Двухпетлевое низкоэнергетическое эффективное действие в n = 2, d = 3 суперсимметричной абелевой модели поля Черна-Саймонса с материей Текст научной статьи по специальности «Математика»

B. S. Merzlikin. Two-loop effective action in N =2, d =3 supersymmetric Abelian Chern-Simons ...

UDC 530.1; 539.1

TWO-LOOP EFFECTIVE ACTION IN N =2, D = 3 SUPERSYMMETRIC ABELIAN CHERN-SIMONS-MATTER MODEL WITH ONE CHIRAL SUPERFIELD

Tl. S. Merzlikin

Department of Theoretical Physics, Tomsk State Pedagogical University, Kievskava str., 60, 634061 Tomsk, Russia. Department of High Mathematics and Mathematical Physics, National Research Tomsk Polytechnic University, pr.

Lenina, 30, 634050 Tomsk, Russia.

E-mail: [email protected]

Using a slowly-varying gauge superfield background we compute two-loop low-energy effective action in three dimensional N = 2 supersymmetric Abelian Chern-Simons-matter model with one chiral superfield up to four-derivative order.

Keywords: effective, action, extended supersymmetrv, Chern-Simons field models.

1 Introduction

The modern interest to three-dimensional supersymmetric field models is partly motivated by progress in constructing field theories describing multiple M2 branes in the ACIS4/CFT3 correspondence. These are N = 8 and N = 6 three-dimensional superconformal models of Chern-Simons gauge fields interacting with matter, known as the BLG [1-3] and ABJM [4] ones. As was mentioned by John Schwarz [5], it is important to study the low-energy effective action in these models to check the conjecture that it describes the dynamics of probe A12 brane in the

4

Following this general motivation we study the low-energy effective action in the simplest three-

N = 2

Simons-matter theory with one chiral superfield.

N=

2, d = 3 superspace [6-10] we compute two-loop low-energy effective action in this model up to the four-derivative order. Because of the superconformal symmetry two-loop contributions to the effective action is strongly restricted. The one-loop effective action in gauge superfield sector (supersymmetric one-loop Euler-Heisenberg effective action) was obtained in [8].

We base our consideration on the work [11].

with one chiral superfield has the form

S [V, Q] = k f dJzVG -J d7z Q e2V Q . (1)

The main goal for us is to study the low-energy effective action in the model under consideration (1) in the gauge superfield sector. In general, it is given by a functional of superfield strengths G, Wa, Wa and their derivatives, Naß = DaWß, Naß = DaWß,

r = J d7z L(G, Wa, Wa, Naß, Naß,...), (2)

where dots denote higher-order derivatives of the superfield strengths. It is hard problem to find the effective action (2) taking into account all derivatives of the fields. Thus, we restrict our consideration on the terms with no more than four space-time derivatives of component fields. For this aim it is enough to consider the following constraint on the superfield strengths:

(i) Supersymmetric Maxwell equations,

DaWa = 0, D aWa = 0; (3)

(ii) Superfield strengths are constant with respect to the space-time coordinates,

dmG = dmWa = dmWa =0 . (4)

2 General structure of effective action

In three-dimensions, the N = 2 gauge superfield V has not only Grassmann-odd superfield strengths Wa and Wa, but also the Grassmann-even scalar superfield strength G.

The action of three-dimensional N = 2 supersymmetric Abelian Chern-Simons matter model

The equations (3) and (4) single out the slowly-varying gauge superfield background. Also we note that the superfields Nap and and not independent under the constraints (3) and (4),

Naß = -N

aß ■

(5)

Therefore, in what follows we keep only Nap and discard Na@.

TSPU Bulletin. 2014. 12 (153)

Under the constraints (3) and (4) the effective action (2) which contains the terms with no more than four derivatives is given by

After summarizing one- and two-loop contributions, we get the parity-even part of the two-loop effective action in the superconformai form,

r = J d7z [fi(G) + f2(G)WaWßNaß

+ fs(G)W2W2] , with some functions fi(G), i = 1, 2, 3.

re

(6)

= r(1) +r(2) = — fd7zG ln G 4n J

(H)

+

1

15

128n y 2keff

1

•d?z (DaDa ln G)2 G

3 Two-loop contributions to effective action

The one-loop contributions to the functions fi(G) (6) for the model (1) were found in [8],

f(1) = — G ln G, f2i; =0, f3i; =

1

i)

i)

11

(7)

4n 256nG5 '

Our aim now is to compute two-loop corrections to this result, i.e., to find f(2).

The two-loop contribution to the effective action has the form

r(2) = __

15

512nkeff

d7z-

W 2W 2

G5

(8)

Here keff is the effective Chern-Simons level which includes one-loop correction to the classical value.

The effective action (8) corresponds to the following values of the functions f2) in (6),

f(2) = f fi = f2

(2)

0,

(2)

15

1

512nkeff G5 '

(9)

Since the model (1) is superconformai [8], the two-loop effective action (8) can be represented in a superconformai form by adding the terms with DaWa and DaWa. The action (8) can be rewrite as follows

r(2) =

15

256nkeff

'd7z (DaDa ln G)2 G

(10)

The effective action (11) is represented in the superconformai form and represents the parity-even part of the low-energy effective action in the model (1) up to the four-derivative order.

4 Conclusion

In the present paper we considered the model (1) in which the gauge superfield is described by the Chern-Simons action. In these models we computed two-loop low-energy effective action (11) up to four-derivative order in the gauge superfield sector. We demonstrate, that superconformai invariance restrict the possible contributions to the effective action. It should be noted that any superconformai effective action for the N =2 gauge superfield can be expressed in terms of superconformai invariants classified in [8].

Acknowledgement

I am grateful to I.L. Buchbinder and I.B. Samsonov for discussions. The work is supported in part by the RFBR grant No. 14-02-31201-mol, by the grant for LRSS project No. 88.2014.2. and the grant of Russian Ministry of Education and Science, project TSPU-122.

References

[1] Bagger J. and Lambert N. 2007 Phys. Rev. D 75 04502.

[2] Bagger J. and Lambert N. 2009 Phys. Rev. D 79 025002.

[3] Gustavsson A. 2009 Nucl. Phys. B 811 66.

[4] Aharony 0., Bergman O., JefFeris D. L., Maldacena J. 2008 JEEP 0810 091.

[5] Schwarz J. H. 2014 JEEP 1401 088.

[6] Buchbinder I. L., Pletnev N. G. 2011 JEEP 1111 085.

[7] Buchbinder I. L., Merzlikin B. S., Samsonov I. B. 2013 JEEP 1307 012.

[8] Buchbinder I. L., Pletnev N. G„ Samsonov I. B. 2010 JEEP 1004 124.

[9] Buchbinder I. L., Pletnev N. G„ Samsonov I. B. 2011 JEEP 1101 121.

[10] Buchbinder I. L., Pletnev N. G„ Samsonov I. B. 2013 Phys. Part. Nucl. 44 234.

[11] Buchbinder I. L., Merzlikin B. S., Samsonov I. B. 2013 Nucl. Phys. B 881.

В. S. Merzlikin. Two-loop effective action in N = 2, d =3 supersymmetric Abelian Chem-Simons

Received, Ц.11.20Ц

В. С. Мерзликин

ДВУХПЕТЛЕВОЕ НИЗКОЭНЕРГЕТИЧЕСКОЕ ЭФФЕКТИВНОЕ ДЕЙСТВИЕ В N = 2, d = 3 СУПЕРСИММЕТРИЧНОЙ АБЕЛЕВОЙ МОДЕЛИ ПОЛЯ ЧЕРНА-САЙМОНСА С

МАТЕРИЕЙ

Для случая медленно меняющегося фонового калибровочного суперполя вычислено двухпетлевое низкоэнергетическое эффективное действие в трехмерной N =2 суперсимметричной абелевой модели ноля Черна-Саймонса, взаимодействующего с одним киральным суперполем, с точностью до производных четвертого порядка от компонентных полей.

Ключевые слова: эффективное действие, расширенная суперсимметрия, модели с полями Черна-Саймонса.

Мерзликин Б. С., кандидат физико-математических наук. Томский государственный педагогический университет. Ул. Киевская, 60, 634061 Томск, Россия. Томский политехнический университет.

Пр. Ленина, 30, 634050 Томск, Россия. E-mail: [email protected]