Two-dimensional Yang–Mills theory
In mathematical physics, two-dimensional Yang–Mills theory is the special case of Yang–Mills theory in which the dimension of spacetime is taken to be two. This special case allows for a rigorously defined Yang–Mills measure, meaning that the (Euclidean) path integral can be interpreted as a measure on the set of connections modulo gauge transformations. This situation contrasts with the four-dimensional case, where a rigorous construction of the theory as a measure is currently unknown.
An aspect of the subject of particular interest is the large-N limit, in which the structure group is taken to be the unitary group
Background
Interest in the Yang–Mills measure comes from a statistical mechanical or constructive quantum field theoretic approach to formulating a quantum theory for the Yang–Mills field. A gauge field is described mathematically by a 1-form
where
The measure
The Yang–Mills measure for two-dimensional manifolds
Study of Yang–Mills theory in two dimensions dates back at least to work of A. A. Migdal in 1975.[1] Some formulas appearing in Migdal's work can, in retrospect, be seen to be connected to the heat kernel on the structure group of the theory. The role of the heat kernel was made more explicit in various works in the late 1970s, culminating in the introduction of the heat kernel action in work of Menotti and Onofri in 1981.[2]
In the continuum theory, the Yang–Mills measure
Dana S. Fine[9][10][11] used the formal Yang–Mills functional integral to compute loop expectation values. Other approaches include that of Klimek and Kondracki[12] and Ashtekar et al.[13] Thierry Lévy[14][15] constructed the 2-dimensional Yang–Mills measure in a very general framework, starting with the loop-expectation value formulas and constructing the measure, somewhat analogously to Brownian motion measure being constructed from transition probabilities. Unlike other works that also aimed to construct the measure from loop expectation values, Lévy's construction makes it possible to consider a very wide family of loop observables.
The discrete Yang–Mills measure is a term that has been used for the lattice gauge theory version of the Yang–Mills measure, especially for compact surfaces. The lattice in this case is a triangulation of the surface. Notable facts[16][17] are: (i) the discrete Yang–Mills measure can encode the topology of the bundle over the continuum surface even if only the triangulation is used to define the measure; (ii) when two surfaces are sewn along a common boundary loop, the corresponding discrete Yang–Mills measures convolve to yield the measure for the combined surface.
Wilson loop expectation values in 2 dimensions
For a piecewise smooth loop
were computed in the above-mentioned works.
If
If
where now
As an example for higher genus surfaces, if
with
There is an extensive physics literature on loop expectation values in two-dimensional Yang–Mills theory.[18][19][20][21][22][23][24][25] Many of the above formulas were known in the physics literature from the 1970s, with the results initially expressed in terms of a sum over the characters of the gauge group rather than the heat kernel and with the function
The low-T limit
The Yang–Mills partition function is, formally,
In the two-dimensional case we can view this as being (proportional to) the denominator that appears in the loop expectation values. Thus, for example, the partition function for the torus would be
where
where the sum is over all irreducible representations of
Returning to the Yang–Mills measure, Sengupta[33] proved that the measure itself converges in a weak sense to a suitably scaled multiple of the symplectic volume measure for orientable surfaces of genus
The large-N limit
The large-N limit of gauge theories refers to the behavior of the theory for gauge groups of the form
In two dimensions, the Makeenko–Migdal equation takes a special form developed by Kazakov and Kostov. In the large-N limit, the 2-D form of the Makeenko–Migdal equation relates the Wilson loop functional for a complicated curve with multiple crossings to the product of Wilson loop functionals for a pair of simpler curves with at least one less crossing. In the case of the sphere or the plane, it was the proposed that the Makeenko–Migdal equation could (in principle) reduce the computation of Wilson loop functionals for arbitrary curves to the Wilson loop functional for a simple closed curve.
In dimension 2, some of the major ideas were proposed by I. M. Singer,[42] who named this limit the master field (a general notion in some areas of physics). Xu[43] studied the large-
In spacetime dimension larger than 2, there is very little in terms of rigorous mathematical results. Sourav Chatterjee has proved several results in large-N gauge theory theory for dimension larger than 2. Chatterjee[52] established an explicit formula for the leading term of the free energy of three-dimensional
References
- ↑ Migdal, A. A. (1975). "Recursion equations in gauge field theories". Soviet Physics JETP 42: 413–418.
- ↑ Jump up to: 2.0 2.1 2.2 Menotti, P; Onofri, E (1981). "The action of SU(N) lattice gauge theory in terms of the heat kernel on the group manifold". Nuclear Physics B 190 (2): 288–300. doi:10.1016/0550-3213(81)90560-5. Bibcode: 1981NuPhB.190..288M. https://cds.cern.ch/record/134274.
- ↑ Jump up to: 3.0 3.1 3.2 Driver, Bruce K. (1989). "YM2:Continuum expectations, lattice convergence, and lassos". Communications in Mathematical Physics 123 (4): 575–616. doi:10.1007/BF01218586. Bibcode: 1989CMaPh.123..575D. http://projecteuclid.org/euclid.cmp/1104178984.
- ↑ Gross, Leonard; King, Chris; Sengupta, Ambar (1989). "Two dimensional Yang-Mills theory via stochastic differential equations". Annals of Physics 194 (1): 65–112. doi:10.1016/0003-4916(89)90032-8. Bibcode: 1989AnPhy.194...65G.
- ↑ Jump up to: 5.0 5.1 Sengupta, Ambar (1992). "The Yang-Mills Measure for S2". Journal of Functional Analysis 108 (2): 231–273. doi:10.1016/0022-1236(92)90025-E.
- ↑ Jump up to: 6.0 6.1 Sengupta, Ambar N. (1992). "Quantum Gauge Theory on Compact Surfaces". Annals of Physics 220 (1): 157. doi:10.1016/0003-4916(92)90334-I.
- ↑ Sengupta, Ambar N. (1997). "Yang-Mills on Surfaces with Boundary: Quantum Theory and Symplectic Limit". Communications in Mathematical Physics 183 (3): 661–704. doi:10.1007/s002200050047. Bibcode: 1997CMaPh.183..661S. http://projecteuclid.org/euclid.cmp/1158328661.
- ↑ Jump up to: 8.0 8.1 Sengupta, Ambar N. (1997). "Gauge Theory on Compact Surfaces". Memoirs of the American Mathematical Society 126 (600). doi:10.1090/memo/0600.
- ↑ Fine, Dana S. (1990). "Quantum Yang-Mills on the two-sphere". Communications in Mathematical Physics 134 (2): 273–292. doi:10.1007/BF02097703. Bibcode: 1990CMaPh.134..273F. http://projecteuclid.org/euclid.cmp/1104201731.
- ↑ Fine, Dana S. (1991). "Quantum Yang-Mills on a Riemann surface". Communications in Mathematical Physics 140 (2): 321–338. doi:10.1007/BF02099502. Bibcode: 1991CMaPh.140..321F. http://projecteuclid.org/euclid.cmp/1104247984.
- ↑ Fine, Dana S. (1996). "Topological sectors and measures on moduli space in quantum Yang-Mills on a Riemann surface". Journal of Mathematical Physics 37 (3): 1161–1170. doi:10.1063/1.531453. Bibcode: 1996JMP....37.1161F.
- ↑ Klimek, Slawomir; Kondracki, Witold (1987). "A construction of two-dimensional quantum chromodynamics.". Communications in Mathematical Physics 113 (3): 389–402. doi:10.1007/BF01221253. Bibcode: 1987CMaPh.113..389K. http://projecteuclid.org/euclid.cmp/1104160286.
- ↑ Ashtekar, Abhay; Lewandowski, Jerzy; Marolf, Donald; Mourão, José; Thiemann, Thomas (1997). "SU(N) quantum Yang-Mills theory in two dimensions: a complete solution.". Journal of Mathematical Physics 38 (11): 5453–5482. doi:10.1063/1.532146. Bibcode: 1997JMP....38.5453A.
- ↑ Lévy, Thierry (2003). "Yang-Mills Measure on Compact Surfaces". Memoirs of the American Mathematical Society 166 (790). doi:10.1090/memo/0790.
- ↑ Lévy, Thierry (2010). "Two-dimensional Markovian holonomy fields". Astérisque 329.
- ↑ Lévy, Thierry (2005). "Discrete and continuous Yang-Mills measure for non-trivial bundles over compact surfaces". Probability Theory and Related Fields 136 (2): 171–202. doi:10.1007/s00440-005-0478-8.
- ↑ Becker, Claas; Sengupta, Ambar N. (1998). "Sewing Yang-Mills measures and moduli spaces over compact surfaces". Journal of Functional Analysis 152 (1): 74–99. doi:10.1006/jfan.1997.3161.
- ↑ Gross, David; Taylor IV, Washington (1993). "Two-dimensional QCD is a string theory". Nuclear Physics B 400 (1): 181–208. doi:10.1016/0550-3213(93)90403-C. Bibcode: 1993NuPhB.400..181G.
- ↑ Cordes, Stefan; Moore, Gregory; Ramgoolam, Sanjaye (1997). "Large N 2D Yang-Mills theory and topological string theory". Communications in Mathematical Physics 185 (3): 543–619. doi:10.1007/s002200050102. Bibcode: 1997CMaPh.185..543C.
- ↑ Kazakov, K.; Kostov, I. K. (1980). "Nonlinear strings in two-dimensional l
gauge theory". Nuclear Physics B 176 (1): 199–205. doi:10.1016/0550-3213(80)90072-3. - ↑ Migdal, A. A. (1975). "Recursion equations in gauge field theories". Sov. Phys. JETP 42 (3): 2413–418.
- ↑ Jump up to: 22.0 22.1 Makeenko, Yuri M.; Migdal, A. A. (1980). "Self-consistent area law in QCD". Physics Letters B 97 (2): 253–256. doi:10.1016/0370-2693(80)90595-X. Bibcode: 1980PhLB...97..253M.
- ↑ Makeenko, Yuri M.; Migdal, A. A. (1981). "Quantum chromodynamics as dynamics of loops". Nuclear Physics B 188 (2): 269–316. doi:10.1016/0550-3213(81)90258-3. Bibcode: 1981NuPhB.188..269M.
- ↑ Rusakov, Boris (1995). "Lattice QCD as a theory of interacting surfaces". Physics Letters B 344 (1–4): 293–300. doi:10.1016/0370-2693(94)01488-X. Bibcode: 1995PhLB..344..293R.
- ↑ Rusakov, Boris (1997). "Exactly soluble QCD and confinement of quarks". Nuclear Physics B 507 (3): 691–706. doi:10.1016/S0550-3213(97)00604-4. Bibcode: 1997NuPhB.507..691R.
- ↑ Albeverio, Sergio; Høegh-Krohn, Raphael; Holden, Helge (1988). "Stochastic multiplicative measures, generalized Markov semigroups, and group-valued stochastic processes and fields". Journal of Functional Analysis 78 (1): 154–184. doi:10.1016/0022-1236(88)90137-1.
- ↑ Albeverio, Sergio; Høegh-Krohn, Raphael; Kolsrud, Torbjörn (1989). "Representation and construction of multiplicative noise". Journal of Functional Analysis 87 (2): 250–272. doi:10.1016/0022-1236(89)90010-4.
- ↑ Jump up to: 28.0 28.1 Witten, Edward (1991). "On quantum gauge theories in two dimensions". Communications in Mathematical Physics 141 (1): 153–209. doi:10.1007/BF02100009. Bibcode: 1991CMaPh.141..153W. http://projecteuclid.org/euclid.cmp/1104248198.
- ↑ Witten, Edward (1992). "Two-dimensional gauge theories revisited". Journal of Geometry and Physics 9 (4): 303–368. doi:10.1016/0393-0440(92)90034-X. Bibcode: 1992JGP.....9..303W.
- ↑ Forman, Robin (1993). "Small volume limits of 2-d Yang-Mills". Communications in Mathematical Physics 151 (1): 39–52. doi:10.1007/BF02096747. Bibcode: 1993CMaPh.151...39F. http://projecteuclid.org/euclid.cmp/1104252044.
- ↑ King, Christopher; Sengupta, Ambar N. (1994). "An explicit description of the symplectic structure of moduli spaces of flat connections". Journal of Mathematical Physics 35 (10): 5338?5353. doi:10.1063/1.530755. Bibcode: 1994JMP....35.5338K.
- ↑ King, Christopher; Sengupta, Ambar N. (1994). "The semiclassical limit of the two-dimensional quantum Yang-Mills model". Journal of Mathematical Physics 35 (10): 5354–5361. doi:10.1063/1.530756. Bibcode: 1994JMP....35.5354K.
- ↑ Jump up to: 33.0 33.1 Sengupta, Ambar N. (2003). "The Volume Measure for Flat Connections as Limit of the Yang-Mills measure". Journal of Geometry and Physics 47 (4): 398–426. doi:10.1016/S0393-0440(02)00229-2. Bibcode: 2003JGP....47..398S.
- ↑ Liu, Kefeng (1996). "Heat kernel and moduli space". Mathematical Research Letters 3 (6): 743–762. doi:10.4310/MRL.1996.v3.n6.a3. http://www.math.ucla.edu/~liu/Research/wit.pdf.
- ↑ Jeffrey, Lisa; Weitsman, Jonathan; Ramras, Daniel A. (2017). "The prequantum line bundle on the moduli space of flat SU(N) connections on a Riemann surface and the homotopy of the large N limit". Letters in Mathematical Physics 107 (9): 1581–1589. doi:10.1007/s11005-017-0956-9. Bibcode: 2017LMaPh.107.1581J.
- ↑ Jeffrey, Lisa; Weitsman, Jonathan (2000). "Symplectic geometry of the moduli space of flat connections on a Riemann surface: inductive decompositions and vanishing theorems". Canadian Journal of Mathematics 52 (3): 582–612. doi:10.4153/CJM-2000-026-4.
- ↑ Goldman, William M. (1984). "The symplectic nature of fundamental groups of surfaces". Advances in Mathematics 54 (2): 200–225. doi:10.1016/0001-8708(84)90040-9.
- ↑ Huebschmann, Johannes (1996). "The singularities of Yang-Mills connections for bundles on a surface. II. The stratification". Mathematische Zeitschrift 221 (1): 83–92. doi:10.1007/BF02622101.
- ↑ Huebschmann, Johannes (1996). "Poisson geometry of flat connections for SU(2)-bundles on surfaces". Mathematische Zeitschrift 221 (2): 243–259. doi:10.1007/PL00004249.
- ↑ Atiyah, Michael; Bott, Raoul (1983). "The Yang-Mills equations over Riemann surfaces". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences 308 (1505): 523–615.
- ↑ Lévy, Thierry; Norris, James R. (2006). "Large deviations for the Yang-Mills measure on a compact surface". Communications in Mathematical Physics 261 (2): 405–450. doi:10.1007/s00220-005-1450-2. Bibcode: 2006CMaPh.261..405L.
- ↑ Jump up to: 42.0 42.1 Singer, Isadore M. (1995). On the master field in two dimensions. Functional analysis on the eve of the 21st century. 1. pp. 263–281.
- ↑ Xu, Feng (1997). "A random matrix model from two-dimensional Yang-Mills theory". Communications in Mathematical Physics 190 (2): 287–307. doi:10.1007/s002200050242. Bibcode: 1997CMaPh.190..287X.
- ↑ Sengupta, Ambar N. (2008). Traces in two-dimensional QCD: the large-N limit. Traces in number theory, geometry and quantum fields. 1. pp. 193?212.
- ↑ Anshelevich, Michael; Sengupta, Ambar N. (2012). "Quantum free Yang-Mills on the plane". Journal of Geometry and Physics 62 (2): 330–343. doi:10.1016/j.geomphys.2011.10.005. Bibcode: 2012JGP....62..330A.
- ↑ Lévy, Thierry (2017). "The Master Field on the Plane". Astérisque 388.
- ↑ Lévy, Thierry; Maida, Mylene (2010). "Central limit theorem for the heat kernel measure on the unitary group". Journal of Functional Analysis 259 (12): 3163–3204. doi:10.1016/j.jfa.2010.08.005.
- ↑ Driver, Bruce; Hall, Brian C.; Kemp, Todd (2017). "Three proofs of the Makeenko-Migdal equation for Yang-Mills theory on the plane". Communications in Mathematical Physics 351 (2): 741–774. doi:10.1007/s00220-016-2793-6. Bibcode: 2017CMaPh.351..741D.
- ↑ Driver, Bruce; Gabriel, Franck; Hall, Brian C.; Kemp, Todd (2017). "The Makeenko-Migdal equation for Yang-Mills theory on compact surfaces". Communications in Mathematical Physics 352 (3): 967?978. doi:10.1007/s00220-017-2857-2. Bibcode: 2017CMaPh.352..967D.
- ↑ Dahlqvist, Antoine (2016). "Free energies and fluctuations for the unitary Brownian motion". Communications in Mathematical Physics 348 (2): 395–444. doi:10.1007/s00220-016-2756-y. Bibcode: 2016CMaPh.348..395D.
- ↑ Dahlqvist, Antoine; Norris, James R. (2020). "Yang-Mills measure and the master field on the sphere". Communications in Mathematical Physics 377 (2): 1163–1226. doi:10.1007/s00220-020-03773-6. Bibcode: 2020CMaPh.377.1163D.
- ↑ Chatterjee, Sourav (2016). "The leading term of the Yang-Mills free energy". Journal of Functional Analysis 271 (10): 2944–3005. doi:10.1016/j.jfa.2016.04.032.
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