# Quantum vacua of 2d maximally supersymmetric Yang-Mills theory

## Abstract

Here, we analyze the classical and quantum vacua of 2d *N* = (8,8) supersymmetric Yang-Mills theory with SU( *N*) and U( *N*) gauge group, describing the worldvolume interactions of N parallel D1-branes with flat transverse directions R ^{8}. We claim that the IR limit of the SU( *N*) theory in the superselection sector labeled *M* (mod *N*) — identified with the internal dynamics of ( *M*, *N*)-string bound states of the Type IIB string theory — is described by the symmetric orbifold *N* = (8,8) sigma model into (R ^{8})D–1/SD when D = gcd( *M*, *N*) > 1, and by a single massive vacuum when *D* = 1, generalizing the conjectures of E. Witten and others. The full world volume theory of the D1-branes is the U( *N*) theory with an additional U(1) 2-form gauge field B coming from the string theory Kalb-Ramond field. This U( *N*) + B theory has generalized field configurations, labeled by the Z-valued generalized electric flux and an independent Z _{N}-valued ’t Hooft flux. We argue that in the quantum mechanical theory, the ( *M*, *N*)-string sector with *M* units of electric flux has a Z _{N}-valued discrete θ angle specified by *M* (mod *N*)more »

- Authors:

- California Inst. of Technology (CalTech), Pasadena, CA (United States). Walter Burke Inst. for Theoretical Physics

- Publication Date:

- Research Org.:
- California Inst. of Technology, Pasadena, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1501470

- Grant/Contract Number:
- SC0011632

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 11; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Brane Dynamics in Gauge Theories; D-branes; Sigma Models; Supersymmetric Gauge Theory

### Citation Formats

```
Koloğlu, Murat. Quantum vacua of 2d maximally supersymmetric Yang-Mills theory. United States: N. p., 2017.
Web. doi:10.1007/jhep11(2017)140.
```

```
Koloğlu, Murat. Quantum vacua of 2d maximally supersymmetric Yang-Mills theory. United States. doi:10.1007/jhep11(2017)140.
```

```
Koloğlu, Murat. Wed .
"Quantum vacua of 2d maximally supersymmetric Yang-Mills theory". United States. doi:10.1007/jhep11(2017)140. https://www.osti.gov/servlets/purl/1501470.
```

```
@article{osti_1501470,
```

title = {Quantum vacua of 2d maximally supersymmetric Yang-Mills theory},

author = {Koloğlu, Murat},

abstractNote = {Here, we analyze the classical and quantum vacua of 2d N = (8,8) supersymmetric Yang-Mills theory with SU(N) and U(N) gauge group, describing the worldvolume interactions of N parallel D1-branes with flat transverse directions R8. We claim that the IR limit of the SU(N) theory in the superselection sector labeled M (mod N) — identified with the internal dynamics of (M, N)-string bound states of the Type IIB string theory — is described by the symmetric orbifold N = (8,8) sigma model into (R8)D–1/SD when D = gcd(M, N) > 1, and by a single massive vacuum when D = 1, generalizing the conjectures of E. Witten and others. The full world volume theory of the D1-branes is the U(N) theory with an additional U(1) 2-form gauge field B coming from the string theory Kalb-Ramond field. This U(N) + B theory has generalized field configurations, labeled by the Z-valued generalized electric flux and an independent ZN-valued ’t Hooft flux. We argue that in the quantum mechanical theory, the (M, N)-string sector with M units of electric flux has a ZN-valued discrete θ angle specified by M (mod N) dual to the ’t Hooft flux. Adding the brane center-of-mass degrees of freedom to the SU(N) theory, we claim that the IR limit of the U(N) + B theory in the sector with M bound F-strings is described by the N = (8,8) sigma model into SymD(R8). We provide strong evidence for these claims by computing an N = (8,8) analog of the elliptic genus of the UV gauge theories and of their conjectured IR limit sigma models, and showing they agree. Agreement is established by noting that the elliptic genera are modular-invariant Abelian (multi-periodic and meromorphic) functions, which turns out to be very restrictive.},

doi = {10.1007/jhep11(2017)140},

journal = {Journal of High Energy Physics (Online)},

number = 11,

volume = 2017,

place = {United States},

year = {2017},

month = {11}

}

*Citation information provided by*

Web of Science

Web of Science

Works referenced in this record:

##
Elliptic Genera of 2d $${\mathcal{N}}$$ N = 2 Gauge Theories

journal, November 2014

- Benini, Francesco; Eager, Richard; Hori, Kentaro
- Communications in Mathematical Physics, Vol. 333, Issue 3

##
A new supersymmetric index

journal, November 1992

- Cecotti, Sergio; Fendley, Paul; Intriligator, Ken
- Nuclear Physics B, Vol. 386, Issue 2

##
Supersymmetric Yang-Mills theories

journal, March 1977

- Brink, Lars; Schwarz, John H.; Scherk, J.
- Nuclear Physics B, Vol. 121, Issue 1

##
From 0-branes to torons

journal, June 1998

- Guralnik, Z.; Ramgoolam, S.
- Nuclear Physics B, Vol. 521, Issue 1-2

##
DLCQ spectrum of $\mathcal{N}=(8,8)\mathrm{}$ super Yang-Mills theory

journal, October 1998

- Antonuccio, F.; Lunin, O.; Pinsky, S.
- Physical Review D, Vol. 58, Issue 10

##
Elliptic genera and quantum field theory

journal, December 1987

- Witten, Edward
- Communications in Mathematical Physics, Vol. 109, Issue 4

##
2d index and surface operators

journal, March 2014

- Gadde, Abhijit; Gukov, Sergei
- Journal of High Energy Physics, Vol. 2014, Issue 3

##
Notes on theories with 16 supercharges

journal, July 1998

- Seiberg, Nathan
- Nuclear Physics B - Proceedings Supplements, Vol. 67, Issue 1-3

##
Seven-sphere and the exceptional N = 7 and N = 8 superconformal algebras

journal, May 1996

- Günaydin, Murat; Ketov, Sergei V.
- Nuclear Physics B, Vol. 467, Issue 1-2

##
On the Landau-Ginzburg Description of n=2 Minimal Models

journal, October 1994

- Witten, Edward
- International Journal of Modern Physics A, Vol. 09, Issue 27

##
Torons and D-brane bound states

journal, August 1997

- Guralnik, Z.; Ramgoolam, S.
- Nuclear Physics B, Vol. 499, Issue 1-2

##
Two dimensional gauge theories revisited

journal, October 1992

- Witten, Edward
- Journal of Geometry and Physics, Vol. 9, Issue 4

##
Automorphic forms onO s +2,2(R) and infinite products

journal, December 1995

- Borcherds, Richard E.
- Inventiones Mathematicae, Vol. 120, Issue 1

##
Matrix string theory

journal, September 1997

- Dijkgraaf, Robbert; Verlinde, Erik; Verlinde, Herman
- Nuclear Physics B, Vol. 500, Issue 1-3

##
Low-energy spectrum of Script N = 4 super-Yang-Mills on *T* ^{3} : flat connections, bound states at threshold, and *S* -duality

journal, June 2007

- Henningson, Måns; Wyllard, Niclas
- Journal of High Energy Physics, Vol. 2007, Issue 06

##
On moduli spaces of flat connections with non-simply connected structure group

journal, May 1997

- Schweigert, Christoph
- Nuclear Physics B, Vol. 492, Issue 3

##
Bound states of strings and p-branes

journal, February 1996

- Witten, Edward
- Nuclear Physics B, Vol. 460, Issue 2

##
The Yang-Mills Equations over Riemann Surfaces

journal, March 1983

- Atiyah, M. F.; Bott, R.
- Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 308, Issue 1505

##
Elliptic Genera of Two-Dimensional $${\mathcal{N} = 2}$$ N = 2 Gauge Theories with Rank-One Gauge Groups

journal, November 2013

- Benini, Francesco; Eager, Richard; Hori, Kentaro
- Letters in Mathematical Physics, Vol. 104, Issue 4

##
D-branes in a topologically nontrivial $B$-field

journal, January 2000

- Kapustin, Anton
- Advances in Theoretical and Mathematical Physics, Vol. 4, Issue 1

##
$\theta $ -angle monodromy in two dimensions

journal, May 2012

- Lawrence, Albion
- Physical Review D, Vol. 85, Issue 10

##
On the Algebras of BPS States

journal, October 1998

- Harvey, Jeffrey A.; Moore, Gregory
- Communications in Mathematical Physics, Vol. 197, Issue 3

##
Elliptic Genera of Symmetric Products and Second Quantized Strings

journal, April 1997

- Dijkgraaf, Robbert; Moore, Gregory; Verlinde, Erik
- Communications in Mathematical Physics, Vol. 185, Issue 1

##
Coupling a QFT to a TQFT and duality

journal, April 2014

- Kapustin, Anton; Seiberg, Nathan
- Journal of High Energy Physics, Vol. 2014, Issue 4

##
Supersymmetries and their representations

journal, March 1978

- Nahm, W.
- Nuclear Physics B, Vol. 135, Issue 1

##
θ-vacua in two-dimensional quantum chromodynamics

journal, June 1979

- Witten, E.
- Il Nuovo Cimento A, Vol. 51, Issue 3

##
(0, 2) trialities

journal, March 2014

- Gadde, Abhijit; Gukov, Sergei; Putrov, Pavel
- Journal of High Energy Physics, Vol. 2014, Issue 3

##
Reading between the lines of four-dimensional gauge theories

journal, August 2013

- Aharony, Ofer; Seiberg, Nathan; Tachikawa, Yuji
- Journal of High Energy Physics, Vol. 2013, Issue 8