Hidden Symmetry and Quantum Phases in Spin 3/2 Cold Atomic Systems
Symmetry Breaking, Phases & Complexiï¬ed Gauge Theory
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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〉
Transcript of Symmetry Breaking, Phases & Complexiï¬ed Gauge Theory
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
Daniel Ferrante
I have made an unusual choice for the sequence of topics in this talk: rather than going from easy-to-hard, I chose the other way around. Not that the topics are "harder" /per se/, but I wanted to give an overall picture, a "bird's eye view". Thus, firstly, I had to set the stage and introduce all of the ingredients. Only then I could go ahead and connect the analogies with objects that are already known to us.So, if you bear with me for the beginning slides, I will connect all the dots when I get to section 4, your "Take-home Message".
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).
Daniel Ferrante
Firstly, i'd like to remark on what does "non-perturbative" mean (as opposed to "perturbative"). As we know all too well, "perturbative" means that we're looking at what happens to an object when we're in the vicinity of it, when whatever we have done to it has *not* changed its main properties and features. For this very reason, "perturbative" is analogous to "local".On the other hand, "non-perturbatively" is naturally associated with "global", i.e., with properties and features which change when you consider the model's global properties.Historically, Physics has followed a path from "local" to "global" tools in what regards QFT: in the beginning we developed perturbation theory and calculation techniques (Feynman graphs, etc); and only much later we started to introduce Cohomology, Morse Theory, Index Theorems, D-modules, etc.Secondly, given the above, the question becomes: "What kinds of 'global tools' could possibly guide the next advent of non-perturbative results?"Well, personally, i chose to appeal to an old friend (albeit with a "face lift"): Symmetry. Nowadays, with these new "global tools", we can find symmetries that we simply couldn't see before. One particular and useful example of this scenario is given by "dualities", e.g.: S-duality, T-duality, Mirror Symmetry, Modular Symmetry, AdS/CFT, Langlands Duality, etc.So, armed with this and Curie's words of wisdom, let's move ahead and see if we can find something new.
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 <∞ ;
Daniel Ferrante
We start with Schwinger's variational principle, which establishes a "transformation theory" (between the fields and sources) for QFT, much along the lines of what von Neumann did for Quantum Mechanics (between position and momentum).The fundamental principle behind this idea is that the Partition Function is analogous to a Fourier Transform, i.e., it's an extension of it, appropriate for the QFT stage (functional integration).It is important, however, to note one thing: If one thinks of the Partition Function as dependent on the parameters of the Action (coupling constants), the convergence of the integral is not guaranteed, and has to be studied on a case-by-case basis — this already happens at the level of the more ubiquitous Fourier Transforms, but it's not common to see this remarked.
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).
Daniel Ferrante
The coupling 'g' is chosen so it resembles the situation in Gauge Theory (as opposed to scalar QFT, as done in this example).» Weak coupling: the mass term dominates the interaction term — the theory is almost free, so the interaction can be treated perturbatively;» Strong coupling: the interaction term dominates the mass term — the strength of the interaction makes it impossible to define the QFT in terms of incoming and outgoing states (c.f., Haag's theorem), the Interaction Picture fails, and it's impossible to build an S-matrix.
Daniel Ferrante
It's important to notice that the integration cycles 'C' (contours) will, in the end of the day, determine the coupling 'g'.Just analogous to what happens in the Schwinger-Dyson case, where the full solution comes with added [integration] constants, which should be understood as part of the coupling 'g', the very same is true in the integral version of the same problem, i.e., if we use a Partition Function formulation, we have to incorporate into 'g' the new constants given by the measure that renders the integral finite.
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].
Daniel Ferrante
The cubic interaction is usually regarded as ill=defined once it's unbounded from below.However, it's nowhere said that the fields themselves have to be Real. We must have Real observables — i.e., a Theory of Measurements that yields Real results —, but the fields themselves can be valued in various different 'objects' (Real scalar fields, Complex scalar fields, Real Vector fields, Complex Vector fields, Matrix fields, Lie algebra-valued fields, etc).
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 .
Daniel Ferrante
This is the most straightforward *non-Abelian* extension of a 0-dimensional (D0-brane) cubic model: the simplest dynamical term to add is "A ^ dA".
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?
Daniel Ferrante
The different cycles (contours) will yield different couplings 'g' — which, in the case of Gauge Theory, amounts to selecting different structure constants [for the representation of the gauge group in question]; c.f. "Toronto Lectures on Physics", by Shlomo Sternberg, 2008-Jan-03, Section 2.3: http://www.fields.utoronto.ca/programs/scientific/07-08/geomanalysis/sternberglectures.pdf .Thus, changing the cycles (contours) changes the couplings which, ultimately, affects the link invariant.
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.
Daniel Ferrante
We start with Schwinger's variational principle, which establishes a "transformation theory" (between the fields and sources) for QFT, much along the lines of what von Neumann did for Quantum Mechanics (between position and momentum).The fundamental principle behind this idea is that the Partition Function is analogous to a Fourier Transform, i.e., it's an extension of it, appropriate for the QFT stage (functional integration).It is important, however, to note one thing: If one thinks of the Partition Function as dependent on the parameters of the Action (coupling constants), the convergence of the integral is not guaranteed, and has to be studied on a case-by-case basis — this already happens at the level of the more ubiquitous Fourier Transforms, but it's not common to see this remarked.
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).
