Yang-Mills theory and the large-N limit

Yang-Mills theory and the large-N limit

Brian C. Hall

Northwestern, November 2017

Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 1 / 41

Credits

Joint work with Bruce Driver and Todd Kemp of UCSD andFranck Gabriel of Warwick:

“Three proofs of the Makeenko–Migdal equation for Yang–Millstheory on the plane” [Driver, Hall, Kemp], Comm. Math. Phys. 351(2017), 741–774

“The Makeenko–Migdal equation for Yang–Mills theory on compactsurfaces” [Driver, Gabriel, Hall, Kemp], Comm. Math. Phys. 352(2017), 967-978

“The large-N limit for two-dimensional Yang-Mills theory” [Hall],arXiv:1705.07808 [hep-th]

Related results: “Yang-Mills measure and the master field on thesphere” [A. Dahlqvist and J. Norris], arXiv:1703.10578 [math.PR]


PART 1: GAUGE THEORIES


Electromagnetism as a U(1) gauge theory

The electromagnetic field on R4 can be thought of as a 2-form F :

F = B1 dy ∧ dz + B2 dz ∧ dx + B3 dx ∧ dy

+ E1 dx ∧ dt + E2 dy ∧ dt + E3 dz ∧ dt,

where B is the magnetic field and E is the magnetic field

Maxwell’s equations imply that F is closed, hence

F = dA,

where A is the vector potential


Geometric interpretation of A and F

The A is non-unique up to gauge transformation

A 7→ A+ df

Geometrically, interpret F as the curvature of a connection for aprincipal U(1) bundle over R4

Interpret A as the connection 1-form


Physical significance of A?

Classical particle moving in an E-M field only “sees” F

Schrodinger equation for quantum particle involves A:

1

i h

∂ψ

∂t= − h2

2m(∇− A)2ψ + Vψ

where ∇− A is the covariant derivative.

Aharonov–Bohm effect says that in a nonsimply connected domain,two different A’s with same F have different results


Aharonov–Bohm effect


Aharonov–Bohm, cont’d

Quantum interference effects from paths on either side of tube

Particle moves only outside the tube, where F = 0 but A 6= 0

Connection A (not just F !) affects the phase of the wavefunction

Only the holonomy of A around the tube matters:

hol(L) = e i∫L A

Idea: in quantum world, A matters, but only gauge-invariantfunctions of A arise


Yang–Mills = nonabelian gauge theories

Let K be a connected compact Lie group, e.g. U(N)

Let P be a principal K -bundle over a manifold M

Let A be a connection on P, let F be the curvature of A

If P = M ×K , then A is Lie-algebra valued 1-form on M and

F = dA+ A∧ A

(quadratic function of A)


Holonomies and gauge transformations

A is a Lie-algebra valued 1-form

In commutative case,

holL(A) = e∫L A

(as in Arahanov–Bohm!)

In noncommutative case

holL(A) = limn→∞

e∫ L(t1)

L(t0)Ae∫ L(t2)

L(t1)A · · · e

∫ L(tn)L(tn−1)

A

Holonomy is (almost) invariant under gauge transformations (likeA 7→ A+ df )


Quantum Yang–Mills theory

If K = U(1), have E-M force: quantum theory involves photonsIf K = SU(3), then A fields represent the strong nuclear forceAssociated particles called gluons bind quarks together


PART 2: QUANTIZING YANG–MILLS


Quantizing Yang–Mills theory

Let A be the space of all connections

In (Euclidean) quantum YM, we consider a heuristic path-integral

C∫Af (A)e−‖FA‖2DA,

describing a random connection

f is gauge-invariant function of connection (e.g. trace of holonomy)

‖FA‖2 is square of L2 norm—quartic function of A

DA is the (formal) Lebesgue measure on the space of connections

Clay Millenium Prize for rigorous interpretation in 4 dimensions!


Quantizing Yang–Mills theory, cont’d

Make a discrete approximation to space-time

Discretized path integral becomes classical statistical mechanicsmodel

Attempt to take the continuum limit with renormalization

Very complicated and difficult!


Rigorous version on plane (Gross–King–Sengupta [1989]and Driver [1989])

Take M = R2 (interpreted as space-time)

SoA = {A1(x , y) dx + A2(x , y) dy}

Use a gauge fixing to subspace A0 of A:

A0 = {A1(x , y) dx |A1(x , 0) = 0}

Every connection can be gauge transformed into A0

For A ∈ A0, the A∧ A term vanishes, so

FA = dA = −∂A1

∂ydx ∧ dy

is a linear function of A.


Rigorous version on plane, cont’d

For gauge-invariant functions f , integral reduces to

C∫A0

f (A)e−‖FA‖2DA

Since FA is linear in A, this is a Gaussian measure, which hasmathematically precise interpretation

The curvature F can be thought of as a Lie-algebra-valued whitenoise

Equation of parallel transport is stochastic differential equation (Ais not smooth)

Theory is invariant under area-preserving diffeomorphisms


PART 3: MAKEENKO–MIGDAL EQUATION ANDLARGE-N LIMIT ON PLANE


Planar Makeenko–Migdal equation of Kazakov–Kostov

Take K = U(N) (unitary group)

Take loop L with simple crossings

Use normalized trace tr(X ) := 1N trace(X ) and define

f (A) = tr(holA(L)),

Compute Wilson loop functional

E{tr(hol(L))} := C∫A0

tr(holA(L))e−‖FA‖2 DA



Answer depends only on areas of faces of graphFix one crossing and let t1, t2, t3, and t4 be the areas of the adjacentfacesMM equation reads(

∂

∂t1− ∂

∂t2+

∂

∂t3− ∂

∂t4

)E{tr(hol(L))}

= E{tr(hol(L1))tr(hol(L2))}where L1 and L2 are:



Four faces don’t have to be distinct

If Fi is the unbounded face, omit the ti -derivative

Good news: the loops L1 and L2 are simpler than L

Bad news: RHS is the expectation of the product of traces notproduct of expectations


Large-N limit of YM on plane

For Yang–Mills for U(N) in any dimension, expect theory to simplifyin large-N limit

Path integral gives random connection; physicists predict thatconnection becomes nonrandom in large-N limit

In limit, there is only one connection (modulo gauge transformations),called the master field

Rigorous results by M. Anshelevic–A. Sengupta and Th. Levy inthe plane case


Planar Makeenko–Migdal in the large-N limit

Since connection becomes nonrandom in the limit, all covariancesvanish

No distinction between expectation of a product and product of theexpectations

Large-N version of MME reads:(∂

∂t1− ∂

∂t2+

∂

∂t3− ∂

∂t4

)τ(hol(L))

= τ(hol(L1))τ(hol(L2))

where τ(·) is the limiting value of E{tr(·)}.


“Master field”: Characterizing the large-N limit on plane

Levy shows that large-N limit of YM on plane is characterized by:

1) The large-N MM equation

2) The unbounded face condition:

d

d |F |τ(hol(L)) = −1

2τ(hol(L)) (F adjoins unbounded face)


“Master field”: Characterizing the large-N limit on plane

Above loop satisfies

τ(hol(L)) = e−t3/2e−t1e−t2(1− t1)(1− t2)


Driver’s formula for YM on plane

Arguments of Makeenko–Migdal and Kazakov–Kostov use formalmanipulations with path integral

Goal: Prove MME for U(N) rigorously (then take a limit)

Use Driver’s formula [1989]

One variable in U(N) for each edge

Put heat kernel (distribution of Brownian motion) for each boundedcomponent

Evaluate with time = area and space variable = holonomy aroundcomponent

Integrate over all edge variables


Driver’s formula: example

Need ρt1(a) for inner region, ρt2(ba−1) for crescent region

t1, t2 = areasHolonomy around whole loop is abIntegrate:

E{tr(hol(L))} =∫U(N)

∫U(N)

tr(ab)ρt1(a)ρt2(ba−1) da db


Levy’s proof of planar MME

General strategy: Differentiate under the integral, use heatequation, integrate by parts

Integrate by parts on a sequence of faces, ending at unbounded face

Cancellation after alternating sum

Later proof by Dahlqvist using loop variables; broadly similar strategy


New proof on plane (with Driver, Kemp)

Local: Only use four faces surrounding the crossing

Main part of proof is less than two pages

Local nature of proof allows extension to compact surfaces!(Unbounded face is not needed.)


PART 4: MAKEENKO–MIGDAL ON SURFACES


New result: MME on compact surfaces (with Driver,Gabriel, Kemp)

Sengupta’s formula: Products of heat kernels on holonomies, withnormalization factor

Example on S2:

E{tr(hol(L))} = 1

Z

∫U(N)

∫U(N)

tr(ab)ρt1(a)ρt2(ba−1)ρt3(b) da db

where t3 is area of “outside” and Z is a normalizing constant.


Makeenko–Migdal for U(N) Yang-Mills on surfaces

Statement is exactly same as on plane:(∂

∂t1− ∂

∂t2+

∂

∂t3− ∂

∂t4

)E{tr(hol(L))}

= E{tr(hol(L1))tr(hol(L2))}

Our proofs for R2 go through almost without change, because proofsare local

One extra heat kernel does not affect the argument

Proofs on R2 by Levy and Dahlqvist both required unbounded face


PART 5: LARGE-N LIMIT ON A SPHERE


Large-N limit on surfaces

Existence of large-N limit on arbitrary surface is unknown

If large-N limit exists, no unbounded face condition, so MMEwon’t completely characterize theory

Best we can hope for: Use MME to reduce arbitrary curve tosimple closed curves


Large-N limit on 2-sphere

Two phases to analysis

Phase 1: Try to use MME to reduce arbitrary loop to simple closedcurve

Phase 2: Compute Wilson loop for simple closed curve (in N → ∞limit)


Phase 1 on sphere

Daul and Kazakov claim reduction is always possible

They only analyze two examples—need a systematic procedure!

a

bc

d

e

a+c

b-c

d

e+c

0 a+c

b-c+d

e+c

ϵ


Phase 1 on sphere

Not so easy to simplify more complicated loops!


Phase 1 on sphere

Systematic procedure: Shrink all but two of faces to zero!

a

bc

d

e

+

+--

++-

-+

+-

-

t(b+c-d)/2

t(b-c+d)/2t(-b+c+d)/2

a+(b+c+d)/2

00

0

e+(b+c+d)/2


Phase 1 on sphere

Reduce arbitrary loop to one that winds n times around a simpleclosed curve

Not hard to reduce that to simple closed curve

Similar method developed independently by Dahlqvist and Norris

a b 0 0 0 c


Phase 2 on sphere: Direct method

Wilson loop for simple closed curve described by Brownian bridge inU(N)

Lifetime of bridge = area of sphere; time-parameter = area enclosed

Eigenvalue process described by nonintersecting Brownian bridgeson circle

Liechty and Wang have studied large-N limit


Phase 2 on sphere: Method of “most probablerepresentation”

Write Wilson loop in terms of heat kernels, expand in terms ofcharacters of representations

See which representations contribute in limit—optimization problem

Rigorous treatment using “β-ensembles” by Dahlqvist and Norris

Phase transition when area(S2) = π2

Result: Rigorous description of large-N limit on S2!


Thank you for your attention!


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Yang-Mills theory and the large-N limit

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