Aspects of the Stringy Instanton Calculus: Part I

Aspects of the stringy instanton calculus: part I

Alberto Lerda

U.P.O. “A. Avogadro” & I.N.F.N. - Alessandria

Vienna, October 7, 2008

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 1 / 42 Plan of this talk

1 Introduction and motivation

2 Branes and instantons in ﬂat space

3 Instanton classical solution from string diagrams

4 The stringy instanton calculus

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 2 / 42 String theory S-matrix elements =⇒ vertices and effective actions in ﬁeld theory

Introduction and motivation

String theory is a very powerful tool to analyze ﬁeld theories, and in particular gauge theories.

Behind this, there is a rather simple and well-known fact: in the ﬁeld theory limit α0 → 0, a single string scattering amplitude reproducesa sum of different Feynman diagrams

α0→0 −→ ++ ...

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 3 / 42 Introduction and motivation

String theory is a very powerful tool to analyze ﬁeld theories, and in particular gauge theories.

Behind this, there is a rather simple and well-known fact: in the ﬁeld theory limit α0 → 0, a single string scattering amplitude reproducesa sum of different Feynman diagrams

α0→0 −→ ++ ...

String theory S-matrix elements =⇒ vertices and effective actions in ﬁeld theory

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 3 / 42 In general, a N-point string amplitude AN is given schematically by Z A = ··· N Vφ1 VφN Σ Σ where

I Vφi is the vertex for the emission of the ﬁeld φi : Vφi ≡ φi Vφi

I Σ is a Riemann surface of a given topology

I ... Σ is the v.e.v. with respect to the vacuum deﬁned by Σ.

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 4 / 42 In general, a N-point string amplitude AN is given schematically by Z A = ··· N Vφ1 VφN Σ Σ where

I Vφi is the vertex for the emission of the ﬁeld φi : Vφi ≡ φi Vφi

I Σ is a Riemann surface of a given topology

I ... Σ is the v.e.v. with respect to the vacuum deﬁned by Σ.

The simplest world-sheets Σ are:

spheres for closed strings and disks for open strings

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 4 / 42 I For any open string ﬁeld φ open, one has

Vφ open disk = 0 ⇒ φ open disk = 0

I spheres and disks describe the trivial vacuum around which ordinary perturbation theory is performed

I spheres and disks are inadequate to describe non-perturbative backgrounds where ﬁelds have non trivial proﬁle!

I For any closed string ﬁeld φ closed, one has

V = ⇒ φ = φ closed sphere 0 closed sphere 0

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 5 / 42 I spheres and disks describe the trivial vacuum around which ordinary perturbation theory is performed

I spheres and disks are inadequate to describe non-perturbative backgrounds where ﬁelds have non trivial proﬁle!

I For any closed string ﬁeld φ closed, one has

V = ⇒ φ = φ closed sphere 0 closed sphere 0

I For any open string ﬁeld φ open, one has

Vφ open disk = 0 ⇒ φ open disk = 0

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 5 / 42 I spheres and disks are inadequate to describe non-perturbative backgrounds where ﬁelds have non trivial proﬁle!

I For any closed string ﬁeld φ closed, one has

V = ⇒ φ = φ closed sphere 0 closed sphere 0

I For any open string ﬁeld φ open, one has

Vφ open disk = 0 ⇒ φ open disk = 0

I spheres and disks describe the trivial vacuum around which ordinary perturbation theory is performed

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 5 / 42 I For any closed string ﬁeld φ closed, one has

V = ⇒ φ = φ closed sphere 0 closed sphere 0

I For any open string ﬁeld φ open, one has

Vφ open disk = 0 ⇒ φ open disk = 0

I spheres and disks describe the trivial vacuum around which ordinary perturbation theory is performed

I spheres and disks are inadequate to describe non-perturbative backgrounds where ﬁelds have non trivial proﬁle!

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 5 / 42 However, after the discovery of D-branes, the perspective has drastically changed, and nowadays we know that also some non-perturbative properties of ﬁeld theories can be analyzed using perturbative string theory!

The solitonic brane solutions of SUGRA with RR charge have a perturbative description in terms of closed strings emitted from disks with Dirichlet boundary conditions

φclosed φclosed =⇒ =⇒ hφclosedidisk 6= 0

source D-brane

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 6 / 42 I We will exploit this idea to describe instantons in (supersymmetric) gauge theories using open strings and D-branes.

I We will see that instantons arise as (possibly wrapped) Euclidean branes

I We will show that in addition to the usual ﬁeld theory effects, this stringy realization of the instanton calculus provides a rationale for explaining the presence of new types of non-perturbative terms in the low energy effection actions of D-brane models.

In this lecture

I We will extend this idea to open strings by introducing “mixed disks” (i.e. disks with mixed boundary conditions) such that

φ open mixed disk 6= 0

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 7 / 42 I We will see that instantons arise as (possibly wrapped) Euclidean branes

In this lecture

I We will extend this idea to open strings by introducing “mixed disks” (i.e. disks with mixed boundary conditions) such that

φ open mixed disk 6= 0

I We will exploit this idea to describe instantons in (supersymmetric) gauge theories using open strings and D-branes.

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 7 / 42 I We will show that in addition to the usual ﬁeld theory effects, this stringy realization of the instanton calculus provides a rationale for explaining the presence of new types of non-perturbative terms in the low energy effection actions of D-brane models.

In this lecture

I We will extend this idea to open strings by introducing “mixed disks” (i.e. disks with mixed boundary conditions) such that

φ open mixed disk 6= 0

I We will exploit this idea to describe instantons in (supersymmetric) gauge theories using open strings and D-branes.

I We will see that instantons arise as (possibly wrapped) Euclidean branes

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 7 / 42 In this lecture

I We will extend this idea to open strings by introducing “mixed disks” (i.e. disks with mixed boundary conditions) such that

φ open mixed disk 6= 0

I We will exploit this idea to describe instantons in (supersymmetric) gauge theories using open strings and D-branes.

I We will see that instantons arise as (possibly wrapped) Euclidean branes

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 7 / 42 In phenomenological applications of string theory, instanton effects are important for various reasons, e.g. • they may generate non-perturbative contributions to the effective superpotentials and hence play a crucial rôle for moduli stabilization

• they may generate perturbatively forbidden couplings, like Majorana masses for neutrinos, ...

• ...

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 8 / 42 In phenomenological applications of string theory, instanton effects are important for various reasons, e.g. • they may generate non-perturbative contributions to the effective superpotentials and hence play a crucial rôle for moduli stabilization

• they may generate perturbatively forbidden couplings, like Majorana masses for neutrinos, ...

• ... Instanton effects in string theory have been studied over the years from various standpoints, mainly exploiting string duality:

Witten, Becker2+Strominger, Harvey+Moore, Beasley+Witten, Antoniadis+Gava+Narain+Taylor, Bachas+Fabre+Kiritsis+Obers+Vanhove, Kiritis+Pioline, Green+Gutperle, + ...

• they may generate perturbatively forbidden couplings, like Majorana masses for neutrinos, ...

• ... Only recently concrete tools have been developed to directly compute instanton effects using (perturbative) string theory:

Green+Gutperle, Billò+Frau+Fucito+A.L.+Liccardo+Pesando, Billò+Frau+Fucito+A.L., Blumenhagen,Cvetic+Weigand, Ibañez+Uranga, Akerblom+Blumenhagen+Luest+Plauschinn+Schmidt-Sommerfeld, + ...

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 8 / 42 Credits Florea + Kachru + McGreevy + Saulina, 2006 Bianchi + Kiritsis, 2007 Cvetic + Richter + Weigand, 2007 Argurio + Bertolini + Ferretti + A.L. + Petersson, 2007 Bianchi + Fucito + Morales, 2007 Ibanez + Schellekens + Uranga, 2007 Akerblom + Blumenhagen+ Luest + Schimdt-Sommerfeld, 2007 Antusch + Ibanez + Macri, 2007 Blumenhagen + Cvetic + Luest + Richter + Weigand, 2007 Billó + Di Vecchia + Frau + A.L. + Marotta + Pesando, 2007 Aharony + Kachru, 2007 Camara + Dudas + Maillard + Pradisi, 2007 Ibanez + Uranga, 2007 Garcia-Etxebarria + Uranga, 2007 Petersson, 2007 Bianchi + Morales, 2007 Blumenhagen + Schmidt-Sommerfeld, 2008 Argurio + Ferretti + Petersson, 2008 Cvetic + Richter + Weigand, 2008 Kachru + Simic, 2008 Garcia-Etxebarria + Marchesano + Uranga, 2008 Billò + Ferro + Frau + Fucito + A.L. + Morales, 2008 Uranga, 2008 Amariti + Girardello + Mariotti, 2008

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 9 / 42 Branes and instantons in ﬂat space

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 10 / 42 I The instanton sector of charge k is realized by adding k D(–1) branes (D-instantons).

String description of SYM theories and their instantons

The effective action of a SYM theory can be given a simple and calculable string theory realization:

I The gauge degrees of freedom are realized by open strings attached to N D3 branes. N D3-branes

I From the disk amplitudes of open string massless ﬁelds one recovers the standard Super Yang-Mills action.

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 11 / 42 String description of SYM theories and their instantons

The effective action of a SYM theory can be given a simple and calculable string theory realization:

I The gauge degrees of freedom are realized by open strings attached to N D3 branes.

N D3-branes

k D-instantons

I The instanton sector of charge k is realized by adding k D(–1) branes (D-instantons).

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 11 / 42 I Instanton solutions of SU(N) gauge theories with charge k correspond to k D-instantons inside N D3-branes. [Witten 1995, Douglas 1995, Dorey 1999, ...]

0 1 2 3 4 5 6 7 8 9 D3 −−−− ∗∗∗∗∗∗ D(−1) ∗∗∗∗ ∗∗∗∗∗∗

Instantons and D-instantons

I Consider the effective action for a stack of N D3 branes Z 1 D. B. I. + C4 + C0 Tr F ∧ F D3 2 The topological density of an instanton conﬁguration corresponds to a localized source for the R-R scalar C0, i.e., to a distribution of D-instantons inside the D3’s.

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 12 / 42 Instantons and D-instantons

I Instanton solutions of SU(N) gauge theories with charge k correspond to k D-instantons inside N D3-branes. [Witten 1995, Douglas 1995, Dorey 1999, ...]

0 1 2 3 4 5 6 7 8 9 D3 −−−− ∗∗∗∗∗∗ D(−1) ∗∗∗∗ ∗∗∗∗∗∗

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 12 / 42 Open string degrees of freedom

In this D-brane system there are different open string sectors N D3-branes

D3/D3

k D-instantons D3/D(−1) D(−1)/D(−1)

• D3/D3 strings: gauge theory ﬁelds • D(–1)/D(–1) strings: neutral instanton moduli • D3/D(–1) strings: charged instanton moduli

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 13 / 42 Moduli vertices and instanton parameters

Open strings with at least one end on a D(–1) carry no momentum: they are moduli, rather than dynamical ﬁelds.

The D(–1)/D(–1) open strings have DD boundary conditions in all directions and the spectrum is:

moduli ADHM Meaning Vertex Chan-Paton

0 µ −ϕ NS aµ centers ψ e adj. U(k) . m −ϕ(z) . χm aux. ψ e . . c ν µ . Dc Lagrange mult. η¯µν ψ ψ .

. αA − 1 ϕ . R M partnersS αSA e 2 . . α˙ A − 1 ϕ . λα˙ A Lagrange mult.S S e 2 .

where µ, ν = 0, 1, 2, 3; m, n = 4, 5, ..., 9; α, α˙ = 1, 2 and A = 1, 2, 3, 4.

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 14 / 42 In the D3/D(–1) sector the string coordinates X µ and ψµ (µ = 0, 1, 2, 3) satisfy mixedND or DN boundary conditions ⇒ their moding is shifted by 1/2 so that • the lowest state of the NS sector is a bosonic spinor of SO(4) • the lowest state of the R sector is a fermionic scalar of SO(4)

moduli ADHM Meaning Vertex Chan-Paton

α˙ −ϕ NS wα˙ sizes ∆S e k × N

α˙ −ϕ w¯α˙ sizes ∆S e N × k

A − 1 ϕ R µ partners ∆SA e 2 k × N . A . − 1 ϕ µ¯ . ∆SA e 2 N × k

∆ and ∆ are the twist ﬁelds whose insertion modify the boundary conditions from D3 to D(–1) type and viceversa.

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 15 / 42 Disk amplitudes

ﬁeld theory limit α0 → 0 D(–1) disks Effective actions

D3 disks D(–1) and mixed disks

Mixed disks SYM action instanton action (ADHM)

Disk amplitudes and effective actions D3 disks

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 16 / 42 Disk amplitudes

ﬁeld theory limit α0 → 0

Effective actions

D3 disks D(–1) and mixed disks

Mixed disks SYM action instanton action (ADHM)

Disk amplitudes and effective actions D3 disks

D(–1) disks

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 16 / 42 Disk amplitudes

ﬁeld theory limit α0 → 0

Effective actions

D3 disks D(–1) and mixed disks

SYM action instanton action (ADHM)

Disk amplitudes and effective actions D3 disks

D(–1) disks

Mixed disks

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 16 / 42 Disk amplitudes and effective actions D3 disks Disk amplitudes

ﬁeld theory limit α0 → 0 D(–1) disks Effective actions

D3 disks D(–1) and mixed disks

Mixed disks SYM action instanton action (ADHM)

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 16 / 42 An example of mixed disk amplitude

Consider the following mixed disk diagram w¯

which corresponds to the following amplitude Z Q DD EE i dzi n α˙ Ao VλVw¯ Vµ ≡ C0 × Vλ(z1)Vw¯ (z2)Vµ(z3) = ... = trk i λ A w¯α˙ µ dVCKG

2 2 where C0 = 8π /g is the disk normalization.

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 17 / 42 The instanton moduli action Collecting all diagrams D(–1) and mixed disk diagrams with insertion of all moduli vertices, we can extract the instanton moduli action n i i S = tr − [a , χm]2 − MαA[χ , MB] + χmw¯ w α˙ χ + µ¯AµBχ 1 µ 4 AB α α˙ m 2 AB ˙ c ¯ α˙ c β c µ ν − iD w (τ )α˙ wβ˙ + iη¯µν[a , a ] α˙ A A µ βA o + iλA µ¯ wα˙ + w¯α˙ µ + σβα˙ [M , aµ]

m where χAB = χm (Σ )AB.

I S1 is just a gauge theory action dimensionally reduced to d = 0 in the ADHM limit.

I The last two lines in S1 correspond to the bosonic and fermionic ADHM constraints.

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 18 / 42 There are two types of solutions: m m 12×2 χ 6= 0 , wuα˙ = 0 χ = 0 , wuα˙ = ρ 0(N−2)×2

χ D3

D3 D(–1) ρ ∼ instanton size D(–1)

Take for simplicity k = 1 (−→ [ , ] = 0). The bosonic “equations of motion” m ¯ c α˙ β˙ wuα˙ χ = 0 , wα˙ u(τ ) wuβ˙ = 0 determine the classical vacua.

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 19 / 42 m 12×2 χ = 0 , wuα˙ = ρ 0(N−2)×2

ρ ∼ instanton size D(–1)

Take for simplicity k = 1 (−→ [ , ] = 0). The bosonic “equations of motion” m ¯ c α˙ β˙ wuα˙ χ = 0 , wα˙ u(τ ) wuβ˙ = 0 determine the classical vacua.

There are two types of solutions:

m χ 6= 0 , wuα˙ = 0

D3 D(–1)

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 19 / 42 Take for simplicity k = 1 (−→ [ , ] = 0). The bosonic “equations of motion” m ¯ c α˙ β˙ wuα˙ χ = 0 , wα˙ u(τ ) wuβ˙ = 0 determine the classical vacua.

There are two types of solutions: m m 12×2 χ 6= 0 , wuα˙ = 0 χ = 0 , wuα˙ = ρ 0(N−2)×2

χ D3

D3 D(–1) ρ ∼ instanton size D(–1)

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 19 / 42 I The other neutral (anti-chiral) fermionic zero-modes

λα˙ A

appear linearly in S1 and are the Lagrange multipliers for the fermionic ADHM constraints. They have dimensions of (length)−3/2.

I The bosonic charged moduli

w¯α˙ , wα˙ describe the instanton size and its orientation (in the SU(N)). They carry dimensions of (length).

I The neutral zero-modes x µ = traµ and θαA = trMαA

are the Goldstone modes of the broken (super)translations. S1 does not depend on them. They play the role of the superspace coordinates.

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 20 / 42 I The bosonic charged moduli

w¯α˙ , wα˙ describe the instanton size and its orientation (in the SU(N)). They carry dimensions of (length).

I The neutral zero-modes x µ = traµ and θαA = trMαA

are the Goldstone modes of the broken (super)translations. S1 does not depend on them. They play the role of the superspace coordinates.

I The other neutral (anti-chiral) fermionic zero-modes

λα˙ A

appear linearly in S1 and are the Lagrange multipliers for the fermionic ADHM constraints. They have dimensions of (length)−3/2.

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 20 / 42 I The neutral zero-modes x µ = traµ and θαA = trMαA

are the Goldstone modes of the broken (super)translations. S1 does not depend on them. They play the role of the superspace coordinates.

I The other neutral (anti-chiral) fermionic zero-modes

λα˙ A

appear linearly in S1 and are the Lagrange multipliers for the fermionic ADHM constraints. They have dimensions of (length)−3/2.

I The bosonic charged moduli

w¯α˙ , wα˙ describe the instanton size and its orientation (in the SU(N)). They carry dimensions of (length).

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 20 / 42 To understand better all this, let us look at the instanton classical solution

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 21 / 42 Instanton classical solution

I The mixed disks are the sources for a non-trivial gauge ﬁeld whose proﬁle is exactly that of the classical instanton!! [Billó et al. 2002,...]

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 22 / 42 Instanton classical solution

I The mixed disks are the sources for a non-trivial gauge ﬁeld whose proﬁle is exactly that of the classical instanton!! [Billó et al. 2002,...]

Let us consider the following mixed-disk amplitude:

w¯

Ac (p) ≡ V c µ Aµ(p) mixed disk

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 22 / 42 Instanton classical solution

I The mixed disks are the sources for a non-trivial gauge ﬁeld whose proﬁle is exactly that of the classical instanton!! [Billó et al. 2002,...]

Using the explicit expressions of the vertex operators, for SU(2) with k = 1 one ﬁnds

V c ≡ V ¯ V c (p) V Aµ(p) mixed disk w Aµ w

ν c α˙ −i p·x0 c = − i p η¯µν w¯ wα˙ e ≡ Aµ(p; w, x0)

I On this mixed disk the gauge vector ﬁeld has a non-vanishing tadpole!

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 22 / 42 I In fact (x − x )ν Ac (x) = 2 ρ2 η¯c 0 µ µν 2 2 2 instanton (x − x0) (x − x0) + ρ (x − x )ν ρ2 = ρ2 η¯c 0 − + ... 2 µν 4 1 2 (x − x0) (x − x0)

c I Taking the Fourier transform of Aµ(p; w, x0), after inserting the free propagator 1/p2, we obtain

Z d 4p 1 (x − x )ν c ( ) ≡ c ( ; , ) i p·x = ρ2 η¯c 0 Aµ x 2 Aµ p w x0 2 e 2 µν 4 (2π) p (x − x0) where we have used the solution of the ADHM constraints so that α˙ 2 w¯ wα˙ = 2 ρ .

I This is the leading term in the large distance expansion of an SU(2) instanton with size ρ and center x0 in the singular gauge!!

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 23 / 42 c I Taking the Fourier transform of Aµ(p; w, x0), after inserting the free propagator 1/p2, we obtain

I This is the leading term in the large distance expansion of an SU(2) instanton with size ρ and center x0 in the singular gauge!!

I In fact (x − x )ν Ac (x) = 2 ρ2 η¯c 0 µ µν 2 2 2 instanton (x − x0) (x − x0) + ρ (x − x )ν ρ2 = ρ2 η¯c 0 − + ... 2 µν 4 1 2 (x − x0) (x − x0)

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 23 / 42 I The subleading terms in the large distance expansion can be obtained from mixed disks with more insertions of moduli.

I For example, at the next-to-leading order we have to consider the following mixed disk which can be easily evaluated for α0 → 0

α0 → 0

I Its Fourier transform gives precisely the 2nd order in the large distance expansion of the instanton proﬁle (x − x )ν c ( )(2) = − ρ4 η¯c 0 Aµ x 2 µν 6 (x − x0)

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 24 / 42 Summary

I Mixed disks are sources for open strings

φopen −→ −→ Vφ open mixed disk 6= 0

sourcemixed disk

I The gauge ﬁeld emitted from mixed disks is precisely that of the classical instanton

VAµ mixed disk ⇔ Aµ instanton

I This procedure can be easily generalized to the SUSY partners of the gauge boson.

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 25 / 42 The stringy instanton calculus

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 26 / 42 I The combinatorics of the disk diagrams w¯

≡ + + ... λ

2 µ α0→0 8π k ' − − S1(Mc(k)) g2 is such that they exponentiate, leading to the instanton partition function Z (k)

[Polchinski 1994, ..., Dorey et al. 1999, ...]

The instanton partition function

I The crucial ingredient is the moduli action S1: it is given by (mixed) disk diagrams; the result depends only on the centered µ αA moduli Mc(k) but not on the center x nor on its super-partners θ

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 27 / 42 The instanton partition function

I The combinatorics of the disk diagrams w¯

≡ + + ... λ

2 µ α0→0 8π k ' − − S1(Mc(k)) g2 is such that they exponentiate, leading to the instanton partition function Z 8π2 k Z − −S1(Mc ) (k) 4 8 g2 (k) 4 8 (k) Z ∼ d x d θ dMc(k) e ∼ d x d θ Zd

[Polchinski 1994, ..., Dorey et al. 1999, ...]

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 27 / 42 I Since we integrate over the instanton moduli M(k), also disconnected diagrams must be considered. For example we must take into account also

2 × = S1(M(k))

I The mixed disk amplitudes giving rise to the moduli action S1(M(k)), i.e.

= − S1(M(k))

from the D3 brane point of view represent a “vacuum contribution”.

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 28 / 42 I The mixed disk amplitudes giving rise to the moduli action S1(M(k)), i.e.

= − S1(M(k))

from the D3 brane point of view represent a “vacuum contribution”.

I Since we integrate over the instanton moduli M(k), also disconnected diagrams must be considered. For example we must take into account also

2 × = S1(M(k))

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 28 / 42 I Of course one should add more disconnected components and take into account the appropriate symmetry factors

I A careful study of the combinatorics of boundaries leads to the exponentiation of the disk amplitudes, namely to [Polchinski 1994] ( ) 1 1 + + 2 × + ...

1 = 1 − S (M )+ S (M )2 + ... 1 (k) 2 1 (k)

8π2 k − S (M ) − − S1(Mc(k)) = e 1 (k) = e g2

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 29 / 42 I Considering all such diagrams one obtains the ﬁeld-dependent moduli action n i S (M ;Φ) = tr w¯ ΦmΦ w α˙ + (Σm) µ¯AΦ µB 2 (k) α˙ m 2 AB m m α˙ α˙ mo + χ w¯α˙ Φm w + w¯α˙ Φm w χ + fermion terms

Field dependent moduli action m I Consider correlators of D3/D3 ﬁelds, e.g. of the scalars Φ , in presence of k D-instantons. They are described by disk diagrams with (at least) one insertion of VΦm . For example we have µB

Φm

µ¯A

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 30 / 42 Field dependent moduli action m I Consider correlators of D3/D3 ﬁelds, e.g. of the scalars Φ , in presence of k D-instantons. They are described by disk diagrams with (at least) one insertion of VΦm . For example we have µB

Φm

µ¯A

I Considering all such diagrams one obtains the ﬁeld-dependent moduli action n i S (M ;Φ) = tr w¯ ΦmΦ w α˙ + (Σm) µ¯AΦ µB 2 (k) α˙ m 2 AB m m α˙ α˙ mo + χ w¯α˙ Φm w + w¯α˙ Φm w χ + fermion terms

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 30 / 42 A k-instanton contribution is then given by the integral over ALL MODULI Z (k) −S1(M(k))−S2(M(k);Φ) Z ∝ dM(k) e Z ∝ d{a, χ, M, λ, D, w, w¯ , µ, µ¯} e−S1−S2

Since xµ = traµ and θαA = trMαA are the Goldstone modes of the broken (super)translations and play the role of the superspace coordinates, it is convenient to separate them and write

Z 8π2 k − −S1(Mc )−S2(Mc ;Φ) (k) 4 8 g2 (k) (k) Z ∼ d x d θ dMc(k) e Z ∼ d 4x d 8θ Zd(k)(Φ)

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 31 / 42 Some simple possibilities:

• D3/D(–1) system on R4 × C3 ⇓ N = 4 SYM + instantons (A = 1, 2, 3, 4)

4 2 • D3/D(–1) system on R × C × C /Z2 ⇓ N = 2 SYM + instantons (A = 1, 2)

4 3 • D3/D(–1) system on R × C /(Z2 × Z2) ⇓ N = 1 SYM + instantons (A = 1)

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 32 / 42 •N = 1 Z Z 4 2 −S1−S2 Z = dx dθ W where W ∝ dMc(k) e

I The integral over the anti-chiral zero modes λα˙ A enforces the fermionic ADHM constraints from S1.

I In general, one must investigate under what conditions these instanton contributions to F or W are non vanishing, and what is their structure (prepotential, superpotential, ...)

In the case of reduced SUSY, some of the θαA will not be present.

•N = 2 Z Z 4 4 −S1−S2 Z = dx dθ F where F ∝ dMc(k) e

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 33 / 42 I The integral over the anti-chiral zero modes λα˙ A enforces the fermionic ADHM constraints from S1.

I In general, one must investigate under what conditions these instanton contributions to F or W are non vanishing, and what is their structure (prepotential, superpotential, ...)

In the case of reduced SUSY, some of the θαA will not be present.

•N = 2 Z Z 4 4 −S1−S2 Z = dx dθ F where F ∝ dMc(k) e

•N = 1 Z Z 4 2 −S1−S2 Z = dx dθ W where W ∝ dMc(k) e

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 33 / 42 In the case of reduced SUSY, some of the θαA will not be present.

•N = 2 Z Z 4 4 −S1−S2 Z = dx dθ F where F ∝ dMc(k) e

•N = 1 Z Z 4 2 −S1−S2 Z = dx dθ W where W ∝ dMc(k) e

I The integral over the anti-chiral zero modes λα˙ A enforces the fermionic ADHM constraints from S1.

I In general, one must investigate under what conditions these instanton contributions to F or W are non vanishing, and what is their structure (prepotential, superpotential, ...)

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 33 / 42 Some explicit examples and applications of this stringy instanton calculus will be discussed in

Marco Billò’s talk

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 34 / 42 The instanton partition function (... again) The k-instanton partition function is the “functional” integral over the instanton moduli: Z (k) −S(Mk ) Z = Ck dMk e

where

•C k is a dimensionful normalization factor which compensates for the dimensions of dMk

• S(Mk ) is the moduli action which accounts for all interactions among the instanton moduli in the limit α0 → 0 at any order of string perturbation theory, i.e. on any world-sheet topology.

( )

−S(Mk ) = limα0→0 + + ...

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 35 / 42 As we have seen before, at the tree-level, we have

≡ + + ...

= h1idisk + hM(k)idisk

2 α0→0 8π k '− − S1(Mc(k)) g2 Thus, −2 0 h1idisk ∼ O(g ) , hM(k)idisk ∼ O(g )

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 36 / 42 Similarly, at the one-loop level, one can show that

=++ + ...

|{z} |{z} h1iannulus hMk iannulus where 0 2 h1iannulus ∼ O(g ) , hM(k)iannulus ∼ O(g )

Thus, in the semi-classical approximation one has [Blumenhagen et al., Akerblom et al. 2006] Z (k) −S(Mk ) Z = Ck dMk e

Z 0 h1idisk + hM(k)idisk + h1i ∼ Ck dMk e annulus

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 37 / 42 I evaluated on a k-instanton background becomes 8π2k S = inst g2

Thus we have the simple relation

The disk vacuum amplitude The YM action 1 Z 1 S = d 4x Tr F 2 g2 2 µν

I evaluated on a constant gauge ﬁeld f becomes V f 2 S(f ) = 4 2 g2

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 38 / 42 Thus we have the simple relation

The disk vacuum amplitude The YM action 1 Z 1 S = d 4x Tr F 2 g2 2 µν

I evaluated on a constant gauge ﬁeld f becomes V f 2 S(f ) = 4 2 g2

I evaluated on a k-instanton background becomes 8π2k S = inst g2

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 38 / 42 The disk vacuum amplitude The YM action 1 Z 1 S = d 4x Tr F 2 g2 2 µν

I evaluated on a constant gauge ﬁeld f becomes V f 2 S(f ) = 4 2 g2

I evaluated on a k-instanton background becomes 8π2k S = inst g2

Thus we have the simple relation

1 S(f )00 S = = inst 2 2 g V4 8π k

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 38 / 42 The disk vacuum amplitude The YM action 1 Z 1 S = d 4x Tr F 2 g2 2 µν

I evaluated on a constant gauge ﬁeld f becomes V f 2 S(f ) = 4 2 g2

I evaluated on a k-instanton background becomes 8π2k S = inst g2

Thus we have the simple relation

00 S(f ) Sinst == ≡ 2 V4 8π

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 38 / 42 The annulus vacuum amplitude A similar relation holds also at one-loop:

I in the constant gauge ﬁeld background we have

V f 2 S(f ) + S1−loop(f ) = 4 2 g2(µ)

where g(µ) is the running coupling constant at scale µ

1 1 b µ2 = + 1 + ∆ 2 2 2 log 2 g (µ) g 16π ΛUV

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 39 / 42 The annulus vacuum amplitude A similar relation holds also at one-loop:

I in the constant gauge ﬁeld background we have

V f 2 S(f ) + S1−loop(f ) = 4 2 g2(µ)

I in the k-instanton background, for a supersymmetric theory we have 8π2k S + S1−loop = inst inst g2(µ)

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 39 / 42 The annulus vacuum amplitude A similar relation holds also at one-loop:

I in the constant gauge ﬁeld background we have

V f 2 S(f ) + S1−loop(f ) = 4 2 g2(µ)

I in the k-instanton background, for a supersymmetric theory we have 8π2k S + S1−loop = inst inst g2(µ) Thus, also at one-loop we have the simple relation

S1−loop(f )00 S1−loop = inst 2 V4 8π k

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 39 / 42 The annulus vacuum amplitude A similar relation holds also at one-loop:

I in the constant gauge ﬁeld background we have

V f 2 S(f ) + S1−loop(f ) = 4 2 g2(µ)

I in the k-instanton background, for a supersymmetric theory we have 8π2k S + S1−loop = inst inst g2(µ) Thus, also at one-loop we have the simple relation

1−loop 00 1−loop S (f ) Sinst == ≡ + 2 V4 8π k

[Abel+Goodsell, Akerblom et al.]

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 39 / 42 The rôle of the annulus amplitude By explicitly computing the annulus diagrams, one ﬁnds

= 0

2 1 0 2 = − 8π k 16π2 b1 log(α µ ) +∆

where the β-function coefﬁcient b1 counts the number of charged (and ﬂavored) ADHM instanton moduli

1 1 b = n − n = # {w, w¯ } − # {µ, µ¯} 1 bos 2 ferm 2 and ∆ are the threshold corrections. [Akerblom et al., Billó et al.]

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 40 / 42 I The k-instanton contributions are

Z 0 (k) h1idisk + hM(k)idisk + h1i Z = Ck dMk e annulus

Z 8π2k 2 − 2 − S(Mc(k);Φ) − 8π k ∆ = Ck dMk e g

k b where Ck ∼ (Ms) 1 . Thus,

2 Z (k) k b1 − 8π k ∆ − S(Mc(k);Φ) Z ∼ Λ e dMk e

where Λ is the dynamically generated scale.

To conclude:

I The k-instanton contributions can be computed from perturbative string diagrams: mixed disks and mixed annuli

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 41 / 42 To conclude:

I The k-instanton contributions can be computed from perturbative string diagrams: mixed disks and mixed annuli

I The k-instanton contributions are

Z 0 (k) h1idisk + hM(k)idisk + h1i Z = Ck dMk e annulus

Z 8π2k 2 − 2 − S(Mc(k);Φ) − 8π k ∆ = Ck dMk e g

k b where Ck ∼ (Ms) 1 . Thus,

2 Z (k) k b1 − 8π k ∆ − S(Mc(k);Φ) Z ∼ Λ e dMk e

where Λ is the dynamically generated scale.

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 41 / 42 Stay tuned for Marco’s talk

Thank you !

Alberto Lerda (U.P.O.) Instantons Vienna, October 7, 2008 42 / 42