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Symmetry Breaking, Phases &Complexified Gauge Theory
From 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⟩
Outline
1 MotivationsGlobal Approach
2 Quantum PhasesZeros of the Partition Function
3 Geometric Langlands Duality
4 Take-Home Message
5 Summary and Speculations
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 asa 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).
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; . . .
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).
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 PhaseTransition: zeroscondense onto Stokes’lines, pinching thecoupling axis.
Im Z
Re Z
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 spaceof 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 ad− bc= 1.
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]
δA− J�
Z [τ, J] = 0 .
Feynman Path Integral: Z [τ, J=0]=1 ,A={gauge orbits} :
Z [τ, J] =�
CeiSτ[A] e−i ⟨ J,A⟩� �� �
Fourier-Mukai transform
DA <∞ ;
Example: 0-dim ϕ4(scalar-valued D0-branes, quartic potential)
Action: S[ϕ] =�
2ϕ2 +
λ
4ϕ4 ; g =
λ
�
�g→ 0
weak−−→ �� λ
g→∞strong−−→ �� λ
;
Schwinger-Dyson:
ϕ (1+gϕ2)� �� ��ϕ+ λϕ3− J = 0
ϕ �→−i∂J−−−−→g= λ/�
Parabolic Cylinder Eq.� �� ��g∂3J + ∂J − J�Z [ J] = 0;
Path Integral: Z [g, J] =
�
Ceiϕ
2 (1+gϕ2) e−i Jϕ dϕ�
Ceiϕ
2 (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).
Example: 0-dim ϕ4(speculation)
0-dim ϕ4 = ultra-local Landau-Ginzburg = ultra-localscalar 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 bydiscontinuous jumps ⇒ catastrophe phenomena(Morse Theory) classification [8].
Example: Scalar D0-branes, Airy Potential
Action: S[ϕ] =ϕ3
3;
Schwinger-Dyson: ϕ2 + J = 0ϕ �→−i∂J−−−−→�∂2J − J�
Z [ J] = 0 : Airy eq.;
Path Integral: Z [ J] =
�
Ceiϕ
3/3 ei Jϕ dϕ�
Ceiϕ
3/3 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].
Example: D0-branes, Airy Potential (extensions)
ϕ3 can be real-, matrix- or Lie algebra-valued: theresults do not change ⇐ Airy functions can beextended to these other fields (Harish-Chandraanalysis; Airy property [7]).Lie algebra-valued fields with cubic interaction ⇒Chern-Simons,
SCS[A] =k4π
�
M
tr�A∧ dA+ 2
3 A∧ A∧ A�.
CS Schwinger-Dyson eqs,
δS
δA=
k
2πF = 0
F =dA+A∧ A = 0A �→−i δJ−−−−→
Dp-branes = BCs [8, 9]
� �� ��−id(δJ)− δJ ∧ δJ
�Z [ J] = 0 .
Open Questions: D0-branes, Airy PotentialCS and Quantum Gravity [8, 9]:✑ What geometries correspond to these complexcycles (saddle-points)?
Knot theory and gauge observables [8]:
Wργ(A) = trρ�Holγ(A)�= trρ�Pexp�
γA�;
�Wρ1
γ1· · ·Wρn
γn
�=
�
CA
Wρ1γ1(A) · · ·Wρn
γn(A)eiSCS[A] DA ;
= L (Lρ1···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 Laffected by the choice of cycles CA?
Conclusions and Future Research
Quantum Phase Transitions⇔ Multiple Solutions toSDEs⇔ Integration Cycle, CC⇔ BoundaryConditions (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: SheafCohomology?Boundary Conditions⇔ Parameters/couplings⇔Geometric Langlands Dualities?
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, SéminaireBourbaki, 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.
Appendix: Langlands Duality (introduction)
Fourier-Mukai Transform [5](Schwinger’s source theory):
A �→ −i δJ ;
Zτ[ J] =�
CeiSτ[A] e−i ⟨ J,A⟩DA <∞ ;
where A is the space of connections modulo gauge transformations; Cis a cycle (multidimensional contour in C) that renders the integral
finite; and τ represents the couplings of the Action.
Gauge Theory:
−i Sτ[A] =1
4g2
�
M4
trFA ∧ �FA� �� �
pure gauge
+iθ
8π2
�
M4
trFA ∧ FA� �� �c2(P) : 2nd Chern class
;
τ =θ
2π+4π i
g2: complex coupling.
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 LGGLn GLnSLn PGLnSp2n SO2n+1Spin2n SO2n/Z2E8 E8
Geometric Langlands duality: invariance underaction of SL2(Z) on N = 4 SYM with gauge groups Gand LG [9, 10]:
✑ Zτ[ J] is a modular form invariant under the actionof SL2(Z) on τ.
Appendix: Langlands Duality (category of branes)
Relevant objects: Branes ≡ boundary conditions,
M4 = 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 ofHiggs G-bundles on X (flat connection ∇ = A+ iϕ) [6];
MH(G) = solution space of�FA − ϕ∧ ϕ = 0 ;
dAϕ = dA � ϕ = 0 .
✑ S-duality ≡ Mirror Symmetry (T-duality) betweenσ-TFTs with targets MH(G) and MH(LG).
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