New Methods in Supersymmetric Quantum Field Theory · It is an SU(2) gauge theory with 12...

New Methods in Supersymmetric Quantum FieldTheory

Zohar Komargodski

Weizmann Institute of Science, Israel

Zohar Komargodski New Methods in Supersymmetric Quantum Field Theory

We are used to the idea that a scalar particle can becomposite, for example the pion π has constituents which canbe resolved if one does experiments at ∼ 1GeV .We are even used to the idea that a massive spin 1 particlecan be composite, e.g. the rho meson ρµ.


It is clearly an outstanding question to understand whether theW ,Z bosons and the Higgs field are fundamental or composite.

But what about the photon?


I am not trying to suggest seriously that this is a phenomenologicalquestion that needs an urgent solution. It is just a theoreticalquestion that I will use as an excuse to survey some recentdevelopments in QFT.


Traditionally, one would argue that a massless gauge field Aµcannot be composite.

Aµ ' Aµ + ∂µΛ. Necessary for unitarity. For example, if twofermions have a massless composite vector bound state, wherewould this gauge symmetry come from?

Weinberg-Witten theorem: The gauge symmetry cannot bethe manifestation of any conserved current in the system:

〈VAC |jµ|Aµ〉 = 0 for all ∂µjµ = 0

Aµ couples to some conserved current (electron minuspositron number). Where would this come from if, accordingto Witten-Witten, Aµ cannot “talk” to any current?


Where would the gauge symmetry come from if it did not exist inthe fundamental theory?

Take a U(1) Goldstone scalar in 2 + 1-dimensional quantum fieldtheory, π ' π + f . It can be transformed to a gauge potential

∂µπ = εµνρ∂νAρ .

This transformation has the inherent ambiguity

Aµ → Aµ + ∂µΛ .

The transformation from the fundamental degrees of freedom tothe low energy degrees of freedom could have an inherentambiguity.


Where does the current to which the gauge field couples comefrom?

By the Weinberg-Witten theorem, it cannot be one of theconserved currents in the theory.

But the current to which a gauge field couples is not a real currentin the theory anyway: If we quantize the theory on M× R withcompact M, then the Hilbert space consists of gauge singlets(Gauss’ law).


There is thus no theoretical obstruction for the compositeness ofmassless spin 1 particles. In nature, it could actually pertain to thephoton and also the W ,Z bosons.


The simplest existence proof is provided by d = 4, N = 1 theories[Seiberg].

The simplest possible case is that we start with SU(4) gaugetheory with 6 fundamental fields QA

i and 6 anti-fundamental fieldsQ i

A. This theory has a negative beta function

β ∼ −3× 3 + 4 = −5 .

It therefore develops strong coupling at some scale ΛQCD andbecomes intractable.

What happens in the infrared?


Seiberg has made a guess that the infrared theory is actuallyweakly coupled but in terms of different variables:

It is an SU(2) gauge theory with 12 fundamentals and 36 neutralscalar fields.

The beta function isβ = −6 + 6 = 0

at one-loop, but it is positive at two loops. So the theory is free inthe infrared.


The SU(2) gauge fields have nothing to do with the original SU(4)gauge theory. The SU(2) gauge fields are new, emergent,weakly-coupled massless composite spin 1 particles.

We often refer to this guess of the low energy degrees of freedomas “duality.” It is a duality in the sense that the originalstrongly-coupled SU(4) description has a more useful,weakly-coupled, description as an SU(2) gauge theory.


Seiberg’s guess is essentially based on ’t Hooft anomaly matchingand comparing the vacua of the two theories.

Recent developments allow to make extremely detailed tests of thisproposal.

The duality is essentially non-perturbative because it involves astrongly coupled description and a weakly coupled description.


Suppose we have a theory with a conserved fermionic charge Qsuch that Q2 = H and Q|boson〉 = |fermion〉 and vice versa.

Suppose H|Ψ〉 = E |Ψ〉, E 6= 0. Then, Q|Ψ〉 = |Ψ′〉 6= 0. Thus,Tr(−1)F = 0 for all the states with E 6= 0.

DefineI = TrH(−1)F

over the whole Hilbert space H. The contributions only come fromH = 0 states (vacua).

I = nB − nF

This is the Witten Index.


E

E=0

B F

I=2−1=1


Since the index does no depend on the renormalization groupscale, we would like to compare the Index I of our SU(4) andSU(2) gauge theories.

The trouble is that it diverges. This is due to the infinitely manySUSY vacua on both sides (flat directions).


We now know how to overcome this longstanding problem! Thefirst steps were done by Romelsberger. We can study the theory onM3 × R rather than on R4 and consider

I (µi ) = TrH((−1)F eµiqi )

for various charges qi that commute with the supercharge.

This is well defined for interesting theories such as our SU(4) andSU(2) theories.


In order to preserve sufficient supersymmetry, M3 must looklocally as S1 × Σ(2) for some Riemann surface Σ(2)

[Dumitrescu-Festuccia-Seiberg, Closset-Dumitrescu-Festuccia-ZK].

A particularly nice example is M3 ∼ S3 (topologically). This iswhat Romelsberger and subsequently many others studied.


We can imagine that we take gYM → 0 and remove the gaugefields. Then, the corresponding gauge chemical potentials ri areobservables.

The effect of gauging with infinitesimal gYM is to integrate over ri .Hence,

Igauged [µi ] ∼∫

[dri ]TrHungauged

((−1)F eµiqi eriQi

)[dri ] is the Haar measure.


For M3 = S3 the computation was done and one finds that ifSU(4) indeed flows to SU(2) with massless emergent gauge fields,then the following identity should hold true (schematically):

(p, p)2(q, q)2

∫ ∏i=1,..,4

[dri ]

∏i ,j≤4 Γ(µi rj , 1/(µi rj), p, q)∏i ,j≤4 Γ(ri/rj , rj/ri , p, q)

=

∏i ,j≤2

Γ(µi/µj , p, q)

∫ ∏i=1,2

[dri ]

∏i ,j≤2 Γ(µi rj , 1/(µi rj), p, q)∏i ,j≤2 Γ(ri/rj , rj/ri , p, q)

Γ(·) is the elliptic hypergeometric gamma function. (·, ·) is theq-Pochhammer symbol.


It appears that mathematicians have independently proved suchidentities quite recently [Spiridonov, Rains, Rahman, van deBult...]. In particular, the identity above holds true!


The elliptic hypergeometric function Γ that appeared above is a“higher version” of the Jacobi theta function Θn(z ; q). The latterare central in complex analysis in two dimensions.

It appears that the computations of

I (µi ) = TrH((−1)F eµiqi )

are closely linked with the theory of complex geometry in fourdimensions:


In order to preserve SUSY on some four-fold M4 the four-foldneeds to be complex. [Klare-Tomasiello-Zaffaroni,Dumitrescu-Festuccia-Seiberg]

If M4 =M3 × S1 then if locally M3 = S1 × Σ(2) we havecomplex structure and a holomorphic Killing vector,guaranteeing 2 supercharges.

The Index is independent of the Hermitian metric on M4, andit depends only on the complex structure.[Closset-Dumitrescu-Festuccia-ZK]

The Index I is thus a function of the complex structureparameters τi and of the moduli of holomorphic vectorbundles. This is the geometric meaning of µi .

The complex structure moduli space of S3 × S1 istwo-complex dimensional [Kodaira-Spencer], accounting forthe parameters p, q that appeared above.


The index only depends on non-metric data, i.e. the complexstructure moduli.

This is somewhat reminiscent of twisting in N = 2, where thetheory becomes completely topological.

We see that one can also “twist” N = 1 theories, but one gets atheory of the complex structure moduli space, rather than atopological theory.


The study of four-dimensional theories on spaces like M3 × S1

opened many new directions but at the end it is a combinatorialobject that counts the states in the Hilbert space of the theory onM3.

Now I would like to briefly describe recent developments of analtogether different sort, where one has access to objects which arenot combinatorial in nature.


Suppose we have a Conformal Field Theory (CFT). Then we canalso place it on the space S4 via the stereographic map.

This is an angle-preserving transformation,

ds2 = dx idx i −→ 1

(x2 + r2)2dx idx i .


Now the theory is free of infrared divergences because space iscompact. The UV divergences are the same as in flat space. Onecan therefore try to compute

ZSd ≡∫

[DX ]e−S[X ;gij ] , gij = δij1

(x2 + r2)2


Assume the theory has various exactly marginal couplingconstants λi . Then we can compute

ZS4(λi )

Is this well defined?


The partition function has various power divergences with the UVcutoff ΛUV of the sort

logZS4 = Λ4UV r

4 + Λ2UV r

2 + ...

which correspond to the counter-terms

Λ4UV

∫d4x√g + Λ2

UV

∫d4x√gR + · · ·

Each of these can be multiplied by an arbitrary function of the λi .and we can have both a log and a finite piece

logZS4 = · · ·+ a(λi ) log(rΛUV )− F (λi ) .


logZS4 = · · ·+ a(λi ) log(rΛUV )− F (λi )

easy to prove that a(λi ) = a. It counts degrees of freedom anddecreases along renormalization group flows

aUV > aIR .

It can also be interpreted as the Entanglement Entropy inside aball.The term F (λi ) is unphysical. It can be removed by thecounter-term ∫

d4xF (λi )E4

with E4 the Euler density.


It would therefore seem that logZS4 only contains the a-anomalyand no other physical information. However, it turns out that thecounter-term ∫

d4xF (λi )E4

is not generally allowed in N = 2 supersymmetric theories.


It is therefore not entirely surprising that for N = 2 theories ind = 4 we find

ZS4 = r−4aeK(λ,λ)/12

with λ the coefficients of exactly marginal operators

δS =∑i

λi∫

d4xd4θOi + c.c.

The Kahler potenitial K is not a holomorphic quantity and so isnot accessible via standard techniques that have been utilized inthe 90s.


With supersymmetric localization [Pestun] we can now computethese sphere partition functions and thus find the metric on thespace of theories.This is a very interesting non-perturbative observable which hasvarious applications.


Consider for instance Seiberg-Witten theory with SU(2),Nf = 4.

ds2 = dτd τ

[3

8(Imτ)2− 135ζ(3)

32π2(Imτ)4− 1575ζ(5)

64π3(Imτ)5+ · · ·

− cos(θ)e−8π2/g2 3

4π(Imτ)3/2+ · · ·

]


These agree with previous perturbative computations. One canalso make contact with the general question of whether instantonsand perturbation theory are linked in QFT. One can prove that theperturbative series is Borel summable (no poles on the positiveBorel plane) [Russo, Honda]. It is therefore unclear how to extractthe instantons from the perturbative series.One can also show that Pade approximations convergeexponentially fast (conjectured in the context of QCD by Karlineret al.).


Various other applications of the ideas above

Many checks of dualities, new dualities...

Wilson loops expectation values (non-perturbative) [Pestun...]

Exact computations of the extremal correlators [Gerchkovitzet al, Baggio et al ... ]

Relations between field theories in different dimensions[Alday-Gaiotto-Tachikawa,...]

Novel tests of AdS/CFT [see e.g. Martelli-Sparks,Cassani-Martelli,...]

Monotonicity of Renormalization Group Flows in d = 3[Jafferis-Klebanov-Pufu-Safdi...]

Many new relations to mathematics


We see that the field of non-perturbative aspects ofsupersymmetric field theories is rapidly evolving and majorlongstanding problems have been recently solved. This has openedthe way for many applications in physics and mathematics.


Thank You For Your Attention


New Methods in Supersymmetric Quantum Field Theory · It is an SU(2) gauge theory with 12...

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