From Wikipedia, the free encyclopedia

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 and then the tends to infinity limit is taken. The large-N limit of two-dimensional Yang–Mills theory has connections to string theory.


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 on a principal -bundle over a manifold taking values in the Lie algebra of the Lie group . We assume that the structure group , which describes the physical symmetries of the gauge field, is a compact Lie group with a bi-invariant metric on the Lie algebra , and we also assume given a Riemannian metric on the manifold . The Yang–Mills action functional is given by

where is the curvature of the connection form , the norm-squared in the integrand comes from the metric on the Lie algebra and the one on the base manifold, and is the Riemannian volume measure on .

The measure is given formally by

as a normalized probability measure on the space of all connections on the bundle, with a parameter, and is a formal normalizing constant. More precisely, the probability measure is more likely to be meaningful on the space of orbits of connections under gauge transformations.

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 was rigorously defined for the case where by Bruce Driver [3] and by Leonard Gross, Christopher King, and Ambar Sengupta. [4] For compact manifolds, both oriented and non-oriented, with or without boundary, with specified bundle topology, the Yang–Mills measure was constructed by Sengupta [5] [6] [7] [8] In this approach the 2-dimensional Yang–Mills measure is constructed by using a Gaussian measure on an infinite-dimensional space conditioned to satisfy relations implied by the topologies of the surface and of the bundle. Wilson loop variables (certain important variables on the space) were defined using stochastic differential equations and their expected values computed explicitly and found to agree with the results of the heat kernel action.

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 on the base manifold and a point on the fiber in the principal -bundle over the base point of the loop, there is the holonomy of any connection on the bundle. For regular loops , all based at and any function on the function is called a Wilson loop variable, of interest mostly when is a product of traces of the holonomies in representations of the group . With being a two-dimensional Riemannian manifold the loop expectation values

were computed in the above-mentioned works.

If is the plane then

where is the heat kernel on the group , is the area enclosed by the loop , and the integration is with respect to unit-mass Haar measure. This formula was proved by Driver [3] and by Gross et al. [3] using the Gaussian measure construction of the Yang–Mills measure on the plane and by defining parallel transport by interpreting the equation of parallel transport as a Stratonovich stochastic differential equation.

If is the 2-sphere then

where now is the area of the region "outside" the loop , and is the total area of the sphere. This formula was proved by Sengupta [5] using the conditioned Gaussian measure construction of the Yang–Mills measure and the result agrees with what one gets by using the heat kernel action of Menotti and Onofri. [2]

As an example for higher genus surfaces, if is a torus, then

with being the total area of the torus, and a contractible loop on the torus enclosing an area . This, and counterparts in higher genus as well as for surfaces with boundary and for bundles with nontrivial topology, were proved by Sengupta. [6] [8]

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 form 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 being the trace in some representation of the group. Expressions involving the heat kernel then appeared explicitly in the form of the "heat kernel action" in work of Menotti and Onofri. [2] The role of the convolution property of the heat kernel was used in works of Sergio Albeverio et al. [26] [27] in constructing stochastic cosurface processes inspired by Yang–Mills theory and, indirectly, by Makeenko and Migdal [22] in the physics literature.

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 is the area of the torus. In two of the most impactful works [28] [29] in the field, Edward Witten showed that as the partition function yields the volume of the moduli space of flat connections with respect to a natural volume measure on the moduli space. This volume measure is associated to a natural symplectic structure on the moduli space when the surface is orientable, and is the torsion of a certain complex in the case where the surface is not orientable. Witten's discovery has been studied in different ways by several researchers. [30] [31] [32] Let denote the moduli space of flat connections on a trivial bundle, with structure group being a compact connected semi-simple Lie group whose Lie algebra is equipped with an Ad-invariant metric, over a compact two-dimensional orientable manifold of genus . Witten showed [28] that the symplectic volume of this moduli space is given by

where the sum is over all irreducible representations of . This was proved rigorous by Sengupta [33] (see also the works by Lisa Jeffrey and by Kefeng Liu [34]). There is a large literature [35] [36] [37] [38] [39] on the symplectic structure on the moduli space of flat connections, and more generally on the moduli space itself, the major early work being that of Michael Atiyah and Raoul Bott. [40]

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 . Thierry Lévy and James R. Norris [41] established a large deviations principle for this convergence, showing that the Yang–Mills measure encodes the Yang–Mills action functional even though this functional does not explicitly appear in the rigorous formulation of the measure.

The large-N limit

The large-N limit of gauge theories refers to the behavior of the theory for gauge groups of the form , , , , and other such families, as goes to . There is a large physics literature on this subject, including major early works by Gerardus 't Hooft. A key tool in this analysis is the Makeenko–Migdal equation.

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- limit of 2-dimensional Yang–Mills loop expectation values using ideas from random matrix theory. Sengupta [44] computed the large-N limit of loop expectation values in the plane and commented on the connection with free probability. Confirming one proposal of Singer, [42] Michael Anshelevich and Sengupta [45] showed that the large-N limit of the Yang–Mills measure over the plane for the groups is given by a free probability theoretic counterpart of the Yang–Mills measure. An extensive study of the master field in the plane was made by Thierry Lévy. [46] [47] Several major contributions have been made by Bruce K. Driver, Brian C. Hall, and Todd Kemp, [48] Franck Gabriel, [49] and Antoine Dahlqvist. [50] Dahlqvist and Norris [51] have constructed the master field on the two-dimensional sphere.

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 lattice gauge theory for any N, as the lattice spacing tends to zero. Let be the partition function of -dimensional lattice gauge theory with coupling strength in the box with lattice spacing and size being n spacings in each direction. Chatterjee showed that in dimensions d=2 and 3, is

up to leading order in , where is a limiting free-energy term. A similar result was also obtained for in dimension 4, for , , and independently.


  1. ^ Migdal, A. A. (1975). "Recursion equations in gauge field theories". Soviet Physics JETP. 42: 413–418.
  2. ^ a b c 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. Bibcode: 1981NuPhB.190..288M. doi: 10.1016/0550-3213(81)90560-5.
  3. ^ a b c Driver, Bruce K. (1989). "YM2:Continuum expectations, lattice convergence, and lassos". Communications in Mathematical Physics. 123 (4): 575–616. Bibcode: 1989CMaPh.123..575D. doi: 10.1007/BF01218586. S2CID  44030239.
  4. ^ Gross, Leonard; King, Chris; Sengupta, Ambar (1989). "Two dimensional Yang-Mills theory via stochastic differential equations". Annals of Physics. 194 (1): 65–112. Bibcode: 1989AnPhy.194...65G. doi: 10.1016/0003-4916(89)90032-8.
  5. ^ a b 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.
  6. ^ a b Sengupta, Ambar N. (1992). "Quantum Gauge Theory on Compact Surfaces". Annals of Physics. 220 (1): 157. doi: 10.1016/0003-4916(92)90334-I.
  7. ^ Sengupta, Ambar N. (1997). "Yang-Mills on Surfaces with Boundary: Quantum Theory and Symplectic Limit". Communications in Mathematical Physics. 183 (3): 661–704. Bibcode: 1997CMaPh.183..661S. doi: 10.1007/s002200050047. S2CID  120492148.
  8. ^ a b Sengupta, Ambar N. (1997). "Gauge Theory on Compact Surfaces". Memoirs of the American Mathematical Society. 126 (600). doi: 10.1090/memo/0600.
  9. ^ Fine, Dana S. (1990). "Quantum Yang-Mills on the two-sphere". Communications in Mathematical Physics. 134 (2): 273–292. Bibcode: 1990CMaPh.134..273F. doi: 10.1007/BF02097703. S2CID  122310649.
  10. ^ Fine, Dana S. (1991). "Quantum Yang-Mills on a Riemann surface". Communications in Mathematical Physics. 140 (2): 321–338. Bibcode: 1991CMaPh.140..321F. doi: 10.1007/BF02099502. S2CID  120616022.
  11. ^ 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. arXiv: hep-th/9504103. Bibcode: 1996JMP....37.1161F. doi: 10.1063/1.531453. S2CID  18159735.
  12. ^ Klimek, Slawomir; Kondracki, Witold (1987). "A construction of two-dimensional quantum chromodynamics". Communications in Mathematical Physics. 113 (3): 389–402. Bibcode: 1987CMaPh.113..389K. doi: 10.1007/BF01221253. S2CID  122234042.
  13. ^ 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. arXiv: hep-th/9605128. Bibcode: 1997JMP....38.5453A. doi: 10.1063/1.532146. S2CID  18153324.
  14. ^ Lévy, Thierry (2003). "Yang-Mills Measure on Compact Surfaces". Memoirs of the American Mathematical Society. 166 (790). doi: 10.1090/memo/0790. S2CID  119143163.
  15. ^ Lévy, Thierry (2010). "Two-dimensional Markovian holonomy fields". Astérisque. 329.
  16. ^ 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. arXiv: math-ph/0501014. doi: 10.1007/s00440-005-0478-8. S2CID  17397076.
  17. ^ 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.
  18. ^ Gross, David; Taylor IV, Washington (1993). "Two-dimensional QCD is a string theory". Nuclear Physics B. 400 (1): 181–208. arXiv: hep-th/9301068. Bibcode: 1993NuPhB.400..181G. doi: 10.1016/0550-3213(93)90403-C.
  19. ^ 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. arXiv: hep-th/9402107. Bibcode: 1997CMaPh.185..543C. doi: 10.1007/s002200050102. S2CID  14684976.
  20. ^ 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.
  21. ^ Migdal, A. A. (1975). "Recursion equations in gauge field theories". Sov. Phys. JETP. 42 (3): 2413–418.
  22. ^ a b Makeenko, Yuri M.; Migdal, A. A. (1980). "Self-consistent area law in QCD". Physics Letters B. 97 (2): 253–256. Bibcode: 1980PhLB...97..253M. doi: 10.1016/0370-2693(80)90595-X.
  23. ^ Makeenko, Yuri M.; Migdal, A. A. (1981). "Quantum chromodynamics as dynamics of loops". Nuclear Physics B. 188 (2): 269–316. Bibcode: 1981NuPhB.188..269M. doi: 10.1016/0550-3213(81)90258-3.
  24. ^ Rusakov, Boris (1995). "Lattice QCD as a theory of interacting surfaces". Physics Letters B. 344 (1–4): 293–300. arXiv: hep-th/9410004. Bibcode: 1995PhLB..344..293R. doi: 10.1016/0370-2693(94)01488-X. S2CID  118908012.
  25. ^ Rusakov, Boris (1997). "Exactly soluble QCD and confinement of quarks". Nuclear Physics B. 507 (3): 691–706. arXiv: hep-th/9703142. Bibcode: 1997NuPhB.507..691R. doi: 10.1016/S0550-3213(97)00604-4. S2CID  119498700.
  26. ^ 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.
  27. ^ 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.
  28. ^ a b Witten, Edward (1991). "On quantum gauge theories in two dimensions". Communications in Mathematical Physics. 141 (1): 153–209. Bibcode: 1991CMaPh.141..153W. doi: 10.1007/BF02100009. S2CID  121994550.
  29. ^ Witten, Edward (1992). "Two-dimensional gauge theories revisited". Journal of Geometry and Physics. 9 (4): 303–368. arXiv: hep-th/9204083. Bibcode: 1992JGP.....9..303W. doi: 10.1016/0393-0440(92)90034-X. S2CID  2071498.
  30. ^ Forman, Robin (1993). "Small volume limits of 2-d Yang-Mills". Communications in Mathematical Physics. 151 (1): 39–52. Bibcode: 1993CMaPh.151...39F. doi: 10.1007/BF02096747. S2CID  123050859.
  31. ^ 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. Bibcode: 1994JMP....35.5338K. doi: 10.1063/1.530755.
  32. ^ 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. arXiv: hep-th/9402135. Bibcode: 1994JMP....35.5354K. doi: 10.1063/1.530756. S2CID  119410229.
  33. ^ a b 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. Bibcode: 2003JGP....47..398S. doi: 10.1016/S0393-0440(02)00229-2.
  34. ^ Liu, Kefeng (1996). "Heat kernel and moduli space". Mathematics Research Letters. 3 (6): 743–762. doi: 10.4310/MRL.1996.v3.n6.a3.
  35. ^ 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. arXiv: 1411.4360. Bibcode: 2017LMaPh.107.1581J. doi: 10.1007/s11005-017-0956-9. S2CID  119577774.
  36. ^ 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. S2CID  123067470.
  37. ^ 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.
  38. ^ 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. S2CID  16857228.
  39. ^ Huebschmann, Johannes (1996). "Poisson geometry of flat connections for SU(2)-bundles on surfaces". Mathematische Zeitschrift. 221 (2): 243–259. arXiv: hep-th/9312113. doi: 10.1007/PL00004249. S2CID  186226623.
  40. ^ 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.
  41. ^ 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. arXiv: math-ph/0406027. Bibcode: 2006CMaPh.261..405L. doi: 10.1007/s00220-005-1450-2. S2CID  2985547.
  42. ^ a b Singer, Isadore M. (1995). On the master field in two dimensions. Functional analysis on the eve of the 21st century. Vol. 1. pp. 263–281.
  43. ^ Xu, Feng (1997). "A random matrix model from two-dimensional Yang-Mills theory". Communications in Mathematical Physics. 190 (2): 287–307. Bibcode: 1997CMaPh.190..287X. doi: 10.1007/s002200050242. S2CID  120011642.
  44. ^ Sengupta, Ambar N. (2008). Traces in two-dimensional QCD: the large-N limit. Traces in number theory, geometry and quantum fields. Vol. 1. pp. 193?212.
  45. ^ Anshelevich, Michael; Sengupta, Ambar N. (2012). "Quantum free Yang-Mills on the plane". Journal of Geometry and Physics. 62 (2): 330–343. arXiv: 1106.2107. Bibcode: 2012JGP....62..330A. doi: 10.1016/j.geomphys.2011.10.005. S2CID  54948607.
  46. ^ Lévy, Thierry (2017). "The Master Field on the Plane". Astérisque. 388.
  47. ^ 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. arXiv: 0905.3282. doi: 10.1016/j.jfa.2010.08.005. S2CID  15801521.
  48. ^ 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. arXiv: 1601.06283. Bibcode: 2017CMaPh.351..741D. doi: 10.1007/s00220-016-2793-6. S2CID  13920957.
  49. ^ 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. arXiv: 1602.03905. Bibcode: 2017CMaPh.352..967D. doi: 10.1007/s00220-017-2857-2. S2CID  14786744.
  50. ^ Dahlqvist, Antoine (2016). "Free energies and fluctuations for the unitary Brownian motion". Communications in Mathematical Physics. 348 (2): 395–444. Bibcode: 2016CMaPh.348..395D. doi: 10.1007/s00220-016-2756-y. S2CID  118973747.
  51. ^ Dahlqvist, Antoine; Norris, James R. (2020). "Yang-Mills measure and the master field on the sphere". Communications in Mathematical Physics. 377 (2): 1163–1226. Bibcode: 2020CMaPh.377.1163D. doi: 10.1007/s00220-020-03773-6. S2CID  18485837.
  52. ^ Chatterjee, Sourav (2016). "The leading term of the Yang-Mills free energy". Journal of Functional Analysis. 271 (10): 2944–3005. arXiv: 1602.01222. doi: 10.1016/j.jfa.2016.04.032. S2CID  119135316.