Pis'ma v ZhETF, vol.90, iss.ll, pp.803-807
© 2009 December 10
Non-conformal limit of AGT relation from the 1-point torus conformai block
V. Alba*+VA And. Morozov*Dl) * Institute for Theoretical and Experimental Physics, 117218 Moscow, Russia + Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia
v Department of General and Applied Physics, Moscow Institute of Physics and Technology 141700 Dolgoprudny, Moscow Reg., Russia
D Physical Department, Moscow State University, 119991 Moscow, Russia
A Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine
03680 Kyiv, Ukraine
Submitted 3 November 2009
Given a 4d M = 2 SYM theory, one can construct the Seiberg-Witten prepotentional, which involves a sum over instantons. Integrals over instanton moduli spaces require régularisation. For UV-finite theories the AGT conjecture favours particular, Nekrasov's way of regularization. It implies that Nekrasov's partition function equals conformai blocks in 2d theories with Wnc chiral algebra (virasoro algebra in our case). For Nc = 2 and one adjoint multiplet it coincides with a torus 1-point Virasoro conformai block. We check the AGT relation between conformai dimension and adjoint multiplet's mass in this case and investigate the large mass limit of the conformai block, which corresponds to asymptotically free 4d M = 2 super symmetric Yang-Mills theory. Though technically more involved, the limit is the same as in the case of fundamental multiplets, and this provides one more non-trivial check of AGT conjecture.
PACS: 11.15.-q, 11.25.Hf
1. Introduction. N = 2 super symmetric Yang-Mills (SYM) theories have attracted attention for rather a long time, because they are ideally suited for the study of interplay between perturbative and non-perturbative effects and for manifestation of various dualities [1-4]. Depending on the fields content, these theories exhibit all types of renormalization behaviour of effective coupling constant g: it may tend to infinity (Landau pole), and to zero (asymptotic freedom with dimensional transmutation in IR) or remain constant (UV-finite).
In N = 2 SYM theory the low-energy effective action is Abelian and its most important part is expressed in terms of the prepotential. Prepotential contains one-loop perturbative contribution and a far more sophisticated non-peturbative part, obtained as a sum over instantons. It was explicitly found by N.Seiberg and E.Witten (SW) [1, 2] with the help of duality arguments, and the answer was soon reformulated in terms of the spectral surfaces and simple integrable systems [5, 6]. The spectral curves were later interpreted in terms of branes. Straightforward evaluation of instanton sums is rather difficult, especially because some of the inte-
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grals over instanton moduli spaces diverge. See [7] for a comprehensive review and references.
A very successful direct caluculation was finally provided by N.Nekrasov [8]. He introduced a new partition function, depending on additional parameters ci and £2, such that the limit ci,c2 —> 0 reproduces SW prepotential.
Recently F.Alday, D.Gaiotto and Y. Tachikawa (AGT) made a ground-breaking conjecture that Nekrasov functions coincide with conformal blocks [9] of 2d Liouville/Toda models, and the e-parameters are needed to allow arbitrary values of the central charge in their chiral Wnc algebras (for Nc = 2 the chiral algebra is just the ordinary Virasoro). AGT suggest a non-trivial association of conformal blocks with UV-finite 4d quiver models. The 4-point tree Virasoro block is associated with the Nc = 2 gauge theory with 2Nc = 4 additional fundamental matter supermultiplets.
If there is instead, a single adjoint matter multiplet which also makes 4d theory UV-finite, the associated conformal block is the toric 1-point function. This claim was made in [10] and partly checked in [11]. We also confirm this relation and check it in one more way. Namely, we consider the limit of the large mass of adjoint mul-
Aext> 1
l-y1' a1> œ
L_y2, A2, 0
Triple vertex with two Virasoro descendants and the 1-point toric conformal block, obtained by taking a trace over Vermat module with a given dimension A. Each line is charaterized by dimension, by Ferrers diagram and external legs are also labeled by the position of the vertex operator on the Riemann surface
tiplet, where it decouples and the 4d theory turns into asymptotically free pure gauge N = 2 SYM. This pure gauge theory can be also obtained as the large-mass limit of the theory with 4 fundamentals, which implies that the corresponding limits of the tree 4-point and the toric 1-point conformal blocks (Figure) should be the same. The first limit has already been studied in [12, 13]. We find the second limit and show that it is indeed the same.
2. AGT relations. AGT hypothesis consists of several statements about relations between 2d CFT and 4d N = 2 SYM theories. One of the statements is that per-turbative part of Nekrasov partition function is equal to the product of DOZZ factors [14, 15], defining dependence of the triple functions in 2d Liouville theory on dimensions. Even more important and interesting is another part of this conjecture: the instanton part of Nekrasov partition function is equal to conformal block in 2d CFT (which depends on the chiral algebra, but not on the other details of 2d conformal model). Many examples were considered in [10] and later discussed in some detail [16-31,11].
A list of many Nekrasov functions is available in numerous papers, starting from original [8]. More difficult is the situation with conformal blocks. Like Nekrasov functions they are formal series; in the simplest cases of interest in the present paper they are in one variable,
B(x) = J2xnB{r'
n=0
(1)
n is called the "level", and particular quantities B^ are built from two kinds of ingredients: Shapovalov form
QM,Y2) =
Fa)
(2)
and two kinds of triple vertices [19]
(L-yMML-yMML-yMco))
7123(^1,^3) =
(^(0)^(1)^(00))
(3)
712,3(^1,^2,^3) =
{L-y3V3 .^(ljL.y^O)}
{v3 Fi(1)F2(0))
(4)
Here V are vertex operators, satisfying operator product expansions
V1(x1)V2(x2) = 5>i - (5)
Operators are made from primaries by the action of Virasoro generators. Virasoro descendants are labeled by Young-Ferrers diagrams Yf. Ferrers diagram is a sequence of integer numbers ki > k2 > k3.... So we define as L^yV =
Using the integral definition of Virasoro operators one can get the following relation:
(¿-„^ |V2(1)V3(0)) = {Vi\v2(l)(LnV3)(0)^ + + <Vi|(JL_iy2)(l)F3(0)) + + (l + n)A2<Vi|F2(l)F3(0)J
(LkV2)(l)V3(0)) , Vn.
1
k> 0
(6)
It is valid for arbitrary fields VJ, not obligatory primary ones [19]. Using this formula we can calculate all needed 7i2;3- The 4-point conformal block was computed already by many authors, because in this case we need only 7i2;3 (0,0,^0, f°r which there is the well known general formula. The 1-point torus conformal block, which is of interest for us here, is made from a more complicated 7i2;3 (0, Y2, Y3), which is not yet known in the general form. Thus we need to compute these vertices one by one.
Writing the correlator of 4 fields and expanding it with the help of (5) and using recently introduced notations we get
B(n) = V 4-point / ^
1^1 = 1^1 = «
li2-AYa)QtiYa,Y0)lm{Y0).
(7)
It is clear that to compute the conformal block one should use 7 instead of each vertex and Qinstead of inner lines.
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L
L
k
From the AGT relation for the 4-point conformal block we obtain
A =
4a2
4ei£2
6e2
c=l+-. (8)
In these relations a is a v.e.v. of the id SYM theory. They were originally obtained in [10] and [18]. Nevertheless we defined one external dimension and power of ^-multiplier.
3. 1-point conformal block on a torus. The formula to calculate 1-point torus conformal block (Figure) is
B(q) = Y,inB(n) =
(9)
= £ q^(L^Y1V1\Vext(i)L-Y2V2(0)}QA1(Y1,Y2).
yi,y3
Besides x it depends on two dimensions, A and Ae3.t and on the central charge c. AGT conjecture identifies this conformal block with analogous expansion of Nekrasov partition function
oo oo
tf(q) = (<r1/2V«)) " E (n) = E , (10)
n= o
n=0
r](q) = q™ JJ(1
n=1
is the Dedekind eta function, q = e2
= A-ïïi/g2
+ 6/2ir is complex coupling constant. depends on
the v.e.v. modulus ft, on adjoint multiplet's mass to and also on ci and e2.
3.1. The First Level. The AGT relation {A, A,;J.(, r} A {ci, to, £i, £2} can be found from equality BW = ffW
at level one. Explicitly
A2
»(1) _ ext
2A
2A
(H)
while
^(1) = (ci — m){e2 to) eie2(e2 - 4ft2)
2e2 - 2CTO + 2to2).
(12)
These quantities coincide provided (1) is suplemented by
Aext —
m(e — to)
v = 1
2 to(to — c)
€l€2
€l€2
(13)
The answer was computed with the help of ad hoc triple conformal correlator with a non primary field [34]. As we already noticed this computation is non trivial
because it involves the vertex 723^ with two non-trivial Young diagrams, see [34] for details.
3.2. The Second Level. The first non-trivial check of AGT conjecture is at level two. We made this check and there is indeed a complete coincidence between conformal block and Nekrasov partition function holds at level two, as already claimed in [11]. Unfortunately, the full formula is too cumbersome to be presented here.
Instead in this paper we concentrate on additional check, which can be extended to all levels: we investigate the limit of large to. According to (13) this is the same as large Aext, and what we need is a new as-ymptotics: (15) Together with (14) this gives an insight: only particular terms dominate in the limit.
4. Large Mass Behaviour. AGT relation is originally formulated for UV-finite gauge theories in Ad. Asymptotically free pure gauge theory arises when masses of additional matter supermultiplets are led to infinity, while the bare coupling x ~ qo is simultaneously led to zero. In the cas
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