A Systematic Study of Instanton Moduli Spaces

Home , E7 (mathematics), F4 (mathematics)

NOPPADOL MEKAREEYA

CERN Max Planck Institute for Physics

DESY September 2013

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 1 / 22 Based on the following works:

[arXiv:1309.1213] with D. Rodriguez-Gomez

[arXiv:1309.0812] with A. Dey, A. Hanany, D. Rodriguez-Gomez, R. K. Seong

[arXiv:1205.4741] with A. Hanany and S. Razamat

[arXiv:1111.5624] with C. Keller, J. Song and Y. Tachikawa

[arXiv:1005.3026] with S. Benvenuti and A. Hanany

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 2 / 22 Summary

Objects of our interest:

I Space of field configurations, known as the moduli space, of instantons in Yang-Mills theory on a flat space or an ALE space.

I Partition function of such a moduli space & instanton partition function.

2 On a ﬂat space C :

I For classical groups ABCD, the instanton solutions can be constructed using linear algebra: ADHM construction. (Atiyah, Drinfeld, Hitchin, Manin ’78).

I Such constructions can be realised from the Dp-D(p + 4) brane system and hence from the Higgs branch of certain 4d N = 2 gauge theories. (Douglas ’95; Witten ’95)

I Instanton partition functions can be computed via the ADHM constructions. (Nekrasov ’02; Nekrasov-Shadchin ’04)

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 3 / 22 Summary (cont.)

For exceptional groups EFG, the ADHM construction is not known!

I For E6, E7 and E8, there are constructions from M5-branes on Riemann spheres with punctures (Gaiotto ’09). The corresponding gauge theories have no known Lagrangians. (Benini-Benvenuti-Tachikawa ’09; Gaiotto-Razamat ’12)

I No such constructions available for F4 and G2.

Instantons partition functions can be computed explicitly and exactly for any simple group ABCDEFG for small instanton numbers

I Using character expansion for Hilbert series and techniques of superconformal indices. (Benvenuti-Hanany-N.M. ’10; Gadde-Rastelli-Razamat-Yan ’11; Gaiotto-Razamat ’12; Hanany-N.M.-Razamat ’12) For one instanton, the inst. partition function admits a closed compact form.

I Matched with a norm of the coherent state (in the twisted sector) of the W-algebra. (Keller-N.M.-Song-Tachikawa ’10)

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 4 / 22 Summary (cont.)

2 On an ALE space of A-type, resolved C /Zn, an analogue of the ADHM construction is known for U and SU instantons: Kronheimer-Nakajima (KN) construction (1990).

I Instanton partition functions have been computed (e.g. Fucito-Morales-Poghossian ’04,’06; Bonelli-Maruyoshi-Tanzini-Yagi ’13; Ito-Maruyoshi-Okuda ’13).

2 For SO and Sp instantons on C /Zn, the KN constructions are much less known.

I Partial constructions are proposed in ’90s (e.g. Douglas-Moore ’96, Intriligator ’97).

I The subject on gauge theory constructions has been revisited recently ; many new models are found. Instanton partition functions for SO and Sp groups are computed. (Dey, Hanany, N.M., Rodiguez-Gomez, Seong ’13)

There are exact relations between instanton partition functions on ALE space and the non-pertubative monopole contributions in gauge theories with ’t Hooft lines.

(Kronheimer ’85; Ito-Okuda-Taki ’11; Gang-Koh-Lee ’12; N.M.-(Rodriguez-Gomez) ’13)

Work in progress: Other types of ALE spaces and exceptional groups.

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 5 / 22 Part I: Instantons on C2

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 6 / 22 The ADHM construction from string theory (Douglas, Witten ’94-’95)

2 k SU(N) instantons on C Can be realised on a system of Dp-branes and D(p + 4)-branes

k Dp-branes on top of N D(p + 4)-branes: k Dp-branes ≡ k instantons in SU(N) gauge theory on the w.v. of D(p + 4).

The w.v. theory of the Dp-branes has 8 SUSYs (e.g. 4d N = 2 for p = 3) and can be represented by a quiver diagram:

SUHNL UHkL

Dp-branes on top of D(p + 4)-branes ←→ Higgs branch of this gauge theory

2 This is identiﬁed with the moduli space of k SU(N) instantons on C

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 7 / 22 The ADHM construction from string theory (Douglas, Witten ’94-’95)

2 k SO(N) or Sp(N) instantons on C Can be realised on a system of k Dp-branes in the background of N D(p + 4)-branes on top of an orientifold plane.

The w.v. theories on the Dp-branes are the following quiver theories:

SOHNL SpHkL A

SpHNL OHkL S

ADHM quiver for G-instantons has the ﬂavour node being G

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 8 / 22 Comments on the ADHM construction

The F and D terms of the ADHM quiver give rise to the moment map equations for hyperK¨ahlerquotients of the instanton moduli spaces

For classical gauge groups, the moment map equations follow from the Langrangian of the corresponding ADHM quiver.

For exceptional gauge groups, no ADHM construction is known!

Even though the ADHM construction is not available, it is still possible to compute instanton partitions function exactly and explicitly!

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 9 / 22 Symmetry of an instanton moduli space

2 The moduli space of k G instantons on C is a singular hyperK¨ahlercone

possesses a symmetry

U(2) 2 × G C

2 where U(2) 2 is a symmetry of , the overall position of the instantons C C can be parametrised by gauge invariant quantities coming from the hypermultiplet of the ADHM quiver, for G = SU(N),SO(N), Sp(N).

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 10 / 22 2 Example 1: One SU(N) instanton on C

SUHNL UHkL

Set k = 1 and translate the ADHM quiver from N = 2 language to N = 1 language.

In N = 1 notation, the quiver looks like

Φ1

Q SUHNL UH1L j Q

Φ2

i i Superpotential W = Qe ϕQi −→ F terms: Qe Qi = 0 Gauge invariants from the hypers:

i i a i φ1, φ2 and M j = QeaQj , with M i = 0 due to the F term

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 11 / 22 2 Example 1: One SU(N) instanton on C (continued)

Global symmetry of the quiver theory: U(1) 2 × SU(2) 2 × SU(N) C C

U(1) 2 SU(2) 2 SU(N) C C

φα +1 2 1 i M j +2 1 Adj

2 The VEVs of free ﬁelds φα parametrise position of the instanton on C . 2 The moduli space of instanton is × M 2 C f1,SU(N),C

I The space M 2 , known as the reduced instanton moduli space, is f1,SU(N),C i i parametrised by M j with M i = 0.

Partition function for gauge invariants on M 2 is known as the Hilbert f1,SU(N),C series of the reduced instanton moduli space:

∞ X 2m g 2 (t, y) = m(highest weight Adj)t e1,SU(N),C m=0

where t keeps track of the U(1) 2 charges. C

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 12 / 22 2 Example 2: One G instanton on C (with any simple group G)

This in fact holds for any simple group G, i.e. the ABCDEFG type groups!

∞ X 2m g 2 (t; y) = m(highest weight of Adj) t e1,G,C m=0

(Benvenuti, Hanany, N.M. ’08)

Reason. A special property of the moduli space of one instanton; it is the orbit of

the highest weight vector in the Lie algebra of GC (e.g. Kronheimer ’90; Vinberg-Popov ’72; Garﬁnkle ’73; Gaiotto, Neitzke, Tachikawa ’08).

2 The expressions for two instantons on C are not this simple, but can still be arranged into lattices of highest weights.

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 13 / 22 2 Example 3: Two Sp(N) instantons on C

The Hilbert series can be computed from the ADHM quiver and can be written in terms of U(2) 2 × Sp(N) character expansion as C

4 ge2,Sp(N)(t, x, y1, . . . , yN ) = f(0; 0,..., 0) + f(0; 0, 1, 0,..., 0)t + [f(1; 2, 0, 0,..., 0) + f(1; 2, 1, 0,..., 0)] t5 , where the function f is deﬁned as

1 ∞ ∞ ∞ ∞ X X X X 2m2+2n2+3n3+4n4 f(a; b1, b2, . . . , bN ) = t × 1 − t4 m2=0 n2=0 n3=0 n4=0

[2m2 + n3 + a; 2n2 + 2n3 + b1, 2n4 + b2, b3, ..., bN ] .

1 2 3 N-1 N

2n2+2n3 2n4

2 The general expressions for two ABCDEF G instantons on C can be found at [arXiv:1205.4741].

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 14 / 22 Hilbert series as instanton partition functions

Symmetry of the moduli space: U(1)2 × SU(2)2 × G. C C 2 2 I t keeps track of U(1) charges, x is a variable for characters of reps of SU(2) , and C C (y1, . . . , yrankG) are variables for characters of reps of G.

4d Nekrasov’s partition function can be obtained as a limit of the Hilbert series:

∨ 1 1 inst 2khG − 2 β(1+2) − 2 β(1−2) −βa Zk,G(1, 2, a) = lim β HSk,G t = e ; x = e ; y = e β→0

Interpretation: Hilbert series is the instanton contribution to the partition function

1 4 of 5d N = 1 pure SYM with gauge group G on Sβ × R .

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 15 / 22 Hilbert, Nekrasov and AGT

One G instanton: In the limit β → 0, the Hilbert series

∞ 1 X 2m g (t, x, y) = m(highest weight of Adj) t , 1,G (1 − tx)(1 − tx−1) m=0

1 1 − β(1+2) − β(1−2) −βai with t = e 2 , x = e 2 , yi = e , reduces to the Nekrasov partition function

inst 1 X 1 Zk=1(1, 2, a) = − Q , 12 (1 + 2 + γ · a)(γ · a) γ∨·α=1, α∈∆(α · a) γ∈∆l

∨ 2γ where ∆ and ∆l are the sets of the roots and the long roots, and γ = γ·γ . (Keller, NM, Song, Tachikawa ’11; thanks to A. Bondal and S. Carnahan)

“AGT relation”: This is equal to the norm of a certain coherent state of the W-algebra. For

non-simply laced G, the coherent state is in the twisted sector of a simply-laced W-algebra.

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 16 / 22 2 Part II: Instantons on C /Zn

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 17 / 22 2 Quiver for U(N) instantons on C /Zn (Kronheimer-Nakajima ’90)

(All nodes are unitary groups with labelled ranks.)

Can be realised from a T-dualisation of Dp-Dp + 4 system on the An−1 ALE space. Pn Instanton data: k = (k1, . . . , kn) and N = (N1,...,Nn), with N = i=1 Ni. Instanton moduli space = Higgs branch of this quiver

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 18 / 22 2 Quiver for SO(N) instantons on C /Zn, n = 2m + 1

This quiver can be obtained by ‘orientifolding’ the Kronheimer-Nakajima quiver.

1 k1, . . . , km ∈ 2 Z≥0

N = 2N1 + ... + 2Nm + Nm+1

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 19 / 22 2 Quiver for Sp(N) instantons on C /Zn, n = 2m + 1

This quiver can be obtained by ‘orientifolding’ the Kronheimer-Nakajima quiver.

k1, . . . , km+1 ∈ Z≥0

N = N1 + ... + Nm+1

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 20 / 22 2 Quivers for SO(N) instantons on C /Zn, n = 2m There are two ways of orientifolding. These give two quivers: 1 I O/O Type for SO(N): k2, . . . , km ∈ 2 Z≥0, N = N1 + 2N2 + ... + 2Nm + Nm+1.

1 I AA Type for SO(2N): k1, . . . , km ∈ 2 Z≥0, 2N = 2N1 + ... + 2Nm.

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 21 / 22 2 Quivers for Sp(N) instantons on C /Zn, n = 2m There are two ways of orientifolding. These give two quivers:

I S/S Type: N = N1 + ... + Nm+1.

I SS Type: N = N1 + ... + Nm.

Noppadol Mekareeya (CERN/MPI) Instanton Moduli Spaces DESY Workshop, September 2013 22 / 22