Erich Poppitz Oronto

Magnetic bions, multiple adjoints, and Seiberg-Witten theory

Erich Poppitz oronto

. . work with Mithat Unsal SLAC/Stanford 1105.3969

also some recent work with Mohamed Anber, Toronto, 1105.0940 ABSTRACT: (as submitted to organizers) We study in detail the magnetic bion conﬁnement mechanism in QCD‐like theories with arbitrary numbers of adjoint Weyl fermions on R**3xS**1. [Anber, EP, 2011]

In the case of one Weyl adjoint flavor, we show how it can be smoothly deformed into the mechanism of confinement in Seiberg‐Witten theory on R**4. This demonstrates quite explicitly that the only analytically controlled examples of confinement in locally 4d & continuum supersymmetric (Seiberg‐Witten) and nonsupersymmetric (QCD(adj)‐a magnetic bion version of Polyakov's mechanism) represent facets of the same phenomenon. [Unsal, EP, 2011]

But, ﬁrst I really need to tell (remind) you what it’s all about. The theme of my talk is about inferring properties of inﬁnite- volume theory by studying (arbitrarily) small-volume dynamics.

The small volume may be most of this talk

or of characteristic size “L”

“Eguchi-Kawai” ... “large-N volume independence”...

long history of stumbling (1980-2008) that I won’t review

some recent (2008+) excitement: remedy by Unsal, Yaffe 2008 EK reduction valid to arbitrarily small L (single-site) if either: periodic adjoint fermions double-trace deformations (more than one Weyl) - no center deform measure to prevent center breaking at inﬁnite-N, deformation does not affect breaking, so EK reduction holds at all L (connected correlators of “untwisted”) observables remedy by Unsal, Yaffe 2008 EK reduction valid to arbitrarily small L (single-site) if either: periodic adjoint fermions double-trace deformations (more than one Weyl) - no center deform measure to prevent center breaking at inﬁnite-N, deformation does not affect breaking, so EK reduction holds at all L (connected correlators of “untwisted”) observables

used for current lattice studies of “minimal walking technicolor” is 4 ...3,5... Weyl adjoint theory conformal or not? small-L(=1) large-N (~20 or more...) simulations (2009-) Hietanen-Narayanan; Bringoltz-Sharpe; Catterall et al small-N large-L simulations (2007-) Catterall et al; del Debbio et al; Hietanen et al... remedy by Unsal, Yaffe 2008 EK reduction valid to arbitrarily small L (single-site) if either: periodic adjoint fermions double-trace deformations (more than one Weyl) - no center deform measure to prevent center breaking at inﬁnite-N, deformation does not affect breaking, so EK reduction holds at all L (connected correlators of “untwisted”) observables

Unsal; Unsal-Yaffe; theoretical studies Shifman-Unsal; Unsal-EP 2007-

used for current lattice studies of fix-N, take L-small: semiclassical studies of “minimal walking technicolor” confinement due to novel strange “oddball” (nonselfdual) topological excitations, whose nature depends on fermion content is 4 ...3,5... Weyl adjoint theory conformal or not? - for vectorlike or chiral theories, with or without supersymmetry small-L(=1) large-N (~20 or more...) simulations (2009-) - a complementary regime to that Hietanen-Narayanan; Bringoltz-Sharpe; Catterall et al of volume independence, which small-N large-L simulations (2007-) requires infinite N - a (calculable!) Catterall et al; del Debbio et al; Hietanen et al... shadow of the 4d “real thing”. remedy by Unsal, Yaffe 2008 EK reduction valid to arbitrarily small L (single-site) if either: periodic adjoint fermions double-trace deformations (more than one Weyl) - no center deform measure to prevent center breaking at infinite-N, deformation does not affect breaking, so EK reduction holds at all L (connected correlators of “untwisted”) observables

Unsal; Unsal-Yaffe; theoretical studies THIS TALK: Shifman-Unsal; Unsal-EP 2007-

used for current lattice studies of ﬁx-N, take L-small: semiclassical studies of “minimal walking technicolor” conﬁnement due to novel strange “oddball” (nonselfdual) topological excitations, whose nature depends on fermion content is 4 ...3,5... Weyl adjoint theory conformal or not? - for vectorlike or chiral theories, with or without supersymmetry small-L(=1) large-N (~20 or more...) simulations (2009-) - a complementary regime to that Hietanen-Narayanan; Bringoltz-Sharpe; Catterall et al of volume independence, which small-N large-L simulations (2007-) requires infinite N - a (calculable!) Catterall et al; del Debbio et al; Hietanen et al... shadow of the 4d “real thing”. Unsal; Unsal-Yaffe; theoretical studies Shifman-Unsal; Unsal-EP 2007-

fix-N, take L-small: semiclassical studies of confinement due to novel strange “oddball” (nonselfdual) topological excitations, whose nature depends on fermion content In 4d theories with periodic adjoint fermions, for small-L, dynamics is semiclassically calculable (including confinement). Polyakov’s 3d mechanism of confinement by “Debye screening” in the monopole-anti-monopole plasma extends to (locally) 4d theories. However, the “Debye screening” is now due to composite objects, the “magnetic bions” of the title. For this talk only consider 4d SU(2) theories with Nw = multiple adjoints Weyl fermions

“applications”: Unsal; Unsal-Yaffe; N w =1 is ~ Seiberg-Witten theory theoretical studies N=1SUSY YM Shifman-Unsal; with soft-breaking mass Unsal-EP 2007- N w =4 - “minimal walking technicolor” - happens to be N=4 SYM fix-N, take L-small: semiclassical studies of without the scalars confinement due to novel strange “oddball” N w =5.5 asymptotic freedom lost (nonselfdual) topological excitations, whose nature depends on fermion content In 4d theories with periodic adjoint fermions, for small-L, dynamics is semiclassically calculable (including confinement). Polyakov’s 3d mechanism of confinement by “Debye screening” in the monopole-anti-monopole plasma extends to (locally) 4d theories. However, the “Debye screening” is now due to composite objects, the “magnetic bions” of the title. “Magnetic bions, multiple-adjoints, and Seiberg-Witten theory” For this talk only consider 4d SU(2) theories with Nw = multiple adjoints Weyl fermions

“applications”: Unsal; Unsal-Yaffe; N w =1 is = Seiberg-Witten theory theoretical studies N=1SUSY YM Shifman-Unsal; with soft-breaking mass Unsal-EP 2007- N w =4 - “minimal walking technicolor” - happens to be N=4 SYM fix-N, take L-small: semiclassical studies of without the scalars confinement due to novel strange “oddball” N w =5.5 asymptotic freedom lost (nonselfdual) topological excitations, whose nature depends on fermion content In 4d theories with periodic adjoint fermions, for small-L, dynamics is semiclassically calculable (including confinement). Polyakov’s 3d mechanism of confinement by “Debye screening” in the monopole-anti-monopole plasma extends to (locally) 4d theories. However, the “Debye screening” is now due to composite objects, the “magnetic bions” of the title. In 4d theories with periodic adjoint fermions, for small-L, confining dynamics is semiclassically calculable.

is now an adjoint 3d scalar Higgs ﬁeld but it is a bit unusual - a compact Higgs ﬁeld: such shifts of A vev absorbed into shift of KK number “n” 4 thus, natural scale of “Higgs vev” is leading to

hence, semiclassical if L << inverse strong scale exactly this happens in theories with more than one periodic Weyl adjoints follows from two things, without calculation: 1.) existence of deconﬁnement transition in pure YM and 2.) supersymmetry in pure YM, at small L (high-T), Veff min at A4 =0 & max at pi/L (Gross,Pisarsky,Yaffe 1980s) in SUSY Veff=0, so one Weyl fermion contributes the negative of gauge boson Veff Q.E.D. Polyakov’s 3d mechanism of conﬁnement by “Debye screening” in the monopole-anti-monopole plasma extends to (locally) 4d theories. However, the “Debye screening” is now due to composite objects, the “magnetic bions” of the title. since SU(2) broken to U(1) at scale 1/L

there are monopole-instanton solutions of ﬁnite Euclidean action, constructed as follows: gauge equivalent vevs v - 2Pi/L v -6Pi/L -4Pi/L -2Pi/L 0 2Pi/L 4Pi/L 6Pi/L

A 4 gauge equivalent vevs v - 2Pi/L v -6Pi/L -4Pi/L -2Pi/L 0 2Pi/L 4Pi/L 6Pi/L

A 4

vev at inﬁnity vev at origin monopole-instanton of action ~ v/g2 3 gauge equivalent vevs v - 2Pi/L v -6Pi/L -4Pi/L -2Pi/L 0 2Pi/L 4Pi/L 6Pi/L

A 4

vev at inﬁnity vev at origin monopole-instanton of action ~ v/g2 3

monopole M trivially embedded in 4d gauge equivalent vevs v - 2Pi/L v -6Pi/L -4Pi/L -2Pi/L 0 2Pi/L 4Pi/L 6Pi/L

A 4

vev at inﬁnity vev at origin

2 monopole-instanton of action ~ |2 Pi/L - v|/g 3

- use a large gauge transformation to make vev at inﬁnity = v - action does not change - x -dependence is induced, hence called “twisted” 4 gauge equivalent vevs v - 2Pi/L v -6Pi/L -4Pi/L -2Pi/L 0 2Pi/L 4Pi/L 6Pi/L

A 4 ......

2 monopole-instanton tower; action ~ |2 k Pi/L - v|/g 3 the lowest action member of the tower can be pictured like this (as opposed to the no-twist): gauge equivalent vevs v - 2Pi/L v -6Pi/L -4Pi/L -2Pi/L 0 2Pi/L 4Pi/L 6Pi/L

A 4 ......

2 monopole-instanton tower; action ~ |2 k Pi/L - v|/g 3 the lowest action member of the tower can be pictured like this (as opposed to the no-twist): KK

“twisted” or “Kaluza-Klein”: monopole embedded in 4d by a twist by a “gauge transformation” periodic up to center - in 3d limit not there! (inﬁnite action) KK

M K. Lee, P. Yi, 1997

magnetic topological suppression M Euclidean D0-brane

KK Euclidean D0-brane

M & KK have ‘t Hooft suppression given by: in SU(N), 1/N-th of the ‘t Hooft suppression factor center-symmetric vev coupling matching in a purely bosonic theory, vacuum would be a dilute M-M* plasma - but interacting, unlike instanton gas in 4d (in say, electroweak theory)

3d Euclidean space-time

physics is that of Debye screening

analogy: electric ﬁelds are screened in a charged plasma (“Debye mass for photon”) in the monopole-antimonopole plasma, the dual photon (3d photon ~ scalar) obtains mass from screening of magnetic ﬁeld:

also by analogy with Debye mass: “(anti-)monopole operators” 2 aka “disorder operators” - not locally expressed in dual photon mass ~ M-M* plasma density terms of original gauge ﬁelds (Kadanoff-Ceva; ‘t Hooft - 1970s)

(for us, v = pi/L)

Polyakov, 1977: dual photon mass ~ conﬁning string tension

“Polyakov model” = 3d Georgi-Glashow model or compact U(1) (lattice) but our theory has fermions and M and KK have zero modes

each have 2N zero modes index theorem w Nye-Singer 2000, disorder operators: M: KK: for physicists: Unsal, EP 0812.2085

M*: KK*:

chiral symmetry

U(1) anomalous, but is not topological shift symmetry is intertwined with exact chiral symmetry

... potential (and dual photon mass) allowed, but what is it due to? Unsal 2007: dual photon mass is induced by magnetic “bions” - the leading cause of conﬁnement in SU(N) with adjoints at small L (including SYM) 3d pure gauge theory vacuum monopole plasma Polyakov 1977

circles = M(+)/M*(-) 4d QCD(adj) fermion attraction M-KK* at small-L Unsal 2007, ....

circles = M(+)/M*(-) squares = KK(-)/KK*(+) 4d QCD(adj) bion plasma at small-L Unsal 2007, ....

circles = M(+)/M*(-) squares = KK(-)/KK*(+) blobs = Bions(++)/Bions*(--) 4d QCD(adj) bion plasma at small-L Unsal 2007, .... 1/g 2(L) Le 4 M + KK* = B - magnetic “bions” -

-carry 2 units of magnetic charge L -no topological charge (non self-dual) (locally 4d nature crucial: no KK in 4d) bion stability is due to fermion attraction balancing Coulomb L/g2 (L) repulsion - results in scales as indicated 4 - bion/antibion plasma screening generates mass for dual photon

“magnetic bion confinement” operates at small-L in any theory with massless Weyl adjoints, including N=1 SYM (& N=1 from Seiberg-Witten theory) it is “automatic”: no need to “deform” theory other than small-L first time confinement analytically shown in a non-SUSY, continuum, locally 4d theory where we left out further subleading, at small ΛL, contributions. Recalling that β0 = (22 can calculate mass gap, string tension...An additional argument, basedUnsal, on effective EP field theory,2009, is appropriate Anber, since EP we are2011 work- − ing at small L. The BPS and leading twisted (KK) monopole solutions that constitute the 4nf )/3, we find: bions of smallest charge and action, i.e. the one most relevant at small L, involve only the lowest Kaluza-Klein modes (with KK numbers 0, 1, see the explicit solutions (3.25)) of the ± fields and can be effectively described by a 3D theory that only involves these lowest modes.14 8 2n The1/ coupling2 in this 3D theory is given by g(L)/L (we do not distinguish between the energy − Wf 2πc˜ log 1 scales π/L or 2π/L here, a difference that will only introduce an inessential correction). Since 3 ( ΛL )the 3D theory is Higgsed at the scale π/L, there is no further running of the 3D coupling M (ΛL) e− and all the physics(less should be expressed relevant in terms of g(L contributions)), obeying the usual (unbroken) 4D , (4.40) Λ ∼ renormalization× group equation. Thus, we argue that the dependence of Zbion on g is given to two-loop order by:

2 8π (1+cg (L)) g2 (L) 2-Loop 1 − 2-Loop strong scale O(1), positive Zbion (g(L)) 14 8n e (4.37) ∼ g − f (L) where we use the positive numberc ˜ =2πc β0/2. 1-Loop where denotes coefficients that play no role in determining the dependence of the dual- ∼ photon mass on the energy scale. We note that in the limit of asymptotically smallM L 1/Λ, where the perturbative calcu- 8 2n 4.4 Dual photon mass and previous small-L “estimates” of conformal window − f lation is justified, the correction to the leadingIn this subsection, semiclassical we determine the dependence of result the photon mass on the S(1 sizeΛ toL two-loop) 3 is dominated order. The dual photon mass is given by eqn. (4.6), which after substituting (4.37), reads: 15 ∼ 4π2c/g(L) by dependence of the bion action on the nonzero8Z Higgsbion(g) mass, e . As the size L =4π − (4.38) M g2L2 ∼ 2 is increased, g(L) increases, hence the exponential1 decreases—and4π 2 the corresponding “Higgs exp 2 (1 + cg2-Loop)+(2nf 4) log g1-Loop(L) , ∼ L −g2-Loop(L) − contribution” to the dual photon mass increases.and g(L) is the runningFor coupling atn thef energy< scale 14/L. Plugging and then appropriatef > loop 4thiseffect does order of (2.2)into (recall that β = (22 4n )/3, β = (136 64n )/3), we obtain: M 0 − f 1 − f 1 1 − not change the leading behavior dictated by the first factorlog log 2 2 on the r.h.s. of (4.40). However, β0 1 β1 Λ L 1 M exp log 1 log ΛL +(4 2nf ) log log Λ ∼ − 4 Λ2L2 − β2 log 1 − − Λ2L2  × 0 Λ2L2 1  β0  for the four Weyl adjoint theory, n = 4, where the1 2 leading dependence of on the S size f 2πc log( ) (1+...) e− ΛL × β1 1/2 4 2nf M β 2 β0 1 4β0 0− 2πc log 1 − − (ΛL) 2 e− 2 ΛL log , (4.39) vanishes, we find that the next leading contribution∼ to ΛML ,shownin(4.40) is an increasing Λ

14Such a theory would be relatively straightforward to obtain via “deconstruction”—see [41] for a construc- function of L. The other terms shown intion ( of4.39 the tower of) twisted and monopole solutions omitted in such a framework. in Since deconstruction (4.40 approximates) do not change this the “extra” dimension only by a finite number of Kaluza-Klein modes, the corresponding tower of twisted conclusion; this is most easily seen from themonopoles fact is also finite. that their dependence on the gauge coupling is power-law, rather than exponential. – 22 – Thus, the dual photon mass (n = 4) increases with increasing L. Since the bion M f plasma density is proportional to the square of the dual photon mass, this means that the topological excitations do not dilute away in the decompactification limit—at least for suffi- ciently small ΛL, where this calculation is valid. Thus, according to the conjecture of [13], which ties conformality on R4 to dilution vs. nondilution of the mass gap on R3 S1 at × increasing L,QCDwithN = 2, and nf = 4 Weyl fermions in the adjoint representation should not exhibit conformal behavior in the large L limit. Taking the “estimate” of [13] at face value means that the conformal window should be 4

5. Summary and discussion

In this paper, we studied in some detail the SU(2) gauge theory with nf massless adjoint Weyl fermions on R3 S1, our main focus being the bion mechanism of confinement of [2]. × We described in detail the tools and approximations involved and discussed the stability of magnetic bions. The relevant scales in the problem at L Λ are shown on Figure 1.Weused methods and approximations familiar from QCD instanton calculations. We also studied the behavior of the mass gap (or string tension) as a function of L at fixed Λ for nf =5, 4, 3, 2. Already the earlier leading-order semiclassical result [13] indicated that the nf = 5 theory is perhaps conformal on R4, with (likely) a weakly-coupled infrared fixed point. The scenario,

15While the analytic expansion of eqn. (4.11) of the non-BPS action is only valid for asymptotically small 3 g mH /mW 10− , see [35], the numerical results for the mH /mW g dependence of the action show that ∼ ∼ at weak coupling, g 1, the action is a monotonically increasing and approximately linear function, hence our ≤ conclusion is valid throughout the weak-coupling regime.

– 23 – where we left out further subleading, at small ΛL, contributions. Recalling that β0 = (22 can calculate mass gap, string tension...An additional argument, basedUnsal, on effective EP field theory,2009, is appropriate Anber, since EP we are2011 work- − ing at small L. The BPS and leading twisted (KK) monopole solutions that constitute the 4nf )/3, we find: bions of smallest charge and action, i.e. the one most relevant at small L, involve only the lowest Kaluza-Klein modes (with KK numbers 0, 1, see the explicit solutions (3.25)) of the ± fields and can be effectively described by a 3D theory that only involves these lowest modes.14 8 2n The1/ coupling2 in this 3D theory is given by g(L)/L (we do not distinguish between the energy − Wf 2πc˜ log 1 scales π/L or 2π/L here, a difference that will only introduce an inessential correction). Since 3 ( ΛL )the 3D theory is Higgsed at the scale π/L, there is no further running of the 3D coupling M (ΛL) e− and all the physics(less should be expressed relevant in terms of g(L contributions)), obeying the usual (unbroken) 4D , (4.40) Λ ∼ renormalization× group equation. Thus, we argue that the dependence of Zbion on g is given to two-loop order by:

2 8π (1+cg (L)) g2 (L) 2-Loop 1 − 2-Loop strong scale O(1), positive Zbion (g(L)) 14 8n e (4.37) ∼ g − f (L) where we use the positive numberc ˜ =2πc β0/2. 1-Loop where denotes coefficients that play no role in determining the dependence of the dual- ∼ photon mass on the energy scale. We note that in the limit of asymptotically smallM L 1/Λ, where the perturbative calcu- 8 2n 4.4 Dual photon mass and previous small-L “estimates” of conformal window − f lation is justified, the correction to the leadingIn this subsection, semiclassical we determine the dependence of result the photon mass on the S(1 sizeΛ toL two-loop) 3 is dominated order. The dual photon mass is given by eqn. (4.6), which after substituting (4.37), reads: 15 ∼ 4π2c/g(L) by dependence of the bion action on the nonzero8Z Higgsbion(g) mass, e . As the size L =4π − (4.38) M g2L2 ∼ 2 is increased, g(L) increases, hence the exponential1 decreases—and4π 2 the corresponding “Higgs exp 2 (1 + cg2-Loop)+(2nf 4) log g1-Loop(L) , ∼ L −g2-Loop(L) − contribution” to the dual photon mass increases.and g(L) is the runningFor coupling atn thef energy< scale 14/L. Plugging and then appropriatef > loop 4thiseffect does order of (2.2)into (recall that β = (22 4n )/3, β = (136 64n )/3), we obtain: M 0 − f 1 − f B 1 1 − not change the leading behavior dictated by the first factorlog log 2 2 on the r.h.s. of (4.40). However, S β0 1 β1 Λ L 1 P M exp log 1 2 1 log ΛL +(4 2nf ) log log Λ ∼ − 4 Λ2L2 − β log − − Λ2L2  × 0 Λ2L2 1  β0  for the four Weyl adjoint theory, n = 4, where the1 2 leading dependence of on the S size f 2πc log( ) (1+...) e− ΛL × β1 1/2 4 2nf M β 2 β0 1 4β0 0− 2πc log 1 − − (ΛL) 2 e− 2 ΛL log , (4.39) vanishes, we find that the next leading contribution∼ to ΛML ,shownin(4.40) is an increasing Λ

14Such a theory would be relatively straightforward to obtain via “deconstruction”—see [41] for a construc- function of L. The other terms shown intion ( of4.39 the tower of) twisted and monopole solutions omitted in such a framework. in Since deconstruction (4.40 approximates) do not change this the “extra” dimension only by a finite number of Kaluza-Klein modes, the corresponding tower of twisted conclusion; this is most easily seen from themonopoles fact is also finite. that their dependence on the gauge coupling is power-law,... rather how than dare exponential. you study non-protected– 22 – quantities? Thus, the dual photon mass (n = 4) increases with increasing L. Since the bion M f plasma density is proportional to the square of the dual photon mass, this means that the topological excitations do not dilute away in the decompactification limit—at least for suffi- ciently small ΛL, where this calculation is valid. Thus, according to the conjecture of [13], which ties conformality on R4 to dilution vs. nondilution of the mass gap on R3 S1 at × increasing L,QCDwithN = 2, and nf = 4 Weyl fermions in the adjoint representation should not exhibit conformal behavior in the large L limit. Taking the “estimate” of [13] at face value means that the conformal window should be 4

5. Summary and discussion

– 23 – An additional argument, basedAn on additional effective field argument, theory, based is appropriate on effective since field we theory, are work- is appropriate since we are work- ing at small L. The BPS anding leading at small twistedL. The (KK) BPS monopole and leading solutions twisted that (KK) constitute monopole the solutions that constitute the bions of smallest charge and action,bions of i.e. smallest the one charge most and relevant action, at small i.e. theL, one involve most only relevant the at small L, involve only the lowest Kaluza-Klein modes (withlowest KK Kaluza-Klein numbers 0, modes1, see the (with explicit KK numbers solutions 0, (3.251, see)) of the the explicit solutions (3.25)) of the ± ± 14 fields and can be effectively describedfields and by can a 3D be theory effectively that onlydescribed involves by a these 3D theory lowest that modes. only involves these lowest modes.14 The coupling in this 3D theoryThe is given coupling by g( inL) this/L (we 3D theory do not is distinguish given by g between(L)/L (we the do energy not distinguish between the energy scales π/L or 2π/L here, a differencescales π that/L or will 2π only/L here, introduce a difference an inessential that will correction). only introduce Since an inessential correction). Since the 3D theory is Higgsed at thethe scale 3D theoryπ/L, there is Higgsed is no atfurther the scale runningπ/L of, there the 3D is no coupling further running of the 3D coupling and all the physics should beand expressed all the in physics terms shouldof g(L), be obeying expressed the in usual terms (unbroken) of g(L), obeying 4D the usual (unbroken) 4D renormalization group equation.renormalization group equation. Thus, we argue that the dependence of Z on g is given to two-loop order by: Thus, we arguebion that the dependence of Zbion on g is given to two-loop order by:

8π2 (1+cg (L)) 8π2 1 g2 (L) 2-Loop (1+cg (L)) − 2-Loop 1 − g2 (L) 2-Loop Zbion (g(L)) e 2-Loop (4.37) 14 8nf Zbion (g(L)) e (4.37) ∼ 14 8nf bion g − (L) bion ∼ − 1-Loop g1-Loop (L) where denotes coefficients thatwhere playdenotes no role in coe determiningfficients that the play dependence no role in of determining the dual- the dependence of the dual- ∼ ∼ BPSphoton massKK on the energyphoton scale. mass on theBPS energy scale.KK M M +where4.4 we Dual left+ out photon further mass subleading, and at previous small ΛL, small- contributions.L “estimates” Recalling that of conformalβ0 = (22 window can calculate mass gap, string4.4 tension... DualAn additional argument, photon basedUnsal, on effective EP field mass theory,2009, is appropriate Anber, and since EP we are2011 work- previous small-− L “estimates” of conformal window ing at small L. The BPS and leading twisted (KK) monopole− solutions that constitute the − 4nf )/3, we find: bions of smallest charge and action, i.e. the one most relevant at small L, involve only the lowest Kaluza-Klein modes (with KK numbers 0, 1, see the explicitfermionic solutions (3.25)) of the ± 14 1 1 L fields and can be effectively described by a 3D theory that only involves thesezero lowest modes. 1 L In this subsection, we8 determine2n theThe1 dependence/ coupling2 in this 3D theory is given by g of(L)/L (we the do not distinguish photon between the energy mass on the S size to two-loop m g f In this1 scales π subsection,/L or 2π/L here, a difference that will we only introduce determine an inessentialmodes correction). Since the dependence of the photon mass on the S size to two-loop H ∼ − W 2πc˜(log ) (ΛL) 3 e ΛL the 3D theory is(less Higgsed at the relevant scale π/L, there is no contributions) further running of the 3D coupling , (4.40) order. TheM dual photon mass− is givenand all the by physics should eqn. be expressed (4.6 in terms of),g(L), which obeying the usual (unbroken) after 4D substituting (4.37), reads: Λ ∼ order.renormalization The× group dual equation. photon mass is given by eqn. (4.6), which after substituting (4.37), reads: Thus, we argue that the dependence of Zbion on g is given to two-loop order by:

2 8π (1+cg (L)) g2 (L) 2-Loop 1 − 2-Loop strong scale O(1), positive Zbion (g(L)) 14 8n e (4.37) L f ∼ g − (L) where we use the positive numberc ˜ =2πc β0/2. 1-Loop g2 8Zbion(g) 2 where 2denotes coefficients that play no role in determining the dependence of the dual- 8π /3g∼ 8Z (g) photon mass on the energy scale. bion We note that in=4 the limitπ of asymptoticallyLe smallM L 1/Λ, where the perturbative calcu- (4.38) 2 2 =4π 8 2n (4.38) 4.4 Dual photon mass and previous small-L “estimates” of conformal2 window2 f M g L M g L − lation is justified, the correction to the leadingIn this subsection, semiclassical we determine the dependence of result the photon mass on the S(1 sizeΛ toL two-loop) 3 is dominated order. The dual photon mass is given by eqn. (4.6), which after substituting (4.37), reads: 15 _ 2 ∼ 4π2c/g(L) by dependence of the bion action on the nonzero8Z Higgsbion(g) mass, e 2 . As the size L Figure 1: The small-L, weak-coupling1 g(L) 41,π hierarchy=4π of scales in the− bion(4.38) plasma: the con- 2 2 2 M g L 1 ∼ 4π 2 exp volume (1 +1 cg4π2-Loop2 )+(2nvolumef 4) log g (L) , is increased, g(L) increases, hence the2 exponentialexp decreases—andsemiclassical(1 + cg )+(2expn 4) log g2 the(L) , corresponding(11-Loop + “Higgs2cg )+(2n 4) log g (L) , 2 2-Loop f 1-Loop 2-Loop f 1-Loop stituent monopole core size is L, the “Higgs cloud” spreads∼ L over−g2-Loop(L) L/g, the− bion 2 size is L/g , and the ∼ L independence−g (L) Abelian independence− 2 2-Loop ∼ L −g (L) − contribution” to thesemiclassical dual8π photonregime mass increases.and g(L) is the runningFor coupling atn thef energy< scale 14/L. Plugging and theregimen appropriatef2-Loop> loop 4thiseffect does order of (2.2)into (recallconfinement that β = (22 4n )/3, β = (136 64n )/3), we obtain: Abelian3 g2 M 0 − f 1 − f typical distance between bions is Le . 1 1 − not change the leading behavior dictated by the first factorlog log 2 2 on the r.h.s. of (4.40). However, β0 1 β1 Λ L 1 confinement M exp log 1 log ΛL +(4 2nf ) log log Λ ∼ − 4 Λ2L2 − β2 log 1 − − Λ2L2  × and g(L) is the running coupling at the energy 0 Λ2L2 scale 1/L. Plugging the1 appropriate loop and g(L) isβ0 the running coupling at the energy scale 1/L. Plugging the appropriate loop for the four Weyl adjoint theory, n = 4, where the1 2 leading dependence of on the S size f 2πc log( ) (1+...) e− ΛL × β1 1/2 4 2nf M β 4 β0 2 2πc 0 log 1 1 − − 4β0 order of (2.2)into (recall that β =− (222 ΛL 4n )/3, β = (136 64n )/3), we obtain: 0 (ΛL) 2 e− log f , 1 (4.39) f vanishes, we find that the next leadingorder contribution of∼ (2.2 )into to ΛML ,shownin((recall that4.40)β is0 an= increasing (22 4nf )/3, β1 = (136 64nf )/3), we obtain: Discussion onM approach to R in refs. - here− only Λ note for 4 and 5− 14 M − − Such a theory would be relatively straightforward to obtain via “deconstruction”—see [41] for a construc- function ofmasslessL. The Weyl other adjoints terms appears shown in thattion ( of4.39 the towerweak of) twisted and monopole coupling solutions omitted in such a framework. IR infixed Since deconstruction (4.40 point approximates) do at not any change this the “extra” dimension only by a finite number of Kaluza-Klein modes, the corresponding tower of twisted monopoles is also finite. 1 at any finite L,conclusion; would thenL, thishence be is mostAbelian that easily abelianization confinement seen from thewith fact and exponentially that abelian their1 dependence small confinement− mass gap on the take gauge place, coupling albeit 1 β0 1 β1 log log 2 2 1 − 1 (1) βΛ0 L 1 β1 log log Λ2L2 1 is power-law,M expand ratherstring tension thanlog exponential. O1 – 22 – log ΛL +(4 2nf ) log log 2 21 Mg2 exp2 1 log 1 2 1 2 2log ΛL +(4 2nf ) log log with an exponentiallyΛ ∼ small− mass4 gap,Λ L e− − ,whereβ0 log g2 2is the small2−2 fixed point− coupling.Λ L  × 2 2 Thus, the dual photon massL (Λnf ∗=∼ 4) increases−Λ4L with increasingΛ L L.− Sinceβ0 log the bion2 2 − − Λ L  × ∼ M ∗ Λ L Thus, all mass scalesplasma density in the is theory proportional approachβ0 to the zero square as ofL the dual. photon mass, this means that the  1  β0  2 1 topological excitations2πc log( doΛL not) (1+ dilute...) away in the decompactification→∞ 2 limit—at least for suffi- e− 2πc log( ΛL ) (1+...) e− ciently small× ΛL, where this calculation is valid. Thus, accordingβ1 to the conjecture of [13], n 1/25 × 4 2nf β1 β 2 β04 f 1 4β0 1/2 3 41 2nf 0− 2πc log 1 β 2− − β0 1 4β0 which ties conformality2 on R2 toΛ dilutionL vs. nondilution0− 2π ofc thelog mass gap on R 1 S −at − (ΛL) e− log 2 2 , ΛL × (4.39) increasing∼ L,QCDwithN = 2, and n =( 4ΛΛL WeylL) fermionse− in the adjointlog representation , (4.39) mW f∼ ΛL should not exhibit conformal behavior in the large L limit. Taking the “estimate” of [13] at face14 value means that the conformal window should be 4

The new result15While that the weanalytic found expansion here of eqn. concerns (4.11) of the non-BPSn = action 4 theory, is only valid where for asymptotically the leading small semi- 3 f g mH /mW 10− , see [35], the numerical results for the mH /mW g dependence of the action show that ∼ ∼ classical result forat weak the coupling, massg gap1, the is actionL independent. is a monotonically The increasing next-to-leading and approximately linear small- function,ΛL behavior hence our of ≤ the mass gap isconclusion that it is increases valid throughout with the weak-couplingL at fixed regime.Λ. Recent lattice studies, see [42], indicate 4 that the nf = 4 theory is conformal on R , apparently with a small fixed-point coupling and an anomalous dimension of the fermion bilinear at the fixed point in good agreement with one-loop perturbation theory. These results, combined– 23 – with our analysis, then advocate for 3 1 1 the following behavior of the nf = 4 theory on R S . As L is increased, for L Λ− , × the coupling and the mass gap increase. As L further increases past L Λ, the coupling ∼

– 24 – The question about the approach to infinite 4d in the non-SUSY case is very interesting...... but let’s turn to SUSY first: We argued that “magnetic bions” are responsible for confinement in N=1 SYM at small L - a particular case of our Weyl adjoint theory - a “Polyakov like” confinement.

This remains true if N=1 obtained from N=2 by soft breaking. On the other hand, we know monopole and dyon condensation is responsible for conﬁnement in N=2 softly broken to N=1 at large L (Seiberg, Witten `94) So, in different regimes we have different pictures of conﬁnement in softly broken N=2 SYM. (Both regimes are Abelian and quantitatively understood. )

Do they connect in an interesting way? Unsal, EP in progress path A - difﬁcult: two mutually nonlocal descriptions at large L to merge into one at small L

small!u large!u B large!L

A C compactification ! L Poisson duality SW D small!L u Figure 1: Taking diﬀerent paths in the u-L plane. The horizontal direction, u, is the modulus of Seiberg-Witten theory and the vertical, L, is the size of S1. Ref. [2] studied the softly broken =2 path BCD - easier: C, D can be arranged always semiclassical N theory on R3 S1 by using elliptic curves through path A. In this work, we reexamine the same theory × along the path BCD in moduli space. The CD branch always remains semi-classical and allows us to understand the relation between the topological defects responsible for conﬁnement at small-L and large-L in detail.

is perturbed by an = 1 preserving mass term for Φ, it exhibits confinement of electric N charges due to magnetic monopole or dyon condensation. Ref. [2] studied the = 2 SYM and its softly broken = 1 version on R3 S1 by using N N × elliptic curves through path-A. However, if we would like to understand the relation between the topological defects (and field theories) at large and small S1, there are some intrinsic difficulties associated with path-A. In particular, the large-S1 theory is magnetically weakly and electrically strongly coupled, and the small-S1 one is electrically weakly (by asymptotic freedom) and magnetically strongly coupled. This means that, when L 1, u 1, both ∼ | | electric and magnetic couplings are order one, and we do not know how to address this domain in field theory. To avoid this diffuculty, we propose a compactification (path-C) at large-u where the theory is always electrically weakly coupled, regardless of the S1-size L. Path-D is also always weakly coupled, either because the u-modulus is large or because an additional modulus, the Wilson line along S1, is turned on (also note that in the small-L domain the = 2 theory always abelianizes, with long-distance dynamics described by a N three dimensional hyper-Kähler nonlinear sigma model [2]).

1.2 Conclusions We find, by using the techniques of Ref. [2] and of our current work, that a locally four dimensional generalization [3,4] of Polyakov’s 3d instanton mechanism of confinement [5] takes over in the small-L mass-perturbed = 2 theory. To elucidate, note that the theory possesses N 3d instanton (and anti-instanton) solutions, which, when embedded in R3 S1, have magnetic, × 1 Qm = 2 F , and topological, QT = 3 1 F F charges, normalized to (Qm,QT )= 1, . S R S ± 2 There are∞ also twisted instantons (and× anti-instantons), which carry charges (Q ,Q )= m T 1, 1 . The mass gap for gauge fluctuations and confinement in the mass-perturbed =2 ± − 2 N theory arise due to Debye screening by topological defects with charges (Q ,Q )=( 2, 0). m T ± This mechanism of confinement was called the “magnetic bion” mechanism in [3, 4] and we show here that it also takes place in the = 1 mass deformation of Seiberg-Witten theory at N

–3– dyon tower (sum over electric charges) of particles with Euclidean worldlines around S1 = dyon pseudoparticle tower small!u large!u B large!L

A C compactification ! L Poisson duality SW D small!L u Figure 1: Taking diﬀerent paths in the u-L plane.monopole-instantons The horizontal direction, and twistedu , is the modulus of Seiberg-Witten theory and the vertical, L, is the sizemonopole-instantons of S1. Ref. [2] studied the softly broken =2 =“KK tower” described earlier N theory on R3 S1 by using elliptic curves through path A. In this work, we reexamine the same theory × along the path BCD in moduli space. The CD branch always remains semi-classical and allows us to understand the relation between the topological defects responsible for conﬁnement at small-L and large-L in detail.

–3– dyon tower (sum over electric charges) of particles with Euclidean worldlines around S1 = dyon pseudoparticle tower small!u large!u B large!L

A C compactification ! L Poisson duality SW D small!L u Figure 1: Taking diﬀerent paths in the u-L plane.monopole-instantons The horizontal direction, and twistedu , is the modulus of Seiberg-Witten theory and the vertical, L, is the sizemonopole-instantons of S1. Ref. [2] studied the softly broken =2 =“KK tower” described earlier N theory on R3 S1 by using elliptic curves through path A. In this work, we reexamine the same theory × along the path BCDIt turnsin out moduli the sums space. over the The instantonCD branch contributions always of remainsthe two towers semi-classical are and allows us to understand theidentical relation and between are related the by topological Poisson duality: defects responsible for conﬁnement at small-L and large-L in detail.

–3– where S(1,ne) is the action of the dyon (??). Substituting back into (1.2), the generalized Poisson duality is:

4π √(vL)2+(ω+2πn)2 g2 vL iωn e− 4 = cos δ K (S(1,n ))e e π ne 1 e n ne ∈Z ∈Z where S(1,n ) is the action of the dyon (??). Substituting back into (1.2), the generalized e vL π S(1,ne) iωne Poisson= duality is: cos δne e− e (1.7) π 2S(1,ne) where S(1,ne)n ise theZ action of the dyon (??). Substituting back into (1.2), the generalized ∈ 4π 2 2 2 √(vL) +(ω+2πn) vL Poisson duality is: − g iωne e 4 = cos δn K1(S(1,ne))e π π e x In the second line, we used the asymptoticn Z4π expansion√(vL)2+(ω+2πn)2 for K1(nxe )Z e− ,x . In the ∈ g2 vL∈ 2x iωn − 4 ≈ →∞e e = cosvLδne K1(S(1,ne))eπ S(1,ne) iωne semi-classical domain discussed in the bulk of the paper, we= assumedπ cos cosδneδn 1 fore− the low-e (1.7) n ne e ∈Z ∈Z π 2≈S(1,ne) vLne Z π lying dyon band, and used the non-relativistic approximation for∈ the dyon massS(1 of,ne) eqn.iωne (??) = cos δne e− e (1.7) π 2S(1,ne) and the expansion (??) for the twistedIn the second instanton line, we used action. the asymptotic Insertingne Z expansion these expansions, for K (x) onπ e bothx,x . In the ∈ 1 ≈ 2x − →∞ semi-classical domain discussed in the bulk of the paper, we assumedπ x cos δ 1 for the low- sides, we obtain exactly (??) ofIn Section the second?? line,; we the used prefactors the asymptotic also expansion match for exactly.K (x) e ,x ne. In the 1 ≈ 2x − →∞≈ semi-classicallying dyon band,domain and discussed used the in the non-relativistic bulk of the paper, approximation we assumed for cos δ the dyon1 for mass the low- of eqn. (??) Now, one might ask, what quantities in the = 2 theory does a Poisson dualityne ≈ like lyingand dyon the expansion band, and used (??) the forN non-relativistic the twisted instanton approximation action. for Inserting the dyon these mass of expansions, eqn. (??) on both (1.7) relate? In fact, this questionandsides, the was expansionwe obtain already ( exactly??) for addressed the (?? twisted) of Section instanton in [1].??; action. the It prefactors turns Inserting out also these that match expansions, the exactly. r.h.s. on both sides, weNow, obtain oneInstanton exactly might corrections ( ask,??) of what Section quantitiesto ??K, ;in the complex prefactors in the structure: also= 2 match theory exactly. does a Poisson duality like of (1.7) is proportional to the large-L dyon tower contribution to the KählerN potential, where (1.7Now,) relate? one might In fact, ask, this what question quantities was in already the = addressed 2 theory does in [1]. a Poisson It turns duality out that like the r.h.s. 4πω N v = Sqrt[u] the complex moduli v (= A5 +iA(1.76)) relate? and σ In fact,ib = thisσ questioni g2 was, recall alreadyL A (?? addressed), are in used [1]. It to turns parameterize out that the r.h.s. of (1.7) is proportional− to− the4 large-L dyon4 tower contribution to the Kähler potential, where of (1.7) is proportionaldual photon to the large-L dyon tower contribution4πω to the Kähler potential, where the hyper-Kähler manifold. Morethe explicitly, complex moduli thev contributions(= A5 +iA6) and σ of theib = σ dyoni g2 tower, recall to(??), the are Kähler used to parameterize − 4πω− 4 the complex moduli v (= A5 +iA6) and σ ib = σ i g2 , recall (??), are used to parameterize 1 the hyper-Kähler manifold. More explicitly,− − the4 contributions of the dyon tower to the Kähler potential are given by: the hyper-Kählercan be manifold. inferred More from explicitly, solution the by contributions Chen, Dorey, Petunin of the (2010) dyon of tower “wall-crossing” to the Kähler equations potential are given by:1 potential areof given Gaiotto, by: Moore,1 Neitzke (2008): an iterative solution, obtained at weak-coupling v >> Lambda, but arbitrary 2vL: 4π 2 L v +n2+iωn +iσ2n 4π 2 2 e e L4πv m2 +ne+iωne+iσnm g L v +n g+2iωne+iσnm − | |s 4 − | | −g2| |s e 4 1 e 1 1 „ « e se„ 4 « „ « Kdyon = , (1.8) K = Kdyon = 3 3 3 13 1 , 1 1, (1.8)(1.8) dyon 3 3 1 2 2 2 1 2 4 √2π√2 L2π2 vL2 v n = 1 n 2 4 √2π 2 L 2 v 2 nm= 1mn2e Z e Z 4 4π 4π2 2 nm= 1 ne | | | | ± ∈± ∈ 2 + n2e + ne Z 4π 2 g g4 | | ± ∈ 2 + n 4 g e large-L sum over electric4 charges of dyon pseudoparticles whilewhile the the Poisson Poisson resummed resummed Kähler Kähler potential, potential, expressed expressed now as now a sum as over a sum the over contributions the contributions of winding solutions, and thus appropriate at small L, is: while the Poisson resummed Kählerof winding potential, solutions,small-L expressed sum and over thus winding appropriate now numbers as a at sum small of twisted overL, is: themonopole-instantons contributions

4πL 2 ω+2πnw 2 v + +iσnm 2 1 g2 4πL( L 2 ) ω+2πnw of winding solutions, and thus appropriate at small L, is: 1 − 4 | | 2 v +( ) +iσnm Kwinding = Kdyon = 2 e q− g | | L . (1.9) Kwinding = KdyonπL= v e 4 q . (1.9) n2m= 1 nw |π|L v ± ∈Z nm= 1 nw Z 4πL | | 2 ω±+2πn∈w 2 1 2 v +( ) +iσnm 2 Note that Kwinding = Kdyon is precisely− g equivalent| | ourL equation (1.7) in the g 0 limit K = K = e 4 . 4 (1.9)→ 2 winding dyon Note that2 NontrivialKwinding = toK checkdyon istheir precisely qequivalence equivalent by a semiclassical our equation calculation (1.7) in the g4 0 limit (i.e.,π withL cosv δne 1). → (i.e., with cos(fornm δgeneral≈= 1 1).valuenw of the moduli as fermion zero modes are different and only sums are equivalent)... The sum| | overn thee ±≈ dyon∈ towerZ contributions to the Kähler potential was obtained in [1] by solvingThe the sum equations over the [2] dyon obeyed tower by contributionsthe hyper-Kählermetric to the Kähler on the potential moduli space was ofobtained the in [1] 3 1 2 Note that Kwinding = Kdyon isby= precisely solving2 theory the on equivalentR equationsS iteratively [2] our obeyed for equation weak by thecoupling. hyper-Kählermetric (1.7 Thus,) in the the Kählerg4 onpotentials the0 moduli limit (1.8) and space of the N × 3 1 (1.9)= are 2 valid theory in the on R limit Sv iterativelyΛ =2, but for for weak arbitrary coupling. values Thus, of v L the. Note Kähler→ that potentials the regime (1.8) and (i.e., with cos δne 1). N × | | N | | where(1.9) ( are1.8), valid (1.9) in are the valid, limitv vΛ =2Λ 1,, there but for is no arbitrary “wall-crossing,” values of thusv L this. Note is a regimethat the regime ≈ | | N =2 The sum over the dyon towerdifferent contributions from the large L regime to| the| v L KählerN 1 regime potential considered inwas [2]. obtained We also| | note in that [1] in [1], where (1.8), (1.9) are valid, |v |Λ =2 1, there is no “wall-crossing,” thus this is a regime instead of the Kähler potential for| v| andN σib (1.9), the Kähler metric component g was by solving the equations [2] obeyeddifferent by from the the hyper-Kählermetric large L regime v L − 1 regime on the considered moduli in space [2]. We alsoof thevv note∗ that in [1], actually given, but it is a simple matter| | to check their equivalence at weak coupling. 3 1 instead of the Kähler potential for v and σ ib (1.9), the Kähler metric component g was = 2 theory on R S iteratively for weak coupling. Thus, the− Kähler potentials (1.8) and vv∗ N × actually1Up to an given, overall constant, but it is which a simple is the same matter in (1.8 to) and check (1.9). their equivalence at weak coupling. (1.9) are valid in the limit v Λ =2, but for arbitrary values of v L . Note that the regime | | 1N | | where (1.8), (1.9) are valid, v Λ Up=2 to an overall1, there constant, is which no “wall-crossing,” is the same in (1.8) and ( thus1.9). this is a regime | | N different from the large L regime v L 1 regime considered in [2]. We also note that in [1], | | –2– instead of the Kähler potential for v and σ ib (1.9), the Kähler metric component g was − vv∗ actually given, but it is a simple matter to check their equivalence–2– at weak coupling.

1Up to an overall constant, which is the same in (1.8) and (1.9).

–2– A4

A5 a) n m =+1 3d instanton tower Figure 4: 3d-instantons in the magnetic charge +1 tower in the regime a a a /g2.The 4 5 4 4 tower is composed of deformation of BPS monopole instantons and KK twisted anti-instantons. The properties of fermionic zero modes is dictated by the leading a5 dependence for the low winding number a) n m =+1 3d instanton tower instantons. A4 Figure 4: 3d-instantons in the magnetic charge +1 tower in the regime a a a /g2.The 4 5 4 4 tower is composedA5 of deformation of BPS monopole instantons and KK twisted anti-instantons. The properties of fermionic zero modes is dictated by the leading a5 dependence for the low winding number 3.4 3d-instanton/4d-dyon tower Poisson dualityinstantons. The statement of Poisson duality, which has been studied in = 4 gauge theories in [12,13] N a) n =+1 3d instanton tower and recently in the = 2 context in [14], is as follows: the small-L (3.7)m and large-L N 3.4 3d-instanton/4d-dyon tower Poisson duality (3.17) instanton sums, which, at first sight,Figure look 4: 3d-instantons completely di inff theerent, magnetic are in charge fact equivalent +1 tower in the regime a a a /g2.The 4 5 4 4 expressions, and one is the Poisson resummationtower is composed11 ofThe the of statement deformation other: of of Poisson BPS monopole duality, instantons which has and beenKK studied twisted in anti-instantons.= 4 gauge The theories in [12,13] N properties of fermionicand recently zero modes in the is dictated= by 2 thecontext leading ina5 [14],dependence is as follows: for the low the winding small- numberL (3.7) and large-L 4πvL 1 4π 2 2 +iσ 2 2 (ω+2instantons.πnw) N e− g4 e− Lvg4 (four(3.17fermion) instanton operator) sums, which, at first sight, look completely different, are in fact equivalent × − 11 nw expressions, and onea) is n them =+1 3d Poisson instanton tower resummation of the other: ∈ Z 4πvL 2 vLg2 2 +iσ Lvg 1 4 2 4πvL 1 4π 2 2 − g4 Figure4 2 4:4π 3d-instantonsne + ineω in the magnetic+iσ charge +1 tower(ω in+2 theπnw regime) a4 a5 a4/g4.The = e e3.4− 3d-instanton/4d-dyon(four fermiong2 tower operator) Poisson2 Lvg duality2 (3.19) 8πtower2 is composed of deformation× e−− of4 BPS monopolee− instantons4 and KK twisted(four anti-instantons.fermion operator) The ne Z × − ∈ propertiesThe statement of fermionic of zero Poisson modes duality, is dictated which bynw the has leading beena studied5 dependence in for= the 4 gauge low winding theories number in [12,13] ∈ Z instantons. 4πvL 2 vLg2 N and recently in the = 2 context2 +iσ in [14],Lvg is as follows:1 4 2 the small-L (3.7) and large-L Both of these are viewed as instanton sums: the first sum, (3.17),− overg the winding number4 ne + ineω N = e 4 e− 2 4π (four fermion operator) (3.19) (3.17) instantonBut, in an appropriate sums, which, regime, at same first 4-fermi sight, terms look8 appear,π2 completely and the “wall-crossing” different,× are in− fact equivalent incorporates 3d monopole-instantons and twistedconsequence (or winding) K(dyon)=K(winding) monopole-instantons can bene semiclassically and their tested [Chen et al 2010] ∈Z 11 Kaluza-Klein tower. The second, (3.7),expressions, which - should in this and limit, one perhaps Poisson is duality the be Poissoncan called also be more resummation a dyon simply understood, sum, asof sans its the wallXing other:, but no time... 3.4 3d-instanton/4d-dyonBoth of these tower are viewed Poisson as duality instanton sums: the first sum, (3.17), over the winding number salient features are dictated by 4d dyons, is a sum4πvL over the electric1 4π charges of2 the dyon 2 +iσ 2 2 (ω+2πnw) The statemente of−incorporates Poissong4 duality, 3de− monopole-instantons whichLvg4 has been studied(four and in twistedfermion= 4 gauge (or operator) winding) theories monopole-instantons in [12,13] and their tower. In this case, the instantons at large-L are realized through dyon particles× whose−N and recently in theKaluza-Klein=n 2w context tower. in The [14], second, is as follows: (3.7), which the small- shouldL perhaps(3.7) and be large- calledL a dyon sum, as its worldlines wrap around the large circle. As discussed earlier, the∈ Z large-L series converges fast 4NπvL small-L physics2 well 2described by a few twisted +iσ 1 vLg 2 2 (3.17) instanton sums,salientg2 which,2 features at first areLvg sight, dictated4 look by4 completelyn 4d+ in dyons,ω diff iserent, a sum are over in fact theequivalent electric charges of the dyon for Lvg 1 and the small-L series converges fast for− Lvg4 1. monopole-instantons2 4π (ase we’de already done) 4 = e 4 2 e− 11 (four fermion operator) (3.19) expressions, and onetower. is the In Poisson this- case,or resummation an8π infinite the instantons sum overof charged the at other: large-×dyons L −are realized through dyon particles whose ne The origin of the Poisson duality (3.19) can be simply understood∈Z as follows. In the regime 4πvLworldlines wrap1 4π around the2 large circle. As discussed earlier, the large-L series converges fast +iσ (ω+2πnw) when the expansions (3.2) and (3.17) make sense,g2 i.e., at weak2 2 couplingLvglarge-L2 physics and well when describedLv by1, a few dyons 2 Both ofe these− 4 for areLvg viewede− as1 and instanton4 the small- sums:L(fourseries the first convergesfermion sum, operator) (3.17 fast), for overLvg the winding1. number one can interpret the two sums in (3.19) in terms of the semiclassical4 - or expansion an infinite sum× around over twisted− a BPS monopole instantons 4 nw Z incorporates 3d monopole-instantonsThe∈ origin of the Poisson and twisted duality4πvL (or (3.19 winding)) can be monopole-instantons simply understood as and follows. their In the regime monopole-instanton solution of zero charge/zero winding,4πvL respectively,2 of actionvLg2 .This 2 +iσ Lvg 1 4 2 2 Kaluza-Klein− gwhen tower. the The expansions second,4 (3.7 (3.2),) which andne + in (g3.17e4 shouldω ) make perhaps sense, be i.e., called at weak a dyon coupling sum, as and its when Lv 1, = e 4 e− 2 4π (four fermion operator) (3.19) salient features are dictated8 byπ2 4d dyons, is a sum× over− the electric charges of the dyon one cann interprete Z the two sums in (3.19) in terms of the semiclassical expansion around a BPS 11For completeness, in Appendix B.1 we provetower. a more In general this case, relation, the∈ which instantons implies at(3.19 large-). L are realized through dyon particles whose 4πvL monopole-instanton solution of zero charge/zero winding, respectively, of action g2 .This Both of theseThe are moral viewed is asthat instanton the dyons, sums: whose the condensation first sum, (3.17 at large-L), over causes the winding number 4 worldlinesconfinement, wrap around are the related large by circle. Poisson As discussedresummation earlier, to the the twisted large-L series converges fast incorporates2 3d monopole-instantons and twisted (or winding) monopole-instantons2 and their for Lvg4 monopole-instantons1 and11For the completeness, small-L thatseries in form Appendix converges the small-LB.1 fastwe provemagnetic for Lvg a more4 bions general1. - which relation, are which implies (3.19). Kaluza-Klein tower. The second, (3.7), which should perhaps be called a dyon sum, as its The originresponsible of the for Poisson confinement duality (at3.19 small) can L, as be was simply already understood described. as follows. In the regime salientwhen features– the 15 – expansions are dictated (3.2) by and 4d ( dyons,3.17) make is a sumsense, over i.e., the at weak electric coupling charges and of the when dyonLv 1, tower. In this case, the instantons at large-L are realized through dyon particles whose one can interpret the two sums in (3.19) in terms of the semiclassical expansion around a BPS worldlines wrap around the large circle. As discussed earlier, the large-L series converges4π fastvL monopole-instanton2 solution of zero charge/zero winding,2 – respectively, 15 – of action 2 .This for Lvg 1 and the small-L series converges fast for Lvg 1. g4 4 4 The origin of the Poisson duality (3.19) can be simply understood as follows. In the regime 11 when theFor expansions completeness, (3.2 in Appendix) and (3.17B.1)we make prove sense, a more i.e., general at weak relation, coupling which implies and when (3.19Lv). 1, one can interpret the two sums in (3.19) in terms of the semiclassical expansion around a BPS 4πvL monopole-instanton solution of zero charge/zero winding, respectively, of action 2 .This g4 – 15 – 11For completeness, in Appendix B.1 we prove a more general relation, which implies (3.19).

– 15 – 5. Phase diagram, abelian (non-’t Hooftian) large-N limits, and discussion

On R4, confinement in SU(2) gauge theory with = 2 supersymmetry softly broken down to N =1(m 1) is a version of abelian confinement. By this we mean that the long-distance N effective Lagrangian is an abelian U(1) gauge theory, despite the fact that microscopic theory is a non-abelian SU(2). The confinement of the electric charges is due to magnetic monopole or dyon condensation [1]. On the other hand, in the limit m 1, where the adjoint Higgs multiplet decouples and the theory reduces to pure = 1 SYM theory, there exists no description of the gauge N dynamics where abelianization takes place. It is usually believed that there is no phase transition as the mass term is dialed from small to large and the theory moves from a regime of abelian confinement to non-abelian confinement. In the pure = 1 SYM theory, as well as in a large-class of non-supersymmetric gauge N theories which remain center symmetric upon compactification down to small radius, it has been recently understood that the LΛ =1 1 regime also exhibits abelian confinement. To conclude, we haveN found an - albeit indirect - relation between The confinement of the electric charges is now due to the magnetic bion mechanism [3, 4]. Analogousthe 4d tomonopole/dyon the mass-deformed theory, condensation it is usually believed thatconfinement there is no phase transitionof Seiberg and asWitten the radius and is dialed the from small-L small to magnetic large. At large- bion-inducedL, it is expected that “Polyakov-like” a non-abelian confinement should take place. confinement. SU(2) Not a phase boundary SU(N), large−N L ~1 L 1/N4 Confinement by monopole/dyon condensation non!abelian non!abelian confinement confinement

abelian confinement ~1

Confinement by quantum moduli space magnetic bion mechanism abelian 0 0 confinement 1/N no ground state m

At what values of m and L does the metamorphosis from the abelian conﬁnement (which we analytically understand) to the non-abelian conﬁnement (which is not yet understood)

– 26 – 5. Phase diagram, abelian (non-’t Hooftian) large-N limits, and discussion

abelian confinement ~1

Confinement by quantum moduli space magnetic bion mechanism abelian 0 0 confinement 1/N no ground state m

u m a) b) TheFigure magnetic 5: The = 2 bion theory brokenmechanism down to = also 1 exhibits applies confinement. to Atlarge small mclassesand/or of non- N N 4 supersymmetricsmall L (in units of Λ) the dynamicstheories abelianizes and at can large distancesbe used and theto theory study exhibits the abelian approach to R . confinement. The phase diagram in the m-L plane for the small-N theory is shown on the left figure, where the shaded areas indicate the calculable regimes at small and large L.Thethird(u) direction— Doeswhich allows the to smoothlyrelation connect between the topological small excitations and responsible large for L confinement topological at large and excitations in small L via Poisson duality—is also indicated. At large-N, shown on the right figure, the calculable SUSYsemi-classical have confinement anything regime shrinks to teach to a narrow us sliver about both in mnon-SUSYand L, in a correlated dynamics? manner, as explained in the text. ... it is perhaps early to tell, but the Poisson resummation of nonperturbative effects has interesting implications in finite-T YM At what values of m and L does the metamorphosis from the abelian confinement (which we analytically understand) to the non-abelian confinement (whichUnsal, is EP, not 11xx.yyyy yet understood) - or next workshop...

– 26 –