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⟩

http://www.het.brown.edu/people/danieldf/

http://arxiv.org/abs/0904.2205



http://arxiv.org/abs/hep-lat/0602013







http://www.het.brown.edu/people/danieldf/



Outline

1 MotivationsGlobal Approach

2 Quantum Phases

Zeros 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 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).



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 Phase

Transition: 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 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.



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 <∞ ;



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).



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].



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].



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 .



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?



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?



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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http://arsphysica.wordpress.com/2009/10/18/teorias-topologicas-de-campo-e-suas-continuacoes-analiticas/


http://arsphysica.wordpress.com/2009/10/31/grothendieck-sistemas-dinamicos-dualidade-de-langlands-e-higgs-bundles/



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http://dx.doi.org/10.1142/S0129055X99000325

http://dx.doi.org/10.1142/S0129055X99000325






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



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



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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