-1)/D3 strings All vertices have Chan-Patons in the bifundamental of U(k) U(N). D(-1) D3 Neveu-Schwarz sector The vertices surviving the orbifold projection are V w (y) = (2πα ) 1 2 V w (y) = (2πα ) 1 2 g 0 2 w α (y) S α (y) e φ(y), g 0 2 w α (y) S α (y) e φ(y), w and w have dimensions of (length) and are related to the size of the instanton solution.

Moduli spectrum in the N = 1 case The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile D(-1)/D3 strings All vertices have Chan-Patons in the bifundamental of U(k) U(N). D(-1) D3 Ramond sector The vertices surviving the orbifold projection are V µ (y) = (2πα ) 3 4 V µ (y) = (2πα ) 3 4 g 0 2 µ (y) S ( ) (y) e 1 2 φ(y), g 0 2 µ (y) S ( ) (y) e 1 2 φ(y). The fermionic moduli µ, µ have dimensions of (length) 1/2.

The N = 1 moduli action The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile (Mixed) disk diagrams with the above moduli, for α 0 yield { S mod = tr id c (W c + i η µν c [ a µ, a ν ]) iλ α( w u α µ u + µ u w αu + [ a α α, M α ])} where (W c ) i j = wiu α (τ c ) α β w βuj D c and λ α Lagrange multipliers for the (super)adhm constraints.

The N = 1 ADHM constraints The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile The ADHM constraints are three k k matrix eq.s W c + i η c µν[ a µ, a ν ] = 0. and their fermionic counterparts w u α µ u + µ u w αu + [ a α α, M α ] = 0. Once these constraints are satisfied, the moduli action vanishes.

Parameter counting The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile E.g., for the bosonic parameters # a µ 4k 2 w α, w α 4kN ADHM constraints 3k 2 Global U(k) inv. k 2 True moduli 4kN After imposing the constraints, more or less multi-center positions,... w α, w α size, orientation inside SU(N),... a µ

The instanton solution from mixed disks The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile Mixed disks = sources for gauge theory fields. The amplitude for emitting a gauge field is A I µ(p) = V A I µ ( p) m.d = V w V A I µ ( p) V w I µ p w = i (T I ) v u p ν η c νµ( w u α (τ c ) α β w β v ) e ip x 0. w

The classical profile The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile The classical profile of the gauge field emitted by the mixed disk is obtained by attaching a free propagator and Fourier transforming: d A I 4 p µ(x) = (2π) 2 AI µ(p) 1 eip x p 2 = 2 (T I ) v u [ (T c ) u v where (T I ) v u are the U(N) generators and (T c ) u v = w u α (τ c ) α β w β v. ] η µν c (x x 0 ) ν (x x 0 ) 4,

The classical instanton profile The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile In the above solution we still have the unconstrained moduli w, w. We must still impose the bosonic ADHM constraints W c w u α(τ c ) α β w β v = 0.

The classical instanton profile The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile In the above solution we still have the unconstrained moduli w, w. Iff W c = 0, the N N matrices (t c ) u v 1 2ρ 2 ( w u α (τ c ) α β w β v ), where 2ρ 2 = w u α w α u, satisfy an su(2) subalgebra: [t c, t d ] = iɛ cde t e.

The classical instanton profile The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile The gauge vector profile can be written as A I µ(x) = 4ρ 2 Tr (T I t c ) η µν c (x x 0 ) ν (x x 0 ) 4 This is is a moduli-dependent (through t c ) embedding in su(n) of the su(2) instanton connection in large-distance leading approx. ( x x 0 ρ) singular gauge Recall the singular gauge

Additional remarks The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile The mixed disks emit also a gaugino Λ α, I account for its leading profile in the super-instanton solution. Subleading terms in the long-distance expansion of the solution arise from emission diagrams with more moduli insertions. At the field theory level, they correspond to having more source terms.

Additional remarks The set-up The N = 1 gauge theory from open strings The ADHM moduli space of the N = 1 theory The instanton profile Question: Why singular gauge? Instanton produced by a point-like source, the D(-1), inside the D3 singular at the location of the source In the singular gauge, rapid fall-off of the fields eq.s of motion reduce to free eq.s at large distance perturbative solution in terms of the source term non-trivial properties of the instanton profile from the region near the singularity through the embedding S x 0 3 SU(2) SU(N)

The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution

The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Non-anticommutative theories and RR backgrounds C µν RR background: new geometry A class of deformed field theories, recently attracting attention, is that of gauge (and matter) fields in the background of a graviphoton field strength C µν from the Ramond-Ramond sector of closed strings. These turn out to be defined on a non-anticommutative superspace, where the, say, anti-chiral fermionic coordinates satisfy { θ α, θ β } C α β (σ µν ) α βc µν..

The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Non-anticommutative theories and RR backgrounds C µν RR background: new structure The superspace deformation can be rephrased as a modification of the product among functions, which now becomes ( f(θ) g(θ) = f(θ) exp 1 2 α β C θ α θ β ) g(θ). There are also new interactions between the gauge and matter fields: see later in the talk.

The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Non-anticommutative theories and RR backgrounds

The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Non-anticommutative theories and RR backgrounds Plan We shall analyze the deformation of N = 1 pure gauge theory induced by a RR graviphoton C µν, the so-called N = 1/2 gauge theory. [Seiberg, 2003],... We shall discuss how to derive explicitely the N = 1/2 theory from string diagrams (in the traditional RNS formulation). Moreover we will derive from string diagrams the instantonic solutions of this theory and their ADHM moduli space.

The graviphoton background The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution RR vertex in 10D, in the symmetric superghost picture: Bispinor FȦḂ On R 4 with F α β = F β α. FȦḂ SȦe φ/2 (z) SḂe φ/2 ( z). 1-, 3- and a.s.d. 5-form field strengths. R6 Z 2 Z 2, a surviving 4D bispinor vertex is F α β S α S (+) e φ/2 (z) S β S(+) e φ/2 ( z). Decomposing the 5-form along the holom. 3-form of the CY an a.s.d. 2-form in 4D C µν F α β( σ µν ) α β, the graviphoton f.s. of N = 1/2 theories.

The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Inserting graviphotons in disk amplitudes Conformally mapping the disk to the upper half z-plane, the D3 boundary conditions on spin fields read S α S (+) (z) = S α S(+) ( z). z= z C z (opposite sign for S α S( ) ( z)). When closed string vertices are inserted in a D3 disk, S α S(+) ( z) S α S (+) ( z).

Disk amplitudes with a graviphoton The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Start inserting a graviphoton vertex: V Λ V Λ V A V F Λ where F V F (z, z) = F α β S α S (+) e φ/2 (z) S βs (+) e φ/2 ( z). Λ we need two S ( ) charge operators to saturate the

Disk amplitudes with a graviphoton The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution We insert therefore two chiral gauginos: V Λ V Λ V A V F Λ with vertices F V Λ (y; p) = (2πα ) 3 4 Λ α (p) S α S ( ) e 1 2 φ(y) e i 2πα p X(y). Λ Without other insertions, however, S α S βs α S β ɛ α βɛ αβ vanishes when contracted with F α β.

Disk amplitudes with a graviphoton The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution To cure this problem, insert a gauge field vertex: V Λ V Λ V A V F Λ that must be in the 0 picture: F A V A (y; p) = 2i (2πα ) 1 2 Aµ (p) ( ) X µ (y) + i (2πα ) 1 2 p ψ ψ µ (y) Λ e i 2πα p X(y) finally, we may get a non-zero result!

Evaluation of the amplitude The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution We have V Λ V Λ V A V F C 4 i dy idzd z dv CKG VΛ (y 1 ; p 1 )V Λ (y 2 ; p 2 )V A (y 3 ; p 3 )V F (z, z) where the normalization for a D3 disk is C 4 = 1 π 2 α 2 and the SL(2, R)-invariant volume is dv CGK = 1 g 2 YM dy a dy b dy c (y a y b )(y b y c )(y c y a ).

Explicit expression of the amplitude The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Altogether, the explicit expression is Skip details V Λ V Λ V A V F = 8 ) gym 2 (2πα ) 1 2 Tr (Λ α (p 1 ) Λ β (p 2 ) p ν 3A µ (p 3 ) F α β { i dy idzd z Sα (y 1 ) S β (y 2 ) :ψ ν ψ µ : (y 3 ) S α (z) S β( z) dv CKG S ( ) (y 1 ) S ( ) (y 2 ) S (+) (z) S (+) ( z) e 1 2 φ(y 1) e 1 2 φ(y 2) e 1 2 φ(z) e 1 2 φ( z) e i 2πα p 1 X(y 1 ) e i 2πα p 2 X(y 2 ) e i 2πα p 3 X(y 3 ) }.

The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Evaluation of the amplitude: correlators The relevant correlators are:

The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Evaluation of the amplitude: correlators The relevant correlators are: 1. Superghosts e 1 2 φ(y1) e 1 2 φ(y2) e 1 2 φ(z) e 1 2 φ( z) = [ ] 1 4 (y 1 y 2 ) (y 1 z) (y 1 z) (y 2 z) (y 2 z) (z z).

The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Evaluation of the amplitude: correlators The relevant correlators are: 2. Internal spin fields S ( ) (y 1 )S ( ) (y 2 )S (+) (z)s (+) ( z) = (y 1 y 2 ) 3 4 (y1 z) 3 4 (y1 z) 3 4 (y2 z) 3 4 (y2 z) 3 4 (z z) 3 4.

The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Evaluation of the amplitude: correlators The relevant correlators are: 3. 4D spin fields Sγ (y 1 )S δ (y 2 ) :ψ µ ψ ν :(y 3 ) S α (z)s β( z) = 1 2 (y 1 y 2 ) 1 2 (z z) 1 2 ( (σ µν ) γδ ε α β (y 1 y 2 ) (y 1 y 3 )(y 2 y 3 ) ) + ε γδ ( σ µν ) α β (z z) (y 3 z)(y 3 z).

The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Evaluation of the amplitude: correlators The relevant correlators are: 4. Momentum factors e i 2πα p 1 X(y 1) e i 2πα p 2 X(y 2) e i 2πα p 3 X(y 3) on shell 1.

The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Evaluation of the amplitude: SL(2, R) fixing We may, for instance, choose y 1, z i, z i. The remaining integrations turn out to be + dy 2 y2 1 dy 3 ( y 2 2 + 1 ) ( π2 y3 2 = + 1) 2. Symmetry factor 1/2 and other ordering compensate each other.

Final result for the amplitude The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution We finally obtain for V Λ V Λ V A V F the result 8π 2 ) (2πα ) 1 2 Tr (Λ(p 1 ) Λ(p 2 ) p ν 3A µ (p 3 ) F α β ( σ νµ ) α β. g 2 YM This result is finite for α 0 if we keep constant C µν 4π 2 (2πα ) 1 2 F α β α β ( σ µν ) C µν, of dimension (length) will be exactly the one of N = 1/2 theory. We get an extra term in the gauge theory action: i ( d 4 x Tr Λ Λ ( µ A ν ν A µ)) C µν. g 2 YM

Another contribute The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Another possible diagram with a graviphoton insertion is Λ V Λ V Λ V H V F. F H µν Λ

Another contribute The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Another possible diagram with a graviphoton insertion is Λ V Λ V Λ V H V F. F H µν Recall that the auxiliary field vertex in the 0 picture is Λ V H (y; p) = (2πα ) H µν(p) ψ ν ψ µ (y)e i 2πα p X(y) 2

Another contribute The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Another possible diagram with a graviphoton insertion is F Λ Λ H µν V Λ V Λ V H V F. The evaluation of this amplitude paralles exactly the previous one and contributes to the field theory action the term: 1 2g 2 YM ( d 4 x Tr Λ Λ H µν) C µν, having introduced C µν as above.

Another contribute The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Another possible diagram with a graviphoton insertion is F Λ Λ H µν V Λ V Λ V H V F. All other amplitudes involving F vertices either vanish because of their tensor structure; vanish in the α 0 limit, with C µν fixed.

The deformed gauge theory action The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution From disk diagrams with RR insertions we obtain, in the field theory limit α 0 with C µν 4π 2 (2πα ) 1 2 F α β α β ( σ µν ) fixed the action S = 1 gym 2 d 4 x Tr { ( µ A ν ν A µ ) µ A ν + 2i µ A ν [ A µ, A ν] 2 Λ α D/ αβ Λ β + i ( µ A ν ν A µ) Λ Λ C µν + H c H c + H c η c µν( [A µ, A ν] + 1 2 Λ Λ Cµν)}.

The deformed gauge theory action The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Integrating on the auxiliary field H c, we get S = 1 gym 2 d 4 x Tr{ 1 2 F 2 µν 2 Λ α D/ αβ Λ β + i F µν Λ Λ C µν 1 4 = 1 g 2 YM (Λ Λ C µν ) 2 } { ( d 4 x Tr F µν ( ) + i ) 2 2 Λ Λ C 1 µν + 2 F µν µν F } 2 Λ α D/ αβ Λ β, i.e., exactly the action of Seiberg s N = 1/2 gauge theory.

The graviphoton in D(-1) disks The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Inserting V F in a disk with all boundary on D(-1) s is perfectely analogous to the D3 case (but we have non momenta). The only possible diagram is F M D V M V M V D V F = π2 ( 2 2πα ) 1 2 tr M M ) D c η µνf c α β α β ( σ νµ ) = 1 ( 2 tr M M D c )C c, M where C c = 1 4 ηc µν C µν.

The graviphoton in mixed disks The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution We can also insert V F in a disk with mixed b.c. s. There is a possible diagram F µ µ D V µ V µ V D V F We have different b.c.s on the two parts of the boundary, but the spin fields in the RR vertex V F have the same identification on both: S α S (+) (z) = S α S(+) ( z). z= z

The graviphoton in mixed disks The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution We can also insert V F in a disk with mixed b.c. s. There is a possible diagram F µ D V µ V µ V D V F This is because we chose D(-1) s to represent instantons with self-dual f.s. and F µν to be antiself-dual. µ

The graviphoton in mixed disks The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution We can also insert V F in a disk with mixed b.c. s. There is a possible diagram F µ µ D V µ V µ V D V F The µ, µ vertices contain bosonic twist fields with correlator (y 1 ) (y 2 ) (y 1 y 2 ) 1 2.

The graviphoton in mixed disks The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution We can also insert V F in a disk with mixed b.c. s. There is a possible diagram F µ µ D V µ V µ V D V F Taking into account all correlators, the SL(2, R) gauge fixing, the integrations and the normalizations, we find the result π2 ( ) 2 (2πα ) 1 2 tr µ u µ u D c η µνf c α β ( σ νµ α β ) = 1 ( 2 tr µ u µ u D c )C c.

The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Effects of the graviphoton on the moduli measure No other disk diagrams contribute in our α 0 limit. The two terms above are linear in the auxiliary field D c deform the bosonic ADHM constraints to W c + i η µν[ c a µ, a ν ] + i 2 (M M + µ u µ ) u C c = 0. This is the only effect of the chosen anti-self-dual. graviphoton bckg. Had we chosen a self-dual graviphoton, we would have no effect.

The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution The emitted gauge field in presence of C µν In the graviphoton background, we have the extra emission diagram I µ p F µ µ V µ V A I µ ( p) V µ V F = 2π 2 (2πα ) 1 2 (T I ) v u p ν ( σ νµ ) α β F α β µ u µ v e ip x 0 = 1 2 (T I ) v u p ν η c νµ µ u µ v C c e ip x 0, No other diagrams with only two moduli contribute to the emission of a gauge field.

The classical solution The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution Taking into account also undeformed emission diagram discussed before, the emission amplitude is [ ] A I µ(p) = i (T I ) v u p ν η νµ c (T c ) u v + (S c ) u v e ip x 0, where (T I ) v u are the U(N) generators and (T c ) u v = w u α (τ c ) α β w β v, (S c ) u v = i 2 µu µ v C c. From this we obtain the profile of the classical solution d A I 4 p µ(x) = (2π) 2 AI µ(p) 1 eip x p 2 [ ] = 2 (T I ) v u (T c ) u v + (S c ) u v η µν c (x x 0 ) ν (x x 0 ) 4.

The classical solution The N = 1/2 gauge theory The deformed ADHM moduli space The deformed instanton solution The above solution represents the leading term at long distance of the deformed instanton solution in the singular gauge. However, above appeared the unconstrained moduli µ, µ, w, w. We need to enforce the deformed ADHM contraints, for k = 1: W c + i 2 (M M + µ u µ u ) C c = 0, w u α, µ u + µ u w αu = 0.