stringy instantons partition function - scuola...

Stringy Instantons Partition function

Giulio Bonelli (SISSA & ICTP)

May 17, 2013

with A. Sciarappa, A. Tanzini and P. VaskoBased on papers in preparation

(SU and FI – Pisa 17 May 2013) May 17, 2013 1 / 22

Outline

1 Finite size corrections to 4d N = 2 gauge theories

2 The ADHM gauged linear sigma model

3 Checks of the results

4 Finite size corrections to Nekrasov’s PF

5 Conclusions


Introduction: 4d N = 2 theories

D = 4 N = 2 gauge theories have an exactly solvable moduli space of orderparameters while the RG-flow in the coupling constants is still non trivial.

The IR effective dynamics of asymptotically free theories can be computed fromsome spectral data (Seiberg-Witten curve & differential) defining an integrablesystem.

The microscopic derivation of the SW solution has been obtained by Nekrasovusing Equivariant Localization.

Equivariant Localization is a technique to exactly compute BPS saturatedobservables in supersymmetric quantum field theories by making explicit the nonrenormalization properties of protected sectors. In the last years it proved to be avery powerful computational method.

The Ω-background – a specific gravitational coupling – provides a regularizationof the theory.

The instanton part of the Nekrasov partition function computes the regularizedvolume of the moduli space of the gauge theory and encodes the data of thecohomology of the ADHM instanton moduli space.

Its perturbative part is still crucial to reproduce the SW geometry.


Introduction: Equivariant Localization in a nut-shellLet

< O >:=

∫D[Φ]e−S[Φ]O[Φ] (1)

be a path integral which admits a scalar Grassmann odd symmetry Q (that iseverything is invariant under the infinitesimal redefinition δΦ = QΦ).Then the path integral is supported at the fixed loci of the odd symmetry Q. (Proof:away from fixed loci the field space can be parameterized by an odd parameter θdescribing the Q-flow, but the integrand is θ-independent and

∫dθ[const ] = 0.)

Q2 = R is then a bosonic symmetry.One can consider the deformation by a Q-exact term S → S + tQV , where V is anodd R-invariant fuctional Q2V = RV = 0

< O >t :=

∫D[Φ]e−S[Φ]−tQV [Φ]O[Φ]. (2)

This is t-independent. (Proof: − ddt < O >t :=

∫D[Φ]e−S[Φ]−tQV [Φ]O[Φ]QV [Φ] =∫

D[Φ]Q[e−S[Φ]−tQV [Φ]O[Φ]V [Φ]

]= 0.)

Therefore one can evaluate

< O >= limt→∞ < O >t (3)

via semiclassical approximation (as if ~ = 1t ). [This is well defined if V satisfies some

technical conditions]• Different choices of V corresponds to different presentations of the same object.


D-brane engineering of finite size effects in d4 N = 2SYMD=4 N = 2 can be engineered via D3-branes at the tip of the singular conifold C2/Z2.Instantons are described by D(-1)-branes.To avoid the singularity it is cleaner to define the theory on the resolved conifold T ∗P1

via D5-branes and D1-branes wrapping the P1 and then to consider the theory in thesmall radius limit.We will study the finite r effectsD3/D(-1) at the tip of the conifold⇒ D5/D1 wrapping the P1

C

C2

C

2CZ2

r0

C

C2

T*P1

r


Introduction: the deformed theory

The Nekrasov partition function computation of the D=4 theory is essentiallygiven by evaluating the gauge theory from the D-brane engineering point of view.

The D = 6 gauge theory on P1 × C2 can be computed from this same point ofview.

The perturbative part from the D5/D5 open string sector & the non-perturbativepart from the D1/D1 and D1/D5 open string sector.

Question 1: How does the Nekrasov partition function gets modified?

Question 2: Which new aspects of the ADHM moduli are probed by thedeformed theory?


Abstract of the Talk

We perform the exact computation in r of the gauged linear sigma modelassociated to the D1/D5 brane system on T ∗P1.

This is described by the embedding of an effective spherical world-sheet in theADHM instanton moduli space.

The computation can be done by supersymmetric localization on the blown-uptwo-sphere.

On the singular geometry C2/Z2, corresponding to the point-particle limit r → 0,the spherical partition function reduces to the Nekrasov partition function.

For finite radius we obtain a tower of world-sheet instanton corrections, that weidentify with the equivariant Gromov-Witten invariants of the instanton modulispace.

From the mathematical viewpoint, we show that the D1-D5 system under studycaptures the equivariant quantum cohomology of the ADHM instanton modulispace in the Givental formalism. (J -function)


The ADHM gauged linear sigma modelConsider the type IIB background R1,3 × T ∗P1 × R2 with the k D1-branes wrappingthe P1 and space-time filling N D5-branes wrapping the P1 too.The dynamics of the D1-branes is described by a two-dimensional N = (2, 2) gaugedlinear sigma model flowing in the infrared to a non-linear sigma model with targetspace the ADHM moduli space of instantonsMk,N .

Λ B1 B2 I JD-brane sector D1/D1 D1/D1 D1/D1 D1/D5 D5/D1

gauge U(k) Adj Adj Adj k kflavor U(N)× U(1)2 1(−1,−1) 1(1,0) 1(0,1) N(0,0) N(1,1)

twisted masses ε −ε1 −ε2 −ai aj − εR-charge 2− 2q q q q + p q − p

The superpotential is W = Trk Λ ([B1,B2] + IJ). The twisted masses corresponds tothe maximal torus in the global symmetry group U(1)N+2 acting onMk,N which wedenote as (aj ,−ε1,−ε2). The R-charges are assigned as the most general ones whichensures R(W ) = 2 and full Lorentz symmetry at zero twisted masses.The D5-D5 sector on C2 × P1 is treated separately and produces the finite rcorrections to the perturbative gauge theory sector.


Evaluation of the partition functionThe computation of the partition function of the gauged linear sigma model on thetwo-sphere can be performed via equivariant localization.The path integral localization is performed with respect to the superchargeQ = Q + Q†, where Q = εαQα and Q† = −(ε†C)αQ†α with C being the chargeconjugation matrix. ε is a particular solution to the Killing spinor equation chosen asε = ei φ2 (cos θ

2 , i sin θ2 ). The supercharges Q,Q† form a su(1|1) subalgebra of the full

superalgebra, up to a gauge transformation Λ,Q,Q†

= M +

R2

+ iΛ, Q2 =(

Q†)2

= 0, (4)

where M is the generator of isometries of the sphere infinitesimally represented by theKilling vector v =

(ε†γaε

)ea = 1

r∂∂φ

and R is the generator of the U(1)R symmetry.The Killing vector field generates SO(2) rotations around the axis fixed by the northand south pole. Finally, the localizing supercharge Q satisfies

Q2 = M +R2

+ iΛ. (5)

The fact that M generates a U(1) isometry with the north and south poles as fixedpoints will play an important rôle later.


Evaluation of the partition function

Multiplets & actions

(2,2) vector multiplet:(Aµ, σ, η, λ, λ,D

)(2,2) chiral multiplet:

(φ, φ, ψ, ψ,F , F

).

(6)

The action isS =

∫d2x (LYM + LFI+top + Lmatter + LW ) , (7)

where

LbosYM =

1g2 Tr

12

(F12 −

η

r

)2+

12

(D +

σ

r

)2+

12

DµσDµσ +12

DµηDµη − 12

[σ, η]2(8)

Lbosmatter = DµφDµφ+ φσ2φ+ φη2φ+ iφDφ+ FF +

iqrφσφ+

q(2− q)

4r 2 φφ (9)

LFI+top = −iξD + iθ

2πF12 , LW =

∑j

∂W∂φj

Fj −∑j,k

12∂2W∂φj∂φk

ψjψk (10)

with r the radius of the sphere and q the R-charge of the chiral multiplet.



Equivariant LocalizationTo localize on field configurations corresponding to the Coulomb branch the followingQ exact deformation of the action was chosen

δS =

∫d2x (LYM + Lψ) , (11)

where

LYM = QTr(Qλ)λ+ λ†(Qλ†)

4, Lψ = Q (Qψ)ψ + ψ†(Qψ†)

2. (12)

This procedure reduces the path integral to an ordinary integral over the constantmodes of the scalar field σ and a sum over the non trivial fluxes of the gauge field onthe two-sphere.The non trivial part of the computation is the precise evaluation of the one-loopdeterminants with twisted masses and fluxes turned on.



Equivariant LocalizationThe S2 partition function reads

Z S2

k,N =1k !

∑~m∈Zk

∫Rk

k∏s=1

dσs

2πe−4πiξσs−iθms

k∏s<t

(m2

st

4+ σ2

st

)ZIJ Zadj (13)

where the one-loop determinants of the matter contributions are given by

ZIJ =k∏

s=1

N∏j=1

Γ(−iσs + iraj − ms

2

)Γ(1 + iσs − iraj − ms

2

) Γ(iσs − ir (aj − ε) + ms

2

)Γ(1− iσs + ir (aj − ε) + ms

2

)Zadj =

k∏s,t=1

Γ(1− iσst − irε− mst

2

)Γ(iσst + irε− mst

2

) Γ(−iσst + irε1 − mst

2

)Γ(1 + iσst − irε1 − mst

2

) Γ(−iσst + irε2 − mst

2

)Γ(1 + iσst − irε2 − mst

2

)with ε = ε1 + ε2.


Evaluation of the partition functionpoles analysis: these are classified by Young tableaux, just like for the Nekrasovpartition (each YT has a KK-tower on top)In order to compute (13), we close the contour integral in the lower half plane. Therelevant poles are at the points

σs = −ids + ims

2+ iraj (14)

with ds ≥ 0, ms ∈ Zk and s = 1, . . . , k .This implies the equivalent expression of the S2-partition function

Z S2

k,N =1k !

∮ k∏s=1

d(rλs)

2πi(zz)−rλs Z1lZvZav (15)

where z = e−2πξ+iθ and the (anti-)Vortex contributions and the 1-loop measure aregiven as follows: define the Pochhammer symbol (a)d as

(a)d =

∏d−1

i=0 (a + i) for d > 01 for d = 0∏d

i=11

a− ifor d < 0

(16)



Z1l =

(Γ(1− irε)Γ(irε1)Γ(irε2)

Γ(irε)Γ(1− irε1)Γ(1− irε2)

)k k∏s=1

N∏j=1

Γ(rλs + iraj )Γ(−rλs − iraj + irε)Γ(1− rλs − iraj )Γ(1 + rλs + iraj − irε)

k∏s 6=t

(rλs − rλt )Γ(1 + rλs − rλt − irε)Γ(rλs − rλt + irε1)Γ(rλs − rλt + irε2)

Γ(−rλs + rλt + irε)Γ(1− rλs + rλt − irε1)Γ(1− rλs + rλt − irε2)

Zv =∑

d1,...,dk ≥ 0

((−1)Nz)d1+...+dk

k∏s=1

N∏j=1

(−rλs − iraj + irε)ds

(1− rλs − iraj )ds

k∏s<t

dt − ds − rλt + rλs

−rλt + rλs

(1 + rλs − rλt − irε)dt−ds

(rλs − rλt + irε)dt−ds

(rλs − rλt + irε1)dt−ds

(1 + rλs − rλt − irε1)dt−ds



Zav =∑

d1,...,dk ≥ 0

((−1)N z)d1+...+dk

k∏s=1

N∏j=1

(−rλs − iraj + irε)ds

(1− rλs − iraj )ds

k∏s<t

dt − ds − rλt + rλs

−rλt + rλs

(1 + rλs − rλt − irε)dt−ds

(rλs − rλt + irε)dt−ds






The resulting structureTherefore the 2-sphere partition function factorizes as the pairing of the equivariantvortex countings corresponding to the North and South pole locations. So we read

Z S2=< 0|x |0 > (17)

that is the pairing of the vacuum wave functions.• The wave function is the equivariant Shadchin’s (anti-)vortex partition functionsdepending on the boundary value of the scalar field in the vector bundle (that’s theanalogous of Nekrasov’s on C).• This structure is generic on SD partition functions.It has a clean interpretation from the IR NLSM point of view (see later)It has been argued that the spherical partition function (squashing independence)computes the vacuum amplitude of the NLSM in the IR

< 0|0 >= e−K (18)

where K is the quantum Kähler potential of the target space X .

=

= x


Reduction to Nekrasov’s at r ∼ 0

Let’s expand the ADHM sphere partition function at small r (point particle limit).Since Zv = 1 + O(r), the non trivial contribution comes from the 1-loop part.Expanding in r the factors of (cfr. the 1-loop part)

Γ(rX )

Γ(1− r X )=

1rX

Γ(1 + rX )

Γ(1− r X )=

1rX

(1 + O(r)) (19)

we find thatZ S2

k,N =1

r 2Nk × Z Nekk,N × (1 + O(r)) (20)

where Z Nekk,N is given by

Z Nekk,N =

1k !

εk

(2πiε1ε2)k

∮ k∏s=1

dσs

P(σs)P(σs + ε)

k∏s<t

σ2st (σ

2st − ε2)

(σ2st − ε2

1)(σ2st − ε2

2)(21)

with P(σs) =∏N

j=1(σs − aj ).


Checks of some limiting cases

There are two important cases where we can check our results against known ones.

Abelian theory, that is k point-like instantons in C2 andMk,1 =˜[

(C2)N /SN

],

called also Hilbert scheme of k points of over C2.

1 instanton, of the U(N) gauge theory where the moduli spaceM1,N = C2 × T ∗PN−1.

What should we compare it to?


Checks of some limiting casesThe vacuum wave function that we computed is a quite known object in topologicalstrings.The vacuum wave function satisfies in general a set of differential equations. Let thechiral ring of observables be φaφb = Cc

abφc , then the flat sections of the Gauss-Maninconnection span the vacuum bundle. These solve(

~Daδcb + Cc

ab)

Vc = 0. (22)

and are all generated by the vacuum wave function J = V0 as Va = −~DaV0.The observables φa provide a basis of the chiral ring H0(X )⊕ H2(X ) witha = 0, 1, . . . , b2(X ), φ0 being the identity operator. V0 defines the vacuum at a givenpoint in the coupling space. It is called "equivariant small Givental function" andcomputes equivariant Gromov-Witten invariants.The metric on the vacuum bundle is given by a symplectic pairing of the flat sectionsgab = 〈a|b〉 = V t

aEVb and in particular the vacuum-vacuum amplitude, can be writtenas the symplectic pairing [r = 1

~ ]

〈0|0〉 = J tEJ =

∮dλZpertZvZav (23)

Since the ADHM moduli space is non-compact, the world-sheet instanton correctionsare non-trivial only in presence of the Omega background. (ε1 + ε2 matters).[Equivariant cohomology of the target space H•T (X ) where T is the torus acting on X . ]


Checks of some limiting cases

We compared then the vortex partition functions we got factorizing the sphericalpartition functions with the equivariant Givental’s small J -functions in the two limitingcases finding perfect agreement.From a mathematical viewpoint our result then is a conjectural formula for theJ -function of the general ADHM moduli spaceMk,N (which is unknown).Let’s discuss now our result on the gauge theory side...


Finite size corrections to Nekrasov’s PFRemind that we still have to compute the D5-D5 sector of the theory to get the fullresult.This amounts to compute the finite size corrections to the perturbative part of theNekrasov’s PFThis has to be done by reducing to the P1 (instead that to a point) the perturbativeexcitations of the U(N) gauge theory on T ∗P1 × C2

ε1,ε2 × C.The one-loop measure is then computed in the ζ-function regularization from theequivariant character of the relevant complex on P1.

logZD5D5 = −∫R+

dtt

Tr [etD]P1 Tr [e−tD]P1 (1− etε1 )(1− etε2 ) (24)

where D = ∂ + Φ is the holomorphic covariant derivative on the P1 twisted by thescalar Φ.[The determinant is materially obtained by index theorem of the relevant complex as∑α cαewα(m) → [wα(m)]cα by reading the Euler character from the Chern one.]

• In the point particle limit Tr [e±tD]P1 → Tre±tΦ and we get back the Nekrasovperturbative partition function as as

ZD5D5 =∏i 6=j

Γ(r)2 (aij ) → Z pert

Nek =∏i 6=j

Γ2(aij )

where Γ(r)2 is a particular known deformation of the Barnes Γ2.


Conclusions

Explore the properties of the solution we found

Relation with integrable systems (Calogero-Sutherland type deformed by aquantum parameter to identify with r )

It is crucially linked to AGT (quantum cohomology of ADHM)

Relation with higher rank DT theory on C2 × P1 (it seems we have the solution )

Reproduce it from open string world-sheet arguments (in the local O(−2)P1

manifold which is not an orbifold).

Generalize to richer theories (add matter, several gauge groups... quiverize it!)

Generalize to curved space-times


Thank mou!


stringy instantons partition function - scuola...

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