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8/8/2019 Symmetry Breaking, Phases & Complexified Gauge Theory :: Miami 2010
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Symmetry Breaking, Phases &
Complexified Gauge TheoryFrom Quantum Phases to Langlands Duality
D. D. Ferrante
Brown University
Miami, 2010-Dec-14
⟨arXiv:0904.2205 [hep-th]⟩
⟨arXiv:0809.2778 [hep-th]⟩
⟨arXiv:0710.1256 [hep-th]⟩
⟨arXiv:hep-lat/0602013⟩
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Outline
1 MotivationsGlobal Approach
2 Quantum Phases
Zeros of the Partition Function
3 Geometric Langlands Duality
4 Take-Home Message
5 Summary and Speculations
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Motivations
Symmetry Something happens which is notobservable: [Pierre] Curie’s more phenomenologicalapproach led him to emphasize the role of “non-symmetry”, rather than symmetry, so that he wasthe first to appreciate the role of symmetry breaking as
a necessary condition for the existence of phenomena,
�C’est la dissymétrie qui crée le phénomène.( “It is dissymetry that creates the phenomenon.” );Pierre Curie, 1908, Œuvres (Gauthier-Villars, Paris).
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Modern Global Approach
Use of global methods to attack problems that cannot bereached via standard perturbative methods:
Higgs Bundles, Moduli Spaces; . . .
Dualities, Branes; . . .
Topological Quantization, Jacobi stability; . . .
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Quantum Phases
Condensed Matter: quantum states of matter(ground states) at zero temperature [1];
High Energy Theory: inequivalent vacua [8]:
partition function depends on the coupling constants;
different ranges of the coupling constants(inequivalent partition functions for the sameLagrangian) ⇒ distinct vacua (ground states);
each ground state (vacuum) is a quantum phase!Quantum Phase Transition ⇔ Symmetry Breaking(vacuum selection).
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Zeros of the Partition Function
Lee-Yang: zeros in R [2];
Fisher: analytic continuation of L-Y ⇒ zeros in C [3];Stokes’ Phenomena: domains of analyticitybounded by Stokes’ lines [4].
✑ Thermodynamicfunctions’ singularities;
✑ Classical Phase
Transition: zeroscondense onto Stokes’lines, pinching thecoupling axis.
Im Z
Re Z
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Langlands Duality (summary)
Langlands Duality: relates seemingly differenttheories (distinct sets of C-couplings τ ) ⇐ S- &
T-duality (mirror symmetry) equivalent ⇐ Modularsymmetry;
Objects in one model ⇔ objects in the dual model:same algebra of observables, inequivalent reps;
Their cohomologies may be interpreted as the space
of vacua in these theories, hence they should beisomorphic.
Geometric Langlands duality: invariance underaction of SL2(Z) on N = 4 SYM with gauge groups Gand LG [9, 10]:
✑ Z [τ , J] is a modular form invariant under theaction of SL2(Z) on τ : τ � = aτ +b
cτ +d
, where a d− b c = 1.
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Quantum Phases in Gauge Theory [8]
Integro-Differential Problems:
Differential Formulation Integral FormulationSchwinger-Dyson eqs Feynman Path IntegralBoundary Conditions Measure (all appropriate cycles)
Schwinger-Dyson Eqs: A→−i δ / δ J :
�δSτ [
BCs = Dp-branes (= knots) −i δ / δ J]
δ A− J
Z [τ , J] = 0 .
Feynman Path Integral: Z [τ , J=0]=1 ,A={gauge orbits} :
Z [τ , J] =
C
ei Sτ [ A] e−i ⟨ J, A⟩
Fourier-Mukai transform
D A <∞ ;
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Example: 0-dim ϕ4(scalar-valued D0-branes, quartic potential)
Action: S[ϕ] = �
2
ϕ2 + λ
4
ϕ4 ; g = λ
� g → 0
weak−−→ � λ
g →∞
strong
−−→ � λ
;
Schwinger-Dyson:
ϕ (1+gϕ2)
� ϕ + λ ϕ3− J = 0
ϕ →−i∂ J−−−−→
g= λ / �
Parabolic Cylinder Eq.
g∂3
J+ ∂ J − J
Z [ J] = 0;
Path Integral: Z [g, J] =
C
eiϕ2 (1+gϕ2) e−i Jϕ dϕ
C eiϕ2 (1+gϕ2) dϕ
=
U(g,J) ,
V(g,J) ,
W(g,J) .
Three different solutions (cycles, BCs): perturbative(symmetric), broken-symmetric (non-perturbative) andinstanton (monopole).
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Example: 0-dim ϕ4(speculation)
0-dim ϕ4 = ultra-local Landau-Ginzburg = ultra-local
scalar QED = scalar D0-branes in a quartic potential;
Different phases: L-G and σ -model over Calabi-Yausurfaces [9];
Lee-Yang zeros of L-G = Fisher zeros of σ -C-Y:
analytic continuation of each other; This is obtained via the 3 distinct cycles: C1, C2, C3 —one cycle is the analytic continuation of the otherone; the third yields a monopole;
Different cycles C ≡ distinct couplings g:weak/strong coupling depend on C;
Stokes lines: domains of analyticity are separated by
discontinuous jumps ⇒ catastrophe phenomena(Morse Theory) classification [8].
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Example: Scalar D0-branes, Airy Potential
Action: S[ϕ] =ϕ3
3;
Schwinger-Dyson: ϕ2 + J = 0ϕ →−i ∂ J
−−−−→∂2
J− JZ [ J] = 0 : Airy eq.;
Path Integral: Z [ J] =
C
eiϕ3 / 3 ei Jϕ dϕ
C
eiϕ3 / 3 dϕ
≡
Ai( J) / Ai(0)
Bi( J) / Bi(0).
Two solutions: Ai and Bi; differ by a π / 2 phase;
ϕ3: ill defined over R-cycle (CR), but well definedover C-cycles (CC) — analytic continuation [8, 9].
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Example: D0-branes, Airy Potential (extensions)
ϕ3 can be real-, matrix- or Lie algebra-valued: theresults do not change ⇐ Airy functions can be
extended to these other fields (Harish-Chandraanalysis; Airy property [7]).
Lie algebra-valued fields with cubic interaction ⇒Chern-Simons,
SCS[ A] = k 4π
M
tr
A∧ d A + 23 A∧ A∧ A
.
CS Schwinger-Dyson eqs,
δS
δ A=
k
2π F = 0
F = d A + A∧ A = 0A →−iδ J
−−−−→
Dp-branes = BCs [8, 9]
−i d(δ J)− δ J ∧ δ J Z [ J] = 0 .
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Open Questions: D0-branes, Airy Potential
CS and Quantum Gravity [8, 9]:✑ What geometries correspond to these complex
cycles (saddle-points)?
Knot theory and gauge observables [8]:
W ρ
γ
( A) = tr ρHolγ( A) = tr ρPexp γ A ;
W ρ1
γ1· · · W ρn
γn
=
CA
W ρ1γ1
( A) · · · W ρn
γn( A) ei SCS[ A]D A ;
= L (L ρ1 ··· ρn ) ;
G invariant L
U(1) winding numberSU(2) Kauffman bracketSU(n) HOMFLY polySO(n) Kauffman poly
(where ρ is a representation of the gauge group; γ is a 1-cycle; P is the path-ordering operator;A is the space of gauge orbits (connections modulo gauge transformations); and L is aninvariant of the link L.)
✑ How are the representations ρi and invariants L
affected by the choice of cycles CA?
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Conclusions and Future Research
Quantum Phase Transitions ⇔Multiple Solutions toSDEs ⇔ Integration Cycle, CC ⇔ Boundary
Conditions (Dp-branes);“Generalized” field configurations (Liealgebra-valued fields, CC, etc) ⇔ “extended”coupling constants;
Lee-Yang Zeros, Fisher Zeros, Stokes Lines ⇔coalescing of quantum phases, lines of analyticitybreakdown, and vacuum selection;
Boundary Conditions ⇔ Local Systems ⇔ HiggsBundle;
Boundary Conditions ⇔ Bundle Topology: Sheaf Cohomology?
Boundary Conditions ⇔ Parameters/couplings ⇔Geometric Langlands Dualities?
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ReferencesS. Sachdev: Quantum Phase Transitions.
Lee-Yang Zeros: Lee-Yang Theorem, Comm. Math. Phys. 33: 145–164, Rev. Math. Phys., Volume:
11, Issue: 8(1999) pp. 1027–1060.
Fisher Zeros: The Nature of Critical Points.
Stokes’ Phenomenon: Pub. Math. IHÉS, 68 (1988), p. 211–221, Proc. R. Soc. Lond. A, Vol. 422,
No. 1862 (Mar. 8, 1989), pp. 7–21.
Fourier-Mukai: Nagoya Math. J. Volume 81 (1981), 153–175, Heuristics of Fourier-Mukai.
Higgs bundles: Proc. Lond. Math. Soc. 1987 s3-55(1):59–126, Publ. Math. IHÉS, 75 (1992), p.
5–95. What is a Higgs Bundle?.Airy functions extensions: arXiv:0707.3235 [math-ph], arXiv:0901.0190 [math-ph],
Comm. Math. Phys. Vol 147, N 1 (1992), 1–23, Am. J. Math. 79, (1957) 87–120, Séminaire
Bourbaki, 4 (1956-1958), Exposé No. 160, 8 p.
D. D. Ferrante, G. S. Guralnik, et al.: arXiv:hep-lat/0602013; arXiv:0710.1256 [hep-th];
arXiv:0809.2778 [hep-th]; arXiv:0904.2205 [hep-th]; arXiv:0912.5525 [hep-lat].
Analytic Continuation of TFTs, Grothendieck, Dynamical Systems, Langlands Duality & HiggsBundles.
E. Witten, et al.: arXiv:hep-th/9301042, arXiv:hep-th/0604151, arXiv:hep-th/0612073,
arXiv:0706.3359 [hep-th], arXiv:0710.0631 [hep-th], arXiv:0712.0155 [hep-th],
arXiv:0809.0305 [hep-th], arXiv:0812.4512 [math.DG], arXiv:0905.2720 [hep-th],
arXiv:0905.4795 [hep-th], arXiv:1001.2933 [hep-th], arXiv:1009.6032 [hep-th].
E. Frenkel: arXiv:0906.2747 [math.RT], arXiv:hep-th/0512172.
di l d li
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Appendix: Langlands Duality (introduction)
Fourier-Mukai Transform [5](Schwinger’s source theory):
A → −iδ J ;
Z τ [ J] =
C
ei Sτ [ A] e−i ⟨ J, A⟩D A <∞ ;
where A is the space of connections modulo gauge transformations; C
is a cycle (multidimensional contour in C) that renders the integralfinite; and τ represents the couplings of the Action.
Gauge Theory:
−i Sτ [ A] =
1
4 g2 M 4 tr F A
∧ �
F A pure gauge
+
iθ
8π 2 M 4 tr F A
∧F A
c2(P ) : 2nd Chern class
;
τ =θ
2π
+4π i
g
2: complex coupling.
A di L l d D lit
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Appendix: Langlands Duality (S-duality ≡ Langlands duality)
S-duality: N = 4 SYM, simply-laced compactconnected simple Lie group G;
(G,τ ) ←→ (LG, Lτ = −1 / τ )
(G,τ + 1)
G LG
GLn GLn
SLn PGLn
Sp2n SO2n+1
Spin2n SO2n / Z2
E8 E8
Geometric Langlands duality: invariance under
action of SL2(Z
) on N = 4 SYM with gauge groups Gand LG [9, 10]:
✑ Z τ [ J] is a modular form invariant under the action
of SL2(Z) on τ .
A di L l d D lit
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Appendix: Langlands Duality (category of branes)
Relevant objects: Branes ≡ boundary conditions,
M 4 = Σ×X ⇒
Σ : Riemann surface, ∂Σ = ∅ ;
X : closed Riemann surface .
Compactification on X : 2-dim TFT σ -model:
✑ Target Space: MH(G) = Hitchin moduli space of Higgs G-bundles on X (flat connection ∇ = A + iϕ) [6];
MH(G) = solution space of
F A − ϕ∧ ϕ = 0 ;
d Aϕ = d A � ϕ = 0 .
✑ S-duality ≡ Mirror Symmetry (T-duality) betweenσ -TFTs with targets MH(G) and MH(LG).
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