# Seiberg-Witten geometry of four-dimensional $\mathcal{N}=2$ SO–USp quiver gauge theories

## Abstract

We apply the instanton counting method to study a class of four-dimensional N=2 supersymmetric quiver gauge theories with alternating SO and USp gauge groups. We compute the partition function in the Ω-background and express it as functional integrals over density functions. Applying the saddle point method, we derive the limit shape equations which determine the dominant instanton configurations in the flat space limit. The solution to the limit shape equations gives the Seiberg-Witten geometry of the low energy effective theory. As an illustrating example, we work out explicitly the Seiberg-Witten geometry for linear quiver gauge theories. Our result matches the Seiberg-Witten solution obtained previously using the method of brane constructions in string theory.

- Authors:

- Publication Date:

- Research Org.:
- Rutgers Univ., Piscataway, NJ (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1579599

- Alternate Identifier(s):
- OSTI ID: 1579556

- Grant/Contract Number:
- SC0010008

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 100 Journal Issue: 12; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Supersymmetric field theories

### Citation Formats

```
Zhang, Xinyu. Seiberg-Witten geometry of four-dimensional N = 2 SO–USp quiver gauge theories. United States: N. p., 2019.
Web. doi:10.1103/PhysRevD.100.125015.
```

```
Zhang, Xinyu. Seiberg-Witten geometry of four-dimensional N = 2 SO–USp quiver gauge theories. United States. doi:10.1103/PhysRevD.100.125015.
```

```
Zhang, Xinyu. Tue .
"Seiberg-Witten geometry of four-dimensional N = 2 SO–USp quiver gauge theories". United States. doi:10.1103/PhysRevD.100.125015.
```

```
@article{osti_1579599,
```

title = {Seiberg-Witten geometry of four-dimensional N = 2 SO–USp quiver gauge theories},

author = {Zhang, Xinyu},

abstractNote = {We apply the instanton counting method to study a class of four-dimensional N=2 supersymmetric quiver gauge theories with alternating SO and USp gauge groups. We compute the partition function in the Ω-background and express it as functional integrals over density functions. Applying the saddle point method, we derive the limit shape equations which determine the dominant instanton configurations in the flat space limit. The solution to the limit shape equations gives the Seiberg-Witten geometry of the low energy effective theory. As an illustrating example, we work out explicitly the Seiberg-Witten geometry for linear quiver gauge theories. Our result matches the Seiberg-Witten solution obtained previously using the method of brane constructions in string theory.},

doi = {10.1103/PhysRevD.100.125015},

journal = {Physical Review D},

number = 12,

volume = 100,

place = {United States},

year = {2019},

month = {12}

}

DOI: 10.1103/PhysRevD.100.125015

Works referenced in this record:

##
ABCD of Instantons

journal, September 2004

- Nekrasov, Nikita; Shadchin, Sergey
- Communications in Mathematical Physics, Vol. 252, Issue 1-3

##
Tinkertoys for the twisted D-series

journal, April 2015

- Chacaltana, Oscar; Distler, Jacques; Trimm, Anderson
- Journal of High Energy Physics, Vol. 2015, Issue 4

##
Splitting of surface defect partition functions and integrable systems

journal, January 2019

- Jeong, Saebyeok
- Nuclear Physics B, Vol. 938

##
Quantum Geometry and Quiver Gauge Theories

journal, January 2018

- Nekrasov, Nikita; Pestun, Vasily; Shatashvili, Samson
- Communications in Mathematical Physics, Vol. 357, Issue 2

##
BPS/CFT correspondence IV: sigma models and defects in gauge theory

journal, July 2018

- Nekrasov, Nikita
- Letters in Mathematical Physics, Vol. 109, Issue 3

##
BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters

journal, March 2016

- Nekrasov, Nikita
- Journal of High Energy Physics, Vol. 2016, Issue 3

##
A note on instanton counting for *N* =2 gauge theories with classical gauge groups

journal, May 2004

- Marino, Marcos; Wyllard, Niclas
- Journal of High Energy Physics, Vol. 2004, Issue 05

##
M-theory and Seiberg-Witten curves: orthogonal and symplectic groups

journal, October 1997

- Brandhuber, A.; Sonnenschein, J.; Theisen, S.
- Nuclear Physics B, Vol. 504, Issue 1-2

##
Quantum Integrability and Supersymmetric Vacua

journal, January 2009

- Nekrasov, Nikita; Shatashvili, Samson
- Progress of Theoretical Physics Supplement, Vol. 177

##
Loop and surface operators in $ \mathcal{N} = 2 $ gauge theory and Liouville modular geometry

journal, January 2010

- Alday, Luis F.; Gaiotto, Davide; Gukov, Sergei
- Journal of High Energy Physics, Vol. 2010, Issue 1

##
Instantons on Noncommutative ℝ 4 , and (2,0) Superconformal Six Dimensional Theory

journal, November 1998

- Nekrasov, Nikita; Schwarz, Albert
- Communications in Mathematical Physics, Vol. 198, Issue 3

##
N = 2 supersymmetric gauge theories, branes and orientifolds

journal, December 1997

- Landsteiner, Karl; Lopez, Esperanza; Lowe, David A.
- Nuclear Physics B, Vol. 507, Issue 1-2

##
Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD

journal, December 1994

- Seiberg, N.; Witten, E.
- Nuclear Physics B, Vol. 431, Issue 3

##
Seiberg-Witten Prepotential from Instanton Counting

journal, January 2003

- Nekrasov, Nikita A.
- Advances in Theoretical and Mathematical Physics, Vol. 7, Issue 5

##
BPS/CFT Correspondence III: Gauge Origami Partition Function and qq-Characters

journal, December 2017

- Nekrasov, Nikita
- Communications in Mathematical Physics, Vol. 358, Issue 3

##
Surface operators and separation of variables

journal, January 2016

- Frenkel, Edward; Gukov, Sergei; Teschner, Jörg
- Journal of High Energy Physics, Vol. 2016, Issue 1

##
Instantons on S4 and CP2, rank stabilization, and Bott periodicity

journal, March 2000

- Bryan, Jim; Sanders, Marc
- Topology, Vol. 39, Issue 2

##
Supersymmetric Vacua and Bethe Ansatz

journal, July 2009

- Nekrasov, Nikita A.; Shatashvili, Samson L.
- Nuclear Physics B - Proceedings Supplements, Vol. 192-193

##
Localization with a surface operator, irregular conformal blocks and open topological string

journal, January 2012

- Awata, Hidetoshi; Fuji, Hiroyuki; Kanno, Hiroaki
- Advances in Theoretical and Mathematical Physics, Vol. 16, Issue 3

##
Instanton counting with a surface operator and the chain-saw quiver

journal, June 2011

- Kanno, Hiroaki; Tachikawa, Yuji
- Journal of High Energy Physics, Vol. 2011, Issue 6

##
Intersecting surface defects and two-dimensional CFT

journal, August 2017

- Gomis, Jaume; Le Floch, Bruno; Pan, Yiwen
- Physical Review D, Vol. 96, Issue 4

##
Deformed Seiberg-Witten curves for ADE quivers

journal, January 2013

- Fucito, Francesco; Morales, Jose F.; Pacifici, Daniel Ricci
- Journal of High Energy Physics, Vol. 2013, Issue 1

##
Darboux coordinates, Yang-Yang functional, and gauge theory

journal, July 2011

- Nekrasov, N.; Rosly, A.; Shatashvili, S.
- Nuclear Physics B - Proceedings Supplements, Vol. 216, Issue 1

##
Towards a 4d/2d correspondence for Sicilian quivers

journal, October 2011

- Hollands, Lotte; Keller, Christoph A.; Song, Jaewon
- Journal of High Energy Physics, Vol. 2011, Issue 10

##
{\mathcal N}=2 S-duality via outer-automorphism twists

journal, April 2011

- Tachikawa, Yuji
- Journal of Physics A: Mathematical and Theoretical, Vol. 44, Issue 18

##
Intersecting surface defects and instanton partition functions

journal, July 2017

- Pan, Yiwen; Peelaers, Wolfger
- Journal of High Energy Physics, Vol. 2017, Issue 7

##
Orientifolds, branes, and duality of 4D gauge theories

journal, November 1997

- Evans, Nick; Johnson, Clifford V.; Shapere, Alfred D.
- Nuclear Physics B, Vol. 505, Issue 1-2

##
The omega deformation, branes, integrability and Liouville theory

journal, September 2010

- Nekrasov, Nikita; Witten, Edward
- Journal of High Energy Physics, Vol. 2010, Issue 9

##
BPS/CFT correspondence II: Instantons at crossroads, moduli and compactness theorem

journal, January 2017

- Nekrasov, Nikita
- Advances in Theoretical and Mathematical Physics, Vol. 21, Issue 2

##
From SO/Sp instantons to W-algebra blocks

journal, March 2011

- Hollands, Lotte; Keller, Christoph A.; Song, Jaewon
- Journal of High Energy Physics, Vol. 2011, Issue 3

##
Classification of 4d $ \mathcal{N} $ =2 gauge theories

journal, December 2013

- Bhardwaj, Lakshya; Tachikawa, Yuji
- Journal of High Energy Physics, Vol. 2013, Issue 12

##
Cubic curves from instanton counting

journal, March 2006

- Shadchin, Sergey
- Journal of High Energy Physics, Vol. 2006, Issue 03

##
Six-dimensional *D* _{ N } theory and four-dimensional SO-USp quivers

journal, July 2009

- Tachikawa, Yuji
- Journal of High Energy Physics, Vol. 2009, Issue 07

##
Tinkertoys for the D N series

journal, February 2013

- Chacaltana, Oscar; Distler, Jacques
- Journal of High Energy Physics, Vol. 2013, Issue 2

##
Affine SL(2) Conformal Blocks from 4d Gauge Theories

journal, September 2010

- Alday, Luis F.; Tachikawa, Yuji
- Letters in Mathematical Physics, Vol. 94, Issue 1

##
Instantons on Quivers and Orientifolds

journal, October 2004

- Fucito, Francesco; Morales, Jose F.; Poghossian, Rubik
- Journal of High Energy Physics, Vol. 2004, Issue 10

##
Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory

journal, September 1994

- Seiberg, N.; Witten, E.
- Nuclear Physics B, Vol. 426, Issue 1

##
Saddle point equations in Seiberg-Witten theory

journal, October 2004

- Shadchin, Sergey
- Journal of High Energy Physics, Vol. 2004, Issue 10

##
Construction of instantons

journal, March 1978

- Atiyah, M. F.; Hitchin, N. J.; Drinfeld, V. G.
- Physics Letters A, Vol. 65, Issue 3