The free energy in the weak-coupling phase of two-dimensional Yang-Mills
theory on a sphere for SO(N) and Sp(N) is evaluated in the 1/N expansion using
the techniques of Gross and Matytsin. Many features of Yang-Mills theory are
universal among different gauge groups in the large N limit, but significant
differences arise in subleading order in 1/N.Comment: 10 pages; no figures; LaTe

B. Rusakov

D. Boulatov

D. Gross

E. Witten

Howard J. Schnitzer

J. Daul

J. Minahan

M. Blau

M. Crescimanno

M. Douglas

Michael Crescimanno

P. Horava

S. Naculich

S. Ramgoolam

Stephen G. Naculich

W. Fulton

English

arXiv

The free energy in the weak-coupling phase of two-dimensional Yang-Mills theory on a sphere for [Formula presented] and [Formula presented] is evaluated in the [Formula presented] expansion using the techniques of Gross and Matytsin. Many features of Yang-Mills theory are universal among different gauge groups in the large [Formula presented] limit, but significant differences arise in subleading order in [Formula presented]. © 1996 The American Physical Society

Crescimanno, Michael

Naculich, Stephen G.

Schnitzer, Howard J.

Bowdoin College

Bowdoin College Bowdoin Digital Commons Physics Faculty Publications Faculty Scholarship and Creative Work 1-1-1996 Evaluation of the free energy of two-dimensional Yang-Mills theory Michael Crescimanno Berea College Stephen G. Naculich Bowdoin College Howard J. Schnitzer Brandeis University Follow this and additional works at: https://digitalcommons.bowdoin.edu/physics-faculty-publications Recommended Citation Crescimanno, Michael; Naculich, Stephen G.; and Schnitzer, Howard J., "Evaluation of the free energy of two-dimensional Yang-Mills theory" (1996). Physics Faculty Publications. 153. https://digitalcommons.bowdoin.edu/physics-faculty-publications/153 This Article is brought to you for free and open access by the Faculty Scholarship and Creative Work at Bowdoin Digital Commons. It has been accepted for inclusion in Physics Faculty Publications by an authorized administrator of Bowdoin Digital Commons. For more information, please contact mdoyle@bowdoin.edu, a.sauer@bowdoin.edu. Evaluation of the free energy of two-dimensional Yang-Mills theoryMichael Crescimanno*Department of Physics, Berea College, Berea, Kentucky 40404Stephen G. Naculich†Department of Physics, Bowdoin College, Brunswick, Maine 04011Howard J. Schnitzer‡Department of Physics, Brandeis University, Waltham, Massachusetts 02254~Received 22 January 1996!The free energy in the weak-coupling phase of two-dimensional Yang-Mills theory on a sphere forSO(N) and Sp(N) is evaluated in the 1/N expansion using the techniques of Gross and Matytsin. Manyfeatures of Yang-Mills theory are universal among different gauge groups in the largeN limit, but significantdifferences arise in subleading order in 1/N. @S0556-2821~96!00114-2#PACS number~s!: 11.15.Pg, 11.10.Kk, 12.38.CyI. INTRODUCTIONTwo-dimensional~2D! Yang-Mills theories have beenused as a laboratory to uncover general nonperturbative fea-tures of gauge theories@1–10#. It has been shown that the1/N expansion of these theories may be represented as aformal string theory, for SU(N) and U(N) gauge groups@2,3# as well as for SO(N) and Sp(N) @4#. It is useful tocompare these various string theories in order to learn whichstructures are generic, and what one might expect in a four-dimensional string theory of QCD.Certain features of 2D Yang-Mills theories are universal,i.e., independent of the gauge group, in the largeN limit @8#.For example, the normalized vacuum expectation values~VEV’s! of Wilson loops on arbitrary surfaces do not dependon the gauge group to leading order in 1/N, a fact mostnaturally understood from the string-theoretic interpretationof these theories@9#. On the other hand, the universality ofgauge theory observables breaks down in subleading ordersof the 1/N expansion. An example of this is the contributionfrom cross caps which appear on the world sheet forSO(N) and Sp(N), but not for SU(N) or U(N) @4#. It isimportant to have a clear understanding of the role of thegauge group in the string interpretation.To further analyze the differences between these theories,in this paper we evaluate the free energy of Yang-Millstheory on the sphere in the small area~weak-coupling! phase,including exponential corrections to the 1/N expansion. Ouranalysis closely parallels that of Gross and Matytsin@10# forU(N), focusing specifically on the gauge groups SO(N) andSp(N). One of the more interesting results of our analysis isthe difference in the double-scaling limit for different gaugegroups@see Eq.~34!ff #.II. THE PARTITION FUNCTIONThe partition function of two-dimensional Yang-Millstheory on the sphere isZ05(R~dimR!2e2 ~l Ā/2N! C2~R!, ~1!where the sum is over all irreducible representationsR of thegauge group, dimR andC2(R) are the dimension and qua-dratic Casimir invariant ofR, Ā is the area of the sphere, andl5e2N, wheree is the gauge coupling. The quadratic Ca-simir invariant is given byC2~R!5 fNF r2U~r !1 T~R!N G ~2!withf5H 1,1/2, U~r !5H r /N for SO~N!,2r /N for Sp~N!, ~3!andT~R!5(i51nni~ni1122i !5(i51k1ni22(j51n1kj2 , ~4!where ni(ki) are the row~column! lengths of the Youngdiagram associated withR, andn is the rank of the gaugegroup.@Our convention is rank Sp(2n) 5n.# Definingl i5ni1n2 i , mi5n2 i for SO~2n!,l i5ni1n2 i112 , mi5n2 i112 for SO~2n11!, ~5!l i5ni1n2 i11, mi5n2 i11 for Sp~2n!,the dimension and quadratic Casimir invariant ofR may beexpressed as@11#* Electronic address: crescima@physics.berea.edu† Electronic address: naculich@polar.bowdoin.edu‡Electronic address: schnitzer@binah.cc.brandeis.edu.PHYSICAL REVIEW D 15 JULY 1996VOLUME 54, NUMBER 2540556-2821/96/54~2!/1809~5!/$10.00 1809 © 1996 The American Physical SocietydimR55 )i, jn~ l i22l j2!~mi22mj2!for SO~2n!,)i, jn~ l i22l j2!~mi22mj2! )i51nl imifor SO~2n11! and Sp~2n!~6!andC2~R!55 (i51nl i22124N~N21!~N22! for SO~N!,12 (i51nl i22148N~N11!~N12! for Sp~N!.~7!These expressions are also valid for spinor representations ofSpin(N), which are associated with Young diagrams withniPZ11/2.The partition function ~1! depends on the area onlythrough the dimensionless combinationA5l f Ā and is givenup to an overall constant byZ0~A,N!}5eb~A,N!X~1 !~a! for SO~2n!,eb~A,N!Y~2 !~a! for SO~2n11!,eb~A,N!Y~1 !~a! for Sp~2n!,eb~A,N!@X~1 !~a!1X~2 !~a!# for Spin~2n!,eb~A,N!@Y~1 !~a!1Y~2 !~a!# for Spin~2n11!,~8!witha5A2N,b~A,N!5H A48~N21!~N22! for SO~N! and Spin~N!,A48~N11!~N12!, for Sp~N!,~9!andX~6 !~a!5 (l 1. . . ..l n>0D2~ l 12 , . . . ,l n2!e2a( j51nl j2,Y~6 !~a!5 (l 1. . . ..l n>0D2~ l 12 , . . . ,l n2!3S )i51nl i2D e2a( j51n l j2, ~10!where thel i are integers inX(1) andY(1) and half integersin X(2) andY(2), andD~ l 12 , . . . ,l n2!5)i, jn~ l i22l j2!5U 1 . . . 1l 12 . . . l n2A  A~ l 12!n21 . . . ~ l n2!n21U ~11!is the van der Monde determinant in the variablesl i2 . Notethat both tensor and spinor representations contribute to thepartition function for Spin(N), while only tensor representa-tions contribute for SO(N). As in the U(N) case@10#, theexpressions~10! are symmetric with respect to the inter-changel j↔l k , and vanish whenl j5l k , so the summa-tions can be extended to2`,l j,` for all l j , yieldingX~6 !~a!512nn! (2`,l 1 , . . . ,l n,`D2~ l 12 , . . . ,l n2!e2a( j51nl j2,Y~6 !~a!512nn! (2`,l 1 , . . . ,l n,`D2~ l 12 , . . . ,l n2!3S )i51nl i2D e2a( j51n l j2, ~12!where, again, thel i are integers inX(1) andY(1) and halfintegers inX(2) andY(2).To further evaluate Eq.~12!, we introduce several sets ofpolynomials in x2, qj(6)(xua)5x2 j1•••, and r j(6)(xua)5x2 j1•••. They are defined to be mutually orthogonal withrespect to the discrete measures(xe2ax2qi~6 !~xua!qj~6 !~xua!5d i j f j~6 !~a!,(xe2ax2x2r i~6 !~xua!r j~6 !~xua!5d i j gj~6 !~a!, ~13!where the sums onx are over integers forq(1) andr (1) andhalf integers forq(2) and r (2). Defining1810 54CRESCIMANNO, NACULICH, AND SCHNITZERqj~6 !~xua!5p2 j~6 !~xua!, f j~6 !~a!5h2 j~6 !~a!,xr j~6 !~xua!5p2 j11~6 ! ~xua!, gj~6 !~a!5h2 j11~6 ! ~a!, ~14!the orthogonality relations~13! reduce to(xe2ax2pi~6 !~xua!pj~6 !~xua!5d i j hj~6 !~a!, xPH Z,Z1 12 .~15!The pj(6)(xua)5xj1••• are the polynomials introduced byGross and Matytsin@10# in their study of U(N). Theyshowed that thehj(6)(a) are given byhj~6 !~a!5h0~6 !~a!)i51jRi~6 !~a!, h0~6 !~a!5(xe2ax2,xPH Z,Z1 12 , ~16!where theRj(6)(a) are defined through the recursion rela-tionsxpj~6 !~xua!5pj11~6 ! ~xua!1Rj~6 !~a!pj21~6 ! ~xua!,R0~6 !~a!50, ~17!and satisfy the differential relations@10#ddalnRj~6 !~a!5Rj21~6 ! ~a!2Rj11~6 ! ~a!,ddah0~6 !~a!52R1~6 !~a!. ~18!Rewriting the van der Monde determinants in Eq.~12! interms of the polynomialsqn(6)(xua) andr n(6)(xua) and usingthe orthogonality relations, we find that the partition functionis given up to a constant by~19!The free energy for the orthogonal and symplectic groups is, therefore,F~A,N!5 lnZ05b~A,N!1FN~A!1const ~20!withFN~A!55 nlnh0~1 !~a!1 (j51n21~n2 j !@ lnR2 j21~1 ! ~a!1 lnR2 j~1 !~a!# for SO~2n!,nln@h0~6 !~a!R1~6 !~a!#1 (j51n21~n2 j !@ lnR2 j~6 !~a!1 lnR2 j11~6 ! ~a!# for HSp~2n!SO~2n11!,~21!to be compared with the result for U(N) @10#:FN~A!5Nlnh0~6 !~a!1 (j51N21~N2 j !lnRj~6 !~a! for HU~N odd!U~N even!. ~22!Using Eq.~18!, we obtain from Eqs.~21! and ~22! the specific heat capacities54 1811EVALUATION OF THE FREE ENERGY OF TWO-DIMENSIONAL . . .d2F~A!dA25514N2@RN~6 !RN21~6 ! # for HSO~N even!SO~N odd!,14N2@RN~1 !RN11~1 ! # for Sp~N!,14N2@RN~6 !~RN11~6 ! 1RN21~6 ! !# for HU~N odd!U~N even!.~23!To obtain more explicit expressions for the free energies, onemay expandRj(6)(a), keeping the leading exponential cor-rection:Rj~6 !~a!5j2a72p2a2e2p2/aGj~a!1•••. ~24!Gross and Matytsin@10# use the recursion relations~18! toshow thatGj~a!5 R dt2p i S 11 1t Dne22p2t/a ~25!with the contour circlingt50 and passing to the right oft521. This can then be used to evaluate the free energy~22! for U(N) below the phase transition1FN52N22lnA62e22p2NA GN~a!1••• for HU~N odd!U~N even!.~26!In the largeN limit, the Gj (a) have the form@10#Gj~a!'~21!j11A j32pnc2S 12 jncD21/43expH 2 2p2NA @g~ j /nc!21#J ,g~x!5A12x2x2lnS 11A12x12A12xD , ~27!nc5p22a.Using Eqs.~24! and~25!, we calculate the free energy forthe orthogonal and symplectic groups~21! below the phasetransitionFN5H S 2 N24 1 N4 D lnA1e2 ~2p2N/A!@6G2n~a!2I 2n~a!#1••• for HSO~N52n!SO~N52n11!,S 2 N24 2 N4 D lnA1e2 ~2p2N/A!@G2n~a!1I 2n~a!#1••• for Sp~N52n!, ~28!whereI 2n~a!522p2a (j51nG2 j21~a!2 j215 R dt2p i S 11 1t D2n e22p2t/a2t11. ~29!In the largeN limit, this yieldsFN55 S 2N241N4 D lnA6S 12 1A12A/p2D e2 ~2p2N/A!GN~a!1••• for H SO~N even!SO~N odd!,S 2 N24 2 N4 D lnA1S 11 1A12A/p2D e2 ~2p2N/A!GN~a!1••• for Sp~N!,~30!1We correct a sign error in Ref.@10# for evenN.1812 54CRESCIMANNO, NACULICH, AND SCHNITZERbut these expressions break down if the areaA nears thecritical areap2. For the Spin(N) groups, theO(e22p2N/A)correction vanishes due to cancellation between the tensorand spinor representations, so that the leading correction isO(e24p2N/A) in that case.Approaching the phase transition from below in thedouble-scaling limit, defined byA→p2 and N→` with N2~p22A!3[gstr225const,~31!Gross and Matytsin@10# show thatRj(6)(a) behaves asRj~6 !5nc2p27~2 ! jnc5/3f 1~x!1O~nc4/3!, x5nc2/3S 12 jncD ,nc→`, ~32!where f 1(x) obeys the Painleve´ II equationf 1924x f1212 p2f 1350. ~33!Using this, we may show that in the double-scaling limit thespecific heat capacity~23! satisfiesd2FNdA25nc44p4N2 F12 2xnc2/32 p42nc2/3 f 12~x!6p2nc2/3 f 18~x!1•••Gx5xNfor HSO~N!Sp~N!, ~34!which has an additional term proportional tof 18(x) comparedwith @10#d2FNdA25nc42p4N2 F12 2xnc2/32 p42nc2/3 f 12~x!1•••Gx5xNfor U~N!. ~35!Equation ~34! gives the one instanton contribution to thespecific heat for SO(N) and Sp(N) in the double-scalinglimit. The computation of the specific heat for Spin(N) ismore complicated due to the contributions to the partitionfunction equation~19! from both tensor and spinor represen-tations.III. CONCLUSIONSMany features of two-dimensional Yang-Mills theory areuniversal in the largeN limit @8,9#, but differ in subleadingorder in 1/N. In this paper, we have explicitly evaluated thefree energy on the sphere in the weak-coupling phase, andshown how it compares among the different gauge groups.The double-scaling limit does not appear to be universal.Any proposed world-sheet action for two-dimensional Yang-Mills string theory must accommodate both the universalbehavior as well as the differences among the gauge groups.Research supported in part by the DOE under grant DE-FG02-92ER40706.@1# E. Witten, Commun. Math. Phys.141, 153 ~1991!; M. Blauand G. Thompson, Int. J. Mod. Phys. A7, 3781~1992!.@2# D. Gross, Nucl. Phys.B400, 161 ~1993!; J. Minahan, Phys.Rev. D47, 3430~1993!; D. Gross and W. Taylor, Nucl. Phys.B400, 81 ~1993!; B400, 395 ~1993!.@3# S. Naculich and H. Riggs, Phys. Rev. D51, 4394~1995!.@4# S. Naculich, H. Riggs, and H. Schnitzer, Mod. Phys. Lett. A8,2223~1993!; Phys. Lett. B319, 466~1993!; Int. J. Mod. Phys.A 10, 2097 ~1995!; S. Ramgoolam, Nucl. Phys.B418, 30~1994!.@5# P. Horava, inConformal Field Theory and String Theory, Pro-ceedings of the NATO Advanced Science Institute, Cargese,France, 1993, edited by L. Baulieut al., NATO ASI Series B:Physics Vol. 328~Plenum, New York, 1995!; ‘‘TopologicalRigid String Theory and Two-dimensional QCD,’’ Report No.hep–th/9507060~unpublished!; S. Cordes, G. Moore, and S.Ramgoolam, ‘‘Large N 2D Yang-Mills Theory and Topologi-cal String Theory,’’ Report No. hep–th/9402107~unpub-lished!.@6# B. Rusakov, Phys. Lett. B303, 95 ~1993!; M. Douglas and V.Kazakov,ibid. 319, 219~1993!; D. Boulatov, Mod. Phys. Lett.A 9, 365~1994!; B. Rusakov, Phys. Lett. B329, 338~1994!; J.Daul and V. Kazakov,ibid. 335, 371 ~1994!.@7# W. Taylor, ‘‘Counting Strings and Phase Transitions in 2DQCD,’’ Report No. MIT–CTP–2297, hep–th/9404175~un-published!; M. Crescimanno and W. Taylor, Nucl. Phys.B437,3 ~1995!.@8# M. Crescimanno and H. Schnitzer, Int. J. Mod. Phys. A11,1733 ~1996!.@9# M. Crescimanno, S. Naculich, and H. Schnitzer, Nucl. Phys.B446, 3 ~1995!.@10# D. Gross and A. Matytsin, Nucl. Phys.B429, 50 ~1994!.@11# W. Fulton and J. Harris,Representation Theory: A FirstCourse~Springer-Verlag, New York, 1991!.54 1813EVALUATION OF THE FREE ENERGY OF TWO-DIMENSIONAL . . .

Evaluation of the free energy of two-dimensional Yang-Mills theory

Crossref

Evaluation of the Free Energy of Two-Dimensional Yang-Mills Theory

Evaluation of the Free Energy of Two-Dimensional Yang-Mills Theory

Abstract

Similar works

Full text

Available Versions

Bowdoin College

Crossref