yang-mills theory and the large-n limit
TRANSCRIPT
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]
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 2 / 41
PART 1: GAUGE THEORIES
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 3 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 4 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 5 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 6 / 41
Aharonov–Bohm effect
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 7 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 8 / 41
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)
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 9 / 41
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 )
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 10 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 11 / 41
PART 2: QUANTIZING YANG–MILLS
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 12 / 41
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!
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 13 / 41
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!
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 14 / 41
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.
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 15 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 16 / 41
PART 3: MAKEENKO–MIGDAL EQUATION ANDLARGE-N LIMIT ON PLANE
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 17 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 18 / 41
Planar Makeenko–Migdal equation of Kazakov–Kostov
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:
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 19 / 41
Planar Makeenko–Migdal equation of Kazakov–Kostov
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 20 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 21 / 41
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(·)}.
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 22 / 41
“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)
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 23 / 41
“Master field”: Characterizing the large-N limit on plane
Above loop satisfies
τ(hol(L)) = e−t3/2e−t1e−t2(1− t1)(1− t2)
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 24 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 25 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 26 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 27 / 41
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.)
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 28 / 41
PART 4: MAKEENKO–MIGDAL ON SURFACES
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 29 / 41
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.
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 30 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 31 / 41
PART 5: LARGE-N LIMIT ON A SPHERE
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 32 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 33 / 41
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)
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 34 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 35 / 41
Phase 1 on sphere
Not so easy to simplify more complicated loops!
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 36 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 37 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 38 / 41
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
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 39 / 41
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!
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 40 / 41
Thank you for your attention!
Brian C. Hall Yang-Mills theory and the large-N limit Northwestern, November 2017 41 / 41