Lefschetz-thimble path integral and its physical application
|
|
- April Ball
- 6 years ago
- Views:
Transcription
1 Lefschetz-thimble path integral and its physical application Yuya Tanizaki Department of Physics, The University of Tokyo Theoretical Research Division, Nishina Center, RIKEN May 21, KEK Theory Center Collaborators: Takuya Kanazawa (RIKEN) Kouji Kashiwa (YITP, Kyoto), Hiromichi Nishimura (Bielefeld) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 1 / 41
2 Motivation Introduction and Motivation Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 2 / 41
3 Motivation Motivation Path integral with complex weights appear in many important physics: Finite-density lattice QCD, spin-imbalanced nonrelativistic fermions Gauge theories with topological θ terms Real-time quantum mechanics Oscillatory nature hides many important properties of partition functions. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 3 / 41
4 Motivation Example: Airy integral Let s consider a one-dimensional oscillatory integration: ( ) dx x 3 Ai(a) = 2π exp i 3 + ax. R RHS is well defined only if Ima = 0, though Ai(z) is holomorphic * cos(x 3 /3+x) exp(-0.5x)*cos(x 3 /3) Figure : Real parts of integrands for a = 1 ( 10) & a = 0.5i Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 4 / 41
5 Motivation Contents How can we circumvent such oscillatory integrations? Lefschetz-thimble integrations [Witten, arxiv: , ] [YT, Koike, Ann. Physics 351 (2014) 250] Applications of this new technique for path integrals Study on phase transitions of matrix models. Lefschetz-thimble method elucidates Lee-Yang zeros. [Kanazawa, YT, JHEP 1503 (2015) 044] General theorem ensuring that Z R. Sign problem in MFA can be solved. Application of the theorem to the SU(3) matrix model [YT, Nishimura, Kashiwa, arxiv: [hep-th], to appear in PRD] Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 5 / 41
6 Lefschetz-thimble integrations Introduction to Lefschetz-thimble integrations Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 6 / 41
7 Lefschetz-thimble integrations Introduction to Lefschetz-thimble integrations Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 7 / 41
8 Lefschetz-thimble integrations Lefschetz-thimble method = Steepest descent integration Oscillatory integrals with many variables can be evaluated using the steepest descent cycles J σ : d n x e is(x) = n σ d R n σ J n z e is(z). σ J σ are called Lefschetz thimbles, and Im[iS] is constant on it. n σ : intersection numbers of duals K σ and R n. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 8 / 41
9 Lefschetz-thimble integrations Example: Airy integral Airy integral: Ai(a) = R ( ) dx x 3 2π exp i 3 + ax. The integrand is holomorphic w.r.t x The contour can be deformed continuously without changing the value of the integration! Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 9 / 41
10 Lefschetz-thimble integrations Rewrite the Airy integral There exists two Lefschetz thimbles J σ (σ = 1, 2) for the Airy integral: Ai(a) = ( ) dz z 3 n σ σ J σ 2π exp i 3 + az. n σ : intersection number of the steepest ascent contour K σ and R. Figure : Lefschetz thimbles J and duals K (a = exp(0.1i), exp(πi)) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 10 / 41
11 Lefschetz-thimble integrations Tips: Airy integral & Airy equation Let us consider the equation of motion : dz d /3+az) 2π dz ei(z3 = 0. This is nothing but the Airy equation: ( ) d 2 da a Ai(a) = 0. 2 Two possible integration contours = Two linearly independent solutions of eom. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 11 / 41
12 Lefschetz-thimble integrations Generalization to multiple integrals Model integral: Z = dx 1 dx n exp S(x i ). R n What properties are required for Lefschetz thimbles J? 1 J should be an n-dimensional object in C n. 2 Im[S] should be constant on J. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 12 / 41
13 Lefschetz-thimble integrations Short note on technical aspects Complexified variables (a = 1,..., n): z a = x a + ip a. Regard x a as coordinates and p a as momenta, so that Poisson bracket is given by {f, g} = [ f g g ] f. x a p a x a p a a=1,2 Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 13 / 41
14 Lefschetz-thimble integrations Short note on technical aspects Hamilton equation with the Hamiltonian H = Im[S(z a )]: ( df(x, p) = {H, f} dz ( ) ) a S dt dt = z a This is Morse s flow equation (= gradient flow): d ( ) 2 dt Re[S] = S z 0 We can find n good directions for J around saddle points! (This is because a π/2-rot. of coord. around a saddle point multiplies ( 1) to an eigenvalue. ) [Witten, 2010] Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 14 / 41
15 Lefschetz-thimble integrations Multiple integral: Lefschetz-thimble method Oscillatory integrals with many variables can be evaluated using the steepest descent cycles J σ : d n x e S(x) = n σ d R n σ J n z e S(z). σ J σ are called Lefschetz thimbles, and Im[S] is constant on it. n σ : intersection numbers of duals K σ and R n. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 15 / 41
16 Gross Neveu-like model Phase transition associated with symmetry Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 16 / 41
17 Gross Neveu-like model 0-dim. Gross Neveu-like model The partition function of our model study is the following: { Z N (G, m) = d ψdψ N exp ψ a (i/p + m)ψ a + G ( N } ) 2 ψ a ψ a. 4N a=1 a=1 ψ, ψ: 2-component Grassmannian variables with N flavors. /p = p 1 σ 1 + p 2 σ 2. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 17 / 41
18 Gross Neveu-like model Hubbard Stratonovich transformation Bosonization (σ ψψ ): with Z N (G, m) = For simplicity, we put m = 0 in the following. S has three saddle points: N dσ e NS(σ), πg R S(σ) σ2 G log[p2 + (σ + m) 2 ]. 0 = S(z) z = 2z G 2z p 2 + z 2 = z = 0, ± G p 2. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 18 / 41
19 Gross Neveu-like model Behaviors of the flow Figures for G = 0.7e 0.1i, 1.1e 0.1i at p 2 = 1 [Kanazawa, YT, arxiv: ]: z = 0 is the unique critical point contributing to Z if G < p 2. All three critical points contribute to Z if G > p 2. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 19 / 41
20 Gross Neveu-like model Stokes phenomenon The difference of the way of contribution can be described by Stokes phenomenon. [Witten, arxiv: , ] Figures of G-plane for ImS(0) = ImS(z ± ) [Kanazawa, YT, arxiv: ]: I G R G Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 20 / 41
21 Gross Neveu-like model Dominance of contribution The Stokes phenomenon tells us the number of Lefschetz thimbles contributing to Z N. However, it does not tell which thimbles give main contribution. Z N # exp( NS(0)) + # exp( NS(z ± )) In order to obtain σ 0 in the large-n limit, z ± should dominate z = 0. ReS(z ± ) ReS(0) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 21 / 41
22 Gross Neveu-like model Connection with Lee Yang zero I G R G Blue line: ImS(z ± ) = ImS(0). Green line: ReS(z ± ) = ReS(0). Red points: Lee-Yang zeros at N = 40. [Kanazawa, YT, arxiv: ] Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 22 / 41
23 Gross Neveu-like model Conclusions for studies with GN-like models 1 Decomposition of the integration path in terms of Lefschetz thimbles is useful to visualize different phases. 2 The possible link between Lefschetz-thimble decomposition and Lee Yang zeros is indicated. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 23 / 41
24 Gross Neveu-like model Preliminary comments on a recent paper [Nishimura, Shimasaki, arxiv: ] from the Lefschetz-thimble viewpoint Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 24 / 41
25 Gross Neveu-like model Singularity of the drift term Model: Z = dx(x + iα) p e x2 /2. Drift term in the complex Langevin equation: S z = z p z + iα. The singularity of the drift term breaks the formal proof for the correctness of CLE at the stage of integration by parts. [Nishimura, Shimasaki, arxiv: ] Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 25 / 41
26 Gross Neveu-like model Expectation values of x 2 for various α CLE breaks down for α 14 at p = 50. [Nishimura, Shimasaki, arxiv: ] Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 26 / 41
27 Gross Neveu-like model Structure of Lefschetz thimbles The Stokes phenomenon happens at α = 4p. y y x (a) α < 2 p x (b) α > 2 p Is this phenomenon related to the breakdown of CLE? Looks interesting. No one can answer yet, though... Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 27 / 41
28 Sign problem in MFA Formal discussion Sign problem in MFA and its solution: Formal discussion Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 28 / 41
29 Sign problem in MFA Formal discussion Motivation At finite-density QCD (in the heavy-dense limit), the Polyakov-loop effective action looks like S eff (l) d 3 x [ e µ l(x) + e µ l(x) ] R. Even after the MFA, the effective potential becomes complex! The integration over the order parameter plays a pivotal role for reality. (Fukushima, Hidaka, PRD75, ) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 29 / 41
30 Sign problem in MFA Formal discussion General set up Consider the oscillatory multiple integration, Z = d n x exp S(x). R n To ensure Z R, suppose the existence of a linear map L, satisfying S(x) = S(L x). L 2 = 1. Let s construct a systematic computational scheme of Z with Z R. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 30 / 41
31 Sign problem in MFA Formal discussion Flow eq. and Complex conjugation Morse s downward flow: dz i dt = Complex conjugation of the flow: dz i dt = ( ( S(z) ) S(z) = z i z i ). ( S(L z) z i therefore z = L z satisfy the same flow eqation: dz i dt = ( S(z ) z i ). ), Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 31 / 41
32 Sign problem in MFA Formal discussion General Theorem: Saddle points & thimbles Let us decompose the set of saddle points into three parts, Σ = Σ 0 Σ + Σ, where Σ 0 = {σ z σ = L z σ }, Σ ± = {σ ImS(z σ ) 0}. The antilinear map gives Σ + Σ. This correspondence also applies to Lefschetz thimbles. (YT, Nishimura, Kashiwa, arxiv: [hep-th]) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 32 / 41
33 Sign problem in MFA Formal discussion General Theorem The partition function: Z = n σ d σ Σ J n z exp S(z) σ 0 + d n z exp S(z). τ Σ + n τ Each term on the r.h.s. is real. J τ +J K τ (YT, Nishimura, Kashiwa, arxiv: [hep-th]) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 33 / 41
34 Sign problem in MFA Application to QCD Sign problem in MFA and its solution: Application to QCD-like theories Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 34 / 41
35 Sign problem in MFA Application to QCD The QCD partition function at finite density The QCD partition function: Z QCD = DA detm(µ qk, A) exp S YM [A], w./ the Yang-Mills action S YM = 1 2 tr β 0 dx4 d 3 x F µν 2 (> 0), and detm(µ, A) = det [ γ ν ( ν + iga ν ) + γ 4 µ qk + m qk ]. is the quark determinant. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 35 / 41
36 Sign problem in MFA Application to QCD Charge conjugation If µ qk 0, the quark determinant takes complex values Sign problem of QCD. However, Z QCD R is ensured thanks to the charge conjugation A A t : det M(µ qk, A) = det M( µ qk, A ) (First equality: γ 5 -transformation, Second equality: charge conjugation) = det M(µ qk, A). Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 36 / 41
37 Sign problem in MFA Application to QCD Polyakov-loop effective model The Polyakov line L: L = 1 3 diag [ e i(θ 1+θ 2 ), e i( θ 1+θ 2 ), e ] 2iθ 2. Let us consider the SU(3) matrix model: Z QCD = dθ 1 dθ 2 H(θ 1, θ 2 ) exp [ V eff (θ 1, θ 2 )], where H = sin 2 θ 1 sin 2 ((θ 1 + 3θ 2 )/2) sin 2 ((θ 1 3θ 2 )/2). Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 37 / 41
38 Sign problem in MFA Application to QCD Charge conjugation in the Polyakov loop model Charge conjugation acts as L L : V eff (z 1, z 2 ) = V eff (z 1, z 2 ). Simple model for dense quarks (l := trl): V eff = h (32 1) ( ) e µ l 3 (θ 1, θ 2 ) + e µ l 2 3 (θ 1, θ 2 ) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 38 / 41
39 Sign problem in MFA Application to QCD Behaviors of the flow Figure : Flow at h = 0.1 and µ = 2 in the Re(z 1 )-Im(z 2 ) plane. The black blob: a saddle point. The red solid line: Lefschetz thimble J. The green dashed line: its dual K. (YT, Nishimura, Kashiwa, arxiv: [hep-th]) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 39 / 41
40 Sign problem in MFA Application to QCD Saddle point approximation at finite density The saddle point approximation can now be performed. Polyakov-loop phases (z 1, z 2 ) takes complex values, so that l, l R. Since Im(z 2 ) < 0 and l 1 3 (2eiz 2 cos θ 1 + e 2iz 2 ), we can confirm that l > l. (YT, Nishimura, Kashiwa, arxiv: [hep-th]) Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 40 / 41
41 Sign problem in MFA Application to QCD Summary for the sign problem in MFA Lefschetz-thimble integral is a useful tool to treat multiple integrals. Saddle point approximation can be applied without violating Z R. Sign problem of effective models of QCD is (partly) explored. Yuya Tanizaki (University of Tokyo, RIKEN) Lefschetz-thimble path integral May 21, KEK 41 / 41
Lefschetz-thimble path integral for solving the mean-field sign problem
Lefschetz-thimble path integral for solving the mean-field sign problem Yuya Tanizaki Department of Physics, The University of Tokyo Theoretical Research Division, Nishina Center, RIKEN Jul 15, 2015 @
More informationIntroduction to Lefschetz-thimble path integral and its applications to physics
Introduction to Lefschetz-thimble path integral and its applications to physics Yuya Tanizaki Department of Physics, The University of Tokyo Theoretical Research Division, Nishina Center, RIKEN February
More informationComplex Saddle Points in Finite Density QCD
Complex Saddle Points in Finite Density QCD Michael C. Ogilvie Washington University in St. Louis in collaboration with Hiromichi Nishimura (Bielefeld) and Kamal Pangeni (WUSTL) XQCD4 June 9th, 24 Outline
More informationLattice QCD at nonzero baryon density
Lattice QCD at nonzero baryon density Gert Aarts Erice, September 2016 p. 1 QCD phase diagram a well-known possibility Erice, September 2016 p. 2 Lattice QCD at nonzero chemical potential partition function/euclidean
More informationIntroduction to Instantons. T. Daniel Brennan. Quantum Mechanics. Quantum Field Theory. Effects of Instanton- Matter Interactions.
February 18, 2015 1 2 3 Instantons in Path Integral Formulation of mechanics is based around the propagator: x f e iht / x i In path integral formulation of quantum mechanics we relate the propagator to
More informationResurgence Structure to All Orders of Multi-bions in Deformed SUSY Quantum Mechanics
Resurgence Structure to All Orders of Multi-bions in Deformed SUSY Quantum Mechanics Toshiaki Fujimori (Keio University) based on arxiv:1607.04205, Phys.Rev. D94 (2016) arxiv:1702.00589, Phys.Rev. D95
More informationPossible higher order phase transition in large-n gauge theory at finite temperature
Possible higher order phase transition in large-n gauge theory at finite temperature RIKEN/BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973 E-mail: hnishimura@bnl.gov Robert D. Pisarski
More informationComplex Langevin dynamics for nonabelian gauge theories
Complex Langevin dynamics for nonabelian gauge theories Gert Aarts XQCD 14, June 2014 p. 1 QCD phase diagram QCD partition function Z = DUD ψdψe S YM S F = DU detde S YM at nonzero quark chemical potential
More informationAmbitwistor strings, the scattering equations, tree formulae and beyond
Ambitwistor strings, the scattering equations, tree formulae and beyond Lionel Mason The Mathematical Institute, Oxford lmason@maths.ox.ac.uk Les Houches 18/6/2014 With David Skinner. arxiv:1311.2564 and
More informationSolitons in the SU(3) Faddeev-Niemi Model
Solitons in the SU(3) Faddeev-Niemi Model Yuki Amari Tokyo University of Science amari.yuki.ph@gmail.com Based on arxiv:1805,10008 with PRD 97, 065012 (2018) In collaboration with Nobuyuki Sawado (TUS)
More informationString / gauge theory duality and ferromagnetic spin chains
String / gauge theory duality and ferromagnetic spin chains M. Kruczenski Princeton Univ. In collaboration w/ Rob Myers, David Mateos, David Winters Arkady Tseytlin, Anton Ryzhov Summary Introduction mesons,,...
More informationQuantum Dynamics of Supergravity
Quantum Dynamics of Supergravity David Tong Work with Carl Turner Based on arxiv:1408.3418 Crete, September 2014 An Old Idea: Euclidean Quantum Gravity Z = Dg exp d 4 x g R topology A Preview of the Main
More informationφ 3 theory on the lattice
φ 3 theory on the lattice Michael Kroyter The Open University of Israel SFT 2015 Chengdu 15-May-2015 Work in progress w. Francis Bursa Michael Kroyter (The Open University) φ 3 theory on the lattice SFT
More informationProperties of monopole operators in 3d gauge theories
Properties of monopole operators in 3d gauge theories Silviu S. Pufu Princeton University Based on: arxiv:1303.6125 arxiv:1309.1160 (with Ethan Dyer and Mark Mezei) work in progress with Ethan Dyer, Mark
More information1/N Expansions in String and Gauge Field Theories. Adi Armoni Swansea University
1/N Expansions in String and Gauge Field Theories Adi Armoni Swansea University Oberwoelz, September 2010 1 Motivation It is extremely difficult to carry out reliable calculations in the strongly coupled
More informationThe Fermion Bag Approach
The Fermion Bag Approach Anyi Li Duke University In collaboration with Shailesh Chandrasekharan 1 Motivation Monte Carlo simulation Sign problem Fermion sign problem Solutions to the sign problem Fermion
More informationGeometry and Physics. Amer Iqbal. March 4, 2010
March 4, 2010 Many uses of Mathematics in Physics The language of the physical world is mathematics. Quantitative understanding of the world around us requires the precise language of mathematics. Symmetries
More informationRandom Matrix Theory
Random Matrix Theory Gernot Akemann Faculty of Physics, Bielefeld University STRONGnet summer school, ZiF Bielefeld, 14-25 June 2011 Content What is RMT about? Nuclear Physics, Number Theory, Quantum Chaos,...
More informationFinite-temperature Field Theory
Finite-temperature Field Theory Aleksi Vuorinen CERN Initial Conditions in Heavy Ion Collisions Goa, India, September 2008 Outline Further tools for equilibrium thermodynamics Gauge symmetry Faddeev-Popov
More informationHigh density QCD on a Lefschetz thimble?
High density QCD on a Lefschetz thimble? Luigi Scorzato (ECT*, Trento) in collaboration with Marco Cristoforetti (ECT*) and Francesco Di Renzo (U.Parma) Oberwölz, 4 September 2012 QCD History and Prospects
More informationAmplitudes & Wilson Loops at weak & strong coupling
Amplitudes & Wilson Loops at weak & strong coupling David Skinner - Perimeter Institute Caltech - 29 th March 2012 N=4 SYM, 35 years a!er Twistor space is CP 3, described by co-ords It carries a natural
More informationMichael CREUTZ Physics Department 510A, Brookhaven National Laboratory, Upton, NY 11973, USA
with η condensation Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 66-85, Japan E-mail: saoki@yukawa.kyoto-u.ac.jp Michael CREUTZ Physics Department
More informationEmergent symmetries in the Canonical Tensor Model
Emergent symmetries in the Canonical Tensor Model Naoki Sasakura Yukawa Institute for Theoretical Physics, Kyoto University Based on collaborations with Dennis Obster arxiv:1704.02113 and a paper coming
More informationThe θ term. In particle physics and condensed matter physics. Anna Hallin. 601:SSP, Rutgers Anna Hallin The θ term 601:SSP, Rutgers / 18
The θ term In particle physics and condensed matter physics Anna Hallin 601:SSP, Rutgers 2017 Anna Hallin The θ term 601:SSP, Rutgers 2017 1 / 18 1 Preliminaries 2 The θ term in general 3 The θ term in
More informationIntroduction to AdS/CFT
Introduction to AdS/CFT D-branes Type IIA string theory: Dp-branes p even (0,2,4,6,8) Type IIB string theory: Dp-branes p odd (1,3,5,7,9) 10D Type IIB two parallel D3-branes low-energy effective description:
More informationNONINTEGER FLUXES, DOLBEAULT COMPLEXES, AND SUPERSYMMETRIC QUANTUM MECHANICS. based on [ ] and [ ] Hannover, August 1, 2011
NONINTEGER FLUXES, DOLBEAULT COMPLEXES, AND SUPERSYMMETRIC QUANTUM MECHANICS based on [1104.3986] and [1105.3935] Hannover, August 1, 2011 WHAT IS WRONG WITH NONINTEGER FLUX? Quantization of Dirac monopole
More informationPNJL Model and QCD Phase Transitions
PNJL Model and QCD Phase Transitions Hiromichi Nishimura Washington University in St. Louis INT Workshop, Feb. 25, 2010 Phase Transitions in Quantum Chromodynamics This Talk Low Temperature Lattice and
More informationQCD and Instantons: 12 Years Later. Thomas Schaefer North Carolina State
QCD and Instantons: 12 Years Later Thomas Schaefer North Carolina State 1 ESQGP: A man ahead of his time 2 Instanton Liquid: Pre-History 1975 (Polyakov): The instanton solution r 2 2 E + B A a µ(x) = 2
More informationHalf BPS solutions in type IIB and M-theory
Half BPS solutions in type IIB and M-theory Based on work done in collaboration with Eric D Hoker, John Estes, Darya Krym (UCLA) and Paul Sorba (Annecy) E.D'Hoker, J.Estes and M.G., Exact half-bps type
More informationLecture 7: N = 2 supersymmetric gauge theory
Lecture 7: N = 2 supersymmetric gauge theory José D. Edelstein University of Santiago de Compostela SUPERSYMMETRY Santiago de Compostela, November 22, 2012 José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric
More informationString/gauge theory duality and QCD
String/gauge theory duality and QCD M. Kruczenski Purdue University ASU 009 Summary Introduction String theory Gauge/string theory duality. AdS/CFT correspondence. Mesons in AdS/CFT Chiral symmetry breaking
More informationNTNU Trondheim, Institutt for fysikk
NTNU Trondheim, Institutt for fysikk Examination for FY3464 Quantum Field Theory I Contact: Michael Kachelrieß, tel. 99890701 Allowed tools: mathematical tables Some formulas can be found on p.2. 1. Concepts.
More informationGravitational Decoupling II:
Gravitational Decoupling II: Picard-Lefschetz Theory Jonathan Brown University of Wisconsin - Madison Great Lakes Strings, University of Chicago April 14, 2018 Based on arxiv:1710.04737 with Alex Cole,
More informationChiral Symmetry Breaking from Monopoles and Duality
Chiral Symmetry Breaking from Monopoles and Duality Thomas Schaefer, North Carolina State University with A. Cherman and M. Unsal, PRL 117 (2016) 081601 Motivation Confinement and chiral symmetry breaking
More informationSupersymmetric Gauge Theories, Matrix Models and Geometric Transitions
Supersymmetric Gauge Theories, Matrix Models and Geometric Transitions Frank FERRARI Université Libre de Bruxelles and International Solvay Institutes XVth Oporto meeting on Geometry, Topology and Physics:
More informationRandom Matrix Theory for the Wilson-Dirac operator
Random Matrix Theory for the Wilson-Dirac operator Mario Kieburg Department of Physics and Astronomy SUNY Stony Brook (NY, USA) Bielefeld, December 14th, 2011 Outline Introduction in Lattice QCD and in
More informationInstanton effective action in - background and D3/D(-1)-brane system in R-R background
Instanton effective action in - background and D3/D(-1)-brane system in R-R background Speaker : Takuya Saka (Tokyo Tech.) Collaboration with Katsushi Ito, Shin Sasaki (Tokyo Tech.) And Hiroaki Nakajima
More informationLecture II: Owe Philipsen. The ideal gas on the lattice. QCD in the static and chiral limit. The strong coupling expansion at finite temperature
Lattice QCD, Hadron Structure and Hadronic Matter Dubna, August/September 2014 Lecture II: Owe Philipsen The ideal gas on the lattice QCD in the static and chiral limit The strong coupling expansion at
More informationBanks-Casher-type relations for complex Dirac spectra
Banks-Casher-type relations for complex Dirac spectra Takuya Kanazawa, a Tilo Wettig, b Naoki Yamamoto c a University of Tokyo, b University of Regensburg, c University of Maryland EPJA 49 (2013) 88 [arxiv:1211.5332]
More informationLecture 12 Holomorphy: Gauge Theory
Lecture 12 Holomorphy: Gauge Theory Outline SUSY Yang-Mills theory as a chiral theory: the holomorphic coupling and the holomorphic scale. Nonrenormalization theorem for SUSY YM: the gauge coupling runs
More informationBaby Skyrmions in AdS 3 and Extensions to (3 + 1) Dimensions
Baby Skyrmions in AdS 3 and Extensions to (3 + 1) Dimensions Durham University Work in collaboration with Matthew Elliot-Ripley June 26, 2015 Introduction to baby Skyrmions in flat space and AdS 3 Discuss
More informationSpectrum of Holographic Wilson Loops
Spectrum of Holographic Wilson Loops Leopoldo Pando Zayas University of Michigan Continuous Advances in QCD 2011 University of Minnesota Based on arxiv:1101.5145 Alberto Faraggi and LPZ Work in Progress,
More informationSpacetime Quantum Geometry
Spacetime Quantum Geometry Peter Schupp Jacobs University Bremen 4th Scienceweb GCOE International Symposium Tohoku University 2012 Outline Spacetime quantum geometry Applying the principles of quantum
More informationarxiv: v1 [hep-th] 3 Sep 2010
PT Symmetry and the Sign Problem Peter N. Meisinger, Michael C. Ogilvie and Timothy D. Wiser Dept. of Physics Washington University St. Louis, MO 63130 USA Abstract Generalized PT symmetry provides crucial
More informationAnomalies and discrete chiral symmetries
Anomalies and discrete chiral symmetries Michael Creutz BNL & U. Mainz Three sources of chiral symmetry breaking in QCD spontaneous breaking ψψ 0 explains lightness of pions implicit breaking of U(1) by
More informationSuperstring in the plane-wave background with RR-flux as a conformal field theory
0th December, 008 At Towards New Developments of QFT and Strings, RIKEN Superstring in the plane-wave background with RR-flux as a conformal field theory Naoto Yokoi Institute of Physics, University of
More informationJuly 2, SISSA Entrance Examination. PhD in Theoretical Particle Physics Academic Year 2018/2019. olve two among the three problems presented.
July, 018 SISSA Entrance Examination PhD in Theoretical Particle Physics Academic Year 018/019 S olve two among the three problems presented. Problem I Consider a theory described by the Lagrangian density
More informationConfinement in Polyakov gauge
Confinement in Polyakov gauge Florian Marhauser arxiv:812.1144 QCD Phase Diagram chiral vs. deconfinement phase transition finite density critical point... Confinement Order Parameter ( β ) φ( x) = L(
More informationSpectral action, scale anomaly. and the Higgs-Dilaton potential
Spectral action, scale anomaly and the Higgs-Dilaton potential Fedele Lizzi Università di Napoli Federico II Work in collaboration with A.A. Andrianov (St. Petersburg) and M.A. Kurkov (Napoli) JHEP 1005:057,2010
More informationHIGHER SPIN CORRECTIONS TO ENTANGLEMENT ENTROPY
HIGHER SPIN CORRECTIONS TO ENTANGLEMENT ENTROPY JHEP 1406 (2014) 096, Phys.Rev. D90 (2014) 4, 041903 with Shouvik Datta ( IISc), Michael Ferlaino, S. Prem Kumar (Swansea U. ) JHEP 1504 (2015) 041 with
More informationFinite Temperature Field Theory
Finite Temperature Field Theory Dietrich Bödeker, Universität Bielefeld 1. Thermodynamics (better: thermo-statics) (a) Imaginary time formalism (b) free energy: scalar particles, resummation i. pedestrian
More informationNon-associative Deformations of Geometry in Double Field Theory
Non-associative Deformations of Geometry in Double Field Theory Michael Fuchs Workshop Frontiers in String Phenomenology based on JHEP 04(2014)141 or arxiv:1312.0719 by R. Blumenhagen, MF, F. Haßler, D.
More informationCalculation of decay constant using gradient flow, towards the Kaon bag parameter. University of Tsukuba, A. Suzuki and Y.
Calculation of decay constant using gradient flow, towards the Kaon bag parameter University of Tsukuba, A. Suzuki and Y. Taniguchi Contents Goal : Calculation of B K with Wilson fermion using gradient
More informationNon-abelian statistics
Non-abelian statistics Paul Fendley Non-abelian statistics are just plain interesting. They probably occur in the ν = 5/2 FQHE, and people are constructing time-reversal-invariant models which realize
More informationEvaluation of Triangle Diagrams
Evaluation of Triangle Diagrams R. Abe, T. Fujita, N. Kanda, H. Kato, and H. Tsuda Department of Physics, Faculty of Science and Technology, Nihon University, Tokyo, Japan E-mail: csru11002@g.nihon-u.ac.jp
More informationFrom confinement to new states of dense QCD matter
From confinement to new states of dense QCD matter From Quarks and Gluons to Hadrons and Nuclei, Erice, Sicily, 17 Sept2011 Kurt Langfeld School of Comp. and Mathematics and The HPCC, Univ. of Plymouth,
More informationMonte Carlo studies of the spontaneous rotational symmetry breaking in dimensionally reduced super Yang-Mills models
Monte Carlo studies of the spontaneous rotational symmetry breaking in dimensionally reduced super Yang-Mills models K. N. Anagnostopoulos 1 T. Azuma 2 J. Nishimura 3 1 Department of Physics, National
More informationChern-Simons Theory and Its Applications. The 10 th Summer Institute for Theoretical Physics Ki-Myeong Lee
Chern-Simons Theory and Its Applications The 10 th Summer Institute for Theoretical Physics Ki-Myeong Lee Maxwell Theory Maxwell Theory: Gauge Transformation and Invariance Gauss Law Charge Degrees of
More informationStrings, gauge theory and gravity. Storrs, Feb. 07
Strings, gauge theory and gravity Storrs, Feb. 07 Outline Motivation - why study quantum gravity? Intro to strings Gravity / gauge theory duality Gravity => gauge: Wilson lines, confinement Gauge => gravity:
More informationMembrane Matrix models. Workshop on Noncommutative Field Theory and Gravity
and non-perturbative checks of AdS/CFT Denjoe O Connor Dublin Institute for Advanced Studies Workshop on Noncommutative Field Theory and Gravity Corfu Summer Institute, Corfu, Sept 22nd 2015 Based on work
More informationTopological zero modes at finite chemical potential
Topological zero modes at finite chemical potential Falk Bruckmann (Univ. Regensburg) Sign 2012, Regensburg, Sept. 2012 with Rudi Rödl, Tin Sulejmanpasic (paper in prep.) Falk Bruckmann Topological zero
More informationThe Phase Diagram of the BMN Matrix Model
Denjoe O Connor School of Theoretical Physics Dublin Institute for Advanced Studies Dublin, Ireland Workshop on Testing Fundamental Physics Principles Corfu2017, 22-28th September 2017 Background: V. Filev
More informationS-CONFINING DUALITIES
DIMENSIONAL REDUCTION of S-CONFINING DUALITIES Cornell University work in progress, in collaboration with C. Csaki, Y. Shirman, F. Tanedo and J. Terning. 1 46 3D Yang-Mills A. M. Polyakov, Quark Confinement
More informationNon-Perturbative Thermal QCD from AdS/QCD
Issues Non-Perturbative Thermal QCD from AdS/QCD * Collaborators: B. Galow, M. Ilgenfritz, J. Nian, H.J. Pirner, K. Veshgini Research Fellow of the Alexander von Humboldt Foundation Institute for Theoretical
More informationContinuity of the Deconfinement Transition in (Super) Yang Mills Theory
Continuity of the Deconfinement Transition in (Super) Yang Mills Theory Thomas Schaefer, North Carolina State University 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0 0.5 0.0 0.5 1.0 0.5
More informationCanonical partition functions in lattice QCD
Canonical partition functions in lattice QCD (an appetizer) christof.gattringer@uni-graz.at Julia Danzer, Christof Gattringer, Ludovit Liptak, Marina Marinkovic Canonical partition functions Grand canonical
More informationHolographic Entanglement Entropy for Surface Operators and Defects
Holographic Entanglement Entropy for Surface Operators and Defects Michael Gutperle UCLA) UCSB, January 14th 016 Based on arxiv:1407.569, 1506.0005, 151.04953 with Simon Gentle and Chrysostomos Marasinou
More informationLectures on Chiral Perturbation Theory
Lectures on Chiral Perturbation Theory I. Foundations II. Lattice Applications III. Baryons IV. Convergence Brian Tiburzi RIKEN BNL Research Center Chiral Perturbation Theory I. Foundations Low-energy
More informationCFT approach to multi-channel SU(N) Kondo effect
CFT approach to multi-channel SU(N) Kondo effect Sho Ozaki (Keio Univ.) In collaboration with Taro Kimura (Keio Univ.) Seminar @ Chiba Institute of Technology, 2017 July 8 Contents I) Introduction II)
More informationAnomaly. Kenichi KONISHI University of Pisa. College de France, 14 February 2006
Anomaly Kenichi KONISHI University of Pisa College de France, 14 February 2006 Abstract Symmetry and quantization U A (1) anomaly and π 0 decay Origin of anomalies Chiral and nonabelian anomaly Anomally
More informationA loop of SU(2) gauge fields on S 4 stable under the Yang-Mills flow
A loop of SU(2) gauge fields on S 4 stable under the Yang-Mills flow Daniel Friedan Rutgers the State University of New Jersey Natural Science Institute, University of Iceland MIT November 3, 2009 1 /
More informationInstanton constituents in sigma models and Yang-Mills theory at finite temperature
Instanton constituents in sigma models and Yang-Mills theory at finite temperature Falk Bruckmann Univ. of Regensburg Extreme QCD, North Carolina State, July 8 PRL (8) 56 [77.775] EPJ Spec.Top.5 (7) 6-88
More informationBPS non-local operators in AdS/CFT correspondence. Satoshi Yamaguchi (Seoul National University) E. Koh, SY, arxiv: to appear in JHEP
BPS non-local operators in AdS/CFT correspondence Satoshi Yamaguchi (Seoul National University) E. Koh, SY, arxiv:0812.1420 to appear in JHEP Introduction Non-local operators in quantum field theories
More informationContact interactions in string theory and a reformulation of QED
Contact interactions in string theory and a reformulation of QED James Edwards QFT Seminar November 2014 Based on arxiv:1409.4948 [hep-th] and arxiv:1410.3288 [hep-th] Outline Introduction Worldline formalism
More information752 Final. April 16, Fadeev Popov Ghosts and Non-Abelian Gauge Fields. Tim Wendler BYU Physics and Astronomy. The standard model Lagrangian
752 Final April 16, 2010 Tim Wendler BYU Physics and Astronomy Fadeev Popov Ghosts and Non-Abelian Gauge Fields The standard model Lagrangian L SM = L Y M + L W D + L Y u + L H The rst term, the Yang Mills
More informationPath Integral Quantization of the Electromagnetic Field Coupled to A Spinor
EJTP 6, No. 22 (2009) 189 196 Electronic Journal of Theoretical Physics Path Integral Quantization of the Electromagnetic Field Coupled to A Spinor Walaa. I. Eshraim and Nasser. I. Farahat Department of
More informationSynopsis of Complex Analysis. Ryan D. Reece
Synopsis of Complex Analysis Ryan D. Reece December 7, 2006 Chapter Complex Numbers. The Parts of a Complex Number A complex number, z, is an ordered pair of real numbers similar to the points in the real
More informationCold atoms and AdS/CFT
Cold atoms and AdS/CFT D. T. Son Institute for Nuclear Theory, University of Washington Cold atoms and AdS/CFT p.1/20 What is common for strong coupled cold atoms and QGP? Cold atoms and AdS/CFT p.2/20
More informationarxiv: v2 [hep-lat] 24 Oct 2014
Prepared for submission to JHEP Some remarks on Lefschetz thimbles and complex Langevin dynamics arxiv:147.29v2 [hep-lat] 24 Oct 214 Gert Aarts, a Lorenzo Bongiovanni, a Erhard Seiler b and Dénes Sexty
More informationt Hooft loop path integral in N = 2 gauge theories
t Hooft loop path integral in N = 2 gauge theories Jaume Gomis (based on work with Takuya Okuda and Vasily Pestun) Perimeter Institute December 17, 2010 Jaume Gomis (Perimeter Institute) t Hooft loop path
More information8.324 Relativistic Quantum Field Theory II
Lecture 6 8.34 Relativistic Quantum Field Theory II Fall 8.34 Relativistic Quantum Field Theory II MIT OpenCourseWare Lecture Notes Hong Liu, Fall α(μ) Lecture 6 5.3.3: Beta-Functions of Quantum Electrodynamics
More informationTWISTOR DIAGRAMS for Yang-Mills scattering amplitudes
TWISTOR DIAGRAMS for Yang-Mills scattering amplitudes Andrew Hodges Wadham College, University of Oxford London Mathematical Society Durham Symposium on Twistors, Strings and Scattering Amplitudes, 20
More informationThe twistor theory of the Ernst equations
The twistor theory of the Ernst equations Lionel Mason The Mathematical Institute, Oxford lmason@maths.ox.ac.uk November 21, 2005 Credits: R.S.Ward, N.M.J.Woodhouse, J.Fletcher, K.Y.Tee, based on classic
More informationPutting String Theory to the Test with AdS/CFT
Putting String Theory to the Test with AdS/CFT Leopoldo A. Pando Zayas University of Iowa Department Colloquium L = 1 4g 2 Ga µνg a µν + j G a µν = µ A a ν ν A a µ + if a bc Ab µa c ν, D µ = µ + it a
More informationLattice QCD study for relation between quark-confinement and chiral symmetry breaking
Lattice QCD study for relation between quark-confinement and chiral symmetry breaking Quantum Hadron Physics Laboratory, Nishina Center, RIKEN Takahiro M. Doi ( 土居孝寛 ) In collaboration with Hideo Suganuma
More informationThe Affleck Dine Seiberg superpotential
The Affleck Dine Seiberg superpotential SUSY QCD Symmetry SUN) with F flavors where F < N SUN) SUF ) SUF ) U1) U1) R Φ, Q 1 1 F N F Φ, Q 1-1 F N F Recall that the auxiliary D a fields: D a = gφ jn T a
More informationQuark-gluon plasma from AdS/CFT Correspondence
Quark-gluon plasma from AdS/CFT Correspondence Yi-Ming Zhong Graduate Seminar Department of physics and Astronomy SUNY Stony Brook November 1st, 2010 Yi-Ming Zhong (SUNY Stony Brook) QGP from AdS/CFT Correspondence
More informationarxiv:hep-lat/ v1 25 Jun 2005
TKYNT-05-16, 2005/June Lee-Yang zero analysis for the study of QCD phase structure Shinji Ejiri Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan (March 1, 2008) Abstract arxiv:hep-lat/0506023v1
More informationarxiv:hep-lat/ v6 11 Dec 2003
ITEP-LAT/2003-32 A hidden symmetry in the Standard Model B.L.G. Bakker a, A.I. Veselov b, M.A. Zubkov b arxiv:hep-lat/0301011v6 11 Dec 2003 a Department of Physics and Astronomy, Vrije Universiteit, Amsterdam,
More informationGinsparg-Wilson Fermions and the Chiral Gross-Neveu Model
Ginsparg-Wilson Fermions and the DESY Zeuthen 14th September 2004 Ginsparg-Wilson Fermions and the QCD predictions Perturbative QCD only applicable at high energy ( 1 GeV) At low energies (100 MeV - 1
More informationYet Another Alternative to Compactification by Heterotic Five-branes
The University of Tokyo, Hongo: October 26, 2009 Yet Another Alternative to Compactification by Heterotic Five-branes arxiv: 0905.285 [hep-th] Tetsuji KIMURA (KEK) Shun ya Mizoguchi (KEK, SOKENDAI) Introduction
More information!onformali" Los# J.-W. Lee D. T. Son M. Stephanov D.B.K. arxiv: Phys.Rev.D80:125005,2009
!onformali" Los# J.-W. Lee D. T. Son M. Stephanov D.B.K arxiv:0905.4752 Phys.Rev.D80:125005,2009 Motivation: QCD at LARGE N c and N f Colors Flavors Motivation: QCD at LARGE N c and N f Colors Flavors
More informationRotor Spectra, Berry Phases, and Monopole Fields: From Antiferromagnets to QCD
Rotor Spectra, Berry Phases, and Monopole Fields: From Antiferromagnets to QCD Uwe-Jens Wiese Bern University LATTICE08, Williamsburg, July 14, 008 S. Chandrasekharan (Duke University) F.-J. Jiang, F.
More informationPhD in Theoretical Particle Physics Academic Year 2017/2018
July 10, 017 SISSA Entrance Examination PhD in Theoretical Particle Physics Academic Year 017/018 S olve two among the four problems presented. Problem I Consider a quantum harmonic oscillator in one spatial
More informationPoS(Confinement X)058
Confining gauge theories with adjoint scalars on R 3 S 1 University of Bielefeld E-mail: nishimura@physik.uni-bielefeld.de Michael Ogilvie Washington University, St. Louis E-mail: mco@physics.wustl.edu
More informationFunctional RG methods in QCD
methods in QCD Institute for Theoretical Physics University of Heidelberg LC2006 May 18th, 2006 methods in QCD motivation Strong QCD QCD dynamical symmetry breaking instantons χsb top. dofs link?! deconfinement
More informationEffective theories for QCD at finite temperature and density from strong coupling
XQCD 2011 San Carlos, July 2011 Effective theories for QCD at finite temperature and density from strong coupling Owe Philipsen Introduction to strong coupling expansions SCE for finite temperature: free
More informationDimensional reduction near the deconfinement transition
Dimensional reduction near the deconfinement transition Aleksi Kurkela ETH Zürich Wien 27.11.2009 Outline Introduction Dimensional reduction Center symmetry The deconfinement transition: QCD has two remarkable
More informationTwistor Strings, Gauge Theory and Gravity. Abou Zeid, Hull and Mason hep-th/
Twistor Strings, Gauge Theory and Gravity Abou Zeid, Hull and Mason hep-th/0606272 Amplitudes for YM, Gravity have elegant twistor space structure: Twistor Geometry Amplitudes for YM, Gravity have elegant
More informationThe Non-commutative S matrix
The Suvrat Raju Harish-Chandra Research Institute 9 Dec 2008 (work in progress) CONTEMPORARY HISTORY In the past few years, S-matrix techniques have seen a revival. (Bern et al., Britto et al., Arkani-Hamed
More information